Opportunities at the interface of network science and metabolic modelling
Varshit Dusad, Denise Thiel, Mauricio Barahona, Hector C. Keun, Diego A. Oyarzún
CChallenges and opportunities at the interface of network science andmetabolic modelling
Varshit Dusad, Denise Thiel, Mauricio Barahona, Hector C. Keun,
3, 4 and Diego A. Oyarz´un
5, 6, 7 Department of Life Sciences, Imperial College London Department of Mathematics, Imperial College London Department of Surgery and Cancer, Imperial College London Department of Metabolism, Digestion and Reproduction, Imperial College London School of Biological Sciences, University of Edinburgh School of Informatics, University of Edinburgh Corresponding author: [email protected] (Dated: 8 June 2020)
Metabolism plays a central role in cell physiology and provides the cellular machinery for building biomoleculesessential for growth. At the genome-scale, metabolism is made up of thousands of reactions interacting withone another. Untangling this complexity is key to understand how cells respond to genetic, environmentalor therapeutic perturbations. Here we discuss the roles of two complementary strategies for the analysis ofgenome-scale metabolic models: constraint-based methods, such as flux balance analysis, and network science.Whereas constraint-based methods estimate metabolic flux on the basis of an optimization principle, network-theoretic approaches reveal emergent properties of the global metabolic connectivity. We highlight how theintegration of both approaches promises to deliver insights on the structure and function of metabolic systemswith wide-ranging implications in basic discovery science, precision medicine and industrial biotechnology.
I. INTRODUCTION
Metabolism comprises the biochemical reactions thatconvert nutrients into biomolecules and energy to sus-tain cellular functions. Advances in high-throughputscreening technologies have enabled the quantitativecharacterization of metabolites, proteins and nucleicacids at the genome-scale, revealling previously un-known links between metabolism and many other cel-lular processes. For example, gene regulation , signaltransduction , immunity and epigenetic modifications have been shown to interact closely with metabolic path-ways. The increasing availability of data and the funda-mental roles of metabolism in various cellular phenotypeshave triggered a surge in metabolic research, togetherwith a revived need for computational tools to untangleits complexity.At the genome scale, metabolism is made of mul-tiple interconnected reactions devoted to the synthesisof specific biomolecules (e.g. proteins, lipids or nucleicacids) and to the production of energy. The notion ofa metabolic pathway is typically employed to organizesets of related reactions into functionally cohesive sub-systems. Thus, lipid pathways, for example, are tradi-tionally studied as distinct subsystems from amino acidor aerobic respiration pathways. Although convenientlydescriptive, such a priori partitioning can obscure thelinks between relevant layers of metabolic organization.Furthermore, metabolic connectivity is not static butactively responds and adapts to extracellular cues. In-deed, through various layers of transcriptional, transla-tional and post-translational regulation, metabolic path-ways can be activated or shutdown depending on ex-ternal perturbations. These metabolic adaptations un-derpin fundamental biological processes, such as micro- bial adaptations to growth conditions or the abilityof pathogens to rewire their metabolism and evade theaction of antimicrobial drugs . Metabolic adaptationsare also thought to modulate the onset of complex dis-eases such as cancer , diabetes, Alzheimer’s, and oth-ers . As a result, there is a growing need for compu-tational methods that go beyond classical pathway def-initions and uncover hidden groupings and interactionsbetween metabolic components.The complexity of metabolism has prompted the de-velopment of a myriad of methods to analyse its con-nectivity. For specific pathways, kinetic models basedon differential equations are widely employed to describethe temporal dynamics of metabolic intermediates andproducts . At the genome scale, however, such kineticmodels face substantial challenges in their constructionand analysis . The most widespread method for genomescale modelling is Flux Balance Analysis (FBA), a pow-erful framework to predict how metabolic fluxes are dis-tributed on the reaction network under an optimizationprinciple, such as maximizing growth. Therefore, theconcept of a metabolic network in FBA refers to thestoichiometric connectivity relating the enzymes and themetabolites they catalyze. Such a definition, however, isat odds with the discipline of network science , in whichcomplex systems are mathematically described throughgraphs that are amenable to computational and mathe-matical analyses.In this paper we discuss the relationship between FBAand graph-based analyses of metabolism, and we high-light the different perspectives they bring to the prob-lem. On the one hand, FBA has been shown to usefullypredict metabolic activity in various environmental andgenetic contexts; on the other, network science can shedlight on the emergent properties of global metabolic con- a r X i v : . [ q - b i o . M N ] J un nectivity. As illustrated in Figure 1, both approachesshare a common basis in that they represent genome-scalemetabolism in terms of a stoichiometric matrix, yet theyoffer different toolkits for analysis. In the following, wediscuss their advantages and caveats, and highlight theneed for integrated methods that combine flux optimisa-tion with the topological and graph-theoretical methodsof network science. II. GENOME-SCALE METABOLIC MODELLING
A widely adopted strategy for genome-scale modellingis constraint-based analysis , an umbrella term for vari-ous algorithms that predict metabolic fluxes using opti-mization principles. Most popular among these is FluxBalance Analysis (FBA), which predicts metabolic fluxesat steady state by solving the following optimizationproblem: max v J ( v )subject to: S v = 0 V min i ≤ v i ≤ V max i , i = 1 , . . . , m, (1)where S is the n × m stoichiometry matrix for a systemwith n metabolites and m reactions; v is a vector contain-ing the m reaction fluxes; and ( V min i , V max i ) are boundson each flux. J ( v ) is the objective function and is suitablychosen to describe the optimization principle assumed tounderpin the physiology of the particular organism un-der study. In microbes, the most common choice for theobjective function is biomass production, in which case J ( v ) = c T · v , i.e., J ( v ) is assumed to be a linear combi-nation of specific biosynthetic fluxes describing biomassoutput as given by the vector c . There exists a broadrange of dedicated FBA software packages and thepopularity of FBA has led to a myriad of extensions thataccount for other complexities of cell physiology, such asgene regulation and dynamic adaptations , amongmany others .Flux Balance Analysis has found applications in di-verse domains, including cell biology , metabolic en-gineering , microbiome studies , and personalizedmedicine . A salient feature of FBA is its abilityto incorporate various types of ‘omics datasets into itspredictions. Various approaches have been developed forthis purpose , most of which incorporate experimen-tal data into the metabolic model through adjustmentsof the stoichiometric matrix S or the flux bounds V min i and V max i in (1).A popular use case of FBA is the identification of es-sential genes , i.e., genes that severely impact cellulargrowth when knocked out. Through simulation of genedeletions, FBA can serve as a systematic tool for in sil-ico screening of lethal mutations, and identification ofbiomarkers and drug targets in disease . A relatedapplication of FBA is the study of metabolic robustness .Since only a fraction of all metabolic reactions are essen-tial in a given environment, knocking out non-essential reactions often has little effect on the phenotype. This isbecause many reactions have functional backups throughother pathways, so as to preserve cellular function in faceof perturbations. By providing insights into the reorgani-zation of fluxes under different conditions, FBA can alsohelp improve our understanding of robustness to geneknockouts , gene mutations and different growthconditions .One limitation of FBA is the crucial importance of theobjective function to be optimized, which needs to bedesigned to represent cellular physiology. In microbes,a common choice is maximization of growth rate, butit is questionable whether this is a realistic cellular ob-jective across organisms or in different growth condi-tions . Although the vast majority of FBA studiesrely on the maximization of cellular growth, other objec-tive functions have been proposed, including maximiza-tion of ATP production and minimization of substrateuptake minimization . III. APPLICATIONS OF NETWORK SCIENCE INMETABOLIC MODELLING
Network science represents complex systems as graphswhere the nodes describe the components of the sys-tem and the edges describe interactions between com-ponents. This general description provides a backbonefor the modelling of large, interconnected systems acrossmany disciplines, including biology, sociology, economicsand others . There have been numerous attempts toformalize the analysis of metabolism under the lens ofnetwork science. Graph-theoretic concepts such as de-gree distributions and centrality measures can revealstructural features of the connectivity of the overall sys-tem, while clustering algorithms can uncover substruc-tures hidden in the network topology. Such tools can becombined with the analysis of perturbations, such as dele-tions of network nodes or edges , which can representchanges in the environment, gene knockouts, or therapeu-tic drugs that target specific metabolic enzymes. UnlikeFBA, in which the analysis depends on the choice of aspecific objective function, network-theoretical methodsrely on the metabolic stoichiometry alone.Metabolic modularity is an area where network sci-ence has shown promising results. Intuitively, a networkmodule is a subset of the network containing nodes thatare more connected among themselves than to the rest ofthe network. Numerous works have studied the modular-ity of metabolic networks, and how the network modulescan be used to coarse-grain the metabolic network intosubunits . The modules identified using networkanalysis have been found to mirror the organization oftextbook biochemical pathways while uncovering novellinks and relationships between them . A recurringtheme in these analyses is the bow-tie topology, wherebya metabolic network can be divided into an input com-ponent, an output component and a strongly connected S v = 0
Genome-scale model robustness genes as nodes x knockouts heterologousexpressionbiomasscarbonsource wasteproducts Flux balance analysis clusteringcentrality nodes % nodes removed
Graph constructionAnalysis tools essentiality genes
Analysis ToolsOptimization x flux distribution FIG. 1.
Strategies for the analysis of metabolism at the genome-scale.
Starting from the metabolic stoichiometry,network science and FBA provide alternative routes for the analysis of metabolic pathways. Common applications include thestudy of metabolic robustness, gene essentiality and the impact of heterologous expression. internal component. This architecture aligns well with anintuitive understanding of metabolism, which comprisesnutrient uptake, waste production and secretion, and alarge number of internal cycles which produce biomassand energy .Despite promising results in the analysis of modularity,network science has achieved mixed success in metabolicresearch. For example, from a network perspective it isnatural to assume that essential genes should be asso-ciated with high centrality scores . This idea drawsparallels from other domains, such as the internet andsocial networks, where highly central nodes are deemedcritical for network connectivity. However, the correla-tion between gene essentiality and node centrality areweak, with various essential metabolites and reactionsexhibiting low centrality scores , possibly as a resultof poorly connected nodes in pathways that supply re-sources essential for growth . Other studies have at-tempted to resolve this problem with new network met-rics specifically tailored to describe important features ofmetabolism .A key challenge for the use of network science inmetabolic modelling is the lack of consensus on how tobuild a graph from a metabolic model. For a networkwith q nodes, the graph is encoded through the q × q ad-jacency matrix A , which has an entry A ij = 0 if nodes i and j are connected, and A ij = 0 otherwise. Dependingon how nodes and edges are defined, one can build dif-ferent graphs for the same metabolic model described bythe stoichiometry matrix S in (1). For example, one canbuild a graph where the nodes are metabolites and theedges are reactions between them . In this case theadjacency matrix is A n × n = ˆ S ˆ S T , (2)where ˆ S is the binary version of the stoichiometry matrix S (i.e. ˆ S ij = 1 when S ij = 0, and ˆ S ij = 0 otherwise). Conversely, a graph where the nodes are reactions andthe edges describe the sharing of metabolites as reactantsor products has an adjacency matrix A m × m = ˆ S T ˆ S . (3)One can also build bipartite graphs, where both metabo-lites and reactions are nodes of different types , oreven hypergraphs where an edge connects a set of reac-tants to a set of products . In addition, all of thesegraphs can be directed/undirected (when the matrix A issymmetric/asymmetric), or weighted/unweighted (wherethe elements A ij can have weights encoding differentproperties). Such modelling choices have a dramatic in-fluence on the results and conclusions drawn from net-work analyses . For example, the existence ofpower law degree distributions and the small-worldproperty , two widespread concepts in network science,have been disputed and attributed to specific waysof constructing the metabolic network graph .A further limitation of graph-based analyses is their ad hoc treatment of pool metabolites, e.g., H O, ATP,NADH and other enzymatic co-factors. Because poolmetabolites participate in a large number of reactions,they distort and dominate the topological properties ofthe network. A common approach to minimize this prob-lem is to prune pool metabolites from the graph; yetthere is no accepted standard on how to do this or howto mitigate the potential loss of information in so do-ing . Another challenge arises from the representationof the reversibility of metabolic reactions in the graph.Although all biochemical reactions are reversible, theytake one direction depending on the physiological condi-tions. Graph-based studies either pre-define a directionfor the flux of the reaction, or they split them into for-ward and backward components . Neither of theseapproaches is ideal: assigning the direction of a reactionbased on one condition may not generalize across otherconditions, whereas incorporating bi-directional edges in-creases the complexity of the analysis.
IV. INTEGRATING FLUX INFORMATION ANDNETWORK SCIENCE
As discussed, network-theoretical models ofmetabolism can be built in multiple ways. Eachparticular choice represents particular modelling as-sumptions and caveats that shape the conclusionsthat can be drawn from them. Recent work, however,strongly suggests that integration of flux informationinto network analyses offers a promising avenue toaddress these challenges.A well-established approach relies on the notion of
Ele-mentary Flux Modes (EFM) . Roughly speaking, EFMsare steady state flux vectors (i.e. satisfying S v = 0) with aminimal, unique set of fluxâĂŘcarrying reactions. EFMsare nonâĂŘdecomposable steady-state pathways, in thesense that if any of its contributing reactions is deleted,the EFM will not be able to carry a steadyâĂŘstateflux . A key result is that every steady state flux distri-bution can be represented as a non-negative linear combi-nation of EFMs. A number of algorithms have been pro-posed for the efficient enumeration of EFMs . Some ofthese algorithms exploit topological connectivity and fluxinformation simultaneously, e.g., by using graph-basedmodels to identify flux balanced pathways ; by enumer-ating the pathways with minimum flux variability ; orby dividing the steady state flux space into smaller, in-dependent sub-modules . A somewhat related strat-egy utilizes the concept of flux coupling , first defined byBurgard and colleagues . Two reactions R i and R j aresaid to be flux coupled if non-zero flux in one impliesa non-zero flux in the other. This can be summarizedin terms of the flux ratio R ij = v i /v j . Depending onminimal and maximal values attainable by R ij , reactionscan be: uncoupled ( R ij ∈ [0 , ∞ )), directionally coupled( R ij ∈ [0 , k ] or R ij ∈ [ k, ∞ ), with k a positive number),or fully coupled ( R ij = k ). A reaction coupling graphcan then be built by adding an edge between R i and R j with the direction of the edge depending on the type ofcoupling. This graph has been successfully employed tofind hierarchical relations between groups of reactions ,as well as to identify driver reactions involved in controlof metabolic activity . Both EFM and flux coupling ap-proaches can be employed to build graphs with stoichio-metric information, but, importantly, they do not requirethe optimization of a cellular objective, as in FBA.Another strand of research has focused on combin-ing FBA solutions with the construction of graph mod-els. Such studies cover a wide range of methodologiesand applications, including the identification of biomark-ers using flux and centrality analyses ; the detectionof metabolic drug targets in cancer with flux similaritygraphs ; the study of metabolic robustness ; and theanalysis of metabolite essentiality . The integra- tion of FBA and network-theoretic analyses can overcomesome of the ambiguities in the construction of graphs torepresent metabolic models. Flux values resolve the di-rection of a reaction in a given physiological state andcan thus be employed to assign weights to the inter-actions between nodes. Various studies have utilizedthis idea to construct graphs with either metabolites asnodes or reactions as nodes . For example,the mass flow graph proposed in Beguerisse et al usesFBA solutions to weigh the edges of a reaction graph. Ifreaction R i produces a metabolite x k that is consumed by R j , then the weight of the edge between both reactionsis w ij = X k (mass flow of x k from R i to R j ) , (4)where the sum acts on all the metabolites that are pro-duced by R i and consumed by R j . The weights (4) aredirectly computed from the stoichiometric matrix S anda flux vector obtained with FBA. Different mass flowgraphs can be then computed for FBA solutions corre-sponding to specific environmental or biological condi-tions. Such mass flow graphs were combined with cen-trality analyses and community detection algorithms toreveal changes in the modular structure of Escherichiacoli metabolism in various growth media, and to iden-tify structural changes in hepatocyte metabolism in ametabolic disease affecting kidney function . V. DISCUSSION
Recent discoveries have led to a renewed interest inthe interplay of metabolism with other layers of the cel-lular machinery . Due to the complexity and scale ofmetabolic reaction networks, computational methods areessential to tease apart the influence of metabolic archi-tectures on cellular function. Here we have discussedthe complementary roles of Flux Balance analysis andnetwork science in the analysis of metabolism at thegenome scale. Although both approaches start from themetabolic stoichiometry (Figure 1), they differ in theirmathematical foundations and the type of predictionsthey produce. FBA predictions can be accurate but theireffectiveness requires high quality ‘omics datasets. Net-work theory, in contrast, requires nothing more than themetabolic stoichiometry, yet can lead to misleading pre-dictions depending on how the network graph is built.As a result, so far FBA has led to more successful con-nections with experimental results than network science.When used in isolation, both FBA and network sciencecan be insufficient to understand changes in metabolicconnectivity triggered by physiological or environmen-tal perturbations. The integration of FBA and network-theoretical methods can close this gap in many applica-tion domains. For example, with the rise of big datain the life sciences, there is a growing interest in us-ing metabolic signatures of patients to tailor treatmentssuited to their individual needs . Computational meth-ods can play a key role in detecting drug targets involvedin metabolic activity, and how their targeting can disruptmetabolic connectivity. A particularly promising area iscancer treatment, where there is considerable interest ondrugs that target specific metabolic enzymes .Another exciting application domain is industrialbiotechnology , where so called “microbial cell facto-ries” are engineered for production of commodity chemi-cals and fine products . In this field, FBA is widely em-ployed for strain design, with the goal of finding combina-tions of genetic interventions that maximize productionof a desired metabolite. A recent trend is to increase pro-duction with synthetic biology tools and dynamic controlof gene expression . This approach needs computa-tional methods that capture the dynamic reallocation ofmetabolic flux. Integrating FBA solutions with networkmodels can provide a versatile tool to identify suitablegenetic modifications for microbial strains with increasedproduction.Further developments at the interface of FBA and net-work science offer a novel way to explore the impact ofperturbations on metabolic connectivity. The flexibilityof FBA allows for the modelling of metabolic perturba-tions of various kinds, including changes in growth condi-tions, deletion of metabolic genes or the action of enzymeinhibitors, whereas the application of graph-theoreticaltools from network theory can bring a broadened under-standing of emergent properties of the overall system.This flexibility is a key advantage and offers promisingpotential to deploy network science tools across a rangeof questions in basic science, biomedicine and industrialbiotechnology.
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