Metabolomic data sets provide a direct read-out of cellular phenotypes and are increasingly generated to study biological questions. Previous work, by us and others, revealed the potential of analyzing extracellular metabolomic data in the context of the metabolic model using constraint-based modeling. With the MetaboTools, we make our methods available to the broader scientific community. The MetaboTools consist of a protocol, a toolbox, and tutorials of two use cases. The protocol describes, in a step-wise manner, the workflow of data integration, and computational analysis. The MetaboTools comprise the Matlab code required to complete the workflow described in the protocol. Tutorials explain the computational steps for integration of two different data sets and demonstrate a comprehensive set of methods for the computational analysis of metabolic models and stratification thereof into different phenotypes. The presented workflow supports integrative analysis of multiple omics data sets. Importantly, all analysis tools can be applied to metabolic models without performing the entire workflow. Taken together, the MetaboTools constitute a comprehensive guide to the intra-model analysis of extracellular metabolomic data from microbial, plant, or human cells. This computational modeling resource offers a broad set of computational analysis tools for a wide biomedical and non-biomedical research community. Omics data are used to determine comprehensively qualitatively or quantitatively cellular components and how they change across different conditions  or from the spent medium of cells in culture (extracellular metabolomic data). Extracellular metabolomic data sets are generated from cultivations of cell lines in order to metabolically characterize them under different experimental conditions  . Additionally, this work provides tools for the analysis of any metabolic model.The tutorials exemplify the workflows for intra-model analysis for 1) quantitative and (2) semi-quantitative extracellular metabolomic data . Matlab code is indicated through >>. Flux units are commonly reported in the Unit mmol/gdryweight/hr (U); however, the unit can be varied depending on data.The first stage provides the basis for the integration of extracellular metabolomic data, i.e., it ensures that a maximal number of metabolites can be integrated with a model. The second stage discusses the application of constraints and the generation of contextualized models. The third stage discusses the quality control of the contextualized models, the computational analysis tools provided by MetaboTools, and finally the validation of the model predictions. Several iterations of the steps in the second and the third stage may be needed to generate high-quality contextualized models and to obtain biologically-plausible model predictions Table . Through6H12O6 as formula.As a first step, the names of detected metabolites need to be associated with the metabolite abbreviations in the model Figure , ChEBI , HMDB W, becauseThe step results in one group of metabolites successfully associated with model metabolite IDs and a second group of metabolites that does not yet exist in the metabolic model. The addition of novel anabolic or catabolic pathways to include the latter group of metabolites into the model can be time-consuming and requires extensive work as well as testing of the model functionality. The necessary steps have been described in detail elsewhere  high-throughput techniques detect more and more comprehensively metabolites in extracellular environments , (ii) these metabolites were outside the scope of previous reconstruction efforts and applications .The correct representation of transport mechanisms is important to accurately simulate cellular metabolism; thus, the metabolic reconstructions are continuously extended  are particularly important for integrating transcriptomic or proteomic data as well as for investigating the effect of genetic alternation in metabolite transporters. After the literature has been mined thoroughly, the new mass- and charge-balanced transport reactions have to be formulated . Alternatively, models may be distributed that mimick with their constraints a particular environmental or genetic condition, e.g., enabling growth on minimal medium under anoxic condition. Hence, the existing constraints of a reconstruction or model may not be a useful starting point for the one's own application. Whereas the directionality of transport reactions and internal reactions is consistent with current biological and thermodynamical data, the definition of the extracellular medium, i.e., the constraints on the exchange reactions . Essential uptake reactions can be identified, e.g., using flux variability analysis (see Appendix).The model requires certain inputs and outputs to have a non-zero value for an objective function. For example, for the production of biomass of a human cell, the uptake of essential amino acids, ions, and other compounds needs to be provided to the model to render the objective function feasible , if it is necessary to prevent that the model is artificially constrained by imposed “infinite” bounds (see Step 16B). If the growth rate, or doubling time, for the given experiment is available, it can be set as constraint on the biomass reaction using the function setMediumConstraints or changeRxnBounds.The function calculateLODs and using the molecular weight of the metabolites . Uptake and secretion profiles for each sample are generated from an input data matrix using the function defineUptakeSecretionProfiles. See Supplementary Tutorial SD. A calculated net change of 5% can be meaningless if the SD is 10%. Hence, caution should be taken when incorporating changes in metabolite abundance, when the change is below or close to the SD. The tutorials illustrate how the constraints can be adjusted to one's data set at the different parts of the workflow.Metabolomic data can be integrated with metabolic models by enforcing exchange rates to uptake or secrete in accordance with the experimental data Figures . To enfocalculateQuantitativeDiffs and integrated with the model using the function setQualitativeConstraints. As a consequence, the flux through an exchange reaction must lay between the smallest detectable and the highest possible flux in case of uptake (defined by the medium composition), and between the minimal detectable efflux and the “infinite” flux value for secreted metabolites . Negative values will be interpreted as uptake and positive values are interpreted as secretion. Based on the input model and user-defined minimal and maximal values, the function prepIntegrationQuant tests whether the uptake and/or secretion of each individual exchange in the input data matrix is feasible, using flux balance analysis , while the upper bound remains unconstrained. Which objective function is chosen should be carefully decided based on the experimental context , is minimized in the resulting model . As the function generateCompactExchModel relies on an approximation, which may not be minimal , the minimization is repeated until the number of added exchange reactions cannot be further reduced. Additional exchanges may be required as untargeted metabolomics methods can still not measure and identify the entire metabolome, despite continuous improvements in the field , which summarizes the numbers of reactions, metabolites, and genes for each contextualized model of the sample set. The information in these variables are extended by the downstream analysis functions (see Step 17).After the minimal set of required exchanges is defined, blocked  reactions are identified  associated with the user-defined set of unexpressed genes. This stringent treatment may lead to an infeasible model, if any metabolite required for the objective cannot be produced anymore by the model. The manual assessment and curation of gene expression data integration has been previously described for a signaling model .Proteomic data can be integrated in a similar way as transcriptomic data. In this case, the protein identifiers need to be matched to the gene IDs in a model, whereby the same attention should be paid to the conversion of protein to gene IDs as for the association of metabolite IDs to the model metabolites. Subsequently, the set of gene IDs that is associated with the absent proteins constitutes the input for in vivo or in vitro but they are not included in the model. Thus, the applied constraints and predicted exchanges need to be manually evaluated against the biological knowledge before proceeding with further computational analysis. If falling below the error range or exceeding the infinite bounds, constraints might need to be scaled . All added exchanges and directions of exchange should be controlled to identify missing or erroneous constraints. For example, in our previous study, the initially generated set of cancer models predicted consumption of superoxide and secretion of oxygen because the constraints had not been appropriately defined . The example illustrates the importance of manual inspection and biological insight when generating condition-specific models from experimental data  to separate upper and lower bound. Even slight differences in the culture conditions may have an impact on the growth performance of the cells.The ability of the contextualized models to grow at the same rate as in the experiment can also be used to validate the contextualized model . The optimization of biomass can be applied to cells in the exponential growth phase, and optimizing for biomass assumes that the cells are thriving toward optimal biomass production  in the model. This can be done manually using the VMH database  that are associated with the list of successfully associated metabolites (*metList*):Alternatively use the function >>  = findRxnsFromMetsCheck also the lower and upper bounds of the transport and exchange reactions to confirm that the directionality corresponds to the reaction formula.To print the reaction list use:>> a = printRxnFormula;*model.rev*) is not updated by all functions in the COBRA toolbox. Most functions rely on the definition of the lower and upper bounds rather than the reversibility vector. Use the following code to make the reversibility vector consistent with the lower and upper bounds:Note that the reversibility vector (>> model.rev = zeros(length(model.rxns));>> model.rev(find(model.lb<0)) = 1;Identify Metabolite Transporter. Manually identify and define as detailed as possible the extracellular metabolite transport systems for all associated metabolites lacking extracellular transport reactions based on the relevant literature . Variations of the key words will provide you with a first impression on the amount of literature that exists on the transport systems of individual metabolites. Transporters vary between compartments. Hence, specifically identify extracellular transport systems. Check for organism, tissue- and cell-type specific metabolite transport mechanisms. Does the literature describe or indicate active or passive transport ? Identify additional substrates and co-substrates , as well as the direction of transport (antiport vs. symport). What is the reaction stoichiometry of the transported compounds? Identify the genes that encode the transport proteins. Consider the existence of isoforms and multiple subunits of protein complexes. Keep track of references to all aspects of the transport. Take notes on contradictory information to enable yourself or others to recapitulate decisions at a later stage. Identify all existing transport mechanisms. Keep the number of diffusion reactions as low as possible by resolving as many transport mechanisms and as detailed as possible. When using information from databases, confirm the correctness of the information based on the primary literature, e.g., confirm that it is extracellular transport and appears in your cell or organism.e Figure ). Start,Add ReactionsReconstruct the transport reactions based on your notes and add them to the model using rBioNet  setMediumConstraints to impose the condition-specific sets of constraints to a model (*model*). When data values exceed the infinite bounds, increase the current bounds (*current_inf*) such that the infinite bounds no longer act as constraints (*set_inf*). Based on cell concentration (*cellConc*), the duration (*t*) of the cultivation, and the cell dry weight (*cellWeight*) fluxes are calculated from metabolite concentrations given in mM (*met_Conc_mM*) for a defined set of metabolites exchange reactions (*medium_composition*), which were part of the defined cell culture medium, and the model is constrained with these fluxes. Uptake of additional compounds, e.g., ions or vitamins can be restricted . The optional input variable *customizedConstraints* allows individual definition of additional reaction constraints, e.g., growth rates.Use the function >>  = setMedium ConstraintsTransform the LOD from ng/ml to mM(6) *theo_mass*) of the metabolites and the respective exchange reaction for the metabolites (*ex_RXNS*) to transform the LOD from ng/ml (*lod_ngmL*) to mM (*lod_mM*).Use the molecular weight ;Define Uptake and Secretion Profiles (No Absolute Quantification)(7) *cond1_uptake* and *cond2_uptake*) and released (*cond1_secretion* and *cond2_secretion*) based on the individual data matrices (*input_A* and *input_B*) and the corresponding exchange reactions (*data_RXNS*) using the function defineUptakeSecretionProfiles for sample pairs. Manually define the threshold (*tol*) to accept a change as uptake or secretion. The function contains further options to manually tailor the uptake  and secretion profiles .Define sets of metabolites that are consumed ;Calculate Semi-Quantitative Differences(8) calculateQuantitativeDiffs to add the LODs to the uptake and secretion profiles and to obtain the values for the relative adjustment of constraints  based on the *slope_Ratio* of the commonly consumed and released metabolites ;Set Qualitative Constraints(9) setQualitativeConstraints to apply individual uptake (*cond_uptake*) and secretion (*cond_secretion*) profile to a model by using the detection limits to enforce uptakes and secretions . Execute this function for each model individually:Use the function >> [modelLOD] = setQualitativeConstraints;*ambiguous_metabolites* to keep exchanges open and let the model freely consume or release the associated metabolites, e.g., if direction of exchange is undefined or differs between biological replicates. All lower bounds of exchange reactions, apart from those defined as ambiguous, mediumCompounds, or from the uptake and secretion profiles, will be constrained to zero. The output model *modelLOD* has the same size of the input model.Use the input variable Apply Semi-Quantitative Differences(10) setSemiQuantConstraints to apply the semi-quantitative differences to two models *modelA* and *modelB*, which were previously defined by the function calculateQuantitativeDiffs . This step is performed simultaneously for the two models.Use the function >>  = setSemiQuantConstraints;Obtain Uptake and Secretion Profiles [Integration of Absolute Concentration Changes], Trouble Shooting(11) prepIntegrationQuant to generate individual uptake and secretion profiles from a data matrix (*metData*) of fluxes, where the rows are the metabolite exchanges (*exchanges*) and the columns are the conditions (*samples*). Negative values are interpreted as uptake and positive values are interpreted as secretion. The individual exchange profile is saved to the user-defined location (*path*). The minimal and maximal flux values  are used to test whether the *model* can consume and secrete the *exchanges*. All fluxes below a user-defined tolerance value (*tol*) are set to zero. The output consists of the flux values to be applied to the upper and lower bounds, which are based user-defined variation  for uptake and secretion, and written to a file named according to the same name to the specified location.Use the function >> prepIntegrationQuant;Check the Exchange Profiles(12) checkExchangeProfiles to generate statistics on the number and identity  of uptake and secretions added per sample (*minMax*). Specify the number of metabolites in the data set (*nmets*). The function loads the exchange profiles generated by using the definition of the *path* and the sample names (*samples*).Check if additional metabolites have been removed from the uptake and secretion profiles. Use the function >>  = checkExchangeProfiles;Integrate Quantitative Constraints, Trouble Shooting(13) setQuantConstraints to generate a contextualized model (*ResultsAllCellLines.sample.modelPruned*) for each sample (*samples*). A minimal flux value (*minGrowth*) can be specified for a defined objective function (*obj*). A set of metabolite exchanges (*medium*) can be specified, which will be part of the retained minimal set of exchange reactions in the output model. Allow additional secretion of metabolites . The exchange profile is loaded from the specified location (*path*). The default value of epsilon is 1e−4, see Vlassis et al. ;*ResultsAllCellLines* contains the pruned and unpruned but constrained contextualized model for each sample (*samples*). *OverViewResults* is a table which lists statistics of the generated models.The output structure Generate the Contextualized Subnetworks(14) generateCompactExchModel to generate a submodel. This pruned model is based on the prediction of a minimal set of metabolite exchanges for a given, constrained model, e.g., the model resulting from running setQualitativeConstraints or setSemiQuantConstraints, while permitting a feasible steady state flux distribution. The prediction of the additional metabolite exchanges is based on the minimization if the cardinality of the exchange reactions . Define the minimal flux value (*minGrowth*) that should be achieved by a defined objective (*biomassRxn*) in the output model . The use of fast flux variability analysis  is optional.Use >>  = generateCompactExchModel;extractConditionSpecificModel to extract the submodel using the approach as in Aurich et al. (*theshold*) for calling a flux value zero .Alternatively, use h et al. . Specify>> [modelPruned] = extractConditionSpecificModel;Integrate Gene Expression Data(15) integrateGeneExpressionData to integrate transcriptomic data (set of genes found to be unexpressed (*dataGenes*)) with the *model*.Use the function >> [modelGE] = integrateGeneExpressionData;Validation of the Model, Trouble Shooting Perform a quality control of the models by checking constraints against biological insight. The steps 16E–F can only be performed if the data exists.(16) Create an expectationIf not done already, review the literature to learn about the biological background of your cells, organisms, and what to expect from the analysis: Identify metabolic traits and inabilities of the target cell and what is known about the respective set of environmental conditions and its response to perturbations. Validate the generated models based on these biological insights, e.g., by asking questions such as those given in Table Check constraints, Trouble ShootingCheck lower and upper reaction bounds in the generated models. Check that all constraints have been applied as intended, e.g., typos in the exchange reaction names might have prevented that constraints were applied. Are all exchange reaction constrained to zero that should have been closed? Do constraints exceed the infinite bounds? Are any constraints below the threshold and thus equal to zero? Are there any constraints that do not concur with the expectation, e.g., uptake of oxygen?Analyze added exchanges, Trouble ShootingstatisticsAddedExchanges and mkTableOfAddedExchanges:Summarize the set of added exchanges using the functions >> [Ex_added_all_unique] = statisticsAddedExchanges;>> [Added_all] = mkTableOfAddedExchanges;Check the additional exchanges that were added to the model. Are the metabolites predicted to be consumed part of the medium composition? Was serum added to the culture medium? For all metabolites, which the model needs to secrete, could these metabolites have been detected? Were they targeted by the metabolomic analysis? Could they have been below the limit of detection? Does untargeted metabolomic data support the presence of these metabolites in the extracellular medium?Check metabolic functions, Trouble ShootingAfter confirming that the constraints are accurately set, check the functionality of the model against the expectation build earlier. Does the model include metabolic pathways known to be operated by the cell and in the considered condition? Which pathways are in the model that are not expected to be present in the modeled cell type? Keep in mind that the model was optimized to explain the uptake and secretion profile and that the internal reaction redundancy remains, if no additional omics data is included. Check, using flux variability analysis, if the unexpected pathways have to carry flux. What imposed exchanges are responsible for the flux through such a pathway?Validate the contextualized model based on experimental dataUse, e.g., intracellular metabolomic data. Check to which extend metabolites reported in intracellular metabolomic data are captured by the contextualized model . Do the models reflect sample-specific differences?Test for condition-specific growth ratessetConstraintsOnBiomassReaction to apply constraints to the biomass reaction based on doubling times (*dT*). The upper and lower bounds of the biomass objective function (*of*) are adjusted with a *tolerance* around the calculated growth rate .If growth rates were not applied as constraints during the model building, use the function >> [modelBM] = setConstraintsOnBiomassReaction;changeRxnBounds to apply growth rates to the model. Make sure the objective is set in the model (use the function changeObjective to change the objective in the model).Alternatively, use the function optimizeCbModel) to see if the model can produce biomass at experimental rates.If no doubling times or growth rates exist for the data set, literature derived rates can provide an approximation for the expected growth rates, as growth rates can vary depending on the culture conditions . Use flux balance analysis  Choose the most suitable analysis  based on the individual biological question to gain insight into the use of the most relevant pathways, reactions, or metabolites by individual or groups of models.Summarize and illustrate the set of essential genesanalyzeSingleGeneDeletion to define and to analyze the set of essential genes for single or large set of models . Define through the input variable *heat* if a heat map should be generated, illustrating the differences across the model set .Use the function t Figure . Save th>>  = analyzeSingleGeneDeletion ;*genes* contains a table that lists per unique gene the number of models with “no effect,” “partial effect,” essential genes or “KO,” and in how many models the gene is not present. Additionally, the number of essential genes is added to *OverViewResults* and the output of the single gene deletion is added to *ResultsAllCellLines*.The output variable Identify the essential reactions driving gene essentiality, Trouble ShootingcheckEffectRxnKO to identify, which reactions are essential. Provide the set or subset of models (*samples_to_test*) from *samples* that should be tested. The functions tests all individual reactions associated with a set of specified genes (*genes_to_test*). The variable *fill* defines what to add into the output matrix if a reaction is not in the model, e.g., 100, num (“NAN”).Use the function >>  = checkEffectRxnKO;*FBA_Rxns_KO* contains the FBA results for constraining one reaction at a time to zero. The second output *ListResults* lists the reactions associated with the input gene list in the same order as *FBA_Rxns_KO*.The output variable Generate intersect and union modelmakeSummaryModels to find the shared and superset of reactions, metabolites and genes of a set of models by creating a union (*unionModel*) and an intersect model (*intersectModel*). The models are generated based on the generic model (*model*) from which the subnetworks have been generated.Use the function >>  = makeSummaryModels;*unionModel*, *intersectModel*) and differential reaction sets  are generated by default.Both output models (Predict flux splitspredictFluxSplits to predict either production or consumption (*dir*) of metabolites  for the models in *ResultsAllCellLines* defined by *samples*. Define the objective function (*obj*) that should be used to generate the flux vector, from which the reaction contributions to production or consumption of the considered metabolite are calculated. Define a full sized model (*model*), as the set of models can use distinct subsets of the reactions of the full model to produce or consume the metabolite of interest. A default value (eucNorm = 1e−6) is set for the parsimonious flux balance analysis. Set a quadratic programming solver [changeCobraSolver] before running predictFluxSplits.Use the function >>  = predictFluxSplits;*transportRxns* = ), and subsequently, exclude unwanted contributors by adding them to *transportRxns*.Exclude reactions, which do not produce or consume the metabolite, e.g., reactions that transport the metabolite between compartments. Define all reactions that produce/consume the metabolite in a first run (*ATPprod* = 1). With this option, the *ATPyield* is calculated, which relates the total ATP production to a user-defined carbon source .In case that the production of ATP is predicted, additional statistics on the use of glycolysis, the citric acid cycle, and the oxidative phosphorylation can be automatically generated  to simplify the analysis over large groups of samples (and metabolites of interest). Detailed results of the analysis are added to *ResultsAllCellLines*.The output lists the reaction with highest flux value for producing or consuming a defined metabolite across analyzed samples ;summarizeSamplingResults:The function is based on COBRA toolbox, refer to the corresponding protocol for details ;*path*, Figure Appendix), generate more sample points by using points from the last file as starting point . Analyze the results: Are the distributions shifted apart for any reactions or pathways that were expected or unexpected for a certain condition or group of samples? Make sure that your interpretations do not rely on unbounded reactions  at a time and for one or more submodels (*samples* in *ResultsAllCellLines*). Define the range of flux rates (uptake or secretion defined in *direct*) through the variables *step_size* (flux units) and number of steps *step_num*. This can be individually defined for each exchange reaction. The output is added to *ResultsAllCellLines*.Use the function >> [ResultsAllCellLines] = performPPP;illustrate_ppp.Illustrate your results with heat-maps using the function >> illustrate_ppp;*path*. The function uses flux variability analysis  needs to be specified. The function allows some additional inputs, i.e., label and font size , to customize the illustrations. Analyze the output, e.g., by manually grouping the models based on the similarity of the planes in the heat maps .The heat maps are saved to the defined Validate the Predicted, Metabolic Phenotypes(18) Compare to experimental dataCompare the results of the previous steps to data sets, e.g., cell culture and perturbation experiments, or compare the predictions to the results of the analysis of additional omics data, e.g., mRNA. Does regulation in metabolic genes support predicted metabolic differences between cells or conditions?Compare to literatureCheck if the results of the predictions generated in the previous step agree with what is reported in the literature. For example, has the cell line been previously reported to heavily rely on an alternative pathway, e.g., reverse flux through the citric acid cycle? Are the essential genes already known to be essential in this cell type, or under the experimental conditions, such as hypoxia?Step 1: Metabolites Cannot Be Associated with Model MetabolitesAppendix; Thiele and Palsson, Appendix;  may remain part of the steady-state solution space. However, no wrong flux distributions are introduced! Another solution would be to add demand reactions . Thus, the model should be tested for leakage before proceeding to further steps  of the existing reaction agrees with your notes. Add the reaction with a new name corresponding to the transported metabolite. If the transport mechanism is different from the reaction mechanism already present in the model (particularly for diffusion reactions), review whether the original reaction needs to be updated based on the novel literature clues. The mechanism might have been better resolved since the original reconstruction . Correct the original reaction in the model based on the result of the literature review. Remove diffusion reactions if no longer adequate.Add, replace, or modify reactions if literature provides novel, convincing clues on the metabolite transport mechanisms. If diffusion is unlikely for a metabolite, consider removing the diffusion reaction that allow the transport “for free.” Add a reaction if the metabolite constitutes an additional substrate to an existing transporter. Make sure that the new substrate is transported in the same way as the original one before duplicating the reaction. Does the reaction mechanism correspond to that in the model? Check if the reaction stoichiometry, directionality, co-substrates, genes, and Gene-protein-reaction association Step 5: Identify Minimal Set of Exchanges for a Model and ObjectiveUse flux variability analysis (FVA) to identify mandatory metabolite uptakes .Step 7: Detection Limits Are UnavailableAn arbitrary low value may be used to set the upper bounds on uptake fluxes and lower bounds on secretion fluxes that is close to the cutoff value .Step 13: Generate Metabolic Fluxesconc2rate to convert concentration changes (*metConc*) measured over the time course of the experiment (*t*), for a certain number of cells (*cellConc*), and dry weight (*cellWeight*) into flux (*flux*).Use the function >> [flux] = conc2Rate;Step 16B: Constraints Are Too Small and Fall Below the Error Range and/or Exceed the Infinite Bounds (Cobra Toolbox Default Is 1e−8)setMediumConstraints).Multiply or divide all reaction bounds (except infinite constraints) with the same factor. Adjust the unit the fluxes accordingly using the same factor. Scale the unit the fluxes are reported in with the same factor. Repeat the previous steps to apply the reconciled constraints. If scaling is not possible because the range of fluxes is too wide , the infinite bounds can be increased .Close exchange reaction in the starting model and repeat the model building by Step 16D: The Model Contains Unexpected PathwaysThe contextualized models miss central features of the cells or certain metabolic functions cannot be performed. Note that the models constructed herein are condition-specific. Are the reported pathways really active in the considered experimental condition? Identify whether there is an exchange connected to the pathway, which is not defined by the consumed metabolites. Could these be missing from set of targeted metabolites? You can add the metabolite to the medium composition  to prevent that it is removed during model building. Iterate until models are conform to the expectation.Keep in mind that the integration of metabolomic data only indirectly causes the removal of metabolic pathways. Pathways are only removed if they become blocked as a consequence of exchanges being closed. An additional reduction of the internal metabolic network can be achieved by the integration of additional omics data sets, if the imposed flux rates are not deterministic enough. Step 17: Generate the VariablesResultsAllCellLines andOverViewResultsto Collect the Information of the Downstream Analysis for Models Not Generated Herein*model*, *model2*, … loaded in the workspace, execute:For models of name >> samples = {‘model’, ‘model2’, …};>> for i=1:length(samples)>> model_name= samples{i}>> eval([‘ResultsAllCellLines.’ model_name '.modelMin =' model_name]);>> eval([‘ResultsAllCellLines.’ model_name '.modelPruned =' model_name]);>> end*modelMin* has all reactions, whereas the pruned model is the reduced model. If only one exist, both can be equal. In the workflow the *modelMin* is used for the single gene deletion, instead of the *modelPruned*.The difference between the models is that the OverViewResults, execute:To generate the variable >> samples = {‘model’, ‘model2’};>> cntO=1;>> OverViewResults{1,cntO} = ‘samples’;>> for i=1:length(samples)>> OverViewResults{i+1,cntO} = samples{i};>> end*samples*.Some of the analysis require the specification of a generic model, which is a superset of the models specified in Step 1, Map metabolite identifiers to Recon IDs, Timing: ~hours.Step 2, Find which metabolites cannot be transported in the model, Timing: ~minutes.Step 3, Identify missing metabolite transporter, Timing: depends on number ~ hours—days.Step 4, Add reactions, Timing: ~hours.Step 5, Define medium constraints, Computational Time: ~seconds.Step 6, Calculate detection limits, Computational Time: ~seconds.Step 7, Define uptake and secretion profiles, Computational Time: ~seconds.Step 8, Calculate semi-quantitative differences, Computational Time: ~seconds.Step 9, Set qualitative constraints, Computational Time: ~seconds.Step 10, Apply semi-quantitative differences, Computational Time: ~seconds.Step 11, Obtain uptake and secretion profiles (flux rates), Computational Time: ~seconds.Step 12, Check the exchange profiles, Computational Time: ~seconds.Step 13, Integrate quantitative constraints and generate minimal exchange model, Computational Time: ~minutes per model.Step 14, Generate the subnetwork by pruning model, Computational Time: ~minutes.Step 15, Integrate gene expression data, Computational Time: ~seconds.Step 16, Validate the model, Timing: ~ days–weeks.Literatue review to create an expectation, Timing: ~ days to weeks.Check constraints, Timing: ~ minutes to hours.Analyze added exchanges, Timing: ~ minutes to hours.Check metabolic functions, Timing: ~ minutes to hours.Comparison to experimental data, Timing: ~ days to weeks.Test growth rates, Computational Time: ~ seconds.Step 17, predict metabolic phenotypes, Timing: ~ hours to weeks.Define essential genes, Computational Time: ~minutes.Identify essential reactions behind the gene essentiality, Computational Time: ~seconds.Generate intersect and union model, Computational Time: ~seconds to minutes.Predict flux splits, Computational Time: ~seconds per model.Sampling the steady-state solution space, Computational Time: ~hours to days per model.Predict behavior to environmental changes, Computational Time: ~minutes to hours per model.Step 18, Validate predicted, metabolic phenotypes, Timing: ~ days to weeks (or months)Contributed code to MetaboTools: IT, MA, RF. Designed the tutorials: MA. Wrote the paper: MA, RF, and IT.This work is supported by the Luxembourg National Research Fund (FNR) through the National Centre of Excellence in Research (NCER) on Parkinson's disease, by the FNR ATTRACT program grant . RF is funded by the U.S. Department of Energy, Offices of Advanced Scientific Computing Research and the Biological and Environmental Research as part of the Scientific Discovery Through Advanced Computing program, grant #DE-SC0010429. This project has also received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 668738.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Flux Balance Analysis is a theoretically elegant, computationally efficient, genome-scale approach to predicting biochemical reaction fluxes. Yet FBA models exhibit persistent mathematical degeneracy that generally limits their predictive power.We propose a novel objective function for cellular metabolism that accounts for and exploits degeneracy in the metabolic network to improve flux predictions. In our model, regulation drives metabolism toward a region of flux space that allows nearly optimal growth. Metabolic mutants deviate minimally from this region, a function represented mathematically as a convex cone. Near-optimal flux configurations within this region are considered equally plausible and not subject to further optimizing regulation. Consistent with relaxed regulation near optimality, we find that the size of the near-optimal region predicts flux variability under experimental perturbation.Accounting for suboptimal solutions can improve the predictive power of metabolic FBA models. Because fluctuations of enzyme and metabolite levels are inevitable, tolerance for suboptimality may support a functionally robust metabolic network. Natural selection, acting within the laws of physics and chemistry, produces systems well adapted to local conditions. This basic evolutionary insight is formalized in constraint-based modeling. A biological system might be understood as the solution to a precisely formulated mathematical problem of optimization under constraint ,2.E. coli, for which nearly complete knowledge of the metabolic reaction network is available [a priori prediction of all cellular metabolic fluxes.A constraint-based approach is often used to model metabolism in vailable ,4. The evailable ,6. With vailable . This opThe objective most often chosen for microbial models is the maximization of growth rate or yield, an approach we call simply Flux Balance Analysis (FBA). The utility of FBA relies on the assumption that growth rate approximates overall fitness and is the primary focus of selection. Numerous successful applications attest to the value and versatility of FBA for predicting growth rates in a variety of contexts -11. HoweIn practice, FBA generally cannot predict a unique rate for all fluxes. A solution which maximizes growth rate is typically mathematically degenerate, describing a region in flux space rather than a single point. Solution degeneracy is a well-described problem in systems in systems like metabolism which are flexible, internally redundant, and underdetermined by data . ApplicaB. subtilis, some of which cause increased growth rates and biomass yields relative to wild type [In principal, metabolism cannot function only to maximize growth rate. This is evidenced by deletions of metabolic genes in ild type . Selectiild type . Thus weWe address both difficulties with a revision of the metabolic objective at the heart of genome-scale metabolic optimization models. We propose that microbial metabolism is better represented as a cloud of nearly optimal flux distributions, rather than a single ideal and fixed solution. Under this hypothesis, regulation drives cellular fluxes to within a degenerate optimal region. Solutions within this region are not further optimized and considered equally plausible. We present a mathematical formulation of this theory that preserves the properties of continuous and convex optimization that make standard FBA mathematically elegant and computationally quick.E. coli. Our model attributes central metabolism with significant flexibility to negotiate the trade-off between optimizing growth rate and matching a target flux vector. This is particularly relevant for metabolic engineering applications when genetically modified cells must adapt to suboptimal flux profiles.Our model allows stronger predictions in some cases, despite strictly weakening the optimality assumption that grounds standard FBA. The Perturbed Solution Expected Under Degenerate Optimality (PSEUDO) outperforms comparable methods in predicting the redistribution of central carbon fluxes that occurs in metabolic mutants of The success of the PSEUDO method suggests that degenerate optimality may be a basic organizing principle for metabolism. In support of this hypothesis, we will show that reported measures of flux variability correlate with the dimensions of the degenerate optimal region. More degenerate fluxes, which theoretically do not require precise values for fast growth, indeed exhibit more variation. Tolerance for numerous optimal and near-optimal flux configurations would be an asset to microbial metabolism, enabling robust growth in the face of perturbations to the network. A degenerate organization of metabolism may be an essential adaptation given fundamental physical constraints on the ability of cells to control their internal and external environments .Figure pgi gene in E. coli eliminates phosphoglucose isomerase activity and constrains flux through that reaction to be zero. A mutant model is then re-solved to predict a new growth rate optimum  . A mutanin silico for sets that improve production of a metabolite of interest.Both FBA and MOMA use convex objective functions and convex constraints. Applications of these models can therefore access a powerful suite of convex programming algorithms . A flux Although solutions to FBA and MOMA problems are guaranteed to be globally optimum, they are not guaranteed to be unique. In practice, many different flux profiles allow equally optimum growth  is immediately diagnostic for this degenerate case. A second form of degeneracy is possible in our model if the near-optimal and mutant polytopes align such that multiple solutions share the same non-zero Euclidean distance. This case is similar to classical FBA, in that an optimal objective value is defined but does not correspond to a unique solution.As detailed in the Methods section, we can reformulate the objective ║p-q║ as a problem for either quadratic or conic convex programming. Therefore this formulation retains the desirable computational properties of FBA and MOMA, i.e. a guaranteed global optimum can be computed rapidly.E. coli deletion mutants from the Keio collection in defined glucose medium [Genome-scale optimization methods have a well-established utility for predicting growth rates of mutant strains under a variety of conditions. We compared growth rate predictions using the PSEUDO method to predictions from the FBA and MOMA techniques, both of which are commonly used for this purpose . Predicte medium . We compρ = 0.55, 0.54, 0.55, respectively). This general similarity was expected. Mutations which stoichiometrically block the production of biomass must be lethal by all three methods. Mutations in pathways that are not active in these media conditions will not affect growth predictions in any model.The PSEUDO method was able to correctly predict the lethality phenotype for 88% of the deletion mutants examined. Results were comparable with FBA and MOMA  and consistent with previous reports . A deletHowever, PSEUDO growth predictions were found to differ by more than 5% from either FBA or MOMA predictions for 41 of the examined mutants Figure A. This sWithout exception, the PSEUDO growth prediction falls between those of FBA and MOMA. This pattern highlights an important feature of the PSEUDO objective function. FBA, by definition, calculates the highest possible growth rate consistent with thermodynamics and the conservation of mass. MOMA and PSEUDO growth predictions will both fall short of this maximum if there is a trade-off between matching the optimal growth rate and matching the rest of the wild-type flux vector. PSEUDO relaxes this trade-off by considering a range of targets within the degenerate optimal region and therefore PSEUDO growth rate predictions generally exceed those of MOMA.p-value: 0.2). In contrast, FBA systematically overestimated growth rates by 0.48 on average . MOMA predictions were lower than predicted values by 0.13 on average, and were significantly biased to underestimation .The growth predictions that differ among the three models are plotted against measured growth data in Figure E. coli metabolic deletion mutants as determined by 13C-tracer experiments were reported by Ishii and [We next sought to compare the biochemical flux predictions derived from our model to empirical flux measurements. The Metabolic flux rates of 31 central carbon reactions in 24 shii and . We founFigure p-value: 2.4·10-12, Meng’s Z-test). Because measured flux values span several orders of magnitude, we also compared predictions to data using rank correlation coefficients that are less influenced by numerical outliers. An overall Spearman rank correlation of 0.82, 0.80, and 0.87 was obtained with FBA, MOMA and PSEUDO respectively. The higher coefficient from PSEUDO was again significant by bootstrap resampling .For a global view of the metabolic behavior predicted by the FBA, MOMA and PSEUDO objective functions, we aggregated and examined predictions for all 320 measured fluxes across 12 mutants Figure ABC. Acrop-value: 0.01). Thus, both the absolute level of TCA cycle flux and flux variations within the TCA cycle were better predicted by PSEUDO. Relatively high error in TCA cycle predictions when using growth as an objective in carbon-limited conditions has been reported in other models [Prediction accuracy was found to vary substantially for different pathways within central carbon metabolism. The histograms in Figure r models ,27. We fzwf mutant as a case study . This increase in total carbon oxidation is consistent with the higher TCA cycle flux predicted by PSEUDO. It may be explained by the relaxation in PSEUDO of strict optimality requirements used in the other methods. PSEUDO is not explicitly driven, like FBA, to optimize biomass production. Nor does it produce excess metabolites and secrete them, like MOMA. With less carbon flux dedicated to these objectives, the PSEUDO model retains more carbon to fully oxidize.The PSEUDO objective function predicts the highest CO2 output on the behavior of each model. We constrained the total CO2 output flux to a series of specific values near the WT optimum and re-solved the zwf mutant model to determine the fluxes to growth and organic secretion. The resulting plots reveal the trade-offs confronted in each objective function when assigning central metabolic fluxes near optimality .To further characterize the assignment of TCA cycle fluxes in PSEUDO, we extended our analysis to examine the COe Figure EFG. We ozwf mutant shows no oxidative PPP activity in published observations [zwf gene product (together with pgl) supplies 6-phosphogluconate to the oxidative reactions of the PPP. In the zwf mutant, the FBA and MOMA models alternately supply this molecule through the action of glucose dehydrogenase and glucokinase. While E. coli is capable of oxidizing glucose directly to gluconate, the activity appears only under a narrow range of conditions not found in laboratory cultures [In fact, the rvations . During cultures .zwf models . The modified bounds b’L and b’U are applied to the subset of fluxes eliminated by mutation, mMUT.The stoichiometric matrix, p and q. Constraints are added to place p within the region of degenerate optimality and impose a mutation on q.The PSEUDO method finds the minimum Euclidean distance between two high-dimensional flux polytopes, S·p = 0 and S·q = 0. This form is standard for most solver algorithms.Note that (5) is identical to (2) from the main text, except we have concatenated the conservation-of-mass constraints In order to be analyzed with the powerful tools of convex programming, an optimization problem must be formulable with a convex objective under convex constraints . The sysMinimizing a positive radicand is equivalent to minimizing the root, so we neglect the radical. The objective then expands to:I representing the identity matrix:Finally we reformulate the above as a linear algebraic expression, with The square matrix in the above expression is symmetric and positive semidefinite, with eigenvalues of only 2 or 0. This is sufficient to guarantee convexity and polynomial solvability .x to our variable set with the linear constraint that xi = pi - qi. The x vector and a new variable, zDIST, are further constrained to lie within a convex quadratic cone C such that zDIST is at least the Euclidean distance between p and q. Minimizing zDIST under these constraints produces the shortest distance from p to q.It is also possible to formulate PSEUDO as conic optimization problem. Conic optimization is among the most general forms of nonlinear convex optimization treated by commercial solvers. In this case we add a vector All convex programming was implemented in MATLAB and solved using MOSEK optimization software.E. coli metabolic simulations were implemented using the iAF1260 model, incorporating stoichiometric and thermodynamic but not regulatory constraints [-1 hr-1, with ammonia, phosphate, sulfate and trace minerals available in excess. Growth-associated ATP consumption was set to 59.81 mmol gDw-1, with 8.39 mmol ATP gDW-1 hr-1 required for growth-independent cellular maintenance. Model conditions correspond to a standard media formulation as used in the experimental data set [All straints . Media pdata set . Flux vaThe MOMA solver requires a wild-type flux distribution as input. In all cases, this was calculated as an FBA prediction for the wild type strain with the same media conditions and biomass composition. Empirically measured fluxes were not an input to any solver.A secondary objective function was used with the FBA and MOMA techniques to minimize the 1-norm of a growth-optimal solution vector. For the FBA method, this secondary objective was applied directly to the mutant solution. For MOMA, the secondary optimization was applied to the FBA-derived wild-type flux vector.Unless otherwise indicated, the PSEUDO solutions reported were calculated using a growth threshold parameter of 90% the WT maximum, as calculated with standard FBA.2 output or phosphogluconate dehydrogenase to a range of exact values and re-solving each model. Molar carbon secretion fluxes were obtained as the sum of all output fluxes weighted by the number of carbon atoms in each output molecule.Sensitivity analysis was performed by constraining the fluxes corresponding to CORandom flux configurations were generated by Monte Carlo sampling .We used Briefly, a random set of initial points was generated within the degenerate optimal region by solving a linear programming objective to maximize a random vector chosen on the unit sphere. These solutions sample the extreme vertices. The hit-and run algorithm is initialized from the center of these extrema, which by the definition of convexity must also lie within the degenerate optimal region. A direction is chosen as the difference between a random extreme point and the center point. Then a line is drawn in this direction from the initial point to the region boundary and a new point is selected randomly from this line. After a sufficiently large number of such movements, the point reached is as drawn from a uniform distribution.et al., we define a mixed fraction for a set of points that allowed us to determine when our algorithm has reached an approximately uniform distribution [Following Bordbar ribution . We partThe authors declare that they have no competing interests.EW and TL constructed the theory and derived the model predictions. EW, TL and PS analyzed the results and drafted the manuscript. All authors read and approved the final manuscript.Degeneracy in Metabolic Flux Analysis.  Two alternate flux distributions in central carbon metabolism that support equally optimal growth. Configuration A uses the pentose phosphate cycle to oxidize glucose. Configuration B runs the pentose phosphate cycle in reverse only as a source of precursor metabolites. While A produces more NADPH, B produces more pyruvate for the TCA cycle. Both options identically supply the ATP, reducing equivalents and carbon skeletons needed for growth.Click here for fileFluxes vary widely within the degenerate optimal range. As described in the main text, the degenerate range is the gap between the minimum and maximum possible values for each flux that support above a threshold near the maximum value.  The degenerate range in these three panels is calculated under the constraint of 99.9%, 99% and 95% maximal growth, respectively. The flux magnitude is calculated for the wild type system using FBA. Fluxes above the grey dotted line can vary by more than 100 times their predicted value without compromising growth. Fluxes on the red dotted line can vary by exactly 1%. Infinite flux variability is possible for fluxes in futile cycles. Zero flux variability indicates a metabolic reaction that cannot occur under these media conditions. Example enzymes from various metabolic pathways are indicated. (D) Flux variability for selected reaction classes. For nonzero fluxes, fold degeneracy is the degenerate range divided by the FBA-predicted flux magnitude. The median fold degeneracy is plotted for each class, with bars indicating the 10th and 90th quantiles.Click here for filep-value less than 0.05.Correlations of flux predictions by three methods. 31 Measured flux values from the Tomita data set were compared to predictions using the FBA, MOMA and PSEUDO objective functions. Reported values are Pearson correlation coefficients. Meng's Z-test was used to test the hypothesis that the PSEUDO-derived correlations were higher than those from each other method. Similar significance results were obtained by bootstrap resampling. Entries in bold and marked with an asterix indicate that the PSEUDO method was not more predictive than both other methods at a Click here for fileFlux variability under perturbation and the shape of the degenerate optimal region. As described in the main text, flux measurements were collected from two published data sets. The Tomita data set consists of 31 fluxes measured in 24 deletion mutants for central carbon processing enzymes. The Sauer data set includes 24 fluxes from 91 deletion mutants for regulatory transcription factors. The standard deviation, σ, divided by the mean flux value yields the coefficient of variation, CV, a unitless measure of flux variability. The Markov Chain Monte Carlo method was used to generate 3000 random points uniformly distributed within the degenerate optimal region of the iAF1260 FBA model of E. coli. Each of these points corresponds to a possible flux profile sustaining growth at at least 90% of the theoretical maximum. The mean and standard deviation of the MCMC-generated points yield an estimated CV representing the variation possible for each flux while still maintaining near-optimal growth. As depicted in Figure Click here for file


Microbial population dynamics in bioreactors depend on both nutrients availability and changes in the growth environment. Research is still ongoing on the optimization of bioreactor yields focusing on the increase of the maximum achievable cell density.Saccharomyces cerevisiae cultured on glucose as carbon and energy source. The model considers the main metabolic routes of glucose assimilation (fermentation to ethanol and respiration) and the occurrence of inhibition due to the accumulation of both ethanol and other self-produced toxic compounds in the medium. Model simulations reproduced data from classic and new experiments of yeast growth in batch and fed-batch cultures. Model and experimental results showed that the growth decline observed in prolonged fed-batch cultures had to be ascribed to self-produced inhibitory compounds other than ethanol.A new process-based model is proposed to describe the aerobic growth of The presented results clarify the dynamics of microbial growth under different feeding conditions and highlight the relevance of the negative feedback by self-produced inhibitory compounds on the maximum cell densities achieved in a bioreactor. Saccharomyces cerevisiae, which is a reference model biological system [Microbial cell populations growing in a closed vessel (batch culture), under suitable environmental and substrate conditions typically show an initial exponential proliferation followed by a decline in growth rate and transition to stationary phase . Such dyl system  and a mil system –7.S. cerevisiae may follow two different pathways: aerobic respiration to CO2 and H2O or fermentation to ethanol. Furthermore, S. cerevisiae is a glucose-sensitive yeast [Glucose catabolism of the yeast ve yeast , which mve yeast , or glucve yeast   is added to the growth medium according to the bioreactor feeding process. Glucose uptake is followed by the glycolytic process, which produces different products (P), from glucose-6-phosphate to pyruvate. These products can follow two distinct pathways for the construction of new cellular material (CM), either respiration or fermentation. Ethanol (E) produced by fermentation can also be used, instead of glucose, as carbon source for the respiratory pathway.Glucose (I), representing growth-associated by-products, which accumulates in the growth medium. Ethanol and the inhibitor are assumed to separately exert a negative feedback on cell growth in a concentration-dependent way.The main feature of the model is the inclusion of an inhibitor (R), and (4) induction of mortality with accumulation of dead cells (D).The second essential assumption is the key role of the glycolytic pathway in the regulation of yeast cell metabolism. High levels of glycolytic products are assumed to be responsible for (1) the activation of aerobic fermentation due to overflow metabolism, (2) repression of respiration (“glucose effect”), (3) accumulation of reserve materials (Feeding) is defined according to the different experimental settings. Metabolic processes listed in Table Uptake, RespirationP, RespirationE, Fermentation and Accumulation) are modelled based on Michaelis–Menten kinetics with respect to a given substrate and with a first order dependency on the active biomass pool (B = P + CM) modified according to the model assumptions made to represent S. cerevisiae physiology. The uptake of glucose (Uptake) and the consequent formation of glycolytic intermediates are also limited by product (P) saturation and the inhibitory effect of ethanol accumulation in the growth medium [RespirationP, RespirationE) are inhibited by the extracellular accumulation of both ethanol and the inhibitor, and also by elevated intracellular concentrations of glycolytic intermediates (glucose effect) [Fermentation) is inhibited by ethanol and the inhibitor, while it is only activated at elevated intracellular concentrations of glycolytic products (metabolic overflow [Accumulation of reserve molecules (mainly glycogen) is activated at high intracellular concentrations of glycolytic intermediates (mainly glucose-6-phosphate [R) saturation.The setup of the glucose supply to the reactor (h medium . The modh medium . The res effect) . Similaroverflow ). Accumuhosphate ) and is I (Secretion) is assumed to be related to anabolic pathways hence it is expressed as a proportion of the respiration and fermentation fluxes. Cell death is assumed to be induced by elevated concentrations of glycolytic products  of the cellular components P, CM and R respectively.Moreover, the production of the inhibitory compound hosphate , 52) andBased on the above model description, the following mass balance equations have been written:The mathematical equations of this model are described in detail in Tables S. cerevisiae.Model simulations were compared to classic experiments of yeast growth in both batch  and fed-S. cerevisiae LBG H 1022 strain growing on glucose as carbon and energy source in a classical experiment by von Meyenburg [The model has been used to simulate the aerobic batch culture of eyenburg . Figure 2 = 0.984; Glucose R2 = 0.987; Ethanol R2 = 0.947) describing yeast proliferation during the diauxic growth in this aerobic batch culture. In such conditions (short term culture), no evidence of inhibitory effect exerted by either ethanol or self-produced inhibitors was observed.The calibrated model produced a very good agreement between the experimental data and the simulated curves  of 3 h, followed by a constant feeding. In the early phase of feeding (1.8 h) glucose uptake was lower than expected due to adaptation of yeast cells to the culture conditions as reported by the authors themselves. In this adaptation phase, glucose was not consumed and accumulated in the medium. Then, growth accelerated and glucose drastically decreased . In this study case, neither experimental data nor simulation results showed evidence of growth decline related to any inhibitory effect.With the exception of some underestimation of the maximum level of ethanol concentration, the model was able to fit the experimental data at very significant levels , a 30% of the total yeast population resulted non-viable in accordance to the simulation (data not shown).In the case of the prototroph ics Fig. . Feeding2 = 0.991; Glucose R2 = 0.885; Ethanol R2 = 0.934). This result was achieved only if the effect of the negative feedback by the inhibitor was considered in the simulations . The profiles of cell mass, residual glucose and ethanol during both the batch and the feeding phases were similar to those of the prototroph . In this case, a 50% of the total yeast population resulted non-viable (data not shown).Also in the case of the auxotroph -1C Fig.  the modeoph Fig. . However−1). These experiments represent a strong validation of the model since a highly significant fitting of cell mass (R2 = 0.983) between experimental data and simulations was maintained under all the examined feeding conditions  used as SFR. This theoretical feeding profile was used to set a new fed-batch experiment with S. cerevisiae CEN.PK 2-1C strain.Moreover, an additional simulation was performed to assess feeding conditions that could avoid the switch to a fermentative metabolism during the fed-batch phase, while keeping high biomass yields. Such goal was achieved by adjusting the reactor feeding according to a logistically decreasing specific growth rate  Fig. .Fig. 7MeThe mathematical model presented has been developed to simulate yeast growth on glucose as carbon and energy source in aerated batch and fed-batch cultures, and it is based on two main assumptions. First, a central metabolic hub is assumed to regulate (1) the shift between the respiratory and fermentative pathways and (2) cell death. Second, self-inhibiting by-products and ethanol are assumed to act as concentration-dependent inhibitors of cell growth.It is well known that pyruvate plays a central role in yeast metabolism , represeP), which at high concentrations progressively trigger the activation of the fermentative pathway (“overflow metabolism”), the repression of respiration (“glucose effect”) and finally cell death.Taking into account the abovementioned findings, the model attributes the role of central metabolic hub to pyruvate and/or other intermediate metabolites produced along the glycolytic pathway . On the other hand, it is worth noting that in the experiment characterized by a variable feeding, the cells were never exposed to high glucose concentrations .Ethanol, the main end-product of glucose fermentation, exerts its toxic effect at multiple levels on yeast cells, acting as a non-competitive inhibitor of growth at concentrations higher than 5% v/v. , 64. BotS. cerevisiae during diauxic shift and stationary phases [The model presented was able to reproduce the dynamic behaviour of several yeast strains growing both in batch and fed-batch cultures. It is interesting that a very simplified System Dynamics model has been sufficient to capture the major dynamics of yeast metabolism and proliferation under different feeding conditions. From the applied point of view, the good prediction performance of the model suggests its possible use for the optimization of feeding strategies aimed to maximize biomass yield and glucose saving. From a theoretical point of view, these results support the importance of negative feedback processes in the understanding of microbial growth processes. Further investigation will be necessary to determine the chemical nature of the inhibitory compounds, other than ethanol and acetate, involved in such negative feedback, and the related mechanisms of action. The identification of the inhibitory compounds will be performed through GC- or LC–MS and NMR methodologies and the study of the mechanisms of action will start with the analysis of available “omics” data, especially those concerning gene expression of y phases . MoreoveThe model has been first developed in the SIMILE (Simulistics Ltd) visual modelling environment in order to facilitate the discussion within the multidisciplinary team during the implementation phases. Then, the mathematical equations were integrated using MATLAB R2012b (the MathWorks) with a variable order solver (ode15s) based on the numerical differentiation formulas (NDFs) particularly efficient with stiff problems .1, n2, n3 are the number of samples per observed outputs, ith measured outputs and ith outputs predicted by the model. The minimization was performed by using the fminsearch MATLAB routine which implements a Nelder–Mead simplex algorithm [The model calibration was performed by minimizing the sum of the squared errors (SSE)lgorithm . Units aS. cerevisiae strains, obtained from literature and the new experimental work have been used to test the model. In detail, model simulations were compared to classic experiments of growth in batch [S. cerevisiae, which is largely used in industry and academic research [−1.Data regarding four in batch  and fed-in batch  and to tresearch , 85. The−1 of the two CEN.PK strains). The resulting parameter values are reported in Table −1) and higher (0.2 h−1) SFRs. In order to test the model predictive capability a further validation experiment was performed. Logistically decreasing specific growth rate (µ*) was assumed as follows:1, m2 and m3 are calibration parameters. An optimization procedure was performed in MATLAB to find the values of m1, m2 and m3 to maximise the biomass yield and minimise the ethanol production. Specifically, we used the MATLAB fminsearch routine to minimize the following objective function (OF):t is the number of simulation time steps and max(CM(t)) is the maximum value of CM achieved during the simulation. The resulting values of µ* where used as SFR to calculate the reactor feeding as reported in the next section. The other parameters were kept constant for all validation tests and are reported in Tables The calibration procedure was performed on four of the available datasets  and dead cells (D) in the culture medium volume (V).The following expression:S. cerevisiae CEN.PK 113-7D  and the auxotroph S. cerevisiae CEN.PK2-1C . They were purchased at EUROSCARF collection (www.uni-frankfurt.de/fb15/mikro/euroscarf).The strains used for the experimental work were: the prototroph −1 of casamino acids . In the case of the auxotroph strain the medium was supplemented with uracil, histidine, leucin, triptophan according to [Fed-batch cultures of the CEN.PK strains were performed in a 2.0 l working volume of a stirred fermenter, Bioflo110 (New Brunswick Scientific). The vessel initially contained 1 l of a medium prepared according to Verduyn et al. , with 2%rding to  so to fu590 of 0.04. After an initial batch phase, where glucose was depleted, the reactor was fed with a solution containing 50% w/v glucose and salts, trace elements, glutamic acid and vitamins according to [−1, by supplying the reactor with an exponentially increasing feeding, according to:F(t) is the time-dependent feed rate (m3 h−1), F0 is the initial feed rate and SFR is the specific feed rate (h−1) which, in ideal conditions [µ).The reactor was inoculated with an adequate aliquot of a pre-culture still growing in the exponential phase and prepared in the same medium above described, to give an initial O.D.rding to . The runnditions , is equaµ critical value in correspondence of which ethanol begins to be produced [The three different SFR values were all below the produced .S. cerevisiae CEN.PK2-1C strain was performed. The run was carried in the same cultural conditions as described above, but logistically decreasing the specific growth rate (µ*), as reported in the paragraph “Model description”. The resulting values of µ* where used as SFR to calculate the reactor feeding F(t).To test the predictive capability of the model another fed-batch culture of the 4OH during fed-batch phase. The foam level in the fermenter was controlled by the automatic addition of antifoam B (Sigma Aldrich) solution at 10% v/v.In all the fed-batch cultures, oxygen was supplied to the reactor by air sparging. The dissolved oxygen tension (DOT) was kept at 30% air saturation by the cascade system, by controlling the impeller speed, or, when this reached its maximum value  by enrichment with pure oxygen. The culture pH was maintained at 5.00 by automatic addition of 2N KOH during batch phase, and 10% v/v NH590) and dry weight determination. The calibration curve relating O.D.590 values to biomass density provided a correlation factor of 2.30 O.D.590 per g l−1. Cell viability during fed-batch runs was determined by viable count (in triplicate) on YPD  agar plates incubated at 30°C for 48 h.Cell mass was determined by optical density at 590 nm (O.D.−1). Glucose and ethanol were determined by enzymatic d-glucose assay (GOPOD Format) and Ethanol-enzymatic kit from Megazyme , respectively. All the samples were analysed in triplicate showing a standard deviation always lower than 5%.Samples were hourly withdrawn from cultures, filtered on 0.45 μm GF/A filters  and the filtrates analysed to determine residual glucose and ethanol concentrations in the culture medium (g l


E. coli. It is shown to significantly improve the running time by between two to three orders of magnitude compared to the state-of-the-art, cumomers approach. We expect the tandemer approach to promote broader usage of MS/MS technology in metabolic flux analysis. Implementation is freely available at www.cs.technion.ac.il/~tomersh/methods.htmlMetabolic flux analysis (MFA) is a widely used method for quantifying intracellular metabolic fluxes. It works by feeding cells with isotopic labeled nutrients, measuring metabolite isotopic labeling, and computationally interpreting the measured labeling data to estimate flux. Tandem mass-spectrometry (MS/MS) has been shown to be useful for MFA, providing positional isotopic labeling data. Specifically, MS/MS enables the measurement of a metabolite tandem mass-isotopomer distribution, representing the abundance in which certain parent and product fragments of a metabolite have different number of labeled atoms. However, a major limitation in using MFA with MS/MS data is the lack of a computationally efficient method for simulating such isotopic labeling data. Here, we describe the tandemer approach for efficiently computing metabolite tandem mass-isotopomer distributions in a metabolic network, given an estimation of metabolic fluxes. This approach can be used by MFA to find optimal metabolic fluxes, whose induced metabolite labeling patterns match tandem mass-isotopomer distributions measured by MS/MS. The tandemer approach is applied to simulate MS/MS data in a small-scale metabolic network model of mammalian methionine metabolism and in a large-scale metabolic network model of  This provides additional information on positional labeling beyond that available via the mass-isotopomer distribution of the parent and product fragments separately (describing the abundance of a metabolite having various combinations of specific mass-isotopomers for the parent and product molecules). The relative abundance of all tandemers is referred to as a tandem mass-isotopomer distribution .Information on the positional labeling of metabolites can be obtained by tandem mass-spectrometry (i.e. MS/MS) and was previously shown to significantly improve quantification of metabolic fluxes via MFA , 16–18. Currently, there is no method for efficiently simulating tandemer distributions. Previous applications of MFA given MS/MS data have inefficiently computed the complete isotopomer distributions for all metabolites in the network  in ordemetabolite fragment pair (MFP) of a metabolite A with parent fragment N and product fragment K by M + i] > [m + j] with respect to a MFP We denote a |K| + 1) .tandemer distribution matrixM + i] > [m + j]. As entries in The entire tandemer distribution of A, with respect to the MFP substrate MFP of Under isotopic steady-state, for metabolite B that is produced solely through one biochemical reaction with a single substrate, A, the tandemer distribution matrix ectively . We refe1 and N1 are mapped to atoms in M and N, respectively) and 2 and N2 are similarly mapped to M and N; For metabolite C that is produced solely through one reaction with two substrates, A and B, For example, let us consider the bi-substrate reaction shown in vi is the flux through reaction i.Under isotopic steady-state, a tandemer distribution matrix for the MFP www.cs.technion.ac.il/~tomersh/methods.html.Our goal is to efficiently simulate the tandemer distribution for a pre-defined set of metabolites . We assume that a metabolic network model with reaction atom-mappings (describing the mapping of substrate to product metabolite atoms in each reaction) and candidate fluxes are given. To address this problem, we present the tandemers approach, whose outline is shown in The identification of a minimal set of MFP's in the metabolic network whose tandemer distributions would enable simulating the tandemer distribution of a given set of metabolites is done using a recursive procedure, in a similar manner to that presented in the EMU approach . Specifin). In practice, as we show below, applying this method on various metabolic networks, the number of MFPs found per metabolite is markedly lower than the number of isotopomers, resulting in improved running time compared to existing methods.The total number of MFPs depends on many factors, including the structure of the metabolic network, number of atoms per metabolite, reaction atom mappings, and number of metabolites for which experimental MS/MS data is available as input. In theory, for each metabolite with n carbons, the worst case number of possible MFPs could go up to:In a uni-substrate reaction producing MFP ents see . TherefoFollowing a method proposed by , we divis it see . Hence, Xi is a matrix whose rows represent the tandemer distributions for MFPs in the i’th cluster , Yi is a matrix whose rows represent tandemer distributions and Cauchy product of tandemer distributions computed for previous clusters, and Ai and Bi consist of corresponding fluxes.Tandemer distributions for MFPs in a cluster are linearly dependent on each other, given the corresponding tandemer distributions for MFPs in previous clusters. Notably, the set of balance equations for tandemer distribution matrices for MFPs in the i’th cluster can be formulated as following:Ai, whose number of rows (and columns) is equal to the number of MFPs in the cluster. As the cumomers approach also involves grouping cumomers in clusters and inverting flux matrices  whose size depends on the number of cumomers in each cluster, the running time of cumomers and tandemers approaches can be compared in terms of cumomers and MFPs cluster sizes, and the time needed to invert the corresponding flux matrices. Theoretically, an n fold reduction in the number of MFPs versus cumomers in a cluster should result in n3 improvement in running time. Notably, the number of non-feasible tandemers in each tandemer distribution matrix .Solving a set of balance equations for MFPs in the i’th cluster requires calculating the inverse of In this section we describe the application of the tandemers approach on a toy metabolic network shown in For example, the isotopic balance equation for tandemer distribution matrices in cluster (I) is formulated as following, according to 13C labeling experiments, we applied it on a simplified metabolic network model of mammalian cellular metabolism of methionine (2 prior to the propylamine transfer). Considering that the number of isotopomers of a metabolite with n carbons is 2n, explicitly modeling the entire isotopomer distribution of all five metabolites in this network (as done in the cumomers method) would require 52,306 variables.To demonstrate the applicability of the tandemers method for efficiently computing experimental MS/MS data in thionine , S1 FileTo apply the tandemers method, we utilized experimentally determined fluxes in this network as input . We assuNext, we ran both the tandemers and cumomers methods multiple times, choosing a different subset of metabolites to calculate their tandemer distribution in each run. The parent fragment was assumed to be the intact metabolite and the product fragment was chosen to be the ribose, adenine, propylamine, or the four methionine carbons other than the methyl group. We find an average number of only 33 MFPs per run of the tandemers method, while the cumomers method requiring of 52,306 cumomer variables regardless of assumed input. The average running time of the tandemers method was found to be 0.00046 seconds, while that of the cumomers method was 0.68 seconds, ~1500-fold higher . NotablyE. coli [To further demonstrate the applicability of the tandemers approach, we applied it on a large-scale metabolic network model of E. coli . This isE. coli . The totWe applied the tandemers method 1000 times to compute the tandemer distribution of randomly chosen sets of metabolites (having between 1 to 20 metabolites). The average number of resulting MFPs was 695, with a maximal MFP cluster size of 32. In comparison, applying the cumomers approach resulted in 19,404 cumomers, and a maximal cluster size of 4,016. The average running time of the tandemers and cumomers methods are 0.01 and 3.3 seconds, respectively, representing a ~300-fold improvement by the tandemers method .E. coli network, respectively, is of a major practical importance.For the tandemers methods, the number of variables represents the number of MFP whose tandemer distribution is calculated; for the cumomers approach, it represents the number of cumomers. The number of these variables corresponds to the size of the flux matrices whose inverse is calculated by each method, and is hence proportional to overall running time (see Section 2.3.3). We report the average variable count, clustersize, and running time for the cumomers and tandemers methods in multiple simulations given different sets of metabolites and collisional fragments, as described above. Notably, considering that MFA applications and especially experimental design of isotope tracing experiments require thousands of repeated simulations of metabolite isotopic labeling, the ~1500-fold and 300-fold improvement in running time observed in the mammalian methionine network and on the Tandem MS holds great promise for metabolic flux analysis as it provides information on metabolite positional labeling , 16–18. E. coli, we computed tandemer distributions for MFPs in which the parent fragment was the entire metabolite. This represents the case where no in-source fragmentation occurs during MS ionization, which is typically the situation with LC-MS. However, in-source fragmentation can occur (mostly with GC-MS), leading to measurement of tandemer distributions with parent fragment that is not the entire metabolite. Such in-source fragmentation can provide further useful information on positional labeling, for example, as was recently used to infer all distinct isotopomers of aspartate [In our application of the tandemers method on a metabolic network of mammalian methionine metabolism and for spartate . ObviousIn a recent study, we described a method, Metabolic Flux Analysis/Unknown Fragments (MFA/UF), capable of using MS/MS data to improve flux inference even when the positional origin of fragments is unknown . MFA/UF Considering that a major current complication in utilizing MS/MS data in metabolic flux analysis involves the lack of computationally efficient methods for simulating such experimental measurements, we expect the tandemers approach to promote broader usage of this technology.S1 File(XLSX)Click here for additional data file.


During the transition from a healthy state to a cancerous one, cells alter their metabolism to increase proliferation. The underlying metabolic alterations may be caused by a variety of different regulatory events on the transcriptional or post-transcriptional level whose identification contributes to the rational design of therapeutic targets. We present a mechanistic strategy capable of inferring enzymatic regulation from intracellular metabolome measurements that is independent of the actual mechanism of regulation. Here, enzyme activities are expressed by the space of all feasible kinetic constants (k-cone) such that the alteration between two phenotypes is given by their corresponding kinetic spaces. Deriving an expression for the transformation of the healthy to the cancer k-cone we identified putative regulated enzymes between the HeLa and HaCaT cell lines. We show that only a few enzymatic activities change between those two cell lines and that this regulation does not depend on gene transcription but is instead post-transcriptional. Here, we identify phosphofructokinase as the major driver of proliferation in HeLa cells and suggest an optional regulatory program, associated with oxidative stress, that affects the activity of the pentose phosphate pathway. During the development of cancer, cells undergo major metabolic changes to increase their capacity for proliferation. In many cases, that transition is characterized by an increased usage of fermentation, which is independent of the presence of oxygen and caused by a higher flux through glycolysis and diminished activity of the TCA cycle3468910Therefore, it comes as no surprise that there has been an ongoing effort to combine data from the genome with the metabolome, the concentrations of all of the cells’ metabolites1212While there exists a wide selection of methods to analyze genomic data and infer regulation events in cancer up to the enzyme level, metabolome data are often analyzed solely on the abundance level with only limited ability to extend this information to the entire biological system or even connect metabolome data to the inferences made from genome data. However, there exist some notable exceptions where methods from Systems Biology have been applied successfully16181920212425In this manuscript, we present the first metabolome profiles for the cancerous HeLa and non-cancerous HaCaT cell lines. We use the k-cone formalism to obtain a differential analysis of enzyme activities between the two cell lines, as that formulation can use metabolome data to provide a mathematical transformation between the normal and disease state. Our analysis identifies alterations in the enzyme activities of HeLa cells, many of which are consistent with previously identified changes. Furthermore, we also show that there exists a set of optional enzyme regulations that may help HeLa cells to alleviate oxidative stress without compromising proliferation. Taken together, we propose that differential k-cone analysis, which may integrate genome-scale metabolic reconstructions and metabolome data, is a suitable conceptual scheme to identify and suggest the regulatory mechanisms required to establish the phenotype in cancer cell lines.K-cone analysis is closely related to the steady state, the state of homeostasis for a metabolic system. During the steady state, metabolite concentrations are constant which gives rise to the steady state equationi = kimi(x). This case holds for mass-action kinetics with stoichiometries sjiHere, S denotes the stoichiometric matrix and v the vector of steady state fluxes. The space of all v that fulfill that equation is commonly known as the flux cone. The k-cone is based on the same equation but further assumes a distinct structure of the fluxes, where each flux can be decomposed into a kinetic constant and a metabolic term given by the metabolite concentrations x as vand gives rise to the k-cone equationi(x) on its diagonal24in vivo equilibrium constants .where M denotes a diagonal matrix containing the mass-action terms mants see 24. Becaun or kd, in its respective k-cone, Kn or Kd. In the case where the fluxes are known along with the metabolome one can use the mass-action kinetics, vi = kimi(x), to pinpoint the kinetic constants directly through dividing the fluxes by the corresponding mass-action terms mi(x). Based on this argumentation, one can derive a relation that incorporates flux data and where kd = TWkn , possibly incorporating other genome-scale data, such as gene expression and growth rate measurementsHowever, one particular normal or disease state only occupies one distinct point, kd = TWkn . Here, WMetabolome measurements were obtained for the HaCaT keratinocyte and HeLa cell lines, which are both selected for their ability to proliferate independently and have very similar doubling times28291531The measured metabolite concentrations were mapped to a model of the central carbon metabolism, extending a previously published model to yield one with 100 irreversible reactions assuming mass-action kineticsin vivo equilibrium constants (Keq) obtained from the equilibrator databasehttp://equilibrator.weizmann.ac.il/) which reduced the k-cone to only 40 basis vectors which were then mapped onto two dimensions by PCA  on the fold-changes and their statistics. Thus, we performed all analyses in log-space considering logshown in . This asThere is the possibility that only a subset of the reactions are actually required to maximize proliferation, leaving a large window of variation for fluxes that do not directly influence the growth rate. Due to the identical model, and thus identical flux cone, for both conditions, those differences would not be detectable with standard methods to calculate metabolic fluxes such as sampling from the flux cone or flux balance analysis that would both result in the same fluxes for both conditions in average. Because the accuracy of our approximation of differential enzyme activity is compromised by large changes in the fluxes between the cell lines, we performed flux variability analysis to identify the maximum log-fold change that could be caused by variation in the fluxes. Here, we first added a biomass reaction to the model and maximized its flux. This step was followed by flux variability analysis to obtain upper bounds for the absolute log-fold change for each reaction fluxThe mean log-fold changes for the EDAs in each reaction are shown in  Notably, many of the enzymes with significant EDAs are already known to be altered in cancer. Those enzymes include phosphofructokinase, phosphoglycerate mutase, pyruvate kinase, 6-phosphogluconate dehydrogenase, pyruvate dehydrogenase and aconitase394041424344Fig. 1vice versa; also see Given our predictions of differential enzyme activity based on metabolome data (EDAs), we also investigated how well these data would correlate with differential gene expression in HaCaT and HeLa cells and, thus, whether the observed changes in enzyme activity are due to changes in gene expression. For this purpose, we assembled a data set consisting of 58 microarray samples (20 HaCaT and keratinocytes and 38 HeLa) on a single platform (HGU133Plus 2.0) obtained from the GEO databaseGenomic analysis of human cancers has already shown that metabolic regulation in cancer can be highly heterogeneous and varies greatly among different cancers and even within patients with the same cancerTo identify reactions with specifically increased heterogeneity in cancer, we selected reactions whose variation increased by at least 3-fold in HeLa cells compared to HaCaT cells. For the EDAs, this selection produced a set of 11 reactions involved in the pentose phosphate pathway, glycolysis and respiration as well as increased ATP usage see . Log-folBecause in our results phosphofructokinase up-regulation is the most prominent change necessary for proliferation, we suggest the mechanism illustrated in As we have shown, differential k-cone analysis is capable of suggesting regulated enzymes in the transition to cancer for the central carbon metabolism of HeLa cells. We feel that this method is particularly well-suited to examining the regulation of non-essential enzymes or enzymes that are not visibly affected on the genome level, as it detects regulation on the metabolome level, explicitly including post-transcriptional regulation events. Analysis based on the k-cone, as performed here, combines well with existing methods such as FBA or control theory by integrating data from the metabolome and thus giving a more appropriate description of the phenotypeHowever, it is also important to note the limitations of our results. At least in the analysis we performed here, the method assumed the kinetics to be governed by the mass-action law, which is, at best, an approximation of the most likely more complex underlying kinetics. Another limitation is lacking metabolite measurements for metabolites included in the used model. The analyses, as we performed here, aimed to be conservative, meaning that we treated cases with missing data as non-differential.B. subtilis suggest that large parts of the central carbon metabolism are regulated on a post-transcriptional level rather than on a transcriptional level, which seems to be the case particularly for the pentose phosphate pathway and late glycolysis, which we also find strongly regulated here225354Our results suggest that the activities of many enzymes in the central carbon metabolism of HeLa cells are similar to the activities found in the non-cancerous HaCaT cells. However, there is a small set of enzymes that can alter their activities in the two cell lines, and those changes do not seem to affect the stability of the system. Those regulation events can be further subdivided into a small set of necessary regulations required for proliferation and a slightly larger set of enzymes that can be regulated on a on-demand basis. Here, the up-regulation of phosphofructokinase (PFK) seems to be the major driver for maintaining proliferation, which explains the requirement of the Warburg effect, as a high concentration of lactate, citrate or unused ATP can allosterically inhibit phosphofructokinase4651We also observed that the metabolic differences between cancer and healthy cells are caused by the alteration of only a few enzymes. This result stands in contrast to mRNA measurements, which often suggest that cancer alters the expression of the majority of enzymes in the central carbon metabolism6 cells were harvested at 70% confluence with 2 ml MeOH including the internal standards. Then, 1.6 ml of cell suspension was transferred to microcentrifuge tubes containing 1.6 ml of CHCl3 plus 640 μl of milliQ water, vortexed and then centrifuged at 2300 g and 4 °C for 5 minutes. 1.5 ml of the transferred aqueous layer was filtered through a Millipore 5-kDa cutoff and evaporated to dryness using a centrifugal evaporator. The measurement and quantification of extracted metabolites were performed by a commercial provider using a capillary electrophoresis (CE) connected to an ESI-TOF-MS with an electrophoresis buffer .Measurement of the ionic metabolites was performed using the CE-MS system. The HeLa cell line was provided by the oncology laboratory of the Centro Medico Siglo XXI, which belongs to the Instituto Mexicano del Seguro Social. The HaCaT cell line was donated by the Centro de Investigación Sobre Enfermedades Infecciosas, which belongs to the Instituto Nacional de Salud Pública. Cell lines were cultured in RPMI-advanced as previously described by our group6 cells, assuming an individual cell volume of 1.54 flAbundances were transformed into concentrations by dividing by the total volume of the 10Missing metabolite concentrations required in the model were imputed in multiple steps. First, missing data were imputed from measurements within the same cell line, followed by imputation across the two cell lines. Concentrations for metabolites that could not be detected in either of the two cell lines were obtained from the Human Metabolome database, primarily using cytosolic concentration measurements and falling back to blood measurements if cytosolic measurements were not availablehttps://github.com/cdiener/dycone together with installation instructions (doi: 10.5281/zenodo.49987). A detailed protocol describing the steps taken to generate all figures and results in this paper is given in the https://github.com/cdiener/kcone-paper and can be used to reproduce all analyses in this manuscript.The raw data used for the analysis are provided in The underlying kinetic model was obtained by extending a previously published model that had been validated by experimental data. The model was updated by annotating all reactions with their respective IDs from KEGG and adjusting the hydrogen balances to coincide with the ones reported in KEGG. We added additional reactions summarizing core mechanisms such as the neutralization of peroxide, import of glutamine, as well as the production of ATP and reduction of NADH in the mitochondria. Appropriate exchange reactions were added to molecules that could either be produced or consumed by metabolic processes not included in the model or obtained from the extracellular environment. The complete model specification can be found in et al.https://cran.r-project.org/web/packages/rcdd) on a H-representation with redundancies removed. Basis elements were normalized to unit length. The non-reduced individual k-cones, individual k-cones were calculated as M−1V, where M denotes the diagonal matrix with the mass action terms on its diagonal. The k-cones which where additionally constrained by equilibrium constants were calculated individually by adding the respective equilibrium constraints to the k-cone equation  = log2 (k2/k1) for T = M1M2−1, as derived in the Approximations of changes in enzyme activities were either obtained by the transformation T via calculating its log-diagonal logbm, this procedure gave rise to the following linear programming program for the flux balance analysisi:Necessity for proliferation was performed by first appending a biomass reaction to the model and identifying the maximum permissible flux through that reaction. The biomass reaction was adapted from Recon 2−16 to ensure a non-zero flux for each reaction, as all reactions of the central carbon metabolism should be active in the cell lines used.Here, the upper bound could be chosen arbitrarily, as later calculations used only the flux ratios, which are invariant to the upper bound. Lower bounds for the fluxes were chosen as 10max, the upper and lower flux limits were obtained by solving two linear programming problems for each flux vi:After obtaining the maximum biomass flux vThe largest absolute log-fold change that could be explained by flux variability analysis could now be obtained asmax were deemed necessary for proliferation.Finally, all individual log-fold changes obtained from the EDAs whose absolute value exceeded that of lfcAll log-fold changes, obtained by the transformation T, were handled in the same way. Significance measures were obtained by performing Welch t-tests on the log-transformed mass-action terms after validating their normal distributions . Sample 36Fifty-eight microarray samples were manually selected from the GEO database by selecting for samples from a single platform (HGU133Plus 2.0), in untreated conditions and only for the previously used cell lines . The analysis of gene expression was performed by normalizing the raw data for the 58 samples by Frozen Robust Multiarray Analysis (fRMA), followed by differential analysis using the limma package, particularly its empirical Bayes method36c and the differential standard deviation σd (obtained from the log-fold change samples described before), a reaction was considered differentially heterogeneous if σd/σc > 3, based on the visualization in Standard deviations for log-fold changes of enzyme activities were obtained from the 9 possible Hela/HaCaT sample combinations of log-fold changes as used in the approximation of credible intervals. To obtain an estimate for the standard deviation in untreated conditions we used the control samples obtained from the paired permutations of HaCaT samples, as previously described. Given the control standard deviation σHow to cite this article: Diener, C. et al. The space of enzyme regulation in HeLa cells can be inferred from its intracellular metabolome. Sci. Rep.6, 28415; doi: 10.1038/srep28415 (2016).