Metabolomics has become increasingly popular in the study of disease phenotypes and molecular pathophysiology. One branch of metabolomics that encompasses the high-throughput screening of cellular metabolism is metabolic profiling. In the present study, the metabolic profiles of different tumour cells from colorectal carcinoma and breast adenocarcinoma were exposed to hypoxic and normoxic conditions and these have been compared to reveal the potential metabolic effects of hypoxia on the biochemistry of the tumour cells; this may contribute to their survival in oxygen compromised environments. In an attempt to analyse the complex interactions between metabolites beyond routine univariate and multivariate data analysis methods, correlation analysis has been integrated with a human metabolic reconstruction to reveal connections between pathways that are associated with normoxic or hypoxic oxygen environments.Correlation analysis has revealed statistically significant connections between metabolites, where differences in correlations between cells exposed to different oxygen levels have been highlighted as markers of hypoxic metabolism in cancer. Network mapping onto reconstructed human metabolic models is a novel addition to correlation analysis. Correlated metabolites have been mapped onto the Edinburgh human metabolic network (EHMN) with the aim of interlinking metabolites found to be regulated in a similar fashion in response to oxygen. This revealed novel pathways within the metabolic network that may be key to tumour cell survival at low oxygen. Results show that the metabolic responses to lowering oxygen availability can be conserved or specific to a particular cell line. Network-based correlation analysis identified conserved metabolites including malate, pyruvate, 2-oxoglutarate, glutamate and fructose-6-phosphate. In this way, this method has revealed metabolites not previously linked, or less well recognised, with respect to hypoxia before. Lactate fermentation is one of the key themes discussed in the field of hypoxia; however, malate, pyruvate, 2-oxoglutarate, glutamate and fructose-6-phosphate, which are connected by a single pathway, may provide a more significant marker of hypoxia in cancer.Metabolic networks generated for each cell line were compared to identify conserved metabolite pathway responses to low oxygen environments. Furthermore, we believe this methodology will have general application within metabolomics. Within systems biology, metabolomic studies have been shown to be an essential component for elucidating the interdependency between substrates and products. To date, metabolomics data have been limited to relatively standard statistical techniques including univariate analyses and multivariate analyses such as principal component analysis ,2. Whilset al. recently demonstrated the application of differential correlation analysis to create networks offering an insight into characterising a variety of biological states [The concentrations of metabolites in cells relate in part to the underlying structure of the metabolic network and thus the relative concentrations of metabolites within these networks can exhibit correlations . There hl states . Whilst l states  and the l states  as well l states . The EHMl states . Dependil states .Escherichia coli[In this study a network-based correlation analysis method has been developed as a novel method for the identification of linked metabolic pathways in the context of the entire metabolic network. Pair-wise correlations were mapped onto the EHMN genome-scale metabolic network and the shortest path connecting the two metabolites was extracted and used to construct a network of the cellular response to a physiological perturbation. Since all reactions present in the EHMN are reversible, all correlations were considered reversible and in this way it does not matter in which direction the path was drawn between correlated metabolites. Network biology is an emerging field; however there are complex issues, which remain to be addressed. For example, the shortest path between metabolites has been previously described for chia coli. The avechia coli.in vitro cell models [via. cell-environment interaction). Rather than simply to analyse the data in terms of each single metabolic feature that changes with respect to oxygen level, we have applied our novel correlation mapping onto the EHMN to reveal both correlated metabolites whose concentrations change in unison and anti-correlated discordant metabolites.We shall illustrate this method for investigating tumour hypoxia. Cancer metabolism has fascinated biologists since Warburg’s experiments in the 1920s . Otto Wal models ,17. Herein vitro and exposed to 21% oxygen (normoxia) or 1% oxygen (hypoxia) have been compared to assess the metabolic effect of lowering oxygen availability and to determine whether or not these effects are commonly observed across different cell lines. The cell lines used were the human breast carcinoma cell line MDA-MB-231 and the human colon carcinoma cell line HCT116. Cell lysates were prepared for each cell line exposed to each experimental oxygen condition and metabolites were profiled using gas chromatography–mass spectrometry (GC-MS). For each cell line exposed to each condition, a total of 30 replicates were collected. Over 50 metabolites were detected using GC-MS in each cell line exposed to normoxia or hypoxia  cycle though glutaminolysis. The TCA cycle is part of central carbon metabolism and it is widely known to be one of the main responses to hypoxia . Glutamiy attack ,23. The Although it has not previously been linked with hypoxia, the greatest difference in correlation between HCT116 cells exposed to normoxia and hypoxia occurred in the correlation between galactose/glucose and 4-hydroxyproline. These metabolites are correlated in hypoxia but are not correlated in normoxia and must therefore be a response to hypoxia. Furthermore, the 4-hydroxyproline peak was correlated with many other metabolites in hypoxia and could be potentially considered a central 'hub’ in cellular response to this oxygen condition, such that pathways involving this metabolite are changed in regulation. This highlights the potential importance of this metabolite in a hypoxic response.Many of the metabolites found to be differently correlated between normoxia and hypoxia in both cell lines were features of central carbon metabolism. Such metabolites include glucose, malate and pyruvate. Central carbon metabolism is obviously a feature of hypoxic response that is conserved between cell lines; however, the way the concentrations of these metabolites is affected by hypoxia is to a certain extent cell line specific. Other central carbon metabolites may also have changed in the same way within the system, just that they were not detected. It is possible to measure all central metabolites and they are present in our libraries, however many can be missing from the profile if their concentrations are below the limits of detection .2O and NAD. When computing the shortest network path connecting two nodes the path calculates a shortcut that passes through these highly connected nodes [The metabolic profiles acquired for correlation analysis here were obtained from whole cell lysates of MDA-MB-231 and HCT116 human carcinoma cells. For this reason it was appropriate to use a metabolic reconstruction that did not contain compartmentalisation of metabolites, for example those biochemical reactions that occur in the cytosol and those that are contained in the mitochondria. The EHMN contains approximately 2800 non-compartmentalised reactions . Correlaed nodes . These mvia the shortest route. This means many pathways form interactions between traditional pathways and thus cross over them  supplemented with 10% fetal calf serum  from a single batch and 2 mM glutamine . Cells were incubated in 95% air and 5% CO6 cells/mL into a 10 cm2 Petri dish . For consistency, the same batch of undefined FCS was used for all experiments. Cells were cultured and allowed to adhere in normoxia (21% oxygen) for 24 h prior to experimental analysis in the incubator described above. Subsequently, cells were divided into two groups where one group was retained in the normoxia condition and the other group was placed in a closed vessel though which gas containing 1% O2, 5% CO2 balanced with N2 was flowed at a rate 2 mL/s (hypoxia). This vessel was developed in-house and was constructed from an ADDIS® 5 L air tight container (including removable clip lid) with a male BSPT- female BSPP reducer and male parallel straight adaptor to connect a 6 mm polytube to a 6 mm 2/2 finger tap at either side of the container .8 mL of cells were seeded, in exponential phase, at 1 × 10g for 15 min) and the supernatant was transferred into a fresh Eppendorf tube. The volume of supernatant to be lyophilised was normalised according to the weight of the pellet for intracellular samples. Furthermore, supernatant remaining after normalisation was used to generate QCs which contained an equal proportion from each sample. Supernatant for both the sample and QC were lyophilised.Prior to metabolite extraction, the hypoxic vessel was closed using the taps and placed in an anoxic chamber . Normoxic samples were extracted in air. Extracellular media were decanted and cells were washed three times with 1 mL phosphate buffered saline (PBS). Subsequently, 1 mL methanol (maintained at -48°C) was added to quench cellular metabolism. Cells were scraped from the culture surface and the suspension solution was placed into Eppendorf tubes. A series of three freeze thaw cycles using liquid nitrogen were performed to enhance metabolite extraction before which the solution was centrifuged  to each sample which was then vortexed and heated at 60°C for 30 min. Finally, samples were centrifuged at 17000 × Samples were analysed using an Agilent 6890 GC  coupled to a LECO Pegasus III  EI-ToF-MS. The GC-MS instrument setup used has been previously described ,28. The The processing of raw GC-MS data was performed following the methods described previously, using the LECO ChromaToF v3.25 software package to apply the chromatographic deconvolution algorithm . Identifd4. QC metabolites with a CV greater than 30% were removed from the whole data matrix [Data were normalised to the internal standard succinic acid a matrix . Prior ta matrix . These dThe Pearson’s correlation calculation is given in Equation 1:r is the correlation coefficient calculated for the pair-wise correlation of variables X and Y, X and Y respectively and n is the number of samples.Equation 1: The Pearson’s product–moment correlation equation (as shown in Rodgers and Nicewander (1988) ) where rTo test for a significant correlation, the correlation value was approximately normalised using the inverse transform of Equation 2 which is known as the Fishers C is the correlation coefficient , which approximately normalises the distribution of the correlation coefficient to a Gaussian distribution and is independent on the number of samples.Equation 2: Fishers The significance of the correlation was tested using Equation 3.N), where the distribution is approximately Gaussian and the 95% significance level is given by Equation 3: Determining the significance of the correlation that is dependent on sample number  these correlations were discarded.Since the correlation threshold was relatively low for SE is the standard error, ρ is the correlation coefficient and n is the sample size.Equation 5: The standard error equation where α = 0.05 was tested using Equation 6 to satisfy the For each metabolite detected, the GC peak area was used for correlation. The Pearson’s correlation coefficients between metabolites were computed pair-wise using Equation 1. Subsequently, the difference between the correlations for two sample types  was calculated. To be considered a significant difference, one of the correlations must be > 0.7 and the difference between the correlations coefficients must be > 0.407. This was determined using the Fishers Ci is the Pearson’s rank correlation coefficient for example, 0.7. (b) Permutation test for comparing correlations between metabolites, where z1 and z2 are the values calculated through equation 3a and N1 and N2 are the minimum sample sizes for each metabolite [Equation 6: (a) Fishers tabolite .2O, CO2, O2, orthophosphate and hydrogen [Pair-wise correlations that significantly differed with respect to sample treatment were mapped directly onto the uncompartmentalised EHMN. For this the pair-wise correlations were taken in turn and using graph theory the shortest pathway connecting these two metabolites was computed. This was done using in-house routines in Matlab. The network was imported into Matlab using the SBML toolbox . Subsequhydrogen ,35.A stoichiometric matrix of the metabolic network was constructed using the SBML toolbox. This matrix was used to extract the two separate matrices of metabolites consumed and produced. The consumed network was multiplied by the transpose of the produced to construct a metabolite connectivity network. The matrix was symmetrised to account for reversibility in the network. Graph theory was implemented to calculate the shortest pathway between two metabolites using the bioinformatics toolbox included in Matlab.The authors declare that they have no competing interests.HK and EA carried out data acquisition, data analysis and wrote the manuscript. JA and WB assisted in the data acquisition. KS helped in the initial data analysis. RG and KW participated in the design, data interpretation and reviewing of manuscript. All authors read and approved the final manuscript.GC-MS data after normalisation to internal standard and filtering based on QCs.Click here for file


Saccharomyces cerevisiae.Genome-scale metabolic networks provide a comprehensive structural framework for modeling genotype-phenotype relationships through flux simulations. The solution space for the metabolic flux state of the cell is typically very large and optimization-based approaches are often necessary for predicting the active metabolic state under specific environmental conditions. The objective function to be used in such optimization algorithms is directly linked with the biological hypothesis underlying the model and therefore it is one of the most relevant parameters for successful modeling. Although linear combination of selected fluxes is widely used for formulating metabolic objective functions, we show that the resulting optimization problem is sensitive towards stoichiometry representation of the metabolic network. This undesirable sensitivity leads to different simulation results when using numerically different but biochemically equivalent stoichiometry representations and thereby makes biological interpretation intrinsically subjective and ambiguous. We hereby propose a new method, Minimization of Metabolites Balance (MiMBl), which decouples the artifacts of stoichiometry representation from the formulation of the desired objective functions, by casting objective functions using metabolite turnovers rather than fluxes. By simulating perturbed metabolic networks, we demonstrate that the use of stoichiometry representation independent algorithms is fundamental for unambiguously linking modeling results with biological interpretation. For example, MiMBl allowed us to expand the scope of metabolic modeling in elucidating the mechanistic basis of several genetic interactions in  One of the challenging tasks in systems biology is to quantitatively predict the metabolic behavior of the cell under given genetic and environmental constraints. To this end, genome-scale metabolic reconstructions and simulation tools are indispensable. The choice of the objective function to be used for simulating genome-scale metabolic models is dependent on the biological context and one of the most relevant parameters for successful modeling. Formulation of the intended objective function often requires the use of multiple fluxes, e.g. the sum of fluxes through ATP-producing reactions. We demonstrate that the existing tools confound biological interpretation of the simulations due to undesired dependence on the representation of stoichiometry and propose a new tool – Minimization of Metabolites Balance (MiMBl). MiMBl allows casting of the desired biological objective functions into linear optimization models and gives consistent simulation results when using numerically different but biochemically equivalent stoichiometry representations. We demonstrate relevance of MiMBl for addressing biological questions through improved predictions of genetic interactions within the yeast metabolic network. Genetic interactions imply functional relationship between the genes and therefore allow assessing different hypotheses for the underlying biological principles. MiMBl explains several of the genetic interactions as outcome of flux re-routing for minimal metabolite turnover adjustments. Escherichia coli can be found in Schuetz et al.e.g. a mutant) will be redistributed so as to be as similar as possible to the wild-type.The fundamental role of metabolism within a living cell has become a focal point of study in many disciplines, such as cell biology, physiology, medicine and synthetic biology. The assembly of all reactions and metabolites into a genome-scale metabolic network provides a comprehensive structural framework for integrative data analysis i.e. where linear combination of fluxes is part of the objective function) is sensitive to the representation of the reaction stoichiometry, with results strongly dependent on the adopted scaling of the stoichiometric coefficients. Such dependency confounds the biological interpretation of simulation results, as biochemically equivalent alternative representations of the same network can lead to contradictory predictions upon a given genetic or environmental perturbation. For example, the status of a given gene may change from non-essential to essential while using biochemically equivalent representations of the stoichiometry of the metabolic network (Table S1). As the stoichiometric representation of any reaction is subjective  and a typical genome-scale modeling problem involves hundreds of reactions, there are infinitely many biochemically equivalent ways to represent a given metabolic network. Any simulation algorithm should therefore be independent of the stoichiometry representation.The biological principles exemplified by simulation tools for both wild-type and perturbed networks are undeniably fascinating, which is confirmed by their numerous applications – including prediction of genetic interactions Minimization of Metabolites Balance (MiMBl), which accounts for reaction stoichiometry in the objective function by mapping the flux space into the metabolite turnover space. As intended, MiMBl shows robust predictions independently of the stoichiometry representation. We demonstrate the biological relevance of the new formulation with increased power for predicting genetic interactions in the metabolic network of S. cerevisiae. In a recent study reporting a large genetic interactions dataset covering the S. cerevisiae metabolic network We motivate the need for rethinking the problem formulation for metabolic modeling by illustrating how the current methods lead to incoherent biological predictions when alternatively representing the reaction stoichiometry. Tackling a proper problem formulation, we propose a new methodology for metabolic modeling – S. cerevisiae metabolic network are used, viz. iFF708 iAZ900 Methods).Several of the biological objective functions widely used in metabolic modeling are currently formulated as linear (or quadratic) combination of fluxes. Minimization of sum of intracellular fluxes and minimization of metabolic adjustment belong to this class and are herein used as case studies of biological principles that can be robustly formulated by using MiMBl. Two different genome-scale reconstructions of the iFF708 S. cerevisiae genome-scale metabolic reconstruction Text S1), we illustrate how the use of this objective function leads to inconsistent predictions when using numerically different, but biochemically equivalent, reaction stoichiometry. Linear scaling of all stoichiometric coefficients of a given reaction  preserves the stoichiometry and must not affect the simulation outcome for a correct problem formulation. However, in this case, scaling of a single reaction (RPI1) results in diverting the carbon flow from glycolysis to pentose phosphate pathway, which is one of the most important metabolic branch points , thus representing different biological solutions.Minimization of the sum of intracellular flux is a routinely used objective function for estimating intracellular fluxes h points Fig. 1. A to D via R5. The goal is to predict flux distribution in the mutant lacking R5. The biological principle of minimization of metabolic adjustment dictates rewiring of the flux through R6. However, lMoMA found contradictory optimal solutions, i.e. solutions that re-route the flux via R2–R3–R4 or R6, depending on the stoichiometric representation of R6 . Notably, widely used FBA-like problems (max/minimization of a single flux) are perfectly robust concerning the scaling of the stoichiometric coefficients. As a single flux is used in the objective function, the relative values of all the remaining fluxes (which depend on the stoichiometry representation) does not influence the optimal solution to be found .In order to provide insight into the nature of the problem leading to the susceptibility of the solution towards alternative representation of the stoichiometric matrix, we use a toy-model depicted in on of R6 Fig. 2b.Text S2 and Fig. S3); ii) it requires a reference flux distribution to obtain the normalization factors, making it inappropriate to formulate objective functions such as minimization of overall intracellular flux; and iii) the influence of each flux on the metabolic adjustment would be exclusively due to its fold change, not taking into account that reactions carrying higher fluxes could have a stronger impact on the predicted flux distribution, as implied in the original concept of minimization of metabolic adjustment.The mathematical caveat illustrated above means that the contribution of the desired biological objective function towards the obtained solution is inseparable from that of the artifacts of stoichiometry representation. Importantly, in large metabolic networks the effects of stoichiometric representation of reactions are cumulative. As we herein show, this problem can be solved by proper normalization of the objective function variables with respect to stoichiometric representation of the reactions. To achieve such normalization, we devised two approaches, normalized lMoMA (normlMoMA) and Minimization of Metabolites Balance (MiMBl). In normlMoMA, each variable in the objective function is normalized by its value in the wild-type flux distribution. Albeit being simple, this normalization method has three major drawbacks: i) many reactions often have null fluxes in the wild-type, thus posing a problem for normalization (M). The turnover of a metabolite is the sum of all fluxes producing (or consuming) it, multiplied by the corresponding stoichiometric coefficients (Methods). The objective function for minimization of metabolic adjustment is reformulated to include metabolite turnovers instead of fluxes  or maximization of ATP yield (by maximizing the ATP turnover for a given substrate uptake rate), among others  as a new method for metabolic modeling. The objective function in MiMBl is formulated as a linear combination of metabolite turnovers . Nevertheless, highly connected metabolites ensure a degree of network connectivity, which is sufficient for decreasing the number of situations where alternative flux distributions around the same metabolite are picked by MiMBl. Indeed, we did not find any case in the simulations performed for this study where growth prediction was altered in the second optimization step. An example case where the second optimization step will be more relevant is simulations involving export of metabolites, where the choice of a particular transporter (as in the reference flux distribution) among several alternative options is desired. A more thorough analysis of MiMBl alternative optima in a genome-scale network is presented below . For instance, the mutant lacking YCR012W, coding for a 3-phosphoglycerate kinase (pPGK1), was predicted to be viable when using the as-published representation of the stoichiometric matrix S01 (Methods). Based on such contradictory results, conclusions cannot be taken on whether YCR012W is predicted to be essential or not. As the number of deleted genes (or other network perturbations) increases, cumulative phenotypic effects related to the functional interactions between the genes are expected to take place and examples as the one mentioned above become even more striking. For triple gene knockouts, more than 200,000 triplets were found such that their predicted phenotype switched from lethal to non-lethal (or vice-versa) for the two biochemically equivalent matrix representations (Table S1). From a biotechnological perspective, predictions from genome-scale modeling have direct influence on the choice of gene targets selected for metabolic engineering. By using lMoMA, we identified metabolic engineering strategies  for production of two different compounds in yeast: succinate – a native product, and vanillin-glucoside – a heterologous product. Not only a significant fraction of mutants had divergent predictions for product yield when using two biochemically equivalent stoichiometric matrices, but also several highly ranked strategies in one case were low priority targets in the other (S7). Moreover, we also observed that the number of predicted synthetic lethal pairs differed by more than two-fold when using alternative stoichiometric matrix representations (Table S2). These inconsistencies have immediate implications on the consequent biological interpretation, as well as on the experimental design, and can be successfully overcome by using MiMBl (Table S3).In order to estimate the extent to which the lack of normalization of stoichiometric coefficients within the objective function influences the biological interpretation of simulation results, we used lMoMA for simulating gene knockouts in the Fig. S8a): i) sensitivity of the results towards the initial wild-type flux distribution used as input for minimizing the metabolic distance; and ii) potential non-uniqueness of the linear programming solution while simulating the mutant phenotype, i.e. existence of alternative optima. The sensitivity analysis for MiMBl towards both sources of uncertainty was performed using iAZ900 reconstruction of the yeast metabolic network, as the same reconstruction is subsequently used to study genetic interactions within the yeast metabolism. Both sources of variability also have impact on lMoMA simulation results .The above analysis proved the robustness of MiMBl towards stoichiometric representation of metabolic reactions. However, some degree of uncertainty in the simulation results might still exist, as we shall show here, essentially arising from two main sources . It has been previously shown that the use of alternative optima within the reference flux distribution obtained with FBA can affect the prediction of growth upon gene deletions using quadratic MoMA Methods). Similarly to what was previously observed for quadratic MoMA 13C flux analysis, will be useful for reducing the uncertainty in the input flux distribution and thereby in obtaining more robust predictions.Firstly, we analyzed the sensitivity towards the wild-type (or reference) flux distribution used as input for minimization of metabolic adjustment. Using an accurate reference flux distribution is crucial for obtaining biologically meaningful simulation results. While some metabolite exchange rates are commonly available as experimentally derived constraints for the wild-type, they are usually not sufficient to uniquely estimate the corresponding intracellular fluxes, Fig. S8a), we performed a flux variability analysis i.e. sum of metabolite turnover distance) to its optimal value (Methods). All of the tested fluxes were observed to have very low or no variability (vimin/vimax>0.99) across all single gene deletion phenotypes. Utility of the second step of MiMBl was seen in case of the flux through PGM1 upon deletion of YOR128C . Nevertheless, existence of a unique solution is problem dependent and it should be verified whether the possibility of alternative optima affects the prediction of fluxes of interest. Therefore, we performed an exhaustive analysis of variability of growth prediction across all single gene deletions, as well as all double gene deletions included in the genetic interactions case study. Growth was uniquely predicted in all these cases .In order to assess the variability due to potential non-uniqueness of the optimal solution obtained with MiMBl , the benefit of using MiMBl as a biological objective function became apparent while predicting genetic interactions. This implies that the biological regulatory principle underlying MiMBl is manifested in yeast (under the investigated conditions) at larger network perturbations or less drastic phenotypes than essentiality. When applied for studying genetic interactions, FBA is a conservative method compared to MiMBl, since the parameter used to define and measure genetic interactions is also the objective of optimization, i.e., growth. Within the metabolic network, the existence of several optimal solutions theoretically satisfying maximum biomass formation is often observed. In case of a single/double gene deletion mutant where an alternative optimal pathway exists, FBA will always find such an alternative solution, even though it may not be biologically plausible due to regulatory constraints, and, thereby may miss potential genetic interactions. On the other hand, MiMBl will help in capturing more refined regulatory effects where the loss of growth is a side effect of minimizing the flux rerouting relative to the wild-type.To what extent MiMBl contributes for increasing biological understandings gained from the application of optimization-based metabolic modeling? To address this question, we used one of the most recent and comprehensive Table S4). In order to assess the performance of the different algorithms, we carried out a sensitivity versus precision analysis. Precision was calculated as the fraction of experimentally validated interactions among all predicted interactions, while the sensitivity represents the fraction of the experimentally validated interactions captured by the predictions (Text S1). A computational epistasis score cutoff (εcutoff) was used to call a particular gene pair to be positively interacting (ε>εcutoff), negatively interacting (ε<−εcutoff) or non-interacting (|ε|<εcutoff) (Text S1). The performance of all algorithms  is summarized as ROC  curves for both, positive and negative epistasis . The sensitivity and precision of the FBA predictions obtained in this study are within the same range as previously reported by Szappanos et al.vs ∼4% for positive, vs ∼3% for negative, Fig. S10). As previously reported by Szappanos and co-workers, lMoMA does not improve FBA predictions. This observation further emphasizes that a proper mathematical formulation of the biological principle (objective function) has a major impact on the ability to interpret in vivo observations.The subset of experimental genetic interactions involving non-essential genes from the yeast metabolic model contains 2745 interactions (939 positive and 1806 negative) connecting 520 genes (cutoff| = 0.013) for further analysis of the predicted interactions . For this cutoff, the correctly predicted genetic interactions map contains 142 interactions (73 positive and 69 negative) connecting 86 genes . Consequently, MiMBl also displayed an improved capacity to predict both positive and negative epistasis involving the same gene. More than 80% of the genes display this feature in vivo. Interestingly, 30% of the genes involved in MiMBl predicted epistasis interact both positively and negatively, while FBA predicts that only 14% of the genes do so . MiMBl captured interactions between genes that are significantly more distant than in case of FBA , FBA could not capture this interaction. A similar effect is observed for the negative interaction between isoenzymes SER3 and SER33. In the absence of both genes, FBA predicts the needed supply of serine to be totally fulfilled by rerouting the metabolic fluxes via the glyoxylate shunt and threonine biosynthesis. On the other hand, MiMBl predicts that the supply of serine will be shared between the two alternative pathways, but the rescue cannot be complete, because the corresponding metabolic adjustment cost overweighs the benefit of increased growth. This prediction is in very good agreement with the experimental verification that the double mutant growth is impaired and can be restored by adding glycine to the medium, which is the intermediate for serine synthesis via glyoxylate or threonine Use of MiMBl not only allowed us to expand the range of genetic interactions predicted by FBA, but also the combined use of these two complementary algorithms enabled finding of relevant interactions where only one or both simulation principles apply. For example, the interaction between Overall, our results demonstrate that the use of optimization-based algorithms that are stoichiometry representation independent is fundamental for unambiguously linking modeling results with biological interpretation. To this end, we report a new method for formulating objective functions for metabolic modeling – MiMBl. As a biological case study, we used MiMBl to gain insights into the flux rewiring underlying genetic interactions within the yeast metabolic network. The analysis showed that the combined use of different objective functions is of primary importance in order to achieve a more complete understanding of the operating principles behind complex biological phenomena such as genetic interactions. Indeed, the number of accurately predicted genetic interactions was almost doubled owing to the use of MiMBl, highlighting the impact of metabolic adjustment constraints on the operation of perturbed metabolic networks. In conclusion, MiMBl provides a framework for consistent mathematical formulation of biological objective functions and thereby facilitates unraveling of the genotype-phenotype relations in metabolic networks.iFF708 Text S1), iFF708 showed less flux variability at physiologically important flux nodes as opposed to more recent reconstruction iAZ900 Fig. S1). On the other hand, for studying large-scale genetic interactions in yeast, we used iAZ900 , ilbv and iupv are the lower and upper bounds for the flux of reaction i.Normalized lMoMa was formulated as follows:m is the subset of N producing or consuming metabolite m and αm,i is the stoichiometric coefficient of metabolite m in reaction i. Note that αm,i is always a positive number in the definition above, irrespective of m being a substrate or a product.Metabolite turnover is defined as the sum of all fluxes producing (or consuming) it, multiplied by the stoichiometric coefficients:MiMBl was formulated as two sequential linear programming problems, as follows:st optimization:1nd optimization:2We note that MiMBl integrates reaction-to-metabolite turnover mapping into the model formulation in terms of defining biological objective functions and thereby making metabolite-usage a determinant for the prediction of metabolic phenotypes. This formulation is thus different from metabolite-centric approaches that have been proposed for interpreting simulation results Y, X and Z represent the metabolites participating in reaction r and Yr, αXr, αZrα represent the corresponding stoichiometric coefficients. Note that when the stoichiometry of reaction r is scaled by θ, the corresponding flux value will be scaled by 1/θ for the same optimal solution. For illustrating the impact of linear scaling of the reactions stoichiometry on the internal flux distribution, the reaction RPI1 of iFF708 model was divided by the scalar θ. For illustrating the impact of using alternative stoichiometry representations on the design of metabolic engineering strategies, two biochemically equivalent stoichiometric matrices were used: i) the as-published matrix from the yeast model (0S) and ii) an equivalent matrix (1S) where the stoichiometric coefficients of the reactions SERxtO, PDC6, FUR1, GAP1_21, PNP1_1, and CYSxtO were divided by 100, 100, 0.1, 0.01, 100 and 0.1, respectively. A third equivalent matrix (2S) was generated by dividing the coefficients of a single reaction (PGK1) by 0.1. The results of the comparison between 0S and 2S are presented in Fig. S6.Alternative stoichiometry representations were obtained by multiplying a given reaction (or a set of reactions) by a scalar θ . Consider reaction r e.g. minimization of intracellular flux and lMoMA. Assuming that thj column of matrix scaling matrix) and thj column of matrix The impact of scaling the constraints of a given linear programming problem depends on whether such changes guarantee the optimality conditions after scaling. Consider the problem:optimality condition is not guaranteed.Unless all entries of Corollary 1: When all  entries of Corollary 2: For any positive diagonal scaling matrix Now consider the following MiMBl-like formulated problem:Similarly to the previous problem, the following optimality condition is satisfied, so et al.et al.The sensitivity of MiMBl and lMoMA towards the use of FBA alternative optima for wild-type flux distribution was evaluated by performing single gene deletion simulations while using 500 different flux distributions corresponding to alternative optima of the same FBA solution. FBA alternative optimal solutions were obtained following a Mixed Integer Linear Programming (MILP) routine similar to the one suggested by Lee Figure S1iFF708 and iAZ900. Metabolites uptake and production rates, as well as growth from iFF708 Comparing the variability of predicted internal fluxes of glycolysis and pentose phosphate pathway between the models (TIF)Click here for additional data file.Figure S2S. cerevisiae genome-scale model iFF708 Methods. a) Linear scaling of the reaction FBP1. As FBP1 carries no flux under the simulated conditions, the scaling of this reaction does not affect the objective function value. b) Linear scaling of the reaction RPE1. For the range of θ tested, the objective function value perfectly correlated with the scaling factor of the reaction RPE1, which indicates that all obtained solutions are in fact the same optimal solution . This profile means that there is no pathway alternative to RPE1 that can become part of the optimal solution. c) Linear scaling of the reaction RPI1. For the range of tested θ, at least two slopes are observed when correlating the objective function value with 1/θ, indicating that at least two different optimal solutions were found for the same problem. d) Linear scaling of the reaction NDI1. Similarly to that of RPI1, scaling of NDI1 leads to different optimal solutions. However, in this case, the objective function value stabilizes after a given θ, which means that this flux no longer influences the optimization. Such profile suggests that the optimal solution found after the given value of θ does no longer involve NID1, but an alternative pathway, which became preferred for minimizing the objective function.Profiles obtained for the objective function value  using alternative stoichiometry representations of (TIF)Click here for additional data file.Figure S3a) Toy-model: R1 to R7 and A to D represent reactions and metabolites, respectively. In the wild-type, or reference, flux goes from A to D via R5. R6 and R2–R3–R4 are two alternative pathways for flux re-distribution after deletion of R5. b) Flux through reactions R2 (full symbols) and R6 (open symbols) obtained after simulation with normlMoMA by using alternative representations of reaction R6 . c) Formulation of normlMoMA objective function (Methods). d) Optimal objective function value (distance) obtained for minimization of metabolic adjustment as function of θR6.A toy-model illustrating how, and why, alternative stoichiometry representations influence simulation of minimization of metabolic adjustment by using normalized lMoMA – normlMoMA. (TIF)Click here for additional data file.Figure S40S and 1S, Methods). Number of mutants diverging in their lMoMA-predicted a) succinate and b) vanillin-glucoside yields for the two alternative representations of stoichiometry. The x-axis represents the percentage of deviation of product formation by the mutants relative to 0S. c) Comparison of ranks of lMoMA-predicted metabolic engineering strategies for improving succinate and vanillin-glucoside production, obtained by using 0S and 1S. d) Comparison of ranks of MiMBL-predicted metabolic engineering strategies for improving succinate and vanillin-glucoside production, obtained by using 0S and 1S.Impact of stoichiometry representation on the design of metabolic engineering strategies depending on the nature of the objective function formulation – MiMBL versus lMoMA. Shown is the comparison of predicted succinate and vanillin-glucoside yields for triple gene knockout mutants obtained with two alternative stoichiometric matrices ((TIF)Click here for additional data file.Figure S5S. cerevisiae depending on the nature of the objective function formulation. Shown is the comparison of predicted succinate yield for a) single, b) double and c) triple gene knockout mutants obtained with two alternative stoichiometric matrices . The number of mutants diverging in their lMoMA-predicted succinate yield for the two alternative representations of stoichiometry is represented on the y-axis, while the percentage of deviation of product formation by the mutants relative to 0S is represented on the x-axis. d–f) Comparison of ranks of lMoMA-predicted metabolic engineering strategies for improving succinate production obtained by using 0S and 1S for d) single, e) double and f) triple gene knockout mutants.Stoichiometry representation impacts the design of metabolic engineering strategies for improving succinate production in (TIF)Click here for additional data file.Figure S6S. cerevisiae depending on the nature of the objective function formulation. Shown is the comparison of predicted vanillin-glucoside yield for a) single, b) double and c) triple gene knockout mutants obtained with two alternative stoichiometric matrices . The number of mutants diverging in their lMoMA-predicted vanillin-glucoside yield for the two alternative representations of stoichiometry is represented on the y-axis, while the percentage of deviation of product formation by the mutants relative to 0S is represented on the x-axis. d–f) Comparison of ranks of lMoMA-predicted metabolic engineering strategies for improving vanillin-glucoside production obtained by using 0S and 1S for d) single, e) double and f) triple gene knockout mutants.Stoichiometry representation impacts the design of metabolic engineering strategies for improving vanillin-glucoside production in (TIF)Click here for additional data file.Figure S7S. cerevisiae depending on the nature of the objective function formulation. a–f) Number of mutants diverging in their lMoMA-predicted a–c) succinate and d–f) vanillin-glucoside yields for two alternative representations of stoichiometry, 0S and 2S (Methods). Results for a,d) single, b,e) double and c,f) triple gene knockout mutants are presented. g–l) Comparison of ranks of lMoMA-predicted metabolic engineering strategies for improving g–i) succinate and j–l) vanillin-glucoside production obtained by using 0S and 2S. Results for g,j) single, h,k) double and i,l) triple gene knockout mutants are presented.Stoichiometry representation impacts the design of metabolic engineering strategies for improving succinate and vanillin-glucoside yields in (TIF)Click here for additional data file.Figure S8a) The left side of the panel illustrates the variability due to possible uncertainty in the reference flux distribution, for example, as obtained by FBA simulations. The right hand side of the panel illustrates variability in the simulation result owing to the possibility of alternative optimal solutions of the MiMBL linear programming problem. Deletion of Gene 1 illustrates a case where a unique optimal solution is found, while deletion of Gene 2 depicts a case of alternative optima. b) Flux variability analysis to assess the existence of the alternative optimal solutions for a given reference flux distribution (c) Flux variability analysis for growth flux following single/double gene deletions. MiMBL yielded unique growth prediction for single and double gene deletion mutants. Only double gene deletions relevant for the genetic interactions case study (main text) were simulated.Alternative optima and sensitivity to reference flux distribution. ribution . Shown aribution . c Flux (TIF)Click here for additional data file.Figure S9a) Sensitivity of MiMBL towards the use of different reference flux distributions (Methods). Shown are histograms of the simulated growth  Flux variability analysis to assess alternative optimal solutions for a given reference flux distribution (Methods). Shown are the flux variability ranges of alanine transport and flux through phosphoglucomutase (PGM1) after deletion of YBL045C and YOR128C, respectively. PGM1 represents a case where the 2nd optimization step of MiMBL contributes to reducing of flux variability.Alternative optima and sensitivity to reference flux distribution: lMoMA. (TIF)Click here for additional data file.Figure S10a) Positive interactions. b) Negative interactions.ROC  curves obtained for predicting genetic interactions with MoMA. The ROC curves for the remaining algorithms were kept for reference. Sensitivity reflects the fraction of experimentally validated interactions captured by the algorithm while precision is experimentally validated interactions among all predicted interactions. (TIF)Click here for additional data file.Figure S11a) and negative (b) interactions for FBA and MiMBL. The epistasis score cutoff of |0.13| is represented by a dashed line. The bottom plots present the precision for positive (c) and negative (d) interactions for FBA and MiMBL. The epistasis score cutoff of |0.13| is represented by a dashed line.Sensitivity and precision for predicted genetic interactions versus epistasis score cutoff for FBA and MiMBL. The top plots present the sensitivity for positive ((TIF)Click here for additional data file.Table S1S. cerevisiae that differ between alternative representations of stoichiometry (S1 and S2), relative to S0.Number of lMoMA-predicted lethal gene/reaction knockouts in (DOCX)Click here for additional data file.Table S2S. cerevisiae genome-scale metabolic model.lMoMA-predicted epistatic interactions within (DOCX)Click here for additional data file.Table S3S. cerevisiae genome-scale metabolic model.MiMBL-predicted epistatic interactions within (DOCX)Click here for additional data file.Table S4et al. involving genes contained in iAZ900 model included in the study.All significant genetic interactions among non-essential genes from Szappanos (XLSX)Click here for additional data file.Text S1Supplementary methods. i) Yeast genome-scale metabolic models and simulation conditions; ii) Flux Balance Analysis; iii) Minimization of overall intracellular flux; iv) Minimization of metabolic adjustment – lMoMA; v) Genetic interactions – epistasis score; vi) Metabolic network distance.(DOCX)Click here for additional data file.Text S2Supplementary notes. i) Toy-model; ii) normlMoMA; iii) Impact of scaling stoichiometry on finding the optimal solution for metabolic flux distributions using FBA-like objective functions – Analytical evidence.(DOCX)Click here for additional data file.


Metabolism is a vital cellular process, and its malfunction can be a major contributor to many human diseases. Metabolites can serve as a metabolic disease biomarker. An detection of such biomarkers plays a significant role in the study of biochemical reaction and signaling networks. Early research mainly focused on the analysis of the metabolic networks. The issue of integrating metabolite networks with other available biological data to reveal the mechanics of disease-metabolite associations is an important and interesting challenge.In this article, we propose two new approaches for the identification of metabolic biomarkers with the incorporation of disease specific gene expression data and the genome-scale human metabolic network. The first approach is to compare the flux interval between the normal and disease sample so as to identify reaction biomarkers. The second one is based on the Reaction-Reaction Network (RRN) to reveal the significant reactions. These two approaches utilize reaction flux obtained by a Linear Programming (LP) based method that can contribute to the discovery of potential novel biomarkers.Biomarker identification is an important issue in studying biochemical reactions and signaling networks. Two efficient and effective computational methods are proposed for the identification of biomarkers in this article. Furthermore, the biomarkers found by our proposed methods are shown to be significant determinants for diabetes. Deficiency in essential metabolites can directly cause metabolic diseases. Metabolic diseases profiling is promising in uncovering the mechanism of disease-metabolite associations. Existing research mainly emphasized on the analysis of metabolic networks -3. ModelFlux Balance Analysis (FBA)  is a conFurthermore, we also construct a RRN which provides a platform for ranking the significance of reactions by using the PageRank algorithm. The reaction network can be constructed in such a way that the nodes represent reactions and an edge is placed whenever the reactions share the common metabolite. Note that one reaction can have thousands of edges with the related reactions which makes the network extremely complicated. This issue can be addressed by identifying densely connected subgraphs. The clustering toolbox is therefore employed to find the subnetworks. The graph clustering is based on the assumption that a group of functionally related nodes are likely to highly interact with each other while being more separate from the rest of the network . In 16]16], the Diabetes Mellitus is a group of metabolic diseases that are amongst the major human malnutrition diseases. Risk assessment is one of the possible ways to prevent the disease. Metabolic profiling, an unbiased technique, can potentially trigger the identification of high-risk candidates and therefore it can reduce the related costs .We then integrate the human metabolic network with disease-specific gene expression data to analyze the flux profiles within the network. In the following section, we introduce two methods for metabolite biomarkers discovery. The validity of the two approaches will be further discussed. The identified metabolite biomarkers may have potential applications for disease diagnosis.http://bigg.ucsd.edu/). One can retrieve the reactions and the involved genes using MATLAB. And the RRN can be implemented in the Cytoscape software package which is available at (http://www.cytoscape.org).The genome-scale human metabolic network reconstructed by Duarte et al.  consistsWe introduce two novel methods for integrating gene expression data and the human metabolic network in biomarker discovery.There are several major steps that we have to conduct before the construction of LP model, and eventually allow us for the detection of biomarkers.Expression levels in reactions1. The LP model2. Flux profiles in disease/normal samples3. Identification of significant reactions4. Significant metabolite discovery5. Comparing to the well known model using the human metabolic network to predict metabolic biomarkers of human inborn errors of metabolism , our modIn this section, we propose the second novel approach to identify the metabolite biomarkers based on RRN.• Clustering toolbox for identifying the subnetworks.k-means algorithm " acting as an inhibitor, is very helpful for patients in clinical trials, see for example . Both "nFurthermore, we analyze the significant reactions selected by integrating the flux analysis and the PageRank algorithm based on the subnetworks of the RRN. We filter out five reactions which are reported in Table →" means the reaction is irreversible. The number inside the parentheses (.) is the quantity of the metabolite. For example, in the Reaction 3511 of Table We remark that the symbol "⇆" means the reaction is reversible and "β-oxidation and the variants in "EHHADH" are associated with type 2 Diabetes. In [Considering the genes involved in the reactions, we have identified 9 important genes participating in Diabetes in Table etes. In , it is detes. In . And genetes. In .From the perspective of the metabolites related to the disease, reaction 3115 is an active reaction involved in fatty acid cholesterol metabolism during prostate cancer progression . Here "pOne can see that these two approaches are extremely different, but some of the metabolite biomarkers are the same. It has been shown that "nadh" and "pi" are the important factors for detecting diabetes. We can conclude that both our proposed methods are effective for identifying the metabolite biomarkers for diseases.In this paper, we first develop a computational method to identify significant genes and metabolites for metabolic diseases. LP based strategy is then utilized to obtain flux profiles in disease/normal samples. Gene expression data in two pairs of samples at disease/normal states contributes to discovering genes and metabolites that can be potential biomarkers. We then further present a second novel approach to identify the significant metabolites for the metabolic disease. We also employ the constraint-based flux distribution to analyze the metabolic network. The clustering method makes it possible to identify the subgraphs with the common properties, which is a key step to construct the RRN with Cytoscape. To evaluate the reaction in the network, we propose the PageRank algorithm to evaluate the node. We integrate the flux value and the rank result to select the significant reactions, from which the related metabolites biomarkers can be identified. The integration of genome-scale human metabolic network data with gene expression levels offers a new way for systematically identifying potential biomarkers.The authors declare that they have no competing interests.LL and HJ came up with the idea. WKC, LL, HJ and VSV designed the research. HJ and YQ performed the research and analyzed the results. WKC, HJ, LL, YQ and VSV wrote the paper. All authors read and approved the final manuscript.


From a more theoretical standpoint, the structural properties of metabolic networks can be analysed to explore feasible metabolic behaviours and to define the boundaries of steady state flux distributions. Elementary flux mode analysis is one of the most efficient methods for performing this analysis. In this context, recent approaches have tended to compare experimental flux measurements with topological network analysis.Corynebacterium glutamicum) and a plant species (Brassica napus) for which experimental flux maps were available. The structural properties of each network were then studied using the concept of elementary flux modes. To do this, coefficients of flux efficiency were calculated for each reaction within the networks by using selected sets of elementary flux modes. Then the relative differences - reflecting the change of substrate i.e. a sugar source for C. glutamicum and a nitrogen source for B. napus - of both flux efficiency and flux measured experimentally were compared. For both organisms, there is a clear relationship between these parameters, thus indicating that the network structure described by the elementary flux modes had captured a significant part of the metabolic activity in both biological systems. In B. napus, the extension of the elementary flux mode analysis to an enlarged metabolic network still resulted in a clear relationship between the change in the coefficients and that of the measured fluxes. Nevertheless, the limitations of the method to fit some particular fluxes are discussed.Metabolic networks describing the main pathways of central carbon metabolism were set up for a bacteria species (13C metabolic flux analysis, by allowing the prediction of changes in internal fluxes before carbon labelling experiments.This consistency between EFM analysis and experimental flux measurements, validated on two metabolic systems allows us to conclude that elementary flux mode analysis could be a useful tool to complement  The use of a 13C labelled substrate associated with a steady state MFA is probably the most useful and straightforward approach to quantify fluxes in the central metabolism. It is based on the re-distribution of labelling among intermediate metabolites of the network, measured using either NMR for the determination of carbon enrichment and positional isotopomers, or Mass Spectrometry (MS) for the determination of mass isotopomers. Specialized software is available to quantify fluxes in a complex metabolic network [13C-MFA has been shown to be an efficient tool to model and quantify the functioning metabolism of cultured prokaryotic and eukaryotic cells and even cultured plant tissues. For instance, in metabolic engineering, 13C-MFA has been used to compare the lysine-producing Corynebacterium glutamicum grown either on glucose, fructose [Brassica napus embryos [Arabidopsis cells [Metabolic pathway analysis has become increasingly important to assess inherent network properties in reconstructed biochemical reaction networks . Metabol network ,3. 13C-Mfructose  or sucrofructose . In planfructose  in maizefructose ,8, tomatfructose , Brassic embryos -13, sunf embryos  and Arabis cells .E. coli [in silico to design efficient and robust strains to produce desired products. Recently, a classification method for EFMs based on motif finding (named ACoM) has been developed in order to analyze the results of the EFM analysis [More theoretically, Elementary Flux Mode (EFM) analysis, a constraint-based approach also called structural analysis, is used to identify all genetically independent pathways that are inherent in a metabolic network. EFM analysis provides a rigorous formalism for describing and assessing metabolic pathways at steady state. It involves three basic conditions, pseudo steady state, feasibility, and non-decomposability . In otheE. coli . With imanalysis . In plananalysis , sucroseanalysis  and staranalysis . Howevera priori objective/selection. For instance, in plant cells and tissues, central carbon metabolism is often dedicated to the production of structural compounds  and storage compounds .In most metabolic studies, objective functions (such as ATP production or cell growth maximization) are used for EFMs to steer the process of metabolic engineering. With the aim of overproducing a target compound, objective functions are based on reaction participation, which links the occurrence of a reaction to the production of the target compound. However, while the criterion of overproduction or efficiency is often significant for optimization (or maximization) studies, this criterion is not entirely meaningful in describing cell physiology without Corynebacterium glutamicum, widely used to produce amino acids  for which metabolic fluxes have been determined for three carbon sources  and (2) a plant system, Brassica napus embryos, for which metabolic fluxes have been measured in embryos growing in the presence of either mineral (ammonium/nitrate) or organic  nitrogen, in addition to glucose [C. glutamicum and B. napus, respectively. Then, the robustness of the relationship was tested by enlarging the metabolic network of B. napus to generic network of heterotrophic plant cells. Finally, the limits of this approach to fit some specific fluxes are discussed.In our study, we used EFM analysis to compare the structural properties of the metabolic networks with the fluxes experimentally determined by 13C-MFA. To do this, flux maps already published for two biological systems were used: (1) a microbial system,  glucose . By calc glucose ) from thEFM analysis starts with the network representation of the constraints, including network topology, stoichiometries and reaction reversibility .Corynebacterium glutamicum (strain ATCC21526) growing on three different sources of sugars, glucose, fructose [Brassica napus embryos growing on two nitrogen sources, amino acids and ammonium/nitrate [Data obtained from two biological systems were selected from the literature for their relevant map of fluxes determined by 13C-MFA, namely, (1) the microbial system of fructose  or sucro/nitrate . The expC.glutamicum, three distinct networks for each case  were created to take into account both the differences in the sugar uptake and in the yield of biomass synthesis with respect to the nature of the sugar. These three networks consist of 29, 30 and 31 reactions (thereby 10 reversible reactions) in the presence of glucose, fructose and sucrose respectively. The networks contain 31 metabolites in the presence of glucose and fructose and 32 metabolites in the presence of sucrose  listed in additional file In the case of B. Napus, the influence of the substrate on the metabolic network, either with only glucose (ammonium and nitrate as nitrogen source) or with glucose plus amino acids  was analysed. This metabolic network contains 26 reactions (9 reversible) and 30 metabolites , mainly located in the TCA pathway  and the new implementation of METATOOL algorithm (elmo http://elmocomp.sourceforge.net/elmocomp.shtml) were chosen for computation [From the tools available for EFM computation, CellNetAnalyzer putation .C. glutamicum networks provides three EFMsets, named Sg, Sf and Ss with only glucose, fructose and sucrose respectively as carbon sugar substrate in the medium and containing 212, 365 and 884 EFMs respectively. Note that the size of the Ss EFMset is higher than the size of Sg and Sf as the sucrose is metabolized in both glucose and fructose.The computation step performed on B. napus network led to 51 modes. As this set of EFMs included EFMs involving both glucose and amino acids, two subsets of EFMs were built, selected from only glucose (ammonium and nitrate as nitrogen source) and from both glucose plus amino acids  as a carbon substrate. The first EFMset, named Nm, contained 15 EFMs that use only glucose as a carbon substrate in the medium . The second EFMset, named Naa, contained 50 EFMs in which at least one of the uptake reactions is present .The computation step performed on B. napus network than in the C. glutamicum networks, the number of EFMs is much lower in the B. napus network than in the C. glutamicum networks. This shows the decrease in the level of the complexity of the B. napus network mainly involving the TCA pathway. For both biological systems, each EFMset represents a functional network with specific behaviour in which the coefficients of flux efficiency of the reactions are calculated.Interestingly, although the number of reactions in slightly lower in the N elementary modes (rows) and P reactions (columns). First, the EFM matrix containing the elements i,j e(row i and column j) is transformed to an efficiency matrix Mε (same size as the matrix of EFMs) by assigning a value of efficiency To address the question of whether the function and/or regulation of a complex metabolic network can be predicted by elementary mode analysis, our analysis relies on the calculation of a coefficient called "flux efficiency". The coefficient of flux efficiency is determined from the parameter of 'control-effective fluxes' introduced by  charactej is determined as the average of the flux efficiencies through this reaction in the EFM matrix (size N):Then, the flux efficiency Note that the forward and the backward reactions  are considered separately in order to assess the efficiency of each reaction specifically. The efficiency of a reversible reaction is thus the sum of its positive value (forward reaction) and its negative value (backward reaction).Sg, Sf and Ss for C. glutamicum and Nm and Naa for the B. napus) was calculated. With respect to a reference condition, i.e. glucose condition for C. glutamicum, and organic nitrogen condition for B. napus, the relative difference (Δ) of both parameters, i.e. the measured flux and the flux efficiency, was calculated. The relative difference, quantifying the perturbation due to the change of substrate, was expressed as the ratio between the difference generated by the change in the carbon (C. glutamicum) or nitrogen (B. napus) source, at the numerator, and the absolute value of the reference, at the denominator. The relative difference of measured fluxes (ΔF) was:The coefficient of flux efficiency depends above all both on the occurrence of the reactions in the EFMset and on the size of the EFMs. That means that the coefficient of flux efficiency is equal to zero when a reaction does not happen and, conversely, a highly frequent reaction in relatively short EFMs is affected by a high value of flux efficiency. The coefficient of flux efficiency of each reaction of the five EFMsets  and in the other state, respectively. Similarly, the relative difference of flux efficiencies C. glutamicum, and organic nitrogen for B. napus) and in the other state, respectively.with Corynebacterium glutamicum cultured with three carbon sources  [Brassica napus embryos grown on two nitrogen sources (ammonium/nitrate or amino acids) [In this section, we compare the EFM analysis with the experimental results obtained for two systems, sucrose) ,5 and Bro acids) ,23.Corynebacterium glutamicum (strain ATCC 21526) grown on glucose and fructose [The comparison of the central metabolism of lysine-producing fructose  shows:  provide an additional carbon source for growing tissues. In the Brassica napus embryos, the main difference between the organic  and the inorganic (ammonium/nitrate) nitrogen condition was the redirection of fluxes leading into and out of the TCA cycle [In non-photosynthetic tissues such as CA cycle . When thC. glutamicum [B. napus embryos [C. glutamicum) and the nitrogen source (B. napus) was analysed by comparing ΔF through the respective reaction. These relative ΔF  were plotted in Figure and ΔF) for both the C. Glutamicum and B. napus systems. This result means that at a high ΔF (or a low ΔF) corresponds a high EFM analysis was performed on both biological systems, a bacterial system  and a embryos ,23) and C. glutamicum , fumarase (fum) isocitrate dehydrogenase (Idh), and alpha-keto dehydrogenase (aKGdh), pyruvate dehydrogenase (Pdh) and (2) a positive of aspK) and of all enzymes of the PPP . It is notable that on the one hand PGI exhibits the lowest change in both the measured fluxes and the flux efficiencies, while on the other hand, the PPP fluxes  exhibit the highest change, regardless of the carbon source (fructose and sucrose). Nevertheless, some discrepancies occur for some enzymes , the value of ΔF, both under fructose and sucrose conditions (compared to the glucose condition). Moreover, on fructose (compared to glucose) pyruvate carboxylase (PC) displays a negative ΔF is positive. Finally, on sucrose (compared to glucose) the PFK) is positive while the ΔF was close to zero. In summary, EFM analysis correctly predicts both the sign and the amplitude of the change in fluxes when the sugar source is changed in C. glutamicum.For m Figure  and 3b, m Figure . ConcernPdh) and  a positiB. napus embryos  was not included in our analysis because its flux measurements yield large error bars (more than 300%)  and negative for glucose uptake (Glc_up). Concerning the nine internal fluxes determined around the TCA cycle , phosphoenolpyruvate carboxylase (Vpepc), ATP citrate lyase (Vcl) and citrate synthase (Vcs) and a positive Vgdh), transaminase , fumarase (Vfum) and glutamate synthase (Vgs). Note that the fluxes involved in nitrogen assimilation, i.e. glutamate dehydrogenase (Vgdh) and glutamate synthase (Vgs), exhibit the highest changes in both measured fluxes and flux efficiencies. Among the 16 fluxes measured experimentally , the direction of the change (sign of ΔF ) of 15 fluxes are accurately predicted by the change in flux efficiencies Vme) is not. To resume, EFM analysis correctly predicts both the sign and the amplitude of the change in fluxes when the nitrogen source is changed for B. napus.In the s Figure , the flue Figure , in agreB. napus, giving a large metabolic network describing the central metabolism of heterotrophic plant cells.The robustness of the previous observations was further tested by adding reactions in the core metabolic network of C. glutamicum involve the main pathways of the central carbon metabolism. In these networks, cofactors such as ADP/ATP, NADP/NADPH and NAD/NADH  are balanced, thus constraining the EFMs not only through the carbon balance but also through the redox and energetic status. In contrast, the fluxes of B. napus embryos were measured on a core network centralized around the TCA cycle without balancing ADP/ATP and NAD/NADH. While the EFM analysis fits the experimental results on this core network, we wonder what would happen to the correlation if the whole metabolic network of heterotrophic plant cells was taken into consideration.The three metabolic networks of B. napus embryos . Based on carbon and energy balances, they are assumed not to be limiting in this system. The other small molecules  are not included in the metabolic network. The other 55 metabolites, including cofactors  and ADP/ATP are internal, meaning that they are expected to be balanced at steady state [Vfbp) and the reversible conversion (from F6P to F16bP) is catalyzed by two enzymes, 6-phosphofructokinase (Vpfk) and fructose-6-phosphate phosphotransferase. However, in this network, only one reaction (Vpfk) is recorded in order to consider only once the isoenzyme in the EFM computation.The complete metabolic network of the dy state . The dirLNm, contains 13 481 EFMs that use only glucose as a carbon substrate and the second set, named LNaa, contains 113 700 EFMs in which at least one of the uptake reactions is present . For these two sets the coefficients of flux efficiency were calculated as previously described in the material and method section. The relative difference of measured fluxes (ΔF) was compared with that of the flux efficiency ΔF and Vidh) is omitted in the plot for the same reason as above (large errors in the measured fluxes). The overall changes in experimental fluxes observed with respect to the nitrogen source are confirmed by the change in the flux efficiency of the corresponding reaction in the enlarged network. The same conclusions as for the core network of B. napus can be drawn: (1) almost similar change in the fluxes towards output metabolites; for the uptake fluxes, the change in both, measured fluxes and flux efficiencies are similar; (2) for the internal fluxes, in agreement with experimental results, the EFM analysis displays a negative Vpk), the phosphoenolpyurate carboxylase (Vpepc), the ATP citrate lyase (Vcl) and the citrate synthase (Vcs) and a positive Vgdh), the transaminase , the fumarase (Vfum) and the glutamate synthase (Vgs). Note that the fluxes involved in nitrogen assimilation, i.e. glutamate dehydrogenase (Vgdh) and glutamate synthase (Vgs), have the highest differences in both ΔF. Again, only the change in malic enzyme flux (Vme) is not correctly predicted by the change in flux efficiencies. But in summary, the relationship between the EFM analysis and experimental measurements in B. napus is confirmed using an enlarged metabolic network of heterotrophic plant cells.With the enlarged network described above, the result of the computation is a set of 114 614 EFMs, indicating a large number of potentially active metabolic routes at steady state. Two subsets of EFMs were built: the first, named k Figure . A stati13C-MFA using maps of both a microbial system (C. glutamicum) [B. napus embryos) [C. glutamicum and B. napus, respectively. More precisely, for C. glutamicum, the change in the internal fluxes measured experimentally (14 on fructose and 15 on sucrose) are accurately predicted by the change in flux efficiencies calculated from the EFM analysis performed for the three sugar sources. For B. napus, the main changes in experimental fluxes observed using the two nitrogen sources  are confirmed by the change in the flux efficiencies of the corresponding reaction in both the core network and the enlarged network describing the metabolism of a heterotrophic plant cell.In the present report, an EFM analysis was performed to compare the structural properties of the metabolic networks  with the fluxes experimentally determined by tamicum) ,5 and a embryos) . By calc13C labelling experiments, whatever the biological system, microorganism or plant cell. Thus, to some extent, changes in metabolic activity following environmental changes could be predicted without considering kinetics. This result is in agreement with [B. napus, the number of EFMs utilizing a given carbon substrate shows interesting properties: glucose (13481 EFMs) is used about 3 to 4 times more often than glutamine (3822 EFMs). Moreover, alanine (576 EFMs) is much less used than glucose (13481 EFMs) or glutamine (3822 EFMs). These observations also match biological intuition [These results establishes a (strong) link between the structure of the network, based on thermodynamic and stoichiometric data, and its metabolic activity, measured by ent with  showing ntuition .B. napus. This confirms and strengthens the link between the metabolic activity measured by 13C-labelling experiments and the structure of the network, especially if the network is large and strongly connected. Indeed, the presence of branch points in a network creates multiple routes between different intermediates and this presents one of the largest challenges for EFMs to correlate with 13C-MFA. Nevertheless, several limitations of this approach have to be mentioned. While most of the differences of flux efficiency are well correlated with the experimental fluxes, some of them do not fit the experimental results. For instance, in C. glutamicum, the flux efficiency of the phosphofructokinase (PFK) decreased on sucrose compared to glucose while the experimental flux did not change. Also for the core and the enlarged network of B. napus, the Vme) is negative while the ΔF is positive. Finally, while four fluxes towards output metabolites  contribute to the relationship in the B. napus network, the eight fluxes towards the output metabolites in the C. glutamicum network are not represented in Figure ΔF is similar for lysine, trehalose, lactate and DHA, the amplitude of the change is rarely similar for these fluxes. In the same way, there is no more coherence between ΔF and It is worth noting that the relationship described above is maintained -and even improved- by enlarging the metabolic network of Assuming that the experimental fluxes are properly determined , this suggests that a part of the network is not accurately designed. Several hypotheses can be made: (1) the lack of knowledge about both the activity and the compartmentalization of the enzymes impedes the correct representation of the network (2) the fluxes towards macromolecules, such as proteins, nucleic acids and starch, are simplified or inaccurately represented, (3) the small compounds  which are assumed not to be limiting in the system plays key roles in the network's functionality, (4) the rate of some reactions (enzymes) can be controlled by substrate-dependent kinetics and, in addition, regulated by effectors. The EFM analysis, based on the stoichiometry and the thermodynamic reversibility of reactions, is a useful tool in the structural modelling of a metabolic network at steady state . However13C-MFA by comparing the networks generated under different environmental conditions (i.e. changes in carbon and nitrogen sources). The observed relationship is defined foremost by the internal fluxes - highly connected between each other - but does not include the fluxes towards output metabolites.Most of the EFM approaches have been carried out on a single model, taking into account the intrinsic correlations between fluxes and their optimization with respect to ATP and biomass synthesis ,20. HereC. glutamicum network, the omission of GAPdh, PFK, and aldolase definitely lead to destruction of the network and the omission of PK and AspK lead to a marked decrease in both the size of the EFMset and the number of fluxes making it impossible to evaluate the relationship previously described (data not shown). In the B. napus network, only the omission of glucose uptake leads to a total destruction of the network, as expected since the glucose is the reference condition. For all the other fluxes of the networks, their omission only leads to a lower number of EFMs but the relationship between the change in measured fluxes and the change in flux efficiency is maintained. The remaining relationship when one reaction is removed from the network can be explained by the flexibility (plasticity) of the network. This highlights the extreme degree of redundancy in the system and the difficulty in pin-pointing any one enzyme as a 'key' or 'regulatory' step in the absence of a global analysis of the whole system. The use of the model provides a rational basis for identifying those enzymes.The relationship between the change in measured fluxes and the change in flux efficiency can be  destroyed by removing one reaction from the network. For instance, in the B. napus). However, this work illustrates that EFM analysis -thanks to the coefficient of flux efficiency- could be used as a tool to analyse a large and connected network and, subsequently, to predict changes in metabolic behaviour before undertaking difficult and time-consuming experiments. Similarly, [Arabidopsis thaliana. Indeed, while the development of new experimental techniques, such as sequencing and chip technologies, triggered the boom in omics data, fluxomics does not measure more than a dozen fluxes simultaneously.With the whole metabolic network of heterotrophic plant cells, a robust network of high redundancy is obtained. Achieving the ability to work with large models is primarily a technical challenge. A bigger network is synonymous with a better one when the knowledge about the functioning of the cell is available to accurately design the network. It is also difficult to build a generic network without taking into account the specificity of the cell and the tissue ; '_out': for exogenous amino acids ; '_p': for metabolites located in plastid; '_v': for metabolites located in vacuole.MBA and SC designed the study and the manuscript. BB, and MDN participated in the design of the study and helped draft the manuscript, FV performed calculations of EFMs and AM helped draft the manuscript. All authors read and approved the final manuscript.Metatool file format for EFM analysisThe networks described in .Click here for fileC. glutamicum and B. napusExperimental fluxes and the corresponding coefficients of flux efficiency .Click here for file