Deep epistasis in human metabolismMarcin Imielinski1,a� and Calin Belta2

1Department of Pathology, Massachusetts General Hospital, Harvard Medical School, Boston,Massachusetts 02114, USA2Department of Mechanical Engineering, Department of Systems Engineering,and Department of Bioinformatics, Boston University, Boston, Massachusetts 02215, USA

�Received 3 February 2010; accepted 1 June 2010; published online 30 June 2010�

We extend and apply a method that we have developed for deriving high-order epistatic relation-ships in large biochemical networks to a published genome-scale model of human metabolism. Inour analysis we compute 33 328 reaction sets whose knockout synergistically disables one or moreof 43 important metabolic functions. We also design minimal knockouts that remove flux throughfumarase, an enzyme that has previously been shown to play an important role in human cancer.Most of these knockout sets employ more than eight mutually buffering reactions, spanning mul-tiple cellular compartments and metabolic subsystems. These reaction sets suggest that humanmetabolic pathways possess a striking degree of parallelism, inducing “deep” epistasis betweendiversely annotated genes. Our results prompt specific chemical and genetic perturbation follow-upexperiments that could be used to query in vivo pathway redundancy. They also suggest directionsfor future statistical studies of epistasis in genetic variation data sets. © 2010 American Institute ofPhysics. �doi:10.1063/1.3456056�

Small molecule metabolism mediates the most basic bio-chemical function of living matter: to derive energy andmacromolecules from nutrients in the environment.Metabolic pathways are tied in intricate networks whichin E. coli and S. cereviseae have been shown to display agreat degree of buffering and epistasis. Epistasis has notbeen previously been examined on a genome-scale in hu-man metabolism. We apply a novel in silico approach tosample the depth of epistasis in the human metabolic net-work with respect to a variety of metabolic objectives.

I. INTRODUCTION

Cellular metabolism provides the biochemical machin-ery to transform a small set of simple nutrients into the com-plex building blocks of life. The metabolic network of ahuman consists of thousands of small molecule species intri-cately linked by an even larger set of biochemical reactions.Dysregulation of metabolism underlies rare fatal conditionslike Niemann–Pick disease and pervasive chronic illnesseslike diabetes. The metabolic pathways of normal human cellsare distinctly altered during carcinogenesis.

The connectivity and coupling of metabolic reactionscan be most simply captured in a stoichiometric model. Thefirst such known model was constructed by Shapiro in the1960s, who applied mathematical formalisms from econo-metrics to conduct an “input-output analysis” of E. colimetabolism.1 Over the past ten years, genome-scale stoichi-ometric metabolic models have been built for a variety oforganisms, including E. coli, S. cereviseae, H. pylori, S. au-reus, and H. sapiens.2–14 Such models facilitate analysis ofsystems-level behaviors such as growth or metabolite pro-

duction in the context of nutrient media and chemical/geneticperturbations.14,15

Part of the complexity of metabolism arises from theexistence of backup pathways for a given function. Suchparallelism can obscure the role of seemingly dispensablenetwork components. The role of genes and enzymes in suchrobust functions can only be revealed through the knockoutof multiple genes or reactions.16–21 We refer to a set of reac-tions whose knockout abolishes a given function as a cut setfor that function. A cut set is minimal for a given function ifits knockout abolishes that function, while the knockout ofnone of its subsets abolishes that function. Reactions thatcontribute to minimal cut sets �MCSs� are “epistatically” re-lated, since their simultaneous knockout induces a phenotypethat is not seen with simpler knockout combinations.45 Inprevious work, we have identified over 11 000 MCSs of fiveor more reactions for biomass production in E. coli.19 Deut-scher et al.17 employed in silico multiple knockout analysisto reveal novel essential roles for reactions in the yeast meta-bolic network. These results indicate the presence of robustparallelism and deep epistasis in the metabolic networks ofE. coli and yeast.

Interest in metabolic epistasis is motivated by practicalconsiderations. First, pharmaceutical manipulation of me-tabolism, e.g., for chemotherapy or antibiotics, can be con-founded by the presence of alternative pathways supplyingan unwanted function. Second, complex epistatic relation-ships can mask statistical genetic associations linking germ-line or somatic genetic variation to disease phenotypes, e.g.,diabetes, cancer.22 The size of the search space for “naive”epistatic genetic models not only incurs significant computa-tional expense but significantly reduces the statistical powerto detect complex associations in genetic data sets. Systemsbiology and network analysis can inform statistical geneticsa�Electronic mail: mimielinski@partners.org.

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by providing biologically grounded epistatic models that fol-low from basic biochemical relationships between geneproducts.23,24

To detect high-order epistasis in in silico metabolic net-works, one can apply a brute-force approach to exhaustivelytest all combinations of single, double, triple, etc. mutants.This approach, although applicable to analysis of low-order��4� knockout combinations, is untenable for higher-ordermutant combinations.17 For example, a network of 1000 re-actions would require over 1014 linear programs to exhaus-tively test all possible quintuple-knockout combinations. Toovercome the curse of dimensionality we have recently ex-tended the elegant approach of Klamt and Gilles for identi-fying complex knockouts.19,25 Klamt and Gilles’ method ex-ploits the pathway decomposition of a metabolic network touncover MCSs, or minimal sets of reactions whose removaldisables a particular objective reaction. Our extended ap-proach takes advantage of a “relaxed” pathway definitionthat is actually computable for genome-scale networks.19

In this paper we extend our approach further and apply itto the study of human metabolism using a published networkreconstruction.12 We compute MCSs for 43 important bio-synthetic objectives and a metabolic enzyme known to bemutated in cancer. Our results reveal high-order epistatic re-lationships between diverse components of human metabo-lism and illuminate essential systems-level roles of reactionsin highly redundant and robust metabolic subnetworks.

II. METHOD

A. Theory

1. NotationR is the set of real numbers, R+

n is the set of alln-dimensional vectors with real and positive components,and Rm�n is the set of all m�n matrices with real entries.Given m ,n�N, we use the notation M = �1, . . . ,m� andN= �1, . . . ,n�. For a set C, we use �C� to denote its cardinal-ity. If A�Rm�n and U�M, then AU denotes the submatrix ofA containing the rows with indices in the set U. Therefore, ifx�Rn, i�N, and U�N, then xi and xU�R�U� denote its ithcomponent and the vector formed by taking componentswith indices in set U, respectively. The inequality x�0 isinterpreted componentwise, i.e., xi�0, i=1, . . . ,n, while theinequality x�0 is interpreted as x�0, x�0. Each vectorx�Rn induces a ray, rx= ��x ���0�. We denote the sparsitypattern of a ray rx�Rn as SP�rx�= �j�N �xj�0�. The spar-sity pattern SP�R� of a ray collection R is a collection of sets�SP�rx� �rx�R�.

2. Stoichiometric metabolic modelWe represent a mass-balanced metabolic network of n

chemical reactions involving m metabolites with stoichiom-etry matrix S�Rm�n. In addition to representing the bio-chemical reactions in the cell, S encodes reactions mediatingexchange �uptake and secretion�, growth-based dilution, andconsumption of small molecule species by macromolecularprocesses. Each entry Sij specifies the stoichiometric coeffi-cient for metabolite i in reaction j, which is negative forsubstrates and positive for products. We represent the flux

distribution through the reactions of the network by v�Rn,where a component v j corresponds to flux through reaction j.We constrain all reactions in the network to be irreversible,i.e., v�0, and represent every reversible reaction with twoirreversible reaction fluxes with opposite orientation.46 Theconcentrations of species in the system at time t are denotedby x�t��R+

m. Under these assumptions, the rate of change intime of species concentrations is given by

x = Sv, v � 0. �1�

Metabolic reactions occur at a fast rate with respect tocell regulatory and environmental changes. When modelingat the slower time scale it is reasonable to apply the quasi-steady state assumption,14 which assumes that for the over-whelming majority of time the system obeys

Sv = 0, v � 0. �2�

Under these assumptions, the flux states of the network lie inthe polyhedral set

K = �v � Rn�Sv = 0,v � 0� , �3�

which we refer to as the feasible flux cone of S induced bystoichiometric, steady state, and irreversibility constraints.

3. Minimal cut setsA cut set for a metabolic objective is a set of reactions

whose knockout abolishes that function.25 The objective isgenerally a flux in the network to be disabled. This can rep-resent a single reaction, the producibility of a target metabo-lite, or the consumption of small molecules species by mac-romolecular processes, e.g., biomass production.

Formally, in a stoichiometric model of metabolism S, aset of reactions C�N is a cut set for an objective reactionj�N if and only if v j �0 is feasible in the wild type �i.e.,∃v�K �v j �0� and

vC = 0 → v j = 0, ∀ v � K . �4�

A cut set C is minimal if no proper subset of C is a cut set.MCSs are “elementary failure modes” of metabolicnetworks.26 High cardinality MCSs also represent fundamen-tal units of synergistic epistasis between network compo-nents involved in robust systems-level functions.19

MCSs can be brute-force identified by exhaustively iter-ating through all possible reaction knockout combinations Cusing linear optimization �e.g., flux balance analysis �FBA��and determining feasibility of v j �0. However, in large net-works �i.e., thousands of reactions�, this method is limited tothe discovery of low cardinality �k�4� MCSs.17

Alternatively, MCSs can be constructed as minimal hit-ting sets of extreme pathways �EPs� using the method ofKlamt and Gilles.19,25 EPs arise from a ray decomposition ofthe feasible flux cone K, and can be thought of as both quan-titative flux states and reaction subsets �i.e., pathways� in thenetwork. A MCS for a metabolic function is a minimal set ofreactions that intersects all EPs that supply that function�e.g., biomass production�. Minimal hitting sets can be enu-merated using various approaches, such as the Bergealgorithm.27

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A major bottleneck to the elegant approach of Klamt andGilles is EP computation, which is not usually possible forgenome-scale networks.47 However, in previous work, wehave overcome this bottleneck by generalizing the theoreticalresults of Klamt and Gilles to rays that generate overapproxi-mations of the feasible flux cone K.19 These rays, which wecall pathway fragments, emerge from the application ofsteady state constraints to only a subset of species in thenetwork, yielding a relaxed flux cone K� for which K�K�.

Like EP, pathway fragments can be used to deriveMCSs. A minimal hitting set C of a collection of pathwayfragments feeding objective j is guaranteed to be a cut set forj.19 However, there are two caveats. A cut set C obtained inthis manner is not guaranteed to be minimal. As a result, itmay require reduction to minimality via an optimizationbased post processing step. Second, we are also not guaran-teed to find all cut sets C through the analysis of pathwayfragments. This nonminimality and incompleteness is theprice of the overapproximation of K by K�, but clearly im-proves the more closely K� envelopes K, i.e., the more spe-cies steady state constraints are enforced.

In practice, we have shown that the analysis of pathwayfragments in genome-scale metabolic networks can yieldlarge numbers of complex MCSs. In E. coli, this approachgenerated more than 11 000 MCSs for biomass production.19

Most MCSs were also of high cardinality ��5� and wouldhave been virtually impossible to identify through brute-force approaches. These MCSs clustered into functional re-action modules that illuminated instances of robust parallel-ism and deep epistasis in E. coli metabolism.

In this paper we apply our MCS computation method tothe network of human metabolism, which contains a muchlarger number of reactions and species compartments thanE. coli metabolism. Our method involves three basic stages:�1� pathway fragment generation, �2� minimal hitting setcomputation, and �3� reduction of cut sets to minimality. Toincrease our yield of MCSs, we modify the previously pub-lished method by generating multiple “extended” pathwayfragment collections in stage 1 to feed into stage 2.

B. Genome-scale MCS computation

1. Basic algorithmThe first stage of our algorithm is pathway fragment

generation. The feasible flux cone K of the metabolic net-work S is the set sum of a finite and unique collection ofextreme rays E�K�.2 E�K� are computed iteratively by deter-mining the extreme rays E�Ki� for a series of polyhedralcones Ki�Rn where K0=R+

n and

Ki = �v�SMiv = 0,v � 0�, i � M , �5�

where Mi�M, �Mi�= i, and Mi�Mi+1.In other words, at each iteration i we determine the cone

Ki of flux configurations that constrains i metabolites tosteady state. Following this, we choose a new metabolite foriteration i+1. It follows directly from Eqs. �3� and �5� thatK�Ki, as shown in Fig. 1.

The initial collection of generators E�K0� consists of raysinduced by the Euclidean basis vectors ej �Rn. At each it-eration i�M, a novel collection Etemp�Ki� is generated from

positive linear combinations of rays in E�Ki−1� that are out ofbalance with respect to �i.e., consume or produce� metabolitei. Etemp�Ki� contains many nonextremal rays, i.e., thosewhich lie in the positive span of two or more of their coun-terparts, and must be pruned to yield E�Ki�.

This pruning process comprises the most computation-ally intensive part of each algorithm iteration. Pruning isimplemented through pairwise comparison of the sparsitypatterns of rays r in Etemp�Ki� and removal of any ray r forwhich there exists an r� such that SP�r���SP�r�. Since thesize of E�Ki� can become quite large with increasing i, thisstep of the computation reaches memory limits �or slows to acrawl� at some iteration imax�m for most genome-scale net-works. However, the difference between m and imax is usu-ally quite small even for large networks, representing a smallsubset of only the most connected metabolites.

We refer to rays in each collection E�Ki� as pathwayfragments for S, since the final EPs E�Km�=E�K� lie in theirpositive span �and thus involve a superset of their reactions�.At iteration i= imax, when computational resources are ex-hausted, pathway fragments P=E�Ki� are collected and in-putted into step 2 �minimal hitting set computation� to gen-erate cut sets.

Our formulation is flexible with respect to the order ofmetabolite traversal. The choice of metabolite at each itera-tion i can be thus optimized with the goal of maximizing imax

and producing a pathway fragment collection that captures alarger number of network constraints. As previously de-scribed, we employ a local greedy optimization strategy toachieve this purpose.19,29

The second stage of our algorithm is minimal hitting setcomputation. Because the cone Kimax overapproximates K�Fig. 1�, a flux configuration v that is infeasible in Kimax willalso be infeasible in K. We use this property to generate cutsets for an objective j from the analysis of pathway frag-ments generating Kimax. Given a pathway fragment collectionP=E�Kimax� and objective reaction j, we generate cut setsthrough the analysis of Pj, the collection of j-containing rays

v1

v2

v3

K i

(cone gener ated bypathway f r agments)

K(feasible flux cone)

FIG. 1. A relaxed flux cone Ki is generated by pathway fragments that fulfilla subset of steady state requirements for metabolic network S. With eachnew iteration i of pathway fragment computation, Ki provides an increas-ingly better overapproximation to the system’s actual feasible flux cone K.Since any flux that is infeasible in Ki is guaranteed to be infeasible in K, wecan use the analysis of pathway fragments generating Ki to compute cut setsfor a reaction j.

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in P. We have previously shown that a reaction set C�N isa cut set for j if C intersects all j containing pathway frag-ments in P, i.e., if C is a hitting set of SP�Pj�. Although wecannot ensure that C will be an MCS for j, we can increasethe likelihood of minimality by choosing C that are minimalhitting sets of SP�Pj�. Minimal hitting sets for SP�Pj� arecomputed using a variant of the Berge algorithm.19,27

In the third stage, we reduce minimal hitting sets toMCSs using linear optimization. To determine whether theminimal hitting set C is an MCS it needs to be checked forminimality against the objective j in metabolic network S.This is done by determining feasibility of flux j when each ofthe immediate subsets of C is knocked out.19 For example, if�C�=10, one would perform ten linear optimizations to verifyits minimality. Cut sets that are not minimal are pruned tominimality by recursively testing subsets with additional lin-ear optimizations.

2. Increasing MCS yield by expanding the pathwayfragment collection

In the first step of our previously published MCS algo-rithm, we carry pathway fragment computation to an itera-tion imax. The next iteration imax+1 fails due to the large num-ber of vector comparisons required to remove nonextremalrays from the collection Etemp�Kimax+1�. However,Etemp�Kimax+1� is usually computable, i.e., by taking positivecombinations of rays in E�Kimax� to balance the metabolite atiteration imax+1, although it may not be storable in mainmemory. Despite containing many nonextremal rays, thiscollection is identical in essence to other pathway fragmentcollections; namely, minimal hitting sets of j-containing raysin Etemp�Kimax+1� are cut sets of reaction j. We can thus gatherj-containing pathway fragments from Etemp�Kimax+1� and ex-ploit them for MCS computation.

What is the benefit of extending pathway fragment com-putation a half iteration further? To explain this, we intro-duce the concept of a boundary set B�P��M \Mi of a path-way fragment collection P=E�Ki�. B�P� is the subset ofmetabolites that are consumed or produced by pathway frag-ments in P �i.e., are not in steady state�. For a given objectivej, B�Pj� can be large or small, depending on the connectivityof j to the remainder of the metabolic network and how wellthe pathway fragment computation has discovered that con-nectivity. At i=0 of pathway fragment computation, Pj con-sists of only a single reaction �i.e., the ray induced by ej� andB�Pj� will consist only of the substrates and products of re-action j. As iterations proceed and substrates/products of re-action j are traversed, �Pj� naturally grows and both �B�Pj��and �B�P�� are reduced.

At iteration imax, B�P� is generally equal to M \Mimax, theset of metabolites consumed or produced by one or morepathway fragments in P=E�Kimax�. Because of the localgreedy optimization strategy used in guiding pathway frag-ment computation, B�P� consists mostly of species in centralpathways �e.g., glycolysis, citric acid cycle�, currency mol-ecules �ATP, NADH, CoA�, and promiscuous ions �hydro-gen, water, ammonia�. A reaction j that is buried in the coremetabolic machinery and only involves these species inB�P�=M \Mimax as substrates or products will still be repre-

sented by only a single ray ej in E�Kimax�. This occurs be-cause that pathway fragment ej is by definition balanced withrespect to all previous species Mimax and thus has been ableto escape engaging in positive linear combination with anyray in any other iteration. For such a reaction j, analysis ofPj would only generate a single cut set �j� �since ej is theonly j containing pathway fragment�.

However, if we generate the collection Etemp�Kimax+1� bydeliberately choosing a metabolite in B�Pj� �i.e., in this casea species consumed or produced by reaction j�, then we willderive positive linear combinations of ej with all pathwayfragments in P \ Pj that consume or produce this species. Al-though this metabolite may have been avoided by the greedysearch due to its promiscuity with the existing pathway frag-ments, we force it to be considered with the aim of expand-ing Pj. The resultant rays obtained in this half iteration willconnect reaction j to other reactions that use both the newmetabolite and at least one species from Mimax. These reac-tions will now take part in cut sets for flux j, potentiallydramatically increasing the yield of MCSs. This “single me-tabolite extension” can be applied separately to each speciesthat is produced or consumed by flux j, further increasing theyield. In general, we can perform extensions for each speciesin B�Pj� given an objective j associated with pathway frag-ment collection Pj, even when �Pj��1. The resulting ex-tended pathway fragment collections can be shuttled to mini-mal hitting set computation to increase the yield of MCSs forany objective j in the network.

C. Implementation

1. Human metabolic networkWe employed the H. sapiens Recon 1 genome-scale stoi-

chiometric model built from 1496 ORFs �open readingframes� spanning 2004 proteins, 2766 metabolites, and 3311metabolic reactions, including 1078 transport reactions.12 Wereplaced reversible reactions with two irreversible reactions,and supplemented this network with 2362 sink fluxes to cap-ture the dilution of intracellular species via growth and theirconsumption by macromolecular processes.15,30 We simu-lated rich media by making all extracellular species availableto the network. In the interest of reducing metabolite dimen-sionality, we performed this by eliminating rows correspond-ing to extracellular metabolites �as opposed to adding addi-tional source fluxes�. The final network corresponding to S inEq. �2� contained 2362 rows �metabolites� and 7654 columns�reactions�.

2. ComputationAll computation was carried out on MATLAB R2008a us-

ing a Dell 8 Core Precision T7400 �Intel Xeon E5430, 2.66GHz, 4 Gbytes random access memory� running Linux andan HP 8 Core DL145 G3 �AMD Opteron, 2.7 GhZ, 32Gbytes random access memory� running Microsoft WindowsServer 2003 R2 Enterprise x64. Pathway fragment computa-tion, minimal hitting set, and reduction routines were imple-mented as MATLAB scripts and can be made available uponrequest. We computed pathway fragments through imax

=2209 of 2362 metabolites yielding 339 504 pathway frag-ments. During the pathway fragment extension step for ob-

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jective j, we excluded extremely promiscuous metabolites�i.e., those consumed/produced by more than 20 000–40 000rays in Pj in B�Pj� to reduce the computational load�. For theoriginal and each extended pathway fragment collection, wecomputed minimal hitting sets of the j containing pathwayfragments. We limited minimal hitting set computation tosets of cardinality �10 in the interest of reducing computa-tional time. We implemented flux feasibility computationswithin the reduction routine as previously described19 usingthe SeDuMi convex optimization toolbox.31

III. RESULTS

A. Deep epistasis underlies many human biosyntheticfunctions

To examine the robustness of the human metabolic net-work with respect to a variety of biosynthetic objectives, wecomputed MCSs for the producibility of 43 important cellu-lar metabolites spanning four biochemical classes: this com-prised 24 amino acids, five deoxyribonucleotides, ten mem-brane lipids, and four core metabolites �i.e., energymetabolism intermediates�. For all cases we examined MCSsmediating cytoplasmic producibility of the given metabolite,with the exception of deoxyribonucleotides where we exam-ined their producibility in the nucleus.

Before proceeding with a description of our results, weemphasize that our MCS computation approach does notguarantee the discovery of all MCSs. Therefore, the absenceof certain MCSs in our results is not necessarily reflective ofthe �unknown� underlying distribution of MCSs for thatfunction. However, by establishing with certainty the exis-tence of MCSs with certain qualities, we can make someinteresting assertions regarding epistasis in human metabo-lism.

In total, we found 33 328 unique MCSs targeting one ormore of 43 metabolic objectives.32 This included 650, 98,21 187, and 11 393 unique MCSs targeting amino acid,nucleotide, membrane lipid, and core metabolite biosynthe-sis, respectively. Most interestingly, MCSs found using ourmethod were almost all �99.5%� of cardinality greater than 5,with 82% having cardinality of 9–10. A similar MCS cardi-nality distribution was observed across all four metaboliteclasses studied �Fig. 2�. These results suggest a strikingdepth of parallelism in human biosynthetic pathways, irre-spective of metabolite class.

Figure 3 shows the number of MCSs we discovered forthe various individual objectives, grouped by metabolite

class. Among the 24 amino acids, only D-Aspartate,L-Aspartate, L-Tyrosine, L-proline, and L-arginine yieldedsignificant numbers of MCSs. The relatively small yield ofnucleotide-targeting MCSs that we discovered was directedat dGTP and dTTP biosynthesis. In contrast we found mul-tiple MCSs for all four core metabolites tested, which in-cluded 4181 MCSs targeting fumarate synthesis and 6977targeting oxoglutarate. Multiple membrane lipid objectivesyielded large numbers of MCSs, with the most plentifulnumber by far associated �18 074� with sphingomyelin pro-duction.

B. MCSs expose novel essential roles in humanmetabolism

MCSs enable an expanded notion of essentiality calledk-essentiality.19 A reaction is k-essential for a given objectiveif there exists a �possibly complex� MCS containing thatreaction and targeting that objective. This is in contrast to thestandard notion of essentiality, which is assigned to a gene orreaction whose single knockout abolishes a phenotype.k-essential links between genes/reactions and systems-levelfunctions arise from synergistic epistasis between parallelpathways in the network.

Complex MCSs found using our method yield manyk-essential reactions. To quantify novel k-essential links be-tween reactions and objectives, we compared the numbers ofk-essential reactions to the number of 1-essential reactionsobtained from a brute-force single knockout analysis of thehuman metabolic network. Figure 4 shows how many reac-tions were deemed k-essential for each objective, with thenumbers of reactions shown to be 1-essential for the objec-tive shown in parentheses next to the metabolite label. Wefound that for most objectives we were able to associatemany more k-essential reactions with the production of agiven metabolite than were able to be found using a singleknockout analysis. In many cases, this difference was pro-found, such as for sphingomyelin, whose producibility wewere able to epistatically link to 235 reactions in the meta-bolic network.

1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1F

requ

ency

MCS Cardinality

Amino AcidCore MetaboliteMembrane LipidNucleotide

FIG. 2. Histogram showing MCS cardinalities stratified across the four me-tabolite classes.

FIG. 3. Histogram showing number of MCSs discovered for each biosyn-thetic objective examined in our study.

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C. MCSs span multiple compartments and metabolicsubsystems

MCSs discovered by our analysis span a breadth of cel-lular compartments. However, the actual distributions ofcompartment span vary distinctly between specific metabo-lite classes �Fig. 5�. In particular, amino acid-targeting MCSsdiscovered by our method employ the fewest number ofcompartments, drawing from cytoplasmic fluxes alone or acombination of cytoplasmic and mitochondrial reactions.MCSs targeting core metabolites span between two and threecompartments, consisting of primarily cytoplasmic and mito-chondrial reactions, however often also employing peroxiso-mal fluxes. Nucleotide-targeting MCSs sometimes employcytoplasmic reactions only, however more often pull combi-nations of reactions from two or three of the following com-partments: cytoplasm, mitochondria, lysosome, and nucleus.Across all metabolite classes studied, membrane-lipid-targeting MCSs are the most diverse: they harness up to fivecompartment combinations that employ reactions from thecytoplasm, endoplasmic reticulum, Golgi apparatus, nucleus,and peroxisome.

There are also metabolite class differences in the sub-system span of discovered MCSs �Fig. 6�. Nucleotide andamino acid-targeting MCSs span between one and five sub-systems. Meanwhile, membrane lipid and core metabolite-targeting MCSs are more functionally diverse, spanning be-tween three and seven and between five and ten metabolicsubsystems, respectively. MCSs targeting different metabo-

lite subclasses not only differ in their functional diversity, butin the actual metabolic subsystems targeted �Fig. 7�. Many ofthese differences are intuitive: nucleotide-targeting MCSsuses reactions from “nucleotides” and “nuclear transport”subsystems. Amino acid-targeting MCSs preferentially in-volve reactions in “citric acid cycle,” “alanine/aspartate me-tabolism,” “urea cycle/amino group degradation,” “glutamatemetabolism,” and “glycine/serine/threonine metabolism.” Incontrast, MCSs targeting core metabolite and membranelipid biosynthesis employ reactions with a much larger anno-tated functional breadth, both spanning 26 subsystems in to-tal. MCSs targeting core metabolites pull reactions from“IMP �inositol monophosphate� biosynthesis,” “pentosephosphate pathway,” “lysine metabolism,” “tryptophan me-tabolism,” and many others. MCSs targeting membrane lipidbiosynthesis employ reactions from “transport, Golgi appa-ratus,” “R group synthesis,” “carnitine shuttle,” “fatty acidactivation,” “inositol phosphate metabolism,” and “bile acidbiosynthesis” subsystems, among others. Subsystems that arecommon to MCSs across all metabolite classes are “extracel-lular transport” and “mitochondrial transport.”

Overall, we find staggering functional complexity anddiversity in the discovered MCSs, as shown in Fig. 8, whichdisplays individual “subsystem signatures” of MCSs andtheir frequency across the results �Fig. 8�. There are 475unique combinations of subsystems represented across the33 328 MCSs. As can be seen in the sparsity patterns of theplot, the majority of these MCSs combine reactions fromextracellular, mitochondrial, or endoplasmic reticulum �ER�transport with reactions chosen from a diversity of 30 coremetabolic subsystems. The existence of MCSs spanning such

0

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L−A

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L−G

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(1)L−

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spartate(1)

L−A

rginine(1)

Isocitrate(2)

Succinate

(1)F

umarate

(1)2−

Oxoglutarate

(1)P

hosphatidylethanolamine

(1)C

holine(1)

Cholesterol(3)

Triacylglycerol

(2)P

hosphatidylserine(1)

Phosphatidylcholine

(1)P

hosphatidylglycerol(3)

Phosphatidic

acid(1)

Phosphatidylinositol

(2)S

phingomyelin

(3)dC

TP

(2)dA

TP

(2)dU

TP

(5)dG

TP

(2)dT

TP

(2)

Amino AcidCore MetaboliteMembrane LipidNucleotide

FIG. 4. Histogram showing number of k-essential reactions discovered foreach biosynthetic objective tested in our study. A reaction is k-essential foran objective if it contributes to at least one MCS for that objective. Thenumber of reactions found to be 1-essential for each objective �by brute-force optimization� is shown in parentheses next to the metabolite label.

1 2 3 4 50

0.4

0.8

Fre

quen

cy

Number of compartments spanned by MCS

Amino AcidCore MetaboliteMembrane LipidNucleotide

FIG. 5. Histogram showing number of compartments spanned by MCSstargeting the four metabolite classes. Frequencies are calibrated separatelyfor each metabolite class.

1 2 3 4 5 6 7 8 9 100

0.4

0.8

Fre

quen

cy

Number of subsystems spanned by MCS

Amino AcidCore MetaboliteMembrane LipidNucleotide

FIG. 6. Histogram showing frequencies of metabolic subsystems among theMCSs found for the four classes of biosynthetic objectives. Frequencies arecalibrated separately for each metabolite class.

Transport, Extracellular

Transport, Mitochondrial

Transport, Endoplasmic

Reticular

SphingolipidM

etabolism

RGroup

Synthesis

Glycerophospholipid

Metabolism

Transport, Peroxisomal

Triacylglycerol Synthesis

Fattyacid

activation

Citric

AcidCycle

FattyAcid

Metabolism

Nucleotides

Alanineand

AspartateM

etabolism

Cholesterol M

etabolism

Fattyacid

elongation

Fattyacid

oxidation, peroxisome

BileAcid

Biosynthesis

Transport, Golgi Apparatus

Urea

cycle/amino

groupm

etabolism

Riboflavin

Metabolism

Glutam

atem

etabolism

Tyrosinem

etabolism

Transport, Nuclear

Arginineand

ProlineM

etabolism

Taurineand

hypotaurinem

etabolism

Carnitine

shuttle

Glycine, Serine, and

ThreonineM

etabolism

IMP

Biosynthesis

Miscellaneous

Inositol PhosphateM

etabolism

Tryptophanm

etabolism

PentosePhosphate

Pathway

LysineM

etabolism

Valine, Leucine, andIsoleucine

Metabolism

Cysteine

Metabolism

Glycolysis/G

luconeogenesis

SalvagePathw

ay

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fre

quency

Subsystem

Amino Acid

Core Metabolite

Membrane Lipid

Nucleotide

FIG. 7. Histogram showing frequencies of metabolic subsystems employedby MCSs targeting the four metabolic classes analyzed in our study. Fre-quencies are calibrated separately for each metabolite class.

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diverse functional annotation classes establishes the exis-tence of complex buffering relationships between seeminglydisparate parts of the human metabolic network.

D. Case study: Cancer metabolism

The fumarase gene undergoes deletion and loss of func-tion mutation in adrenal, kidney, and smooth muscletumors.33 Fumarase mediates the conversion of fumarate tomalate in the mitochondrion. It is a component of the Krebscycle, which mediates oxidative energy metabolism and is acentral hub for multiple biosynthetic pathways linked toamino acid, lipid, and DNA synthesis. Fumarase activity isthought to play an important tumor-suppressive role in nor-mal noncancerous cells. MCSs for this flux thus representspotential alternative evolutionary targets for a dysplastic cellseeking to achieve a higher degree of malignancy. We alsouse this example to illustrate in more detail the compositionand topologic relationships between different MCSs target-ing the same function.

We computed 97 MCSs for mitochondrial fumarase flux�FUMm� spanning 34 reactions in the network, respectively.MCSs associated with FUMm and the network of reactionsassociated with them are shown in Fig. 9. Abbreviations usedin this figure are shown in Table I. These 97 MCSs contain

only a single low cardinality MCS, which consists of theFUMm reaction itself �an objective reaction is by definitionits own single cardinality MCS, unless it is infeasible�. Wecall this trivial set the “identity MCS.” The remainingFUMm associated MCSs involve the simultaneous knockoutof multiple reactions, as the binary heatmap in Fig. 9�a�shows.

Reactions contributing to FUMm MCSs cluster into asingle robust metabolic network module, shown in Fig. 9�b�.Most of these MCSs inhibit FUMm flux by targeting themitochondrial fumarate �fum_m� pool. This occurs most sim-ply by simultaneously knocking out four mitochondrial fu-marate transport reactions �FUMtm, FUMSO3tm,FUMSO4tm, and FUMTSULtm� and conversion from succi-nate �SUCD1m�. We refer to this set as the “index” MCS asit can be used to understand the remaining FUMm-targetingMCSs, which indirectly inhibits mitochondrial fumaratetransport by knocking out mitochondrial sulfur pools.

As shown in Fig. 9�b�, fumarate import into the mito-chondria is coupled to the outflux of SO3

− �FUMSO3tm�,SO4

2− �FUMSO4tm�, and thiosulfate �FUMTSULtm�. Malateimport is similarly coupled, however, to the influx of thesesulfur molecules via the reactions MALSO3tm, MALSO4tm,and MALTSULtm, respectively. One variation on the index

FIG. 8. Membership map depicting subsystem signatures for MCSs identified in this study and corresponding histogram for each. The sparsity pattern of eachrow in the map represents a unique combination of subsystems, and the histogram on the right depicts how many MCSs exist with that given signature.

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MCS is thus to disable the MALSO4tm transporter instead ofFUMSO4tm. This perturbation indirectly inhibitsFUMSO4tm by knocking out mitochondrial SO4

2− �so4_m�pools, which that fumarate transporter depends on. A similarapproach swaps FUMTSULtm with MALTSULtm in the in-dex MCS, which prevents the influx of thiosulfate into mito-chondria �tsul_m� and thus inhibits thiosulfate dependent im-port. SO3

−, unlike SO42− and thiosulfate, has three routes into

the mitochondria in this model, via MALTSO3, CYANSTm,and 3SPYRSP. As a result, knockout strategies that exclude

FUMSO3tm must suppress all three of these reactions ortheir upstream pathways. The resulting MCSs, shown towardthe bottom of Fig. 9�b�, reach deep into sulfur metabolismpathways targeting reactions linking cysteine �a sulfur con-taining amino acid� degradation with arginine, aspartate, andglutamine pathways. These MCSs are of high cardinality andall employ 8–10 reactions. The most interesting of these tar-get mitochondrial malate dehydrogenase �i.e., replace3SPYRSP with MDHm� to indirectly prevent the formationof 3-sulfonylpyruvate �3snpyr_m�. Additional MCSs target

FIG. 9. �a� Membership map depicting 97 MCS and corresponding network of reactions predicted to be k-essential for mitochondrial fumarase �FUMm� flux.The sparsity pattern of each row in the map represents reaction membership in a single MCS. �b� In the corresponding network diagram, round shaded nodesrepresent species and labeled hyperedges represent network reactions. Abbreviations used in this figure are listed in Table I.

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upstream SO42− pathways that generate cytoplasmic SO4

2−

from ferricytochrome �ficytC_m�.A survey of the MCSs associated with the FUMm objec-

tive demonstrates the complexity of individual MCSs andsuggests an intuitive framework with which to understandthem. Although the “mechanism” of each MCS can be recon-structed with relative ease through browsing of the networktopology, it must be noted that this simplicity only arises posthoc after a considerable computational effort has been ap-plied for MCS identification.

IV. DISCUSSION

A. What is the depth of epistasis in humanmetabolism?

Our analysis reveals a plethora of high cardinality MCSsacross a variety of metabolic objectives in human metabo-lism. Since the knockout of all reactions in an MCS disablesa given objective, while none of its subsets is predicted to,each high cardinality MCS posits a deep buffering relation-ship between multiple genes, mediated through the metabolicnetwork. Our FUMm-targeting MCSs expose such interac-tions in multiple settings: many occur between intercompart-ment transporters, of which there are multiple versions for agiven metabolite �e.g., fumarate, malate� that are differen-

tially coupled to other small molecules in the network. Re-dundancy can also emerge from the multiple forms of el-emental small molecules �e.g., SO3

−, SO42−, and thiosulfate�

that form distinct but interconnected pools that can be differ-entially exploited for targeted knockout. Our analysis of bio-synthetic objectives shows that most MCSs span multiplecellular compartments and metabolic subsystems in impres-sively complex combinations. This pattern is largely inde-pendent of metabolite class, suggesting that epistatic interac-tions link many reactions in metabolism to robust systems-level functions.

The presence of large MCSs suggests that with respect tomultiple objectives, parts of human metabolism resemble atleast a nine or ten-lane highway. Compared to our analysis ofE. coli metabolism done using similar tools, the depth ofepistatic interaction that we see in the human metabolic net-work is comparable.19 This degree of buffering is also com-parable to results obtained in yeast using random sampling.17

How faithful is our estimate of the depth of epistasis inhuman metabolism? Our modeling assumptions are very per-missive: we allow all reactions encoded in the genome to beactive and provide all nutrients for which a transporter exists.In the human body, metabolic genes are turned on and off ina cell and tissue type specific manner. Therefore, the metabo-lism of a specific cell in the human body may exhibit shal-

TABLE I. Flux and metabolite abbreviations in the fumarase MCS network. Suffixes “_c” and “_m” are added in the network diagram to denote cytoplasmicand mitochondrial compartments, respectively.

Fluxabbreviation Name

Metabbreviation Met name

3SALAASPm Cysteinesulfinate-aspartate mitochondrial shuttle 3sala 3-Sulfino-L-alanine3SALATAim 3-sulfino-alanine transaminase �irreversible�, mitochondrial 3snpyr 3-Sulfinopyruvate3SPYRSPm 3-sulfinopyruvate hydrolase �spontaneous reaction�, mitochondrial adp ADPASNNm L-asparaginase �mitochondrial� akg 2-OxoglutarateASNtm L-asparagine transport, mitochondrial asn-L L-AsparagineASPGLUm Aspartate-glutamate mitochondrial shuttle asp-L L-AspartateASPTAm Aspartate transaminase atp ATPCYANSTm Cyanide sulfurtransferase, mitochondrial cyan Hydrogen cyanideCYANt Cyanide transport via diffusion �extracellular to cytosol� cys-L L-CysteineCYANtm Cyanide transport via diffusion �mitochondrial� ficytC Ferricytochrome cCYOOm3 Cytochrome c oxidase, mitochondrial complex IV focytC Ferrocytochrome CCYSO Cysteine oxidase fum FumarateFUMm Fumarase, mitochondrial glu-L L-GlutamateFUMSO3tm Fumarate:sulfite antiport, mitochondrial h2o2 Hydrogen peroxideFUMSO4tm Fumarate:sulfate antiport, mitochondrial hco3 BicarbonateFUMtm Fumarate transport, mitochondrial mal-L L-MalateFUMTSULtm Fumarate:thiosulfate antiport, mitochondrial nh4 AmmoniumMALSO3tm Malate:sulfite antiport, mitochondrial o2 O2MALSO4tm Malate:sulfate antiport, mitochondrial o2s Superoxide anionMALTSULtm Malate:thiosulfate antiport, mitochondrial oaa OxaloacetateMDHm Malate dehydrogenase, mitochondrial pyr PyruvateO2Stm Superoxide anion transport via diffusion �mitochondria� so3 SulfitePCm Pyruvate carboxylase so4 SulfateSPODMm Superoxide dismutase succ SuccinateSUCDlm Succinate dehydrogenase tcynt ThiocyanateSULFOX Sulfite oxidase tsul ThiosulfateTCYNTt Thiocyanate transport via diffusion �cytosol to extracellular�TCYNTtm Thiocyanate transport via diffusion �mitochondrial�TSULt4_3 Thiosulfate transport via sodium symport

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lower buffering if it has many metabolic genes turned off andhas access to poor nutrient media. One could partially evalu-ate this conjecture by testing MCSs generated in our study ina model equipped with additional cell type specific gene ex-pression based constraints. The actual complexity of in vivo“nutrient media” in a human is harder to estimate; however,in most well-vascularized compartments it is reasonable toassume that extracellular fluid is quite “rich” in nutrientcontent.

Our estimate of epistatic depth may also be low for sev-eral reasons. First, if the metabolic model is incomplete andthere exist additional unannotated reactions that introducenovel alternative pathways to a given objective, then the ac-tual depth of epistasis will be greater than our estimate. Wehave also imposed a relatively arbitrary upper cardinalitybound in our MCS search of ten �due to computational re-strictions�; however, the domination of our results by MCSswith nine or ten reactions suggests that the current networkharbors higher cardinality combinations that we may detectif we increase this limit.34

B. The implications of redundancy in metabolism

The presence of many backup pathways in the metabo-lism of humans and microbes may suggest apparent “dupli-cation” of function at the network level. This could be inter-preted to provide an additional layer of redundancy on top ofgene duplication, which is widespread in multiple organismsand conserved throughout evolution.35,36 However, unlikegene duplication, systems-level redundancy is generally me-diated by genetically distant proteins that provide “distrib-uted robustness” by having one or more shared physiologicalfunctions.37 In this case, the term “redundancy” may be toostrong to describe this phenomenon, since such backup path-ways are rarely identical; rather, one pathway may use adifferent set of nutrients and produce a different set of by-products to achieve an objective. As a result, certain path-ways may be more optimal or efficient in certain environ-ments, allowing a microorganism with a high degree ofdistributed robustness a survival advantage in the face ofchanging environments. Applied to the metabolism of hu-mans �and other higher organisms�, such robustness couldpresumably allow for a wide spectrum of differentiation phe-notypes �e.g., liver versus muscle cells� optimized for agiven cell or tissue level function. The relationship of net-work buffering depth as a function of differentiation state inhuman cells �e.g., comparing embryonic stem cells with ter-minally differentiated liver cells� is an interesting directionfor future investigation.

C. Applications of human metabolic MCSs

1. Using MCSs to guide perturbation experimentsMCSs provide precise predictions of parallelism in the

metabolic network. The degree of actual parallelism in vivodepends on the actual utilization and capacity of individualpathways in the cell. If an individual pathway is able toprovide the needs of a metabolic objective on its own, then itwill be very difficult to detect the role of other reactions inan MCS using low-order knockouts. In the case of perfect

synergistic epistasis, every member of an MCS will representa separate pathway individually capable of sustaining an ob-jective. In this case the role of one gene or enzyme for agiven objective will only be visualizable in the context of theknockout of all other MCS components. However, in realitythe utilization and capacity of various metabolic channelslikely differ, and low-order knockouts may reveal marginaleffects. Such questions can be pursued experimentally bychemical or genetic perturbation experiments guided byMCS predictions. One may begin such an investigation withmarginal knockout experiments that chemically or geneti-cally disable only single enzymes in the MCS. One may alsopursue “pinching” experiments that knockout all but one ofthe members of an MCS, effectively forcing all flux to anobjective through that remaining reaction.

2. Using MCSs to discover epistatic genetic factorsDespite several years of massive genome-wide associa-

tion studies seeking links between common genetic variationand disease susceptibility, much of the heritable risk attrib-uted to human illness �e.g., diabetes, heart disease� or traits�e.g., height� remains unexplained. Although the location ofthis “missing heritability” is a subject of active debate, onepossibility is that it arises from complex interactions betweenmultiple genetic variants.22,38 Given the degree of redun-dancy that we observe at the metabolic network level, itwould be reasonable to assume that many deleterious genetictraits require the simultaneous malfunctioning of parallelpathways. �This relationship between “functional epistasis,”which refers to the synergistic versus alleviating effects ofdifferent mutations on function, and “statistical epistasis”�e.g., in the sense of Fisher�, which refers to the presence ofnonadditive effects in statistical genetic models, has beenpreviously explored.23,24�

The discovery of such interactions through purely statis-tical means is limited by computational complexity and sta-tistical power. One approach to limit the number of hypoth-eses is to use a functional biological framework to guide thesearch for interacting genetic combinations. By capturing thefundamental units of functional redundancy in a system,MCSs provide such a framework. This could be applied forthe study of metabolism as well as signaling networks, whichcan be represented in a stoichiometric constraint-basedframework.39,40 Such models can be similarly subjected toMCS analysis to identify mutually buffering elements andprovide a template upon which to capture complex epistaticeffects.

3. Constraint-based modeling of cancer metabolismAlthough it has been over 80 years since Warburg’s ob-

servation that cancer cells ferment sugar in the presence ofoxygen, major interest in cancer metabolism has only re-cently reemerged.41,42 Our analysis has explored alternativeand complex knockout strategies that would be predicted toreplicate decreases in fumarase flux observed in certain can-cer cells and tumors. The MCSs we have discovered provideinteresting starting points for investigating alternative muta-

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tional patterns that may achieve similar effects in other tu-mor types. Since cancer evolution is a dynamic and stochas-tic process, it is expected that tumor cells may take multipleroutes to achieve the same objective. MCSs derived from ametabolic reconstruction can potentially help us infer thatevolutionary route, especially when it involves the simulta-neous inhibition of multiple enzymes in the network.Constraint-based models can also help us join disparate mu-tational patterns under the banner of a single predicted phe-notype. This can be a potentially powerful framework withwhich to investigate marginal effects and high-order interac-tion patterns in somatic sequence variation data sets such asthat emerging from The Cancer Genome Atlas.43

D. Computational directions

Although our computational approach yields a consider-able number of high cardinality genome-scale MCSs, we areunable to guarantee the discovery of all MCSs below a cer-tain cardinality for a given metabolic objective. This is incontrast to brute-force optimization and bilevel optimizationapproaches, which have shown efficacy in deriving completeMCS collections for a given cardinality but have only beenapplied to yield low-order buffering interactions �i.e.,k�5�.20 The completeness of our results is limited by theability of the pathway fragment collection to connect specificobjectives to sets of reactions that provide their flux. Algo-rithmic improvements that yield a more complete collectionof MCSs may be useful in perturbational experiment designor statistical analysis of genetic variation.

This paper provides an incremental modification to ouroriginal genome-scale MCS algorithm.19 Our approachgreatly increases the yield of MCSs that we obtain for certainobjectives by expanding the pathway fragment collection aniteration further in multiple metabolite directions. Our under-lying metabolite selection approach during pathway frag-ment generation is a local greedy optimization strategy. Al-ternative approaches to metabolite ordering during pathwayfragment generation may be more ideally adapted to gener-ating MCSs for a given objective. These include those thatdifferentially weigh reactions or metabolites of interest, ortake into account compartment structure.

A large limiting factor of MCS yield is the quality of thepathway fragment collection, which can be roughly mea-sured by imax or the number of balanced metabolites. Evenwithout additional algorithmic improvements, parallel imple-mentations of standard pathway fragment computation willserve to increase imax. Our unpublished results suggest thateven small increments in imax yield substantially larger col-lections of more MCSs. Significant parallel computing effortapplied toward generating a human metabolic pathway frag-ment collection may be worthwhile in creating a queryableresource from which to derive more complete and targetedcollections of MCSs.

In the interest of generality, our current MCSs have nottaken into account the effects of gene/protein expression orregulation. Instead we have built MCSs from a “master net-work” in which all reactions are allowed to be simulta-neously active. In reality, only a subset of reactions will beactive due to gene expression, post-translational modifica-

tions, and feedback regulation. One computational approachto model this effect is to superimpose metabolic gene expres-sion data from experiments on flux models as a set of on/offconstraints on reactions. Regulation can be more accuratelymodeled using a mixed integer linear programming frame-work that takes into account transcriptional and signalingcircuitry.44 The addition of either set of additional constraintswould yield potentially more minimal MCSs and providemore accurate predictions for experimental follow-up.

V. CONCLUSION

We have applied MCS analysis to reveal multiple ex-amples of deep epistasis in human metabolism. Our ability togenerate results at the size of the human metabolic networkleverages an important modification to our original genome-scale MCS algorithm.19 The MCSs we generate forge noveland interesting links between multiple metabolic reactionsand distinct functions of human metabolism, including me-tabolite biosynthesis and the maintenance of flux through atumor suppressor enzyme. Our results also suggest a highdegree of redundancy in the human metabolic network,which prompts future experimental and analytic follow-up.This includes applying MCS-guided in vivo chemical or ge-netic perturbation experiments to human cells and statisticalgenetic analyses that use MCSs as templates to probe geneticvariation data sets for high-order epistatic effects.

ACKNOWLEDGMENTS

We thank the referees and editors for helpful suggestionsand critical comments that helped shape this manuscript. Wethank the Enterprise Research IS group at Partners Health-care for the in-depth support and provision of the HPC fa-cilities.

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�2007�.45This particular type of interaction is more technically referred to as “syn-

ergistic” or “aggravating” epistasis.46This commonly used formulation does not result in any loss of generality;

however, the constraint of all fluxes to be non-negative simplifies thegeometry of our solution space.

47In practice, Klamt and Gilles do not compute EP, but elementary fluxmodes �EFMs�, which are a related pathway decomposition. AlthoughEFMs are thought to be a more compact representation of pathway struc-ture �Ref. 28� we use EP in our formulation due to their mathematicalsimplicity. However, we obtained similar results �and limitations� witheither formalism.

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