Metabolic networks represent all chemical reactions that occur between molecular metabolites in an organism’s cells. They offer biological context in which to integrate, analyze, and interpret omic measurements, but their large scale and extensive connectivity present unique challenges. While it is practical to simplify these networks by placing constraints on compartments and hubs, it is unclear how these simplifications alter the structure of metabolic networks and the interpretation of metabolomic experiments.We curated and adapted the latest systemic model of human metabolism and developed customizable tools to define metabolic networks with and without compartmentalization in subcellular organelles and with or without inclusion of prolific metabolite hubs. Compartmentalization made networks larger, less dense, and more modular, whereas hubs made networks larger, more dense, and less modular. When present, these hubs also dominated shortest paths in the network, yet their exclusion exposed the subtler prominence of other metabolites that are typically more relevant to metabolomic experiments. We applied the non-compartmental network without metabolite hubs in a retrospective, exploratory analysis of metabolomic measurements from 5 studies on human tissues. Network clusters identified individual reactions that might experience differential regulation between experimental conditions, several of which were not apparent in the original publications.Exclusion of specific metabolite hubs exposes modularity in both compartmental and non-compartmental metabolic networks, improving detection of relevant clusters in omic measurements. Better computational detection of metabolic network clusters in large data sets has potential to identify differential regulation of individual genes, transcripts, and proteins. Life is inherently unstable, and cellular metabolism acts as its vanguard, continually adapting to sustain balance. This system is complex, requiring cooperation and coordination between many types of biochemical entities . Large, There is a growing need for strategies to investigate metabolic adaptation in human health and disease. Traditional, reductionist biology tends to conceptualize cellular metabolism as a collection of separate pathways or processes (groups of reactions) that perform their own unique functions with little interaction. However, a growing body of work has demonstrated surprising versatility in the metabolic system, especially in human diseases such as obesity, diabetes, and cancer . ConnectBiological networks are abstract projections that are useful for studying these complex, real systems. Interestingly, even the global structures of molecular biological networks are informative; patterns of efficient communication between specialized modules suggest competing mechanisms of stochastic evolution and natural selection . FurtherBecause metabolic networks are abstract projections, their definitions can emphasize different aspects of the metabolic system. This versatility argues for customizability, in particular with regard to compartments and hubs. Subcellular compartmentalization within membranous organelles is an important dimension of metabolism in eukaryotic cells, yet standard metabolomic measurements on bulk samples do not discriminate between pools of metabolites in separate compartments. Hence, for the sake of analyzing and interpreting these measurements, it may or may not be reasonable to simplify cellular metabolism by ignoring compartmentalization . Also, aHere we describe alternative definitions of metabolic networks and their relevance in application to experiments. We hypothesized that alternative representations of metabolism using compartmental or non-compartmental network models with or without metabolite hubs would differentially influence the interpretations of metabolomic experiments. We also explored the potential for algorithms to detect biologically relevant clusters of metabolomic measurements on these networks. Our goal was to define these networks and describe their differences while also providing methods and tools for future use in the community. To this end, we curated and adapted the latest systemic model of human metabolism . We desiSystemic metabolic models summarize all chemical reactions between small-molecular metabolites that occur within an organism. Another name for these models is genome-scale metabolic reconstructions, with major applications in computational simulations to predict broader cellular growth and to resolve finer metabolic flux balance analysis through specific pathways ,17. ThesMetabolomic technologies separate, identify, and quantify small molecules from biological samples. While some studies use nuclear magnetic resonance (NMR), larger studies commonly use chromatography with gas (GC) or liquid (LC) mobile phases that integrate with various forms of mass spectrometry (MS); and combining measurements from multiple technologies increases the breadth of a study. Each type of technology has its own parameters and requirements for processing and analyzing the data. In particular, normalization to total signal in each sample corrects for loss of material and fluctuation in detector sensitivity. Furthermore, measurements commonly lack absolute calibration such that values only represent relative comparisons between samples. Many data sets from metabolomic studies are not publicly available; however, there are initiatives to include more of these data in public repositories, such as the Metabolomics Workbench . WhereasMetaboNet  is our mWe curated the latest systemic model of human metabolism and adapted it to provide biological context in metabolic experiments. In particular, our goal was to filter irrelevant reactions from the model while also optimizing our ability to match metabolomic measurements to metabolites. Steps 1 and 2 of curation both comprised enhancements to the information about metabolites and reactions. We accessed the latest model of human metabolism, Recon version 2M.2 ,41, and To begin our study of human metabolic networks, we designed and developed a tool to visualize the definition and exploration of the human metabolic network. This tool is an experimental prototype that does not intend to replace the broader functionality of major tools in network biology . Rather,DyMetaboNet  is a dynDuring our design and development of DyMetaboNet , we recoWe defined multiple network representations of human metabolism. Networks are abstract simplifications of complex systems, and alternative representations can be reasonable while emphasizing different aspects of the underlying information. We chose to keep some definitions consistent while altering other constraints to evaluate their influence. All of our definitions represent metabolism in a directional bipartite network  with disOur first constraint involves compartmentalization. Compartmental networks Fig.  include Our second constraint involves filters by specific cellular compartments and metabolic processes. These compartments and processes define sets of metabolites and reactions of interest. For example the user might want to consider only reactions and metabolites within the mitochondrion compartment. Similarly, the user might want to consider only reactions and metabolites that participate in the citric acid cycle process Fig. . MetaboNOur third constraint relates to the exclusion of specific metabolites from the metabolic network. This exclusion means that the network does not include nodes to represent these metabolites, and consequently there are also no links to or from them. Regardless of exclusion of nodes and links for a metabolite, reactions themselves still include information about all metabolites that participate as reactants and products.Metabolite hubs are special candidates for exclusion from the metabolic network. A few metabolites are common reactants and products in metabolic reactions, such that they contribute a large proportion of the connectivity in metabolic networks  11]. Th. Th11]. Category 1 metabolite hubs are less relevant to metabolic regulation and experiments. Many of these metabolites are prolifically abundant in the cell. While they are all chemically essential to metabolic reactions, some of these metabolites, such as water, dioxygen, and carbon dioxide, are unlikely to participate in the type of metabolic regulation that is commonly relevant to experiments. Furthermore, perturbations in the abundance of these metabolites would be difficult to interpret, and some of these metabolites are undetectable in metabolomic measurements. Category 1 hubs  include Category 2 metabolite hubs are more relevant to metabolic regulation and experiments. The abundance of these metabolites in the cell fluctuates in metabolic regulation, and they are relevant to many metabolic experiments. However, some of these metabolites participate in so many reactions that they dominate connectivity in the metabolic network. Exclusion of these metabolites from the metabolic network reveals more subtle trends involving the influences of other metabolites that are of greater interest in some contexts. Category 2 hubs  include This constraint for exclusion of specific metabolites is very sensitive and requires customization to the context of each metabolic experiment. Consequently, MetaboNet  makes thWe next set out to describe and compare our metabolic networks both qualitatively and quantitatively. MetaboNet  exports Like many other real systems, metabolism is a small world . The smaentclass1pt{minimat Tables , and botentclass1pt{minimat Tables . This stCompartmentalization confers major structural properties to metabolic networks. In their global visualizations, compartmental Fig.  and 3B aMetabolite hubs dominate connectivity within metabolic networks. In their global visualizations, networks with Fig.  and 3C aTo compare metabolic networks from a complementary perspective, we considered their profiles of prominent metabolite nodes. We noticed that metabolites’ nodes in these networks have degrees that follow roughly exponential distributions Fig.  and 4F, Glutamate is an impressive example of connection and cooperation in metabolism. While not equal to hub status, this metabolite is particularly promiscuous in metabolic reactions. In both the compartmental and non-compartmental networks without hubs, glutamate is the top-ranking metabolite in terms of both its close and long-range influence Fig.  and 4H. Reasonable definitions of metabolic networks differ to the extent that their application to metabolic experiments warrants careful selection. These networks offer potential to facilitate design of experiments and to contextualize functional interpretations of metabolomic measurements. Importantly, our analyses demonstrate that constraints by compartmentalization and metabolite hubs alter the structure of the metabolic network substantially Fig. , Table 2Having selected the non-compartmental metabolic network without hubs, we next prepared to evaluate its application to retrospective analyses on real metabolomic measurements. Because our model and networks represent intracellular human metabolism, we searched for studies on solid human tissues rather than plasma, serum, other body fluids, or excrement. We selected 5 studies with publicly accessible metabolomic measurements  on samplP-values) by the Student t-test and also calculating the base-2 logarithms of fold changes. For each study, we visualized these fold changes and probabilities simultaneously in volcano plots  match to metabolites in our model of human metabolism . We normots Fig. , S4A. Thots Fig.  that werP-values from metabolomic measurements to matching metabolites in the non-compartmental metabolic network without hubs. We then exported this network with measurements to Cytoscape [P-values. Finally, we searched for clusters exhibiting patterns of both accumulation and depletion in proximal metabolites  ,57 descrP-values on clusters of proximal nodes. Our use of this general cluster enrichment method was exploratory, and we acknowledge the potential for novel network enrichment algorithms to account for reaction directionality and patterns of accumulation and depletion in proximal metabolites. Integrating such a metabolism-specific clustering algorithm, together with detection of patterns across functional categories of reactions  [Together, MetaboNet  and DyMertments)  and chemrtments) , might hrtments) .Biological models offer the potential to integrate holistic, functional context in interpretations of omic measurements ,8. As biWe developed the MetaboNet package  as a traMetaboNet requires sources of information from the Recon 2M.2 model of human metabolism , versionWe accessed the latest model of human metabolism Table . We acceWe used the tools and repository of MetaNetX  and versWe curated and enhanced information about metabolites in our model of human metabolism. We made 197 custom curations to information about metabolites, especially to correct and enhance references to external databases. We accessed information for all 114,100 records about metabolites in version 4.0 of HMDB , file “hWe also curated and filtered information about reactions in our model of human metabolism. We made 102 custom curations to information about reactions, especially to clarify names on the basis of their references to genes . We intehttps://tcameronwaller.github.io/dymetabonet/), all necessary source files and code download to the user’s computer, and the entire application runs locally on the user’s computer. The internet browser has a firewall to contain this information from web applications and thereby protect the client’s computer. DyMetaboNet imports information about metabolites, reactions, compartments, and processes that MetaboNet [We designed and developed the DyMetaboNet web application  for basietaboNet  exports etaboNet . DynamicetaboNet . DyMetabWe defined networks to represent human metabolism. We selected a representation as a directional, bipartite network with distinct types of nodes for reactions and metabolites Fig. . In thisWe defined metabolic networks to represent metabolism both with and without compartmentalization Fig.  and 1B. We exert customizable criteria for reactions and metabolites to qualify for representation in the network. In our model of metabolism, reactions specify the compartments in which they occur, and they also specify metabolic processes to which they belong. Hence, these compartments and processes define sets of reactions and metabolites, and the relevance of these sets depends on the context of experiments. Our procedure accommodates customizable lists of compartments and processes to apply as filters. Similarly, the relevance of individual reactions and metabolites depends on the context of experiments. Our procedure also accommodates customizable lists of reactions and metabolites to include or exclude from the network. By default, we exclude metabolite hubs from the network Fig. . To qualTo measure density, we used an algorithm from NetworkX  that norTo measure the centralities of individual nodes, we used algorithms from NetworkX  that calTo measure centralization of the entire network, we implemented our own versions of algorithms that calculate single-mode degree and betweenness centralities and normalize these by comparison to maximum ,76. MetaTo measure cluster coefficients of individual nodes, we used an algorithm from NetworkX  that calTo measure mean cluster coefficient of the entire network, we used an algorithm from NetworkX  that calTo measure the mean path length of the entire network, we implemented our own custom version of an algorithm that calculates the mean of lengths of shortest paths between all single-mode pairs of nodes .To measure the small-world coefficient of the entire network, we adapted the σ (sigma) coefficient  for a biTo measure the degree assortativity coefficient of the entire network, we used algorithms from NetworkX . We firsWe applied algorithms and metrics from graph theory to describe our metabolic networks. Bipartite networks  such as P-values) between measurements for each analyte in samples from each experimental group. These calculations depended on whether a study’s samples were in dependent pairs from the same patient. For pairs of dependent samples, we calculated the mean of base-2 logarithms of fold changes for measurements from each pair, and we calculated the P-value using a 2-sided t-test for dependent populations. For independent samples, we calculated the base-2 logarithm of the fold change between the means of measurements from each group, and we calculated the P-value using a 2-sided t-test for independent populations. Our subsequent analyses used the mean base-2 logarithm of fold change and the P-value to compare each analyte between experimental groups. We visualized these values in custom volcano plots that we implemented using version 3.1.1 of MatPlotLib [We curated and processed public metabolomic measurements for general analyses. We accessed metabolomic measurements from records for projects and studies within the Metabolomics Workbench  Table S. From thtPlotLib ,71.We integrated metabolomic measurements in metabolic networks for further analysis. Most analytes in Metabolomics Workbench  include P-value : platform independentProgramming language: Python 3Other requirements: SciPy, NumPy, NetworkX, MatPlotLib, WordCloudLicense: GNU General Public License version 3Dynamic definition and visual exploration of metabolic networks.Project name: DyMetaboNethttps://github.com/tcameronwaller/dymetabonetProject home page: Operating system(s): platform independentProgramming language: JavaScriptOther requirements: Data-Driven Documents (D3)License: GNU General Public License version 3GigaScience repository, GigaDB [This article’s analyses used v1.0.0 of MetaboNet  and v1.0, GigaDB .Supplementary information: Supplementary Methods and Results are available via the additional file associated with this article.Analysis of metabolomic measurements in setsAnalysis of metabolomic measurements in network clustersTable S1: Definition of metabolite hubsTable S2: Definition of network metricsTable S3: Curation of metabolomic measurementsTable S4: Metabolite set enrichment analysisTable S5: Cluster metabolitesTable S6: Cluster reactionsFigure S1: Integration and analysis of metabolomic measurements on metabolic networks, Study 2Figure S2: Integration and analysis of metabolomic measurements on metabolic networks, Study 3Figure S3: Integration and analysis of metabolomic measurements on metabolic networks, Study 4Figure S4: Integration and analysis of metabolomic measurements on metabolic networks, Study 5giz137_GIGA-D-18-00489_Original_SubmissionClick here for additional data file.giz137_GIGA-D-18-00489_Revision_1Click here for additional data file.giz137_Response_to_Reviewer_Comments_Original_SubmissionClick here for additional data file.giz137_Reviewer_1_Report_Original_SubmissionHeinz-Bernd Schuttler -- 1/2/2019 ReviewedClick here for additional data file.giz137_Reviewer_2_Report_Original_SubmissionShen Tong -- 1/8/2019 ReviewedClick here for additional data file.giz137_Reviewer_2_Report_Revision_1Shen Tong -- 9/27/2019 ReviewedClick here for additional data file.giz137_Supplemental_FileClick here for additional data file.ABP: adenosine 3′,5′-bisphosphate; ADP: adenosine 5′-diphosphate; AMP: adenosine 5′-monophosphate; ATP: adenosine 5′-triphosphate; BCAA: branched chain amino acid; ChEBI: Chemical Entities of Biological Interest; CMP: cytidine 5′-monophosphate; D3: Data-Driven Documents; EC: Enzyme Commission; FAD2+: flavin adenine dinucleotide; FADH2: flavin adenine dinucleotide reduced; GDP: guanosine 5′-diphosphate; HGNC: HUGO Gene Nomenclature Committee; HMDB: Human Metabolome Database; HUGO: Human Genome Organization; JSON: JavaScript Object Notation; KEGG: Kyoto Encyclopedia of Genes and Genomes; NAD1+: nicotinamide adenine dinucleotide; NADH: nicotinamide adenine dinucleotide reduced; NADP1+: nicotinamide adenine dinucleotide phosphate; NADPH: nicotinamide adenine dinucleotide phosphate reduced; NIH: National Institutes of Health; p53: tumor suppressor protein 53; RefSeq: Reference Sequence; SBML: Systems Biology Markup Language; TIGAR: tumor suppressor protein 53–induced glycolysis and apoptosis regulator; UDP: uridine 5′-diphosphate; URL: Uniform Resource Locator; XML: Extensible Markup Language.The authors declare that they have no competing interests.T.C.W. and J.A.B. received support from the NIH Interdisciplinary Training Grant T32 Program in Computational Approaches to Diabetes and Metabolism Research, 1T32DK11096601 to Wendy W. Chapman and Simon J. Fisher. B.E.C. received support from the office of the Senior Vice President for University of Utah Health Sciences. T.C.W. and J.R. received support from the National Cancer Institute grant CA228346 to J.R. J.R. is also an Investigator of the Howard Hughes Medical Institute.Conceptualization: T.C.W., J.R. Supervision: T.C.W., J.A.B., A.L., B.E.C., J.R. Project Administration: T.C.W., J.A.B., A.L., B.E.C., J.R. Investigation: T.C.W. Formal Analysis: T.C.W. Software: T.C.W., J.A.B. Methodology: T.C.W. Validation: T.C.W. Data Curation: T.C.W. Resources: T.C.W., J.A.B., A.L., B.E.C., J.R. Funding Acquisition: T.C.W., J.A.B., A.L., B.E.C., J.R. Writing—Original Draft Preparation: T.C.W. Writing—Review and Editing: T.C.W., J.A.B., A.L., B.E.C., J.R. Visualization: T.C.W.


Escherichia coli and on a metabolic model of human hepatocytes. The flux-dependent graphs under various environmental conditions and genetic perturbations exhibit systemic changes in their topological and community structure, which capture the re-routing of metabolic flows and the varying importance of specific reactions and pathways. By integrating constraint-based models and tools from network science, our framework allows the study of context-specific metabolic responses at a system level beyond standard pathway descriptions.Cells adapt their metabolic fluxes in response to changes in the environment. We present a framework for the systematic construction of flux-based graphs derived from organism-wide metabolic networks. Our graphs encode the directionality of metabolic flows via edges that represent the flow of metabolites from source to target reactions. The methodology can be applied in the absence of a specific biological context by modelling fluxes probabilistically, or can be tailored to different environmental conditions by incorporating flux distributions computed through constraint-based approaches such as Flux Balance Analysis. We illustrate our approach on the central carbon metabolism of  Escherichia coli's core metabolism in different growth conditions, as well as a rare metabolic disease affecting human hepatocytes.Cellular metabolism is the result of a highly enmeshed set of biochemical reactions that is naturally amenable to graph-based analyses. Yet there are multiple ways to construct a graph representation from any given metabolic model. Here an international research team of UK and Spain scientists presents a principled approach to study metabolic models through the lens of network science. They propose a framework to construct graphs for genome-scale metabolic models that resolve various challenges, such as the incorporation of pool metabolites, the preservation of the directionality of metabolic flows, and the capability to incorporate specific flux information. The method can be integrated into pipelines based on flux balance analysis and provides a systematic framework to explore changes in network connectivity as a result of environmental shifts or genetic perturbations. The framework thus allows to interrogate context-specific metabolic responses beyond standard pathway descriptions. The authors illustrate the approach through the analysis of  Yet metabolic reactions are highly interconnected: enzymes convert multiple reactants into products with other metabolites acting as co-factors; enzymes can catalyse several reactions, and some reactions are catalysed by multiple enzymes, and so on. This enmeshed web of reactions is thus naturally amenable to network analysis, an approach that has been successfully applied to different aspects of cellular and molecular biology, e.g., protein-protein interactions,2 transcriptional regulation,3 or protein structure.5Metabolic reactions enable cellular function by converting nutrients into energy, and by assembling macromolecules that sustain the cellular machinery.6 have previously been applied to the analysis of structural properties of metabolic networks, including their degree distribution,10 the presence of metabolic roles,11 and their community structure.15 A central challenge, however, is that there are multiple ways to construct a network from a metabolic model.16 For example, one can create a graph with metabolites as nodes and edges representing the reactions that transform one metabolite into another;18 a graph with reactions as nodes and edges corresponding to the metabolites shared among them;21 or even a bipartite graph with both reactions and metabolites as nodes.22 Importantly, the conclusions of graph-theoretical analyses are highly dependent on the chosen graph construction.23Tools from graph theory1 Existing graph constructions have been useful for developing an intuitive understanding of metabolic complexity. Many of these constructions, however, lead to graphs that do not include directional information that is central to metabolic function.16 In addition, current graph constructions are usually derived from the whole set of metabolic reactions in an organism, and thus correspond to a generic metabolic ‘blueprint’ of the cell. Yet cells switch specific pathways ‘on’ and ‘off’ to sustain their energetic budget in different environments.24 Hence, such blueprint graphs might not capture the specific metabolic connectivity in a given environment, thus limiting their ability to provide biological insights in different growth conditions.A key feature of metabolic reactions is the directionality of flows: metabolic networks contain both irreversible and reversible reactions, and reversible reactions can change their direction depending on cellular and environmental contexts.Normalised Flow Graph (NFG), a weighted, directed graph with reactions as nodes, edges that represent supplier-consumer relationships between reactions, and weights given by the probability that a metabolite chosen at random from all reactions is produced/consumed by the source/target reaction. This graph can be used to carry out graph-theoretical analyses of organism-wide metabolic organisation independent of cellular context or environmental conditions. We then show that this formalism can be adapted seamlessly to construct the Mass Flow Graph (MFG), a directed, environment-dependent, graph with weights computed from Flux Balance Analysis (FBA),25 the most widespread method to study genome-scale metabolic networks.In this paper, we present a flow-based approach to construct metabolic graphs that encapsulate the directional flow of metabolites produced or consumed through enzymatic reactions. The proposed graphs can be tailored to incorporate flux distributions under different environmental conditions. To introduce our approach, we proceed in two steps. We first define the 30 Finally, the Mass Flow Graph incorporates additional biological information reflecting the effect of the environmental context into the graph construction. In particular, since the weights in the MFG correspond directly to fluxes (in units of mass per time), different biological scenarios can be analysed using balanced fluxes  under different carbon sources and other environmental perturbations.32Our formulation addresses several drawbacks of current constructions of metabolic graphs. Firstly, in our flow graphs, an edge indicates that metabolites are produced by the source reaction and consumed by the target reaction, thus accounting for metabolic directionality and the natural flow of chemical mass from reactants to products. Secondly, the Normalised Flow Graph discounts naturally the over-representation of pool metabolites , nicotinamide adenine dinucleotide (NADH), protons, water, and other co-factors) that appear in many reactions and tend to obfuscate the graph connectivity. Our construction avoids the removal of pool metabolites from the network, which can change the graph structure drastically.Escherichia coli metabolism33 reveals the importance of including directionality and appropriate edge weights in the graph to understand the modular organisation of metabolic sub-systems. We then use FBA solutions computed for several relevant growth conditions for E. coli, and show that the structure of the MFG changes dramatically in each case , thus capturing the environment-dependent nature of metabolism. Secondly, we study a model of human hepatocyte metabolism evaluated under different conditions for the wild-type and in a mutation found in primary hyperoxaluria type 1, a rare metabolic disorder,34 and show how the changes in network structure of the MFGs reveal new information that is complementary to the flux analysis predicted by FBA.After introducing the mathematical framework, we showcase our approach with two examples. Firstly, in the absence of environmental context, our analysis of the NFG of the core model of n metabolites Xi  that participate in m reactionsαij and βij are the stoichiometric coefficients of species i in reaction j. Let us denote the concentration of metabolite Xi at time t as xi(t). We then define the n-dimensional vector of metabolite concentrations: x(t) = (x1(t),…,xn(t))T. Each reaction takes place with rate vj, measured in units of concentration per time.35 We compile the reaction rates in the m-dimensional vector: v(t) = (v1(t),…,vm(t))T.Consider a metabolic network composed of n × m matrix S is the stoichiometric matrix with entries Sij = βij − αij, i.e., the net number of Xi molecules produced (positive Sij) or consumed (negative Sij) by the j-th reaction. Figure 32The mass balance of the system can then be represented compactly by the system of ordinary differential equationsS. A common approach16 is to define a unipartite graph with reactions as nodes and m × m adjacency matrixS , in which two reactions (nodes) are connected if they share metabolites, either as reactants or products. Self-loops represent the total number of metabolites that participate in a reaction . Furthermore, the structure of A is dominated by the large number of edges introduced by pool metabolites that appear in many reactions, such as water, ions or enzymatic cofactors. Computational schemes have been introduced to mitigate the bias caused by pool metabolites,27 but these do not follow from biophysical considerations and need manual calibration. Finally, the construction of the graph A from is not easily extended to incorporate the effect of environmental changes.Though widely studied,A, we propose a graph formulation that follows from a flux-based perspective. To construct our graph, we unfold each reaction into two separate directions (forward and reverse) and redefine the links between reaction nodes to reflect producer-consumer relationships. Specifically, two reactions are connected if one produces a metabolite that is consumed by the other. As shown below, this definition leads to graphs that naturally account for the reversibility of reactions, and allows for the seamless integration of biological contexts modelled through FBA. Inspired by matrix formulations of chemical reaction network kinetics,36 we rewrite the reaction rate vector v as:v+ and v− are non-negative vectors containing the forward and backward reaction rates, respectively. Here the m × m matrix diag(r) contains r in its main diagonal, and r is the m-dimensional reversibility vector with components rj = 1 if reaction Rj is reversible and rj = 0 if it is irreversible. With these definitions, we can rewrite the metabolic model in Eq.  is the matrix of absolute values of the corresponding entries of Sm2. Note that each entry of the matrix Xi produced by reaction Rj. Conversely, the entries of Xi consumed by reaction Rj.The unfolding into forward and backward fluxes leads us to the definition of production and consumption stoichiometric matrices:S without any additional biological information, such as metabolite concentrations, reaction fluxes, or kinetic rates. In the absence of such information, the probability that a metabolite molecule Xk chosen uniformly at random from S is produced by reaction Ri and consumed by reaction Rj is:Xk produced and consumed by all reactions that have been accounted for in Sm2. Unlike models in that rely on stochastic chemical kinetics,37 the probabilities in Eq.  is a widely used method to predict environment-specific flux distributions. FBA computes a vector of metabolic fluxes entclass1pt{minimam × 1 vector of production and consumption fluxes asTo incorporate the biological information afforded by FBA solutions into the structure of a metabolic graph, we again define the graph edges in terms of production and consumption fluxes. Similarly to Eq. , we unfok-th entry of Xk is produced and consumed; the equality of the production and consumption fluxes follows from the steady state condition, The Ri and Rj as the total mass flow of metabolites produced by Ri that are consumed by Rj. Assuming that the amount of metabolite produced by one reaction is distributed among the reactions that consume it in proportion to their flux (and respecting the stoichiometry), the flux of metabolite Xk from reaction Ri to Rj is given byTo construct the flow graph, we define the weight of the edge between reactions Xk is 10 mmol/gDW/h, with reaction Ri producing Xk at a rate 1.5 mmol/gDW/h and reaction Rj consuming Xk at a rate 3.0 mmol/gDW/h, then the flow of Xk from Ri to Rj is 0.45 mmol/gDW/h.For example, if the total flux of metabolite Ri and Rj:Summing (10) over all metabolites, we obtain the edge weight relating reactions In matrix form, these edge weights are collected into the adjacency matrix of theR7. Note how the MFG for scenario 2 displays an extra edge between reactions R4 and R7, as well as distinct edge weights to scenario 1 (see Sec. SI 2 for details). These differences illustrate how changes in the FBA solutions translate into different graph connectivities and edge weights.The MFG provides a versatile framework to create environment-specific metabolic graphs from FBA solutions. In Fig. E. coli.33 This model  and our proposed Normalised Flow Graph A graph has 95 nodes and 1,158 undirected edges, whereas the A graph, where edges represent shared metabolites between two reactions, the directed edges of the To examine the effect of flux directionality on the metabolic graphs, we compare the Reaction Adjacency Graph , phosphoenolpyruvate synthase  and ABC-mediated transport of L-glutamine  drop from being among the top 10% most important reactions in the A graph to the bottom percentiles in the The effect of directionality becomes apparent when comparing the importance of reaction nodes in both graphs Fig. , as meas46 The community structure of metabolic graphs has been the subject of multiple analyses.44 However, most community detection methods are applicable to undirected graphs only, and thus fail to capture the directionality of the metabolic graphs we propose here. To account for graph directionality, we use the Markov Stability community detection framework,48 which uses diffusion on graphs to detect groups of nodes where flows are retained persistently across time scales. Markov Stability is ideally suited to find multi-resolution community structure45 and can deal with both directed and undirected graphs49 (see “Menthods” section). In the case of metabolic graphs, Markov Stability can reveal groups of reactions that are closely interlinked by the flow of metabolites that they produce and consume.Community detection is frequently used in the analysis of complex graphs: nodes are clustered into tightly related communities that reveal the coarse-grained structure of the graph, potentially at different levels of resolution.E. coli. For the A graph, Markov Stability reveals a partition into seven communities  is mainly composed of reactions that consume or produce ATP and water. Yet, the biomass reaction (the largest consumer of ATP) is not a member of C 1(A) because, in the standard A graph construction, any connection involving ATP has equal weight. Other communities in A are also determined by pool metabolites, e.g. C 2(A) is dominated by H+, and C 3(A) is dominated by NAD+ and NADP+, as illustrated by word clouds of the relative frequency of metabolites in the reactions within each community. The community structure in A thus reflects the limitations of the RAG construction due to the absence of biological context and the large number of uninformative links introduced by pool metabolites.Figure 2. Community C4A and For the E. coli metabolism in four relevant growth conditions: aerobic growth in rich media with glucose; aerobic growth in rich media with ethanol, anaerobic growth in glucose; and aerobic growth in glucose but phosphate- and ammonium-limited. The results in Fig. To incorporate the impact of environmental context, we construct the Mass Flow Graphs in Eq.  using FBNext we summarise how the changes in the community structure of the MFGs for the four conditions reflect the distinct relationships of functional pathways in response to growth requirements.Mglc) is the carbon-processing community, comprising reactions that process carbon from D-glucose to pyruvate including most of the glycolysis and pentose phosphate pathways, together with related transport and exchange reactions. C2(Mglc) harbours the bulk of reactions related to oxidative phosphorylation and the production of energy in the cell, including the electron transport chain of NADH dehydrogenase, cytochrome oxidase, and ATP synthase, as well as transport reactions for phosphate and oxygen intake and proton balance. C2(Mglc) also includes the growth reaction, consistent with ATP being the main substrate for both the ATP maintenance (ATPM) requirement and the biomass reaction in this growth condition. Finally, C3(Mglc) contains reactions related to the citric acid cycle (TCA) and the production of NADH and NADPH , together with carbon intake routes strongly linked to the TCA cycle, such as those starting from phosphoenolpyruvic acid (PEP).We found a robust partition into three communities with an intuitive biological interpretation Fig. . C 1, which reflects the switch from d-glucose to ethanol as a carbon source. C1(Metoh) contains gluconeogenic reactions (instead of glycolytic), due to the reversal of flux induced by the change of carbon source, as well as anaplerotic reactions and reactions related to glutamate metabolism. In particular, the reactions in this community are related to the production of precursors such as PEP, pyruvate, 3-phospho-D-glycerate (3PG), glyceraldehyde-3-phosphate (G3P), d-fructose-6-phosphate (F6P), and d-glucose-6-phosphate, all of which are substrates for growth. Consequently, the biomass reaction is also grouped within C1(Metoh) due to the increased metabolic flux of precursors relative to ATP production in this biological scenario. The other two reaction communities (energy-generation C2(Metoh) and citric acid cycle C3(Metoh)) display less prominent differences relative to the Mglc graph, with additional pyruvate metabolism and anaplerotic reactions as well as subtle ascriptions of reactions involved in NADH/NADPH balance and the source for acetyl-CoA.The robust partition into three communities that we found for this scenario resembles the structure of ces Fig. . Most saManaero has a robust partition into four communities. C1(Manaero) still contains carbon processing (glucose intake and glycolysis), yet now decoupled from the pentose phosphate pathway. C3(Manaero) includes the pentose phosphate pathway grouped with the citric acid cycle (incomplete) and the biomass reaction, as well as the growth precursors including alpha-D-ribose-5-phosphate (r5p), d-erythrose-4-phosphate (e4p), 2-oxalacetate and NADPH. The other two communities are specific to the anaerobic context: C2(Manaero) contains the conversion of PEP into formate ; C4(Manaero) includes NADH production and consumption via reactions linked to glyceraldehyde-3-phosphate dehydrogenase (GAPD).The profound impact of the absence of oxygen on the metabolic balance of the cell is reflected in drastic changes in the MFG Fig. . Both thoverflow metabolism,51 which occurs when the cell takes in more carbon than it can process. As a consequence, the excess carbon is secreted from the cell, leading to a decrease in growth and a partial shutdown of the citric acid cycle. This is reflected in the reduced weight of the TCA pathway and its grouping with the secretion routes of acetate and formate within C3(Mlim). Hence, C3(Mlim) comprises reactions that are not strongly coupled in favourable growth conditions, yet are linked together by metabolic responses to limited ammonium and phosphate. Furthermore, the carbon-processing community C1(Mlim) contains the glycolytic pathway, yet detached from the pentose phosphate pathway (as in Manaero), highlighting its role in precursor formation. The bioenergetic machinery, contained in community C2(Mlim), includes the pentose phosphate pathway, with a smaller role for the electron transport chain .Under growth-limiting conditions, we found a robust partition into three communities Fig. . The comIn addition to the effect on community structure, Fig. Mglc → Metoh) as carbon source decreases the importance of pyruvate metabolism and oxidative phosphorylation, while increasing the importance of the pentose phosphate pathway. A shift from aerobic to anaerobic growth in glucose (Mglc → Manaero) sees a large reduction in the importance of oxidative phosphorylation and the citric acid cycle, coupled with a large increase in the importance of gluconeogenesis, pyruvate metabolism, and transport and exchange reactions. The effect of growth-limiting conditions in aerobic growth under glucose (Mglc → Mlim) is reflected on the increased importance of pyruvate metabolism and a reduction in the importance of oxidative phosphorylation, citric acid cycle, and the pentose phosphate pathway. The importance of transport and exchange reactions is also increased under limiting conditions. Such qualitative relations between growth conditions and the importance of specific pathways highlights the utility of the MFGs to characterise systemic metabolic changes in response to environmental conditions.In aerobic growth, a shift from glucose to ethanol , thus allowing the exploration of the multiscale organisation of metabolic reaction networks. The modules so detected reflect subsets of reactions where metabolic fluxes tend to be contained.In contrast with methods for undirected graphs, the Markov Stability frameworkMglc, the MFG of E. coli under aerobic growth in glucose. By varying the Markov time t, a parameter in the Markov Stability method, we scanned the community structures at different resolutions. Our results show that, from finer to coarser resolution, the MFG can be partitioned into 11, 7, 5, 3, and 2 communities of high persistence across Markov time ) and high robustness under optimisation (as shown by dips in VI(t)). For further details, see Section 4.2 and Refs.52Figure The Sankey diagram in Fig. Metoh, Manaero, Mlim).Other reaction groupings reflect more specific relationships. For example, the citric acid cycle  appears as a cohesive unit across most scales, and only splits in two in the final grouping, reflecting the global role of the TCA cycle in linking to both glycolysis and oxidative phosphorylation. The pentose phosphate pathway, on the other hand, is split into two groups (one linked to glutamate metabolism and another one linked to glycolysis) across early scales, only merging into the same community towards the final groupings. This suggests a more interconnected flux relationship of the different steps of the penthose phosphate pathway with the rest of metabolism. In Supplementary Figure 34 which extends the widely used HepatoNet1 model53 with an additional 50 reactions and 8 metabolites. This extended model was used in Ref.34 to compare wild-type cells (WT) and cells affected by the rare disease Primary Hyperoxaluria Type 1 (PH1), which lack alanine:glyoxylate aminotransferase (AGT) due to a genetic mutation. AGT is an enzyme found in peroxisomes and its mutation decreases the breakdown of glyoxylate, with subsequent accumulation of calcium oxalate that leads to liver damage.To showcase the applicability of our framework to larger metabolic models, we analyse a model of human hepatocyte (liver) metabolism with 777 metabolites and 2589 reactions,34 we first obtain 442 FBA solutions for different sets of metabolic objectives for both the wild-type (WT) model and the PH1 model lacking AGT (reaction r2541). We then generate the corresponding 442 MFGs for each WT and PH1, and obtain the averages over each ensemble: Following,i, O2 and H2O2 between the peroxisome and the cytosol .Figure Overall, the centrality (PageRank) of most reactions in the MFG is relatively unaffected by the PH1 mutation, as shown by the good correlation between the PageRank percentiles in It is worth remarking that although oxalate and citrate reactions are directly linked to metabolic changes associated with the PH1 diseased state, none of them exhibits large changes in their flux predicted by FBA, yet they show large changes in PageRank centrality.local rearrangement of flows as a consequence of the deletion of reaction r2541. On the other hand, the citrate and oxalate reactions  with large changes in their centrality yet undergo small changes in flux, thus reflecting global changes in the flux structure of the network. Importantly, the transport reactions of O2, H2O2, serine and hydroxypyruvate between cytosol and peroxisome  all undergo large changes both in centrality and flux, highlighting the importance of peroxisome transfer reactions in PH1. We provide a full spreadsheet with these analyses as Supplementary Material for the interested reader.These observations underscore how the information provided by our network analysis provides complementary information to the results from FBA. As shown in Fig. 54 i.e., metabolites linked by arrows representing enzymatic reactions between them as in Fig. 16 Each of those graphs can be directed or undirected and with weighted links computed according to different rules. The choices and subtleties in graph construction are crucial both to capture the relevant metabolic information and to interpret their topological properties.17Metabolism is commonly understood in terms of functional pathways interconnected into metabolic networks,30 Context-specific Mass Flow Graphs (MFGs) can incorporate the effect of the environment, e.g., with edge weights corresponding to the total flux of metabolites between reactions as computed by Flux Balance Analysis (FBA). FBA solutions for different environments can then be used to build different metabolic graphs in different growth conditions.Here we have presented a flux-based strategy to build graphs for metabolic networks. Our graphs have reactions as nodes and directed weighted edges representing the flux of metabolites produced by a source reaction and consumed by a target reaction. This principle is applied to build both ‘blueprint’ graphs (NFG), which summarise probabilistically the fluxes of the whole metabolism of an organism, as well as context-specific graphs (MFGs), which reflect specific environmental conditions. The blueprint Normalised Flow Graph has edge weights equal to the probability that source/target reactions produce/consume a molecule of a metabolite chosen at random from the stoichiometric matrix in the absence of any other information, and can thus be used when this matrix is the only information available. The NFG construction naturally tames the over-representation of pool metabolites without the need to remove them from the graph arbitrarily, as often done in the literature.The proposed graph constructions provide complementary tools for studying the organisation of metabolism and can be embedded into any FBA-based modelling pipeline. Specifically, the NFG relies on the availability of a well-curated stoichiometric matrix, which is produced with metabolic reconstruction techniques that typically precede the application of FBA. The MFG, on the other hand, explicitly uses the FBA solutions in its construction. Both methods provide a systematic framework to convert genome-scale metabolic models into a directed graph on which analysis tools from network theory can be applied.E. coli. Through the analysis of topological properties and community structure of these graphs, we highlighted the importance of weighted directionality in metabolic graph construction, and revealed the flow-mediated relationships between functional pathways under different environments. In particular, the MFGs capture specific metabolic adaptations such as the glycolytic-gluconeogenic switch, overflow metabolism, and the effects of anoxia. The proposed graph construction can be readily applied to large genome-scale metabolic networks.39To exemplify our approach, we built and analysed NFG and MFGs for the core metabolism of 34 Our network analysis of the MFGs revealed a consistent organisation of the reaction graph, which is highly preserved under the mutation. Our analysis also identified notable changes in the network centrality score and community structure of certain reactions, which is linked to key biological processes in PH1. Importantly, network measures computed from the MFGs reveal complementary information to that provided by the sole analysis of perturbed FBA fluxes.To illustrate the scalability of our analyses to larger metabolic models, we studied a genome-scale model of a large metabolic model of human hepatocytes with around 3000 reactions in which we compared the wild-type and a mutated state associated with the disease PH1 under more than 400 metabolic conditions.32 thus opening avenues towards environment-dependent, graph-based analyses of cell metabolism. An area of interest for future research is the use of MFGs to study how network measures of flow graphs can help characterise metabolic conditions that maximise the efficacy of drug treatments or disease-related distortions, e.g., cancer-related metabolic signatures.58 In particular, MFGs can quantify metabolic robustness via graph statistics upon removal of reaction nodes.22Our flow graphs provide a systematic connection between network theory and constraint-based methods widely employed in metabolic modelling,The proposed framework for graph construction for metabolic networks can be extended in different directions. The core idea is the distinction between production and consumption fluxes (5), and how to encode the relationship between produced and consumed mass flows in the weighted links of a graph. This general principle can also be used to build other potentially useful graphs. For example, two other graphs that describe relationships between reactions are:competition and synergy mass flow graphs, which follow directly from  to generate probabilistic reaction fluxes in the sense of stochastic chemical reaction networks.63 Of particular interest to metabolic modelling, we envision that MFGs could provide a route to evaluate the robustness of FBA solutions64 by exploiting the non-uniqueness of the MFG from each FBA solution in the space of graphs. Such results could enhance the interface between network science and metabolic analysis, allowing for the systematic exploration of the system-level organisation of metabolism in response to environmental constraints and disease states.The framework can also be extended to include dynamic adaptations of metabolic activity in different ways: by using dynamic extensions of FBA;32 is a widely-adopted approach to analyse metabolism and cellular growth. FBA calculates the reaction fluxes that optimise growth in specific biological contexts. The main hypothesis behind FBA is that cells adapt their metabolism to maximise growth in different biological conditions. The conditions are encoded as constraints on the fluxes of certain reactions; for example, constraints reactions that import nutrients and other necessary compounds from the exterior.Flux Balance Analysis (FBA)S is the stoichiometry matrix of the model, v the vector of fluxes, c is an indicator vector  = 1 when i is the biomass reaction and zero everywhere else) so that cTv is the flux of the biomass reaction. The constraint Sv = 0 enforces mass-conservation at stationarity, and vlb and vub are the lower and upper bounds of each reaction’s flux. Through these vectors, one can encode a variety of different scenarios.33 The biomass reaction represents the most widely-used flux that is optimised, although there are others can be used as well.65The mathematical formulation of the FBA is described in the following constrained optimisation problem:4 and phosphate intake were fixed at 4.5 mmol/gDW/h and 3.04 mmol/gDW/h respectively (a reduction of 50% compared to a glucose-fed aerobic scenario with no restrictions).In our simulations, we set the individual carbon intake rate to 18.5 mmol/gDW/h for every source available in each scenario. We allowed oxygen intake to reach the maximum needed in to consume all the carbon except in the anaerobic condition scenario, in which the upper bound for oxygen intake was 0 mmol/gDW/h. In the scenario with limited phosphate and ammonium intake, the levels of NH48 This framework uses diffusion processes on the network to find groups of nodes  that retain flows for longer than one would expect on a comparable random network; in addition, Markov Stability incorporates directed flows seamlessly into the analysis.49We extract the communities in each network using the Markov Stability community detection framework.G of the graph , we construct a rate matrix for the process: Kout is the diagonal matrix of out-strengths, kiout, = 1. In general, a directed network will not be strongly-connected and thus a Markov process on M will not have a unique steady state. To ensure the uniqueness of the steady state we must add a teleportation component to the dynamics by which a random walker visiting a node can follow an outgoing edge with probability λ or jump (teleport) uniformly to any other node in the network with probability 1−λ.42 The rate matrix of a Markov process with teleportation is:N × 1 vector a is an indicator for dangling nodes: if node i has no outgoing edges then ai = 1, and ai = 0 otherwise. Here we use λ = 0.85. The Markov process is described by the ODE:L = IN − B. The solution of (17) is x = π, where π is the leading left eigenvector of B.The diffusion process we use is a continuous-time Markov process on the network. From the adjacency matrix C communities can be encoded into the N × C matrix H, where hic = 1 if node i belongs to community c and zero otherwise. The C × C clustered autocovariance matrix of (17) isc,s) of R measures how likely it is that a random walker that started the process in community c finds itself in community s after time t when at stationarity. The diagonal elements of R thus record how good the communities in H are at retaining flows. The Markov stability of the partition is then defined asA hard partition of the graph into t to obtain an optimised partition 66 which is known to give high quality solutions t, i.e. the duration of the Markov process, can be understood as a resolution parameter for the partition into communities.47 In the limit t → 0, Markov stability will assign each node to its own community; as t grows, we obtain larger communities because the random walkers have more time to explore the network.48 We scan through a range of values of t to explore the multiscale community structure of the network. The code for Markov Stability can be found at http://wwwf.imperial.ac.uk/mpbara/Partition_Stability/.The optimised communities are obtained by maximising the cost function (19) over the space of all partitions for every time 45 Firstly, we optimise (19) 100 times for each value of t and we assess the consistency of the solutions found. A relevant partition should be a robust outcome of the optimisation, i.e., the ensemble of optimised solutions should be similar as measured with the normalised variation of information:67t:To identify the important partitions across time, we use two criteria of robustness.VI(t) is small, indicating robustness of the partition to the optimisation. Hence we select partitions with low values (or dips) of VI(t). Secondly, relevant partitions should also be optimal across Markov time, as indicated by a low values of the cross-time variation of information:If all Louvain runs return similar partitions, then VI.52Therefore, we also search for partitions with extended low value plateaux of No new data were generated during the course of this research. The results of the analysis are available in the Supplementary Spreadsheet.http://wwwf.imperial.ac.uk/mpbara/Partition_Stability/All computations performed on MATLAB. Code for Markov Stability available at Supplementary MaterialData Set 1


In the analysis of metabolism, two distinct and complementary approaches are frequently used: Principal component analysis (PCA) and stoichiometric flux analysis. PCA is able to capture the main modes of variability in a set of experiments and does not make many prior assumptions about the data, but does not inherently take into account the flux mode structure of metabolism. Stoichiometric flux analysis methods, such as Flux Balance Analysis (FBA) and Elementary Mode Analysis, on the other hand, are able to capture the metabolic flux modes, however, they are primarily designed for the analysis of single samples at a time, and not best suited for exploratory analysis on a large sets of samples.We propose a new methodology for the analysis of metabolism, called Principal Metabolic Flux Mode Analysis (PMFA), which marries the PCA and stoichiometric flux analysis approaches in an elegant regularized optimization framework. In short, the method incorporates a variance maximization objective form PCA coupled with a stoichiometric regularizer, which penalizes projections that are far from any flux modes of the network. For interpretability, we also introduce a sparse variant of PMFA that favours flux modes that contain a small number of reactions. Our experiments demonstrate the versatility and capabilities of our methodology. The proposed method can be applied to genome-scale metabolic network in efficient way as PMFA does not enumerate elementary modes. In addition, the method is more robust on out-of-steady steady-state experimental data than competing flux mode analysis approaches.https://github.com/aalto-ics-kepaco/PMFA.Matlab software for PMFA and SPMFA and dataset used for experiments are available in Bioinformatics online. Principal component analysis (PCA) is one of the most frequently applied statistical methods in systems biology , is found by selecting a set of reactions which represents a metabolic flux mode which is approximately in steady state and explains most of the data variability. In addition, we propose a sparse variant, called Sparse Principal Metabolic Flux Mode analysis (SPMFA), to further help the interpretation of the principal components.In this paper we propose a novel method to find metabolic flux modes that explains the variance in gene expression or fluxomic data collected from heterogeneous environmental conditions without requiring a fixed set of predefined pathways to be given. The proposed method is called as principal metabolic flux mode analysis (PMFA). Here each principal component, called Our method differs from existing methods in the literature such as Flux Balance Analysis (FBA)  of the mSaccharomyces cerevisiae gene expression data where the performance is measured in terms of normalized variance captured. In the fourth experiment, elaborates on the biological findings obtained using SPMFA to analyze the variance in the mitochondrial subsystem of whole-genome S.cerevisiae metabolic network. We conclude the paper with discussion.The structure of this paper is as follows. The methods section describes the theory and development of a novel method to analysis fluxomic and gene expression data. The section includes the descriptions of data, means and algorithms by which the new method has been benchmarked. In the results section, we report a comparative study on the similarities and differences of PCA, SPCA, FBA, PMFA, SPMFA and PEMA. The study highlights four experiments. In the first experiment, we compare PMFA to PEMA in the retrieval of active elementary flux modes on a dataset for which the ground truth is known. In the second experiment, we study the effect of stoichiometric regularization on the fraction of test set variance explained by PMFA and alternative methods  with Leave-One-Out (LOO) cross-validation. In the third experiment, SPMFA is used for the recovery of sparse flux modes from whole-genome Here we shortly review the existing basic methods for the analysis of fluxomic data.Principal component analysis: We assume N samples and rN reactions, with each entry corresponding to an estimated reaction rate for a particular reaction in a particular experiment. We assume throughout the paper that all variables have been centered to have zero empirical mean. The empirical covariance matrix is then given by l2 norm of the vector 1−w1w1T) .λ is a user defined hyper-parameter which controls the degree of sparsity on PC. However, the principal components extracted by neither method represent metabolic flux modes, and will not in general adhere to thermodynamic constraints on reaction directions.The weights, also called the loadings, of the principal component Stoichiometric modelling: The metabolic balance of the metabolic system is described using the exchange stoichiometric matrix mN internal metabolites, columns present the rN metabolic reactions including transport reactions and each element mth metabolite in the rth reaction: r produces (or consumes) the metabolite m. The value m is not involved in the reaction r. For a flux vector S∈ℝNm×Nr  which coElementary modes: The concept of an elementary mode (EM) : FBA , that combines the PCA and stoichiometric modelling views of metabolism.the weights associated with irreversible reactions should always be positive, i.e. System is in a steady state, where the internal metabolite concentrations do not change, i.e. the metabolite producing and consuming fluxes cancel each other out: Considering (1) and (2) the modified optimization problem for doing PCA with structural constraint is as followingl2 norm: λ imposes the degree of hardness of the steady-state constraint. For λ = 0 the wir > 0. We denote this version of the method as rev-PMFA.To obtain meaningful solutions of steady state flux distributions as PC loading one can impose two additional constraints in PCA formulation:l2 norm on l1 norm regularizer on l1 norm regularizer l1 regularizer and a trade-off parameter λ we get to solve the following optimization problem:λ imposes the degree of hardness of the steady-state constraint. Similarly to λ = 0 the The l1 regularizer on l2 and l1 norm and C is a used defined hyper-parameter which controls the degree of sparsity in principal metabolic flux (PMF) loadings. Similarly to PMFA, Sparse PMFA can also be made to consider all reaction reversible by dropping the inequality constraints The above formulation of PCA with stoichiometric constraint still suffers from the fact that each principal component is typically a linear combination of all possible reaction activities, thus it is often difficult to interpret the results. This problem can be avoided by a variant of PMFA, the sparse principal metabolic flux mode analysis (SPMFA) using an One can apply our method to focus on variance within a subsystem of the whole metabolic network  by restricting the covariance matrix in objective function to the fluxes in the subsystem, while keeping the stoichiometric regularizer the same as before. Similarly, when some flux measurements are missing, one can change the covariance matrix in the objective function to exclude the measurements that are missing.For example, to study the variation within the redox subsystem, let t:The objective function of To find a good local optimum, we repeat the above optimization with different random starting points, and take the best local minimum as the solution. In our experiments we used 100 repetitions.multi-factor PMFA model, i.e. a model containing several PMFs jointly representing the data, we follow a approach similar to some PCA algorithms, namely the deflation of the covariance matrix. However, due to additional stoichiometric constraint here we deal with a sequence of non-orthogonal vectors, d + 1 we first transfer already found principal flux modes Qd. To obtain a a matrix . This aldir denotes the PCA augmented with reaction directionality constraints.PCA: Principal component analysis as given by dir is the SPCA augmented with reaction directionality constraints.SPCA: Sparse PCA corresponding to FBA: Flux balance analysis with an objective of maximizing biomass production given by (3).l2 regularization on the stoichiometric constraint l1 regularization on stoichiometric constraint PMFA: Principal Flux Mode Analysis as described in Section 2.2. l2 regularization on stoichiometric constraint, while l1 regularization on stoichiometric constraint.SPMFA: Sparse Principal Flux Mode Analysis as given by Principal Elementary Mode Analysis (PEMA)  and 44 reactions, yielding a total number of 98 EMs   and area under precision recall curve (AUPR). The precision/recall metrics, widely used in information retrieval, is to assess how well the flux modes computed by PEMA and PMFA correlate with the ground truth active EMs. The PFM loadings are reported in Supplementary File PFMloading.ods in PichiaPastorisResultAndAnalysis.zipIn our first experiment we evaluated the predictive performance the proposed PMFA and PEMA in correctly retrieving underlying active elementary flux modes. We used the P.pastoris metabolic network. We then sort the EMs in descending order of correlation and consider first i, by considering ground truth active EMs within the first i EMs as true positives and other EMs with the top i as false positives. A precision/recall curve can be then plotted by taking the precision/recall values for all is, in the order of the descending correlation in the sorted list. The AUPR is denoted as area under the precision recall curve and AUC is denoted as area under receiver operating characteristic curves  as the models prediction and used those for computing the precision/recall values for each In a PMFA model with In the noise free case, PEMA performs comparatively to PMFA, especially in terms of the AUC metric and when using a high enough number of factors in the model. However, the performance of PEMA deteriorates quickly upon increased noise. The decrease of performance is particularly apparent in the AUPR metric.In this experiment we focused on the ability of PMFA to explain variance on data in a predictive setting, that is, on new data that has not been used for model estimation. We focused on the amount of variance explained in the test set in a Leave-One-Out (LOO) cross-validation setting.S.cerevisiae and P.pastoris). The deviation from the steady-state is controlled by the regularization parameter λ give low deviation from steady-state and vice-versa.We studied the effect of stoichiometric regularization . The transition for rev-PMFA is not as smooth as PMFA with the directionality constraint. It is apparent that the directionality constraint increases the stability of PMFA without reducing much explained variance on test data.By change of the regularization parameter Compared to PEMA, The fraction of variance explained the first PMF from rev-PMFA is higher than 1-, 5- and 10-factor PEMA regardless of the amount of stoichiometric regularization or application of the directionality constraints. The amount of variance explained by the first PMF from PMFA is also much higher than 1-factor PEMA even with high Stoichiometric regularization, while the 5- and 10-factor PEMA reach the level of PMFA for both datasets.S.cerevisiae containing a total of 3494 reactions for, making dense principal components and flux modes difficult to interpret. The SPFM loadings along with the amount of inter-cellular metabolites produced or consumed by SPFM for various degree of steady state constraints are reported in Supplementary File SPFM-geneexpression.ods in SPFMoxygenseriesResultandAnalysis.zip.To quantify the fraction of explained variance normalized by the complexity of the extracted flux mode, we measure the normalized fraction of variance, calculated asl0 norm, i.e. the cardinality of non-zero elements of In this experiment, we evaluated the Sparse Principal Metabolic Flux Mode Analysis, SPMFA, in discovery of sparse flux modes, i.e. only few reactions with non-zero coefficients. We focus on the full genome data, i.e. all steady-state and transient samples of At the maximum, PMFA captures slightly more explained variance than SPMFA at , left. Cl1 stoichiometric regularizer (λ) but switching to SPMFA regime as regularization is increased. This reflects the fact that with small λ the model is not yet sparse but sparsity quickly emerges once λ is increased.The variant It is notable that on this large heterogeneous dataset, all methods fail to capture meaningful amounts of normalized sample variance in the vicinity the steady state  representing a state shortly after the loss of oxygen. The 2nd PMF discriminates among samples with different oxygen level in the environment, with samples correlating with the 2nd PMF the better the less oxygen is available. This is evident in the monotonic decreasing correlation of the steady-state samples based on the oxygen level, as well as the generally increasing correlation of the time-series samples as a function of time (and decreasing oxygen).Six metabolic pathways emerged upon closer inspection of the individual reactions associated with 1st PMF and 2nd PMF. Two of these pathways were associated with the 1st PMF while four pathways were associated with the 2nd PMF. The main reactions and their loadings are shown in PMFAsup.pdf, The Malic pathway (A) associated with the 1st PMF consisted of malate import, dehydrogenation to produce NADPH and/or NADH, pyruvate export and acetolactate synthesis from pyruvate. The ATP pathway (F) associated with the 1st PMF included oxoglutarate import, TCA cycle reactions from oxoglutarate to succinyl-CoA, succinate export and two means to extract the ATP-equivalent stored in succinyl-CoA. The negative loading of Succinyl-CoA: acetate CoA transferase may indicate a switch from this reaction to other reactions generating ATP more explicitly. This hypothesis is supported by the full set of reactions in the Supplementary The Acetaldehyde pathway (B) associated with the 2nd PMF represents the conversion of acetaldehyde to acetate with the generation of NADPH, and the sequestration of the formed acetate to acetyl-CoA and further to hydroxymethylglutaryl-CoA, an intermediate in the mevalonate and the ketogenesis pathways. The 2nd PMF contained the Malate shuttle for generating mitochondrial NADPH. The 2nd PMF also contained the reactions for the electron transport chain and oxidative phosphorylation (D), possibly for the removal of residual oxygen. In (E), a pathway catabolizing pyruvate via Glycine hydroxymethyltransferase, the Glycine cleavage complex, Methylene-THF dehydrogenase and Methenyl-THF cyclohydrolase is captured. The Tetrahydrofolate pathway (E) ended with Methionyl-tRNA formyltransferase, thus producing one NADH and one NADPH per pyruvate catabolized. The four pathways associated with the 2nd PMF appear to capture the generation of mitochondrial NADPH, a vital cofactor for metabolic adaptation by biosynthesis.In this paper we have proposed a novel method for the analysis of metabolic networks, called the Principal Metabolic Flux Analysis, PMFA, through the combination of stoichiometric flux analysis and principal component analysis, finds flux modes that explain most of the variation in fluxes in a set of samples. Unlike most stoichiometric modeling methods, PMFA is not tied to the steady-state assumption, but can automatically adapt—by the change of a single regularization parameter—to deviations from the stoichiometric steady-state, whether they are due to measurement errors, biological variation or other causes. Our experiments showed that the method is more robust to the steady-state violations than competing approaches, and can compactly capture the variation in the data by a few factors. For the analysis of whole-genome metabolic networks, we further proposed Sparse Principal Flux Mode Analysis, SPMFA that allows us to discover flux modes with a small fraction of reactions activated, thus could be interpreted as pathways. Our experiments showed that our methods are more efficient in capturing the variance in sets of experiments than methods based on elementary flux mode analysis or flux balance analysis. The efficient Concave Convex Procedure optimization allows the method to scale up to whole-genome models unlike methods based on search in the space of elementary flux modes.S.cerevisiae showed that PMFA was able to identify six mitochondrial pathways responsive to changes in oxygen availability. In addition, the analysis grouped these pathways in easily interpretable pathways.Analysis of cultivation data on the whole-genome metabolic network of The Malic pathway’s capability to provide the mitochondrion with reducing equivalents in the form of NADPH and NADH, and the ATP pathway’s capability to provide the mitochondrion with ATP were apparently captured by the 1st PMF. The four pathways associated with the 2nd PMF appeared to capture the generation of mitochondrial NADPH, a vital cofactor for metabolic adaptation by biosynthesis.Supplementary DataClick here for additional data file.


The increasing global prevalence of diabetes has been accompanied by a rise in diabetes-related conditions. This includes diabetic cardiomyopathy (DbCM), a progressive form of heart disease that occurs with both insulin-dependent (type-1) and insulin-independent (type-2) diabetes and arises in the absence of hypertension or coronary artery disease. Over time, DbCM can develop into overt heart failure. Like other forms of cardiomyopathy, DbCM is accompanied by alterations in metabolism which could lead to further progression of the pathology, with metabolic derangement postulated to precede functional changes in the diabetic heart. Moreover in the case of type-2 diabetes, underlying insulin resistance is likely to prevent the canonical substrate switch of the failing heart away from fatty acid oxidation toward increased use of glycolysis. Analytical chemistry techniques, collectively known as metabolomics, are useful tools for investigating the condition. In this article, we provide a comprehensive review of those studies that have employed metabolomic techniques, namely chromatography, mass spectrometry and nuclear magnetic resonance spectroscopy, to profile metabolic remodeling in the diabetic heart of human patients and animal models. These studies collectively demonstrate that glycolysis and glucose oxidation are suppressed in the diabetic myocardium and highlight a complex picture regarding lipid metabolism. The diabetic heart typically shows an increased reliance on fatty acid oxidation, yet triacylglycerols and other lipids accumulate in the diabetic myocardium indicating probable lipotoxicity. The application of lipidomic techniques to the diabetic heart has identified specific lipid species that become enriched and which may in turn act as plasma-borne biomarkers for the condition. Metabolomics is proving to be a powerful approach, allowing a much richer analysis of the metabolic alterations that occur in the diabetic heart. Careful physiological interpretation of metabolomic results will now be key in order to establish which aspects of the metabolic derangement are causal to the progression of DbCM and might form the basis for novel therapeutic intervention. Expansion of the global population and an increase in age-standardized prevalence  have togHeart failure is classically defined as an inability of the heart to deliver sufficient blood to the tissues of the body to meet metabolic demands . It can Diabetic cardiomyopathy, however, is considered to differ from other causes of heart failure. Metabolic disturbances precede the first signs of mechanical failure in the diabetic myocardium and are considered likely to be causative . MoreoveThe growing field of metabolomics employs a collection of analytical techniques which have the potential to elucidate this progression of metabolic disturbances in the diabetic heart and may additionally highlight novel plasma-borne biomarkers. In this review, we first give a brief overview of some of the most commonly used metabolomic techniques and models, before considering in detail those studies that have applied these techniques to investigate metabolic changes in the diabetic heart. In particular, we draw attention to the growing field of lipidomics and how alterations to plasma lipid species could serve as putative biomarkers for DbCM. Finally, we highlight how existing and emerging metabolomic technologies could be used to address some of the remaining questions concerning cardiac metabolism in diabetes.The comprehensive study of all metabolites in a biological system is referred to as metabolomics . In comp1H, 13C, and 31P have nuclei with a quantum spin number of ½, and therefore have two energy levels which can be detected by NMR spectroscopy. Helpfully, these nuclei are also ubiquitous across metabolic species in biological systems, making NMR a highly valuable tool in metabolomic studies  can be usefully studied, and this depends on both physicochemical properties  and biological properties  of the molecule  ratio  ratio . Determiz) ratio . It can,z) ratio .Mass spectrometry is often coupled to chromatographic separation, usually either GC or LC, which can greatly expand the number of metabolites that can be detected . SamplesChromatography refers to the separation of different molecular species in a mixture, such as a biofluid or tissue extract. A column is utilized, through which different analytes pass at different rates depending on their chemical properties and that of the column used. The time taken for a particular metabolite to elute from the chromatographic column is referred to as the retention time, and this can be used for molecular identification.Two chromatographic techniques are commonly employed in metabolomic studies: GC and LC, which differ in the phase in which metabolites are separated. GC uses a long column to separate metabolites in the gaseous phase. This can be beneficial when coupled to MS, since metabolites are already volatile when entering the mass spectrometer . LC sepaAlthough chromatography is usually coupled to MS, other techniques for identifying metabolites following chromatographic separation can also be employed. Examples of alternative detectors include FIDs and TCDs, which can be used to identify analytes as they are eluted from GC columns .Metabolomic studies are often subdivided into open-profiling and targeted analyses. Open-profiling involves detecting and measuring all (or at least as many as possible) of the metabolites in a sample, and is therefore sometimes referred to as unbiased metabolomics . Such stA number of animal models of diabetes, obesity and the metabolic syndrome  have beob/ob mouse  mouse which exhibits a diabetic phenotype owing to alterations to cardiac glycerol-phosphate metabolism  or intraperitoneally (i.p.)  are commonly used to induce diabetes in rats or mice through the destruction of pancreatic β-cells . Such coy (i.p.) , and resy (i.p.) .High fat diets are often used to induce obesity and the metabolic syndrome in rodent models . Since i31P-MRS, has allowed measurements to be made of the myocardial PCr to ATP ratio (PCr/ATP) in vivo as an index of cardiac energy reserve. The creatine kinase system acts to temporally and spatially buffer ATP in the myocardium, transferring phosphate groups from ATP to creatine at the mitochondria and from PCr to ADP at the myofibrils  . 31P-MRSss clear . Whilst ss clear , others ss clear .These apparent discrepancies may be explained by the model or population studied. All human studies considered in db/db mouse  acylcarnitine species has been seen , suggestTaken together, these results suggest that glucose metabolism is suppressed in both T1DM and T2DM. Meanwhile, metabolomic studies generally support findings that fatty acid oxidation is elevated in the diabetic heart, although some inconsistencies do highlight the importance of not considering metabolomics in isolation.Beyond FA oxidation, alterations to other pathways of lipid metabolism are evident in DbCM, and may play an important role in the progression of the pathology. Indeed, one proposed mechanism for the development of DbCM holds that despite an initial compensatory increase in FA oxidation capacity, lipid overload results in the accumulation of harmful lipid intermediates, a phenomenon known as lipotoxicity . This, i1H-MRS studies coupled to MRI, which found that increased myocardial TAG concentration correlates with impaired diastolic function  , causinge stress , furtherThe development and progression of DbCM is associated with altered levels of specific lipid species in the myocardium, which has led to the suggestion that lipids might act as plasma-borne biomarkers for the condition. Shotgun lipidomic studies have often been used for this purpose , 2018. P−/−Gpat1 mouse. This mouse was primarily developed as a model of altered cardiac FA metabolism, and in turn it develops some characteristics of diabetes  highlighted several amino acid species  the concentrations of which were altered in the diabetic hearts . AlthougMetabolomic techniques are proving to be valuable in helping to unravel the metabolic disturbances of DbCM. Open profiling methods may lead to the identification of putative biomarkers in animal models of DbCM, whilst targeted approaches have highlighted alterations in lipid, ketone, and amino acid pathways.in vivo  affected by DbCM to suggest mechanisms underpinning dysfunction. Meanwhile, the results of studies demonstrating alterations in cardiac metabolism in diabetes need to be placed in context, and here complementary techniques are useful in indicating how changes in metabolite concentrations or flux correspond to alterations in, e.g., mitochondrial respiratory capacity  or cardiin vivo  or ex vi and metabolites constantly flux through a variety of metabolic pathways. Some techniques have tried to overcome this limitation including hyperpolarized-13C MRS studies measuring, for example, PDH flux (A significant disadvantage of metabolomic techniques is that findings typically reflect metabolite levels at a single time point, whereas PDH flux , and staPDH flux . NeverthPDH flux . It is t18F, 11C, 13N and 15O, are taken up by metabolically-active tissues including the heart, and radioactive decay of the tracer can then be measured, indicating uptake and flux of the tracer in tissues of interest. Although expensive, owing to the cost of producing the tracers and instrumentation, metabolic and cardiovascular parameters such as myocardial blood flow, myocardial oxygen consumption, and substrate specific uptake, utilization, and oxidation (in vivo, making PET a useful, non-invasive tool for investigating metabolism in DbCM. Combining PET with metabolomics (e.g., in vivo changes in flux of targeted metabolites and to identify novel alterations in metabolic pathways in an unbiased manner.One such technique, that has been utilized widely in human studies e.g., , and somxidation  can be mcs e.g.,  thereforin situ (Finally, novel applications of metabolomic techniques may also help to elucidate metabolic changes in the diabetic myocardium. MSI is an application of MS that enables the distribution of tissue metabolites to be analyzed in situ . Unlike in situ . This tein situ  and altein situ . It coulMetabolomic studies have helped to elucidate metabolic derangements that occur in the diabetic heart. Glycolysis and glucose oxidation are depressed in DbCM and it is highly likely that this is accompanied by energetic impairment. Open-profiling lipidomic studies have demonstrated that that there are specific classes of lipid, and individual species within these classes, that are distinctly altered in the diabetic myocardium and some of these species might serve as biomarkers for DbCM, particularly if they relate to alterations in circulating lipid species. There are, however, questions remaining concerning the physiological significance of lipid remodeling and the mechanistic role this might play in the pathology. For instance, whilst it has been postulated that lipotoxicity may be a driving force behind functional impairment, whether changes in individual lipid species are causal in the development and/or progression of DbCM, or a simply consequence of the condition needs further investigation.All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.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.