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RESEARCHARTICLE OpenAccess Gaussiangraphicalmodelingreconstructs pathwayreactionsfromhigh-throughput metabolomicsdata JanKrumsiek 1 ,KarstenSuhre 1,2 ,ThomasIllig 3 ,JerzyAdamski 4 ,FabianJTheis 1,5* Abstract Withtheadventofhigh-throughputtargetedmetabolicprofilingtechniques,thequestionofhowto interpretandanalyzetheresultingvastamountofdatabecomesmoreandmoreimportant.Inthisworkwe addressthereconstructionofmetabolicreactionsfromcross-sectionalmetabolomicsdata,thatiswithoutthe requirementfortime-resolvedmeasurementsorspecificsystemperturbations.Previousstudiesinthisareamainly focusedonPearsoncorrelationcoefficients,whichhoweveraregenerallyincapableofdistinguishingbetween directandindirectmetabolicinteractions. Results: InournewapproachweproposetheapplicationofaGaussiangraphicalmodel(GGM),anundirected probabilisticgraphicalmodelestimatingtheconditionaldependencebetweenvariables.GGMsarebasedonpartial correlationcoefficients,thatispairwisePearsoncorrelationcoefficientsconditionedagainstthecorrelationwithall othermetabolites.Wefirstdemonstratethegeneralvalidityofthemethodanditsadvantagesoverregular correlationnetworkswithcomputer-simulatedreactionsystems.ThenweestimateaGGMondatafromalarge humanpopulationcohort,covering1020fastingbloodserumsampleswith151quantifiedmetabolites.TheGGM ismuchsparserthanthecorrelationnetwork,showsamodularstructurewithrespecttometaboliteclasses,andis stabletothechoiceofsamplesinthedataset.Ontheexampleofhumanfattyacidmetabolism,wedemonstrate forthefirsttimethathighpartialcorrelationcoefficientsgenerallycorrespondtoknownmetabolicreactions.This featureisevaluatedbothmanuallybyinvestigatingspecificpairsofhigh-scoringmetabolites,andthen systematicallyonaliterature-curatedmodeloffattyacidsynthesisanddegradation.Ourmethoddetectsmany knownreactionsalongwithpossiblynovelpathwayinteractions,representingcandidatesforfurtherexperimental examination. Conclusions: provideavaluabletoolfortheunbiasedreconstructionofmetabolicreactionsfromlarge-scalemetabolomics datasets. Background Metabolomicsisanewlyarisingfieldaimingatthemea- surementofallendogenousmetabolitesofatissueor bodyfluidundergivenconditions[1-3].Theresulting metabolome ofabiologicalsystemisconsideredtopro- videareadoutoftheintegratedresponseofcellular processestogeneticandenvironmentalfactors[4]. Understandingthecomplexbiochemicalinterplay betweenhundredsofmeasuredmetabolitespeciesisa dauntingtask,whichcanbeapproachedbycombining advancedcomputationalmethodswithdatafromlarge population-basedstudies.Onthebiochemicallevel, metaboliteconcentrationsaredeterminedbyasetofspe- cificmetabolicenzymes.Variabilitiesinbothenzyme activityandmetaboliteexchangerates-inducedbya continuousspectrumofmetabolicstatesthroughout measuredsamples-giverisetocharacteristicpatternsin themetaboliteprofileswhicharedirectlylinkedthe underlyingbiochemicalreactionnetwork[5,6].Although humanmetabolismhasbeenextensivelycharacterizedin *Correspondence:fabian.theis@helmholtz-muenchen.de 1 InstituteofBioinformaticsandSystemsBiology,HelmholtzZentrum Fulllistofauthorinformationisavailableattheendofthearticle Krumsiek etal . BMCSystemsBiology 2011, 5 :21 http://www.biomedcentral.com/1752-0509/5/21 ©2011Krumsieketal;licenseeBioMedCentralLtd.ThisisanOpenAccessarticledistributedunderthetermsoftheCreative CommonsAttributionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,and reproductioninanymedium,providedtheoriginalworkisproperlycited. thepastdecades[7],thereconstructionofmetabolicnet-worksfromsuchmetabolitepatternsisakeyquestioninthecomputationalresearchfield.PreviousattemptsfocusedonlinearmetaboliteassociationsmeasuredbyPearsoncorrelationcoefficients.Theseincludestudiesutilizingtime-coursemeasurementsandclustering[8],theoreticalapproachesrelatingmetabolitefluctuationstopropertiesofthedynamicalsystem[5]andmetaboliccontrolanalysistoderiveeffectsofenzymevariability[6].Otherreconstructionmethodsrelyonspecificperturba-tionsofthebiologicalsystem,liketheinductionofcon-centrationpulsesforcertainmetabolites[9].Amajordrawbackofcorrelationnetworks,however,istheirinabilitytodistinguishbetweendirectandindirectassociations.Correlationcoefficientsaregenerallyhighinlarge-scaleomicsdatasets,suggestingaplethoraofindirectandsystemicassociations.Forexample,tran-scriptionalcoregulationamongstmanygeneswillgiverisetoindirectinteractioneffectsinmRNAexpressiondata[10].Similareffectscanbeobservedinmetabolicsystemswhich,incontrasttogeneticnetworks,containfastbiochemicalreactionsinanopenmass-flowsystem.Metabolitelevelsaresupposedtobeinquasi-steadystatecomparedtothetimescalesofupstreamregula-toryprocesses[11].Thatis,metaboliteswillfollowchangesingeneexpressionandphysiologicalprocessesontheorderofminutesandhours,butwillappearunchangedontheorderofseconds.Theseproperties,eventhoughsubstantiallydifferentfrommRNAexpres-sionmechanisms,alsogiverisetoindirect,system-widecorrelationsbetweendistantlyconnectedmetabolites.Gaussiangraphicalmodels(GGMs)circumventindir-ectassociationeffectsbyevaluatingdenciesinmultivariateGaussiandistributions[10].AGGMisanundirectedgraphinwhicheachedgerepresentsthepairwisecorrelationbetweentwovari-ablesconditionedagainstthecorrelationswithallothervariables(alsodenotedaspartialcorrelationcoeffi-cients).GGMshaveasimpleinterpretationintermsoflinearregressiontechniques.Whenregressingtworan-domvariablesontheremainingvariablesinthedataset,thepartialcorrelationcoefficientbetweenisgivenbythePearsoncorrelationoftheresi-dualsfrombothregressions.Intuitivelyspeaking,weremovethe(linear)effectsofallothervariablesonandYandcomparetheremainingsignals.Ifthevari-ablesarestillcorrelated,thecorrelationisdirectlydeter-minedbytheassociationofXandYandnotmediatedbytheothervariables.PartialcorrelationshaverecentlybeenappliedtobiologicaldatasetsfortheinferenceofassociationnetworksfrommRNAexpressiondata[12-15],andfortheelucidationofrelationshipsbetweengenomicfeaturesinthehumangenome[16].Onepre-viousstudyusedsecond-orderpartialcorrelationsofgeneticassociationstoelucidategeneticallydeterminedrelationsbetweenmetabolites[17].InthismanuscriptwenowstudythecapabilitiesofGGMstorecovermetabolicpathwayreactionssolelyfrommeasuredmetaboliteconcentrations.First,wedis-cussthequalityofthemethodandpossibleproblemsandpitfallsoncomputer-simulatedsystems.WethenapplyGGMstoalipid-focusedtargetedmetabolomicsdatasetof1020bloodserumsampleswith151mea-suredmetabolitesfromtheGermanpopulationstudyKORA[18,19].TheGGMissparseincomparisontothecorrespondingPearsoncorrelationnetwork,displaysamodularstructurewithrespecttodifferentmetaboliteclasses,andisstabletowardschangesintheunderlyingdataset.Wedemonstratethattop-rankingmetabolitepairsandfurtherdenselyconnectedsubgraphsintheGGMcanindeedbeattributedtoknownreactionsinthehumanfattyacidbiosynthesisanddegradationpath-ways.Inordertosystematicallyverifythisfinding,wemappartialcorrelationcoefficientstothenumberofreactionstepsbetweenallmetabolitepairsbasedonaliterature-curatedfattyacidpathwaymodel.WeobservestatisticallysignificantdiscriminatoryfeaturesofGGMstodistinguishbetweendirectlyandnon-directlyinter-actingmetabolitesinthemetabolicnetwork.Inaddition,low-orderpartialcorrelationsturnedouttobeasuitablealternativetofull-orderGGMsforthepresentdataset.Finally,wewillsummarizeanddiscusstherelevanceofGGMsformetabolomicsdatasets,pointoutlimitationsofthemethodandsuggestfuturesteps.Allmetabolo-micsdatausedinthisstudy,thegeneratedcorrelationnetworks,modelfilesandmetaboliteannotationsareavailableonlineathttp://hmgu.de/cmb/ggm.ResultsandDiscussionGGMsdelineatedirectrelationshipsinartificialreactionComputer-simulatedreactionsystemsareavaluabletoolfortheevaluationofcorrelation-basedmeasurespriortotheirapplicationtorealmetabolomicsdatasets.Pre-viousworksfocusedonthemodelingofbiologicalrepli-cateswithintrinsicnoiseonthemetabolitelevels[5].Incontrast,wehereinvestigatetheeffectsofvariationofenzymaticactivityinahumanpopulationcohort.Suchvariationmightbegeneticallydeterminedor,morelikely,betheresultofdistinctregulatoryeffectsandmetabolicstatesbetweenindividuals.Allreactionsys-temswereimplementedasordinarydifferentialequa-tionswithsimplemass-actionkineticsratelawsandreversibleMichaelis-Menten-typeenzymekinetics(seeMethods).Inordertoaccountfortheabove-mentionedenzymaticvariabilityweappliedalog-normalnoisemodel,whichhasbeenpreviouslydescribedtobearea-sonableapproximationofcellularrateparameteretalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page2of16 distributions[20].Thestandarddeviationwassettoavalueof0.2fortheunderlyingnormaldistribution(notethattheresultsareinsensitivetothemagnitudeofForeachparametersamplewecalculatedthemetabolitesteadystateconcentrationsonlog-scale,andsubse-quentlyestimatedtheGGMbycalculatingpartialcorre-lationcoefficients.Allanalyzedsystemsexhibitsingle,uniquesteadystatesindependentoftherespectivepara-metervalues.ThisfeaturewasstructurallyverifiedusingtheERNESTtoolbox[21]forallnetworksexceptthenegativefeedbacksystem.Forthelatterone,weemployedempiricalinitialstatesamplingtoensuremonostabilityinthegivenparameterrange(seeAddi-tionalfile1,section1).Thefirstnetworkweanalyzedconsistsofalinearchainofthreemetaboliteswithdifferentvariantsofreactionreversibility(Figure1A-C).Weobservehighpairwisecorrelationsformetabolitesinmutualequili-briumduetoreversiblereactions(Figure1A).Thisisinaccordancewithpreviousfindingsfrom[6],wherecor-relation-generatingmechanismsinmetabolicreactionnetworkswereidentified.Furthermore,thissimpleexampledemonstrateshowpartialcorrelationcoeffi-cientsinGGMsdiscriminatebetweendirectlyandindir-ectlyrelatedmetabolites.Ifonlyirreversiblereactionsareemployedinthechain,neitherregularcorrelationnetworksnorGGMscandistinguishbetweendirectandindirecteffects(Figure1B).SpeciesAistheonlyinputmetaboliteinthesystem,andthuscompletelydeter-minesthelevelsofbothBandC.Thisleadstogenerallyhighandnon-distinguishablecorrelationsbetweenthethreemetabolites.However,ifweintroduceexchangereactionsforallspecies,theGGMagaincorrectlydescribesthenetworkconnectivity(Figure1C).Suchexchangemechanismsarelikelytobepresentformostintracellularmetabolites,whichusuallyparticipateinmultiplemetabolicpathways(seee.g.KEGGPATH-WAYonline).Notethatforthisthirdcasebothregular Figure1Evaluationofcorrelationnetworks(CN)andGaussiangraphicalmodels(GGM)onartificialsystems.Linewidthsrepresentrelativeedgeweightsintherespectivenetworks(scaledtothestrongestedges).Linearchainofthreemetaboliteswithreversibleintermediatereactions.WhilethestandardPearsoncorrelationnetwork(CN)isfullyconnected,implyinganoverallhighcorrelationofallmetabolites,theGGMcorrectlydiscriminatesbetweendirectandindirectinteractions.Linearchainwithirreversibleintermediatereactions.NeitherCNnorGGMcandistinguishdirectfromindirecteffects,asmetaboliteAequallydeterminesthelevelsofbothBandC.Linearchainwithirreversiblereactionsandinput/outputreactionsforeachmetabolite.AlthoughtheedgeweightsforbothCNandGGMaregenerallylower,theGGMnowcorrectlypredictsthenetworktopology.Branched-chainfirst-ordernetworksarecorrectlyreconstructedbytheGGM.productinhibitionmodules.Whenmodeledasanopensystem,isdecoupledfromtheothermetabolitesandreconstructionfailsatthispoint.Dashedlinesmarkenzymeinhibitioninteractions,largerarrowstotherightindicatefasterforwardthanbackwardreactions.Cofactor-drivennetworkresemblingthefirstthreereactionsfromtheglycolysispathway.Acorrelationnetworkfailstopredictthecorrectpathwayrelationships.Non-linearsystemwithabi-molecularreaction.TheGGMpredictsonlyaonlyweakinteractionbetweenBandC.Thisisduetocounterantagonisticprocessesofisomerizationandsubstrateparticipationinthesamereaction.etalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page3of16 andpartialcorrelationvaluesarenotablylowerthanforthefirsttwochainvariants.Inadditiontolinearchains,pathwaymodulesconsistingofbranchedtopologieswithfirst-order,reversiblereactionsarecorrectlyrecon-structedbyourmethod(Figure1D).AnoverviewofthereconstructionaccuracyofGGMsonvarioustypesoffirst-ordernetworkswithdifferentvariantsofreactionreversibilitycanbefoundinAdditionalfile1,section2.Interestingly,forsomereactionsetups,theaccuracyofthemethodimprovesdrasticallywithanincreasingamountofexternalnoise.Specifically,ifthemetabolitetransporttowardsapathwayissubjecttohigherfluctua-tions,theGGMedgeweightdifferencebetweendirectlyandindirectlyconnectedmetabolitesbecomeslarger.ForadetaileddiscussionofthisfindingwereferthereadertoAdditionalfile1,section3.Thesecondques-tionweaddressedwithartificialreactionnetworkswastheinfluenceofenzyme-catalyzedreactionsonGGMestimation.Thereforewesetupreactionchainswithfourmetabolitesincorporatingreversibleenzymaticreactions.Forwardmaximalreactionratesmaxweresettwiceasfastasthebackwardreactionsinordertoensureadirectedmassflow.WefoundthattheusageofMichaelis-Menten-typeenzymekineticsinsteadofmass-actionkineticsdoesnotalterourgeneralfindings.Whenforwardreactionratesexceedbackwardreactionsbyfar,theGGMdiscriminationqualityisimpaired.Thisisinlinewiththeobservationthatpurelyirreversiblereactionscannotbedistinguishedinthemass-actioncase(seeabove).Otherspecificparameters,liketheMichaelisconstant,didnotaffectGGMcalculation(Additionalfile1,section4).Anotherimportantaspectofenzyme-catalyzedreactionsareallostericregulationmechanism,likeend-productinhibitionforinstance,whichconstitutesanegativefeedbackfromtheendtothebeginningofapathway[22].Thereconstructionresultsdifferdependingonwhetherexchangereactionsareincludedinthesystemfornot(Figure1E).Iftheinhibitorymodulerepresentsaclosedsystem(noexter-nalfluxesexceptforthefirstandlastmetabolite),theregulatoryinteractiondoesnotininfluenceGGMcalcu-lation.Thenetmetaboliteturnoverspeedmightbedrasticallyaffected,butthetopologicaleffectsofthisreactionchainonthecorrelationstructureremainunchanged.Incontrast,whenexchangereactionsareintroduced(secondexampleinFigure1E),theinhibitiondecouplesAfromtheothermetabolitesandtherecon-structionfailsforthismetabolite.Detailedresultsfordifferentstrengthsoftheinhibitoryinteractionarepre-sentedinAdditionalfile1,section5.Next,westudiedtheinfluenceofcofactor-drivenreac-tionsonthereconstruction.Cofactorsareubiquitoussubstancesusuallyinvolvedinthetransferofcertainmolecularmoietiesorredoxpotentials[23].Weinvestigatedsuchcofactor-coupledreactions(a)becausetheyintroducenon-linearityinthesimulateddynamicalsystems,and(b)becausecofactorsareusuallyinvolvedinmanyreactionsandthusgeneratenetwork-widemetabolitedependencies.Wesetupanetworkresem-blingthefirstthreereactionsfromtheglycolysispath-way.Itconsistsoffourmetabolitesandtwoenergytransfer-relatedcofactors,ATPandADP,involvedintwophosphorylationreactions[24].AgaintheGGMpreciselydescribesmetaboliteconnectivityinthesystem,whereasaregularcorrelationgraphleadstofalseinter-pretationsofthenetworktopology(Figure1F).Cofac-torsweremodeledwithinputandoutputreactionstotherestofthemetabolicsysteminordertoaccountfortheabove-mentionedparticipationofcofactorsinvar-iousreactionsofthesystem.Again,itmakesasubstan-tialdifferencewhethersuchexchangereactionsareincludedinthemodelornot.Sinceourtoymodelonlyrepresentsasmallpartofalargersystem,missingexchangereactionforcofactorswouldcreateafalsemassconservationrelationthatcompromisescorrelationcalculation.Finally,weinvestigatedtheeffectsofratelawswithnon-linearsubstratedependenciesintheabsenceofcofactors.Thereforewemodeledareversible,bimolecularsplitreactionwithisomerizationofthetwosubstrates(Figure1G).Anexampleofsuchareactionnetworkcanbefoundintheglycolysispathwaybetweenfructose-1,6-bisphosphate,glyceraldehyde-3-phosphateanddihydroxyacetonephosphate.OursimulationsdemonstratethatagainaregularPearsoncorrelationnetworkcannotdelineatedirectfromindirectrelation-shipsinthepathway.TheGGMonlydetectsaweakassociationbetweenBandC.Thisisduetocounteran-tagonisticprocessesinthisreactionsetup:isomerizationandotherreversiblereactionsgenerallyinducepositivecorrelations,whereascoparticipationassubstratesinthesamereactioninducesnegativecorrelations.Sucheffectsofcorrelation-generatingmechanismswhichcanceleachotherouthavebeendescribedbefore[6]andposeaproblemtoallreconstructionapproacheswhichrelyonlineardependencies.Thedrawbacksofcorrelation-basedmethodsdis-cussedinthissection,especiallyinhibitorymechanismswithexchangereactionsandantagonisticmechanism,havetobekeptinmindwhenattemptingtoreconstructmetabolicreactionsfromsteadystatedata.Forthepre-sentstudy,however,weassumetheprimarilylinearlipidpathwaysnottocontainsuchproblematicreactionAGGMinferredfromalarge-scalepopulation-baseddatasetdisplaysasparse,modularandrobuststructureInthefollowingweestimatedaGaussiangraphicalmodelusingtargetedmetabolomicsdatafromtheetalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page4of16 GermanpopulationstudyKORA[18]("KooperativeGesundheitsforschunginderRegionAugsburg).Weusedasubsetofthedatasetpreviouslyevaluatedinagenome-wideassociationstudy[19],containing1020targetedmetabolomicsfastingbloodserummeasure-mentswith151quantifiedmetabolites.Themetabolitepanelincludesacyl-carnitines,fourclassesofphospholi-pidspecies,aminoacidsandhexoses(seeMethods).BothregularPearsoncorrelationcoefficientsandpartialcorrelationcoefficients(inducingtheGGM)werecalcu-latedonthelogarithmizedmetaboliteconcentrations.Alledgescorrespondingtocorrelationvaluessignifi-cantlydifferentfromzeronowinducethenetworksdis-playedinFigure2A+B.Inordertoexcludecorrelationeffectsgeneratedbygeneticvariationinthestudycohort,weinvestigatedtheininfluenceofSNPalleledatafrom[19]ontheGGMcalculation.Wefoundgeneticeffectstobeneglectable(seeAdditionalfile2),indicatingthatGGMscaptureintrinsicbiochemicalpropertiesofthesystem.Pearsoncorrelationcoefficientsshowastrongbiastowardspositivevaluesinourdataset(Figure2C);atypicalfeatureofhigh-throughputdatasets,alsoobservede.g.inmicrorarrayexpressiondata,whichcanbeattributedtounspecificorindirectinteractions[10].Weobtain5479correlationvaluessignificantlydifferentfromzerowith 88310 =0.01afterBonfer-ronicorrection),yieldinganabsolutesignificancecorre-lationcutoffvalueof0.1619(seeMethods).Incontrast,theGGMshowsamuchsparserstructurewith417sig-nificantpartialcorrelationsafterBonferronicorrection(Figure2D).Mostvaluescenteraroundapartialcorrela-tioncoefficientofzero,whereasweobserveaclearshifttowardspositivesignificantvalues.Notethatnegativepartialcorrelationsprovideparticularinformationthatwillbediscussedlaterinthismanuscript.TheGGMdisplaysamodularstructurewithrespecttothesevenmetaboliteclassesinourpanel,whiletheclassseparationinthecorrelationnetworkappearsratherblurry(Figure2E+F).Weobserveaclearsepara-tionoftheaminoacidsandacyl-carnitinesfromallotherclasses.Thefourgroupsofphospholipids(diacyl-PCs,lyso-PCs,acyl-alkyl-PCs,andsphingomyelins)stillshowedlocallyclusteredstructures,butarestronglyinterwoveninthenetwork.Thisisprobablyaneffectofthedependenceofallphospholipidsonasimilarfattyacidpooland,subsequently,thebiosynthesispathwayactingonthissubstratepool.Inordertogetanobjec-tivequantificationofthisobservation,wecalculatedthegroup-basedmodularityonallsignificantlypositiveGGMedgesaccordingto[25](seeMethods).Thesamemeasurewascalculatedfor10randomizedGGMnet-works(randomedgerewiring).FortheoriginalGGMweobtainamodularityof=0.488,andtherandomnetworksyield=0.118±0.016,resultinginahighlysignificant-scoreof=23.49.Furthermore,themodu-larityvalueinducedbyusingthemetaboliteclasseswascomparedtoapartitioningoptimizedbysimulatedannealing.Theoptimizedmodularityisonlyslightlyhigherwith=0.557andtheresultingpartitioningisverysimilartothemetaboliteclasses(seeAdditionalfile3).Performingthemodularityanalysiswiththefull,weightedpartialcorrelationmatrixproducesequivalentresults(alsoshowninS3).Animportantquestionforamultivariatestatisticalmeasuresuchaspartialcorrelationsistherobustnesswithrespecttochangesintheunderlyingdataset.Furthermore,thedependenceofthemeasureonthesizeofthedatasetneedstobeaddressed.Toanswerthesequestions,weperformedtwotypesofperturba-tionsofourdataset.First,weappliedsamplebootstrap-pingwith1000repetitionsandcomparedtheresultingpartialcorrelationstotheoriginaldataset(Additionalfile4,FigureS1).Weobservesmallmeandifferenceswithlowstandarddeviation(0.03±8.2·10).Thisindicatesthatforalargedatasetwith=1020samples,GGMsarerobustagainstthechoiceofsamples.Weassumethateachdistinctmetabolicstateinthecohortiscapturedbyabootstrapsample,andthusallinforma-tionrequiredtocalculatetheGGMiscontained.Inadditiontothebootstrapanalysis,weestimatedpartialcorrelationsforcontinuouslydecreasingsamplesizes(Additionalfile4,FigureS2).Foreachdatasetsizewerandomlypickedsamplesfromtheoriginaldatasetandrepeatedtheprocedure100times.TheanalysisshowsthattheGGMisstableevenunderdecreaseofthesam-plenumber.Forinstance,foradatasetcontainingonlyaroundhalfoftheoriginalsamples(=530)wegetapartialcorrelationdifferenceof0.03±6.910.Onlywhenthenumberofsamplesgetsclosetothenumberofvariables(=151)thecorrelationmatrixbecomesill-conditionedandstrongdifferencesfromtheoriginalpartialcorrelationsoccur.Theseproblemsofsmallermetabolomicsstudiescouldbedealtwithbyregulariza-tionapproachesortheusageoflow-orderpartialcorre-lation[26].Takentogether,ourresultsdemonstratethattheanalyzedmetabolomicsdatasetissufficienttorobustlyelucidaterelationshipsbetweenthemeasuredStrongGGMedgesrepresentknownmetabolicpathwayinteractionsThenextstepinouranalysiswasthemanualinvestiga-tionofmetabolitepairsdisplayingstrongpartialcorrela-tioncoefficients.Remarkably,weareabletoprovidepathwayexplanationsformostmetabolitepairsintheetalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page5of16 Figure2Networkpropertiesofthecorrelationnetwork(CN)andGaussiangraphicalmodel(GGM)inferredfromatargetedmetabolomicspopulationdataset(1020participants,151quantifiedmetabolites)Graphicaldepictionofsignificantlypositiveedgesinbothnetworks,emphasizinglocalclusteringstructures.Eachcirclecolorrepresentsasinglemetaboliteclass.Histogramsof 151211325= pairwisecorrelationcoefficients(i.e.edgeweights)forbothnetworks.Greenlinesindicatethemedianvalues,redlinesdenoteasignificancelevelof0.01withBonferronicorrection.TheCNdisplaysageneralbiastowardspositivecorrelationsthroughoutallmetabolites.FortheGGM,themedianvalueliesaroundzeroandweobserveashifttowardssignificantlypositivevalues.Modularitybetweenmetaboliteclassesmeasuredastherelativeout-degreefromeachclass(rows)toallotherclasses(columns).TheGGM(right)showsaclearseparationofmetaboliteclasses,withsomeoverlapsforthedifferentphospholipidspeciesdiacyl-PCs,lyso-PCs,acyl-alkyl-PCsandsphingomyelins.Valuesrangefromwhite(0.0out-degreetowardsthisclass)toblack(1.0).PCs=phosphatidylcholines.etalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page6of16 top20positivepartialcorrelations(Table1).Inthefol-lowing,wewillspecificallydiscussinteresting,high-scor-ingmetabolitepairsalongwiththeirresponsibleenzymesinthemetabolicpathways.Thehighestpartialcorrelationinthedatasetwith0.821isfoundforthetwobranched-chainaminoacidsValineandxLeucine,wherethelattercompoundrepre-sentsbothLeucineandIsoleucine(whichhaveequalmassesandarenotdistinguishablebythepresentmethod).Thethreemetabolitesareincloseproximityinthemetabolicnetworkconcerningtheirbiosynthesisanddegradationpathways.FurtherrelatedaminoacidpairsthatdisplaysignificantpartialcorrelationsareHistidineandGlutamine(=0.383),GlycineandSerine(=0.326)aswellasThreonineandMethionine(=0.298).Clear-cutsignaturesofthedesaturationandelonga-tionoflongchainfattyacidscanbeseenforvarioussphingomyelinsandlyso-PCs(Figure3A).Forexample,SMC18:0andSMC18:1stronglyassociatewith0.767,mostprobablyrepresentingtheinitial9desa-turationstepofthepolyunsaturatedfattyacidbiosynth-esispathwayfromC18:0toC18:1-9bySCD(Steaoryl-CoAdesaturase).ThesimilarlyhighpartialcorrelationbetweenSMC16:1andSMC18:1(=0.765)aswellaslysoPCaC16:1andlysoPCaC18:1(=0.315)canbeattributedtotheELOVL6-dependentelongationfrom9toC18:1-11.Interestingly,thisreactionisnotcontainedinthepublicreactiondatabasesbuthasbeenpreviouslydescribedby[27]. Table1Top20positiveGGMedgeweights(i.e.partialcorrelationcoefficients,PCC)inourdatasetalongwithproposedmetabolicpathwayexplanationsMetabolite1Metabolite2PCCCommentValxLeu0.821Branched-chainaminoacidsSMC18:0SMC18:10.767SCD/SCD5desaturationSMC16:1SMC18:10.765ELOVL6PCaeC34:2PCaeC36:30.7522reactionstepsSM(OH)SM(OH)0.743sphingolipid-specificdesaturation?PCaaC34:2PCaaC36:20.735ELOVL1/ELOVL6elongationC10:0-carnC8:0-carn0.735-oxidationsteplysoPCalysoPCa0.731ELOVL6elongationPCaaC38:6PCaaC40:60.709ACOX1/3+variousELOVLsSM(OH)SM(OH)0.686sphingolipid-specificelongation?PCaaC36:4PCaaC38:40.672ACOX1/3+variousELOVLsPCaaC32:1lysoPCa0.661C16:0/C16:1phospholipidPCaaC38:5PCaaC40:50.653variousELOVLsPCaeC34:3PCaeC36:50.607atleast3reactionstepsPCaaC36:5PCaaC38:50.596ACOX1/3+variousELOVLsSMC24:0SMC24:10.577sphingolipid-specificdesaturation?PCaeC32:1PCaeC32:20.574SCD/SCD5desaturationSM(OH)SMC24:10.567possibleelongationintermediateC18:1-carnC18:2-carn0.561-oxidationintermediateMostmetabolitepairscanbedirectlylinkedtoreactionsinthefattyacidbiosynthesispathway,the-oxidationpathwayoraminoacid-associatedpathways. Figure3BiochemicalsubnetworksidentifiedbytheGGM.Linewidthscorrespondtopartialcorrelationcoefficients.Elongationanddesaturationsignatures,mostlikelymediatedbyELOVL6andSCD,forC16andC18fattyacidsincorporatedinlyso-PCsandsphingomyelins.Top:Diacyl-phosphatidylcholine(PCaa)specieswithelongationandperoxisomal-oxidationassociations.SeveralcombinatorialvariantsofsidechaincompositionsarepossibleforC36:4andC38:4,andthusdifferentenzymescouldmediatethisconnection.Bottom:Alkyl-acyl-phosphatidylcholines(PCae)withsupposedlydistinctsidechaincomposition,givingrisetoalowassociationwithadirectlyconnectedspeciesRecovered-oxidationpathwayfromC18downtoC4.Fourenzymeswithoverlappingsubstratespecificitiescatalyzetherate-limitingreactionsofthispathway.Twohigh-scoringtriads,wheremetabolitepairswithapathwaydistanceoftwoconstitutestrongpartialcorrelations.Thisfeatureofpartialcorrelationsaidsinthereconstructionofthenetworktopologybeyondthedirectneighborhoodofeachmetabolite.etalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page7of16 WeidentifyavarietyofstrongGGMedgesbetweendiacyl-PC(lecithins,PCaa)andacyl-alkyl-PC(plasmalo-gens,PCae)metabolitepairs(Figure3B).Forinstance,PCaaC34:2andPCaaC36:2associatestronglywith=0.735,andPCaaC36:4andPCaaC38:4showapar-tialcorrelationof=0.672.WhilethefirstpaircanbepreciselyexplainedbyanelongationfromC16:0toC18:0byELOVL6,differentcombinatorialvariantscomeintoplayforthePCaaC36:4/PCaaC38:4pair.Ourmass-spectrometrytechniqueonlymeasuresbruttocompositions,thatisthebulksidechaincarboncontentandtotaldegreeofdesaturation.Dependingontheexactcompositionofbothfattyacidresiduesintherespectivelipids,thisassociationcouldbecausedbylong-chainelongations(C14toC16andC16toC18throughfattyacidsynthaseandELOVL6,respectively),byvery-long-chainelongations(C22:4toC24:4throughELOVL2orELOVL5)andevenbyperoxisomal-oxi-dationoffattyacids(throughACOX1orACOX3).AninterestingsituationarisesforthephospholipidsPCaeC34:2,PCaeC36:3andPCaeC36:2.Fromitsbruttoformulathelatterspeciescouldrepresentanintermedi-atestepbetweentheothertwometabolites.However,itassociatespoorlywithbothotherphospholipids,whichinturndisplayastrongpartialcorrelation(=0.752).Thisfindingcanbeexplainedbydistinctfattyacidsidechaincompositions,showingdifferentialincorporationofC18:0,C18:1andC18:2(Figure3B,bottom).Fortheacyl-carnitinegroupweobservearemarkablyhighpartialcorrelationof=0.735forC8-carnandC10-carnandfurtheracyl-carnitinepairswithacarbonatomdifferenceoftwo(Figure3C).Theseassociationscanbeattributedtothe-oxidationpathway,i.e.thecatabolicbreakdownoffattyacidsinthemitochondria[23].Duringthisdegradationprocess,Cunitsarecon-tinuouslysplitofffromtheshrinkingfattyacidchain.acyl-CoAdehydrogenases,ACADS,ACADMandACADL,ACADVL,catalyzetheratelimitingreactions-oxidationfordifferentfattyacidchainlengths[28,29].Ourinterpretationofacyl-carnitinecorrelationsassignaturesofmitochondrial-oxidationisinaccor-dancewith[19],whereweidentifiedassociationsbetweenC8+C10,C12andC4withgeneticvariationintheACADM,ACADLandACADSloci,respectively.Weobserveseveralassociationsthatwerenotdirectlyattributabletoenzymaticinteractionsinthefattyacidbiosynthesisordegradationpathways.Forinstance,lysoPCa18:1andlysoPCa18:2shareastrongGGMedge(=0.543)althoughthe12-desaturationstepfromoleicacidtolinoleicacidisknowntobemissinginhumans[30].Thismissingreactiongivesrisetotheessentialityoffattyacidsinthe-6unsaturatedfattyacidpathway.Afunctionalexplanationcouldbeasys-temicequilibriumbetweenthetwofattyacidsorremodelingprocessesspecificforthelyso-PCmetaboliteclass.FurtherexamplesarehighpartialcorrelationsbetweenthehydroxysphingomyelinsSM(OH)C22:1andSM(OH)C22:2(=0.743)aswellasthesphingo-myelinsSMC24:0andSMC24:1(=0.577).Tothebestofourknowledge,thereisnoevidenceforsuchfattyaciddesaturationreactionsinhumans.ThedetectedassociationsmightthereforerepresentnovelpathwayinteractionsrecoveredbytheGaussiangraphi-calmodel.Negativevaluesplayaparticularroleintheinterpreta-tionofpartialcorrelationscoefficients.Ontheonehand,theyobviouslyoccurwheneverregularnegativecorrelationsareinvolved.Mechanismsgivingrisetonegativecorrelationsare,forexample,coparticipationinthesamereaction(cf.Figure1E),massconservationrelations[6]oropposingregulatoryeffects.Itistobenoted,however,thatnegativecorrelationsarerareinourspecificmetabolomicsdataset(cf.Figure2C).Ontheotherhand,duetothemathematicalpropertiesofpartialcorrelationcoefficientsnegativepartialcorrela-tioncoefficientsoccurwhenevertwometabolitesandhaveastrongcorrelationwithathirdmetabolitebutdonotshareahighcorrelationvaluewitheachother.TwoexamplesfromourdatasetareshowninFigure3D.First,SMC18:0isnegativelypartiallycorre-latedwithSMC16:1,andbothoftheseinturnarehighlypositivelypartiallycorrelatedwithSMC18:1.ThefattyacidsC16:1andC18:0havenodirectconnectioninthepathway,causingthestrongnegativepartialcor-relationvalue.Asimilarsituationcanbefoundforthreediacyl-PCs:PCaaC34:2andPCaaC36:1showahighpartialcorrelationwithPCaaC36:2,butanegativepar-tialcorrelationwitheachother.Again,thereisnopossi-bledirectreactionfromaC34:2lipidspeciestoaC36:1species.Notallmetabolitetriadsinthenetworkshowsuchaone-negative/two-positivemotif.Butifpresent,theyprovideanotherstepinthereconstructionofmeta-bolicpathways(beyondthedirectneighborhoodofeachmetabolite)bydetectingmetaboliteswhichareexactlytwostepsapart.Partialcorrelationcoefficientsdiscriminatebetweendirectlyandindirectlyconnectedmetabolitesinaliterature-curatedfattyacidpathwaymodelTheanalysesfromtheprevioussectionstrengthenedourconceptionthataGGMinferredfrombloodserummetabolomicsdatarepresentstruemetaboliteassocia-tions.TosystematicallyassesshowGGMedgesandpathwayproximitybetweenourlipidmetabolitesarerelated,wegeneratedaliterature-basedmodeloffattyacidbiosynthesis(Figure4A).Thismodelincludesreac-tionsfromthepublicdatabasesBiGG(H.sapiensRecon1)[7],theEdinburghHumanMetabolicNetwork[31]etalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page8of16 andKEGGPATHWAY[29].Wethenmappedthepar-tialcorrelationcoefficientsfromtheKORAdatasetontotheminimalnumberofreactionstepsbetweeneachpairofmetabolites(pathwaydistance).Sinceourmetabolitepanelcontainsfatty-acidbasedlipids,weprojecttherespectivelipidcompositionsontothefattyacidbiosynthesispathway(Figure4B-D).Fortheanaly-sisofacyl-carnitinesweimplementedamodelofthe-oxidationpathway,consistingofalinearchainofC2degradationsteps(C10C6etc.).Weobserveastrongtendencytowardssignificantlypositivepartialcorrelationsforapathwaydistanceofone,i.e.directlyconnectedmetabolitepairs,forallfivemetaboliteclasses(Figure5A).Intotal,86outof130partialcorrelations(66%)forapathwaydistanceofonearesignificantlypositive.Forinstance,forthelyso-PC Figure4FattyacidbiosynthesismodelandpathwaydistancecalculationmethodDenovosynthesisoffattyacidswithinitialSCD-dependentdesaturations(left),andthe-3and-6poly-unsaturatedfattyacidpathways(middleandright).Notethatweomittedthespecificpositionsofeachdouble-bondsincethemass-spectrometrytechniqueinourstudydoesnotresolvepositionalinformation.distancecalculationontwolyso-PCs.Weprojectedlipidsidechaincompositionsontotherespectivefatty-acidbiosynthesisreactions.Reactionreversibilityisnottakenintoaccountinourcalculation,i.e.distancesarealwayssymmetric.Ifnoknownpathwayconnectionbetweentwofattyacidsexists,weassignaformaldistanceofinfinity.Forphospholipidsthatcontaintwofattyacidresiduesweneedtotakeintoaccountallcombinatorialvariants.WehereshowthreevariantsfortheconnectionbetweenPCaaC38:4andPCaaC38:5.Intheseexamples,PCaaC38:4couldeitherconsistofC18:0+C20:4orC16:0+C22:4,whilePCaaC38:5couldbeC18:0+C20:5orC16:0+C22:5.Theshortestpossibledistance,oneinthiscase,willbeusedforfurthercalculations.etalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page9of16 class(Figure5A)nearlyallpartialcorrelationcoeffi-cientsforapathwaydistanceofoneareabovesignifi-cancelevel,whereasmostvaluesforadistanceoftwoorlargerremaininsignificant.Someoutliersfromthisobservation,however,requirecloserinspection:First,forsomemetaboliteclassesweobservenegativepartialcor-relationvaluesformetabolitepairsthatareexactlytwostepsapartinthemetabolicpathway:10of73partialcorrelationsinthediacyl-PCclassand2of2partialcor-relationsinthesphingomyelinclassaresignificantlynegativeforadistanceoftwo.Thesenegativevaluesareeffectsofthecoregulatedmetabolitetriadsdescribedpreviouslyinthistext.Second,wefind91of932(~9:8%)unconnectedmetabolitepairs(pathwaydistance)withapartialcorrelationabovesignificancelevel.Thesepairsrepresentpotentiallynovelpathwaypredic-tions,missinginteractionsinthemodeloreffectsupstreamofthemetabolicnetworklikeenzymeAdirectcomparisonofbothpartialandPearsoncor-relationcoefficientsforthediacyl-phosphatidylcholineclassisshowninFigure5B.Asdescribedearlierinthismanuscript,weobserveageneralover-abundanceofsig-nificantPearsoncorrelationsindependentoftheactualpathwaydistance.Evenforthemetaboliteswithoutaknownpathwayconnection,1394ofatotalof1569Pearsoncorrelationsaresignificant(88.85%,overallclasses),incontrastto131outof1569forthepartialcorrelations(8.35%).Thesignificantlydifferentcorrelationvaluedistribu-tionsbetweendirectlyandindirectlylinkedmetabolites(Figure5A+B)barelyprovideagoodquantificationoftheactualdiscriminationaccuracyofthisfeature.There-foreweassessedthediscriminativepowerofpartialcor-relationstotellapartdirectfromindirectinteractionsbymeansofsensitivityandspecificity.Thesensitivityevaluateswhichfractionofdirectlyconnectedmetabo-litesinthepathwayarerecoveredbysignificantGGMedges,whereasthespecificitystateshowmanyofthesignificantedgesactuallyrepresentadirectconnection.Acommonlyusedtradeoffmeasurebetweensensitivityandspecificityisthescore,whichisdefinedastheharmonicmeanofbothquantities[32](seeMethods).Figure5Clistssensitivity,specificcityandFforall5metaboliteclassesalongwithanevaluationofpartialcorrelationdistributiondifferencesbetweendirectlyandindirectlylinkedmetabolites(determinedbyWilcoxonranksumtest).valuesover0.75andsignificantp-valuesfortheranksumtestindicateastrongdiscrimina-tioneffectofpartialcorrelationcoefficientsconcerningdirectvs.indirectpathwayinteractions.Possiblereasonsfornon-perfectsensitivityandspecificcityvalueswillbediscussedindetailattheendofthistext.Low-orderpartialcorrelationsThedatasetfromourpresentstudycontainedenoughsamplestocalculatefull-orderpartialcorrelations,thatistocalculatepairwisecorrelationsconditionedagainst Figure5Systematicevaluationofpartialcorrelationcoefficientsversuspathwaydistances.DashedlinesinAandBindicateasignificancelevelof0.01withBonferronicorrection.Pathwaydistancesfromourconsensusmodelagainstpartialcorrelationcoefficientsforthefivelipid-basedmetaboliteclassesinourdataset.Weobserveanenrichmentofsignificantpartialcorrelationsforapathwaydistanceofone,whichrapidlydropsforanincreasingnumberofpathwaysteps.ComparisonofpartialcorrelationcoefficientsandPearsoncorrelationcoefficients.Pearsoncorrelationcoefficientsaregenerallyhigh,independentoftheactualpathwaydistance,indicatingforsystemiccoregulationeffectsthroughoutthelipidmetabolism.Wilcoxonranksumtestp-valuesbetweenthepartialcorrelationdistributionsofdirectlyandindirectlyconnectedpairs,andsensitivity/specificity/valuesmeasuringthediscriminatorypowertodistinguishdirectfromindirectpairs.etalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page10of16 allother-2metabolites.However,previousstudiesdemonstratedthatlow-orderpartialcorrelationapproachescanalreadybesufficienttoelucidatedirectinteractions[12,16].Inordertoassesshowthesemea-suresperformincomparisontothefull-orderGGM,wecalculatedfirst-,second-andthirdorderpartialcorrela-tionsusingtheapproachdevelopedby[12]forbothcomputer-simulatednetworksandthemetabolomicsdata(Additionalfile5).Thetoysystemsrevealclearcaseswherelow-orderapproachesfail,forinstanceindiamondmotifdisplayedinFigure1D.Surprisingly,however,especiallyfirst-orderpartialcorrelationsworkedremarkablywellindiscriminatingdirectfromindirectinteractionsintherealdata(FvaluesclosetothosedisplayedinFigure5C).Thisresultprovidestwovaluablepiecesofinformation.First,low-orderpartialcorrelationapproaches,whichrequiremuchlesssam-plestoobtainstableestimates,appeartobeasuitablealternativetoGGMsforthemetabolitepanelusedinthisstudy.Second,thehighrelativescoringoffirst-orderpartialcorrelationsprovidesinsightsintothecor-relationstructuresinthedataset.Inparticular,thisresultindicatesthattheunderlyingmetabolicpathwaysareprimarilycomposedofacyclic,linearchains,whichfitswelltothefattyacidpathwaysdominatingourmea-suredlipidspecies.ConclusionsInthispaperweaddressedthereconstructionofmeta-bolicpathwayreactionsfromhigh-throughputtargetedmetabolomicsmeasurements.Previousreconstructionapproachesemployedpairwiseassociationmeasures,pri-marilystandardPearsoncorrelationcoefficients,toinfernetworktopologyinformationfrommetaboliteprofiles[5,6,8,33].WeheredemonstratedtheusefulnessofGaussiangraphicalmodelsandtheirabilitytodistin-guishdirectfromindirectassociationsbyestimatingthedependencebetweenvariables.GGMsarebasedonpartialcorrelationcoefficients,thatisthePear-soncorrelationbetweentwometabolitescorrectedforthecorrelationswithallothermetabolites.Fromcomputersimulationsofmetabolicreactionnet-workswededucedasetimportantaspectstobeconsid-eredwheninterpretingpartialcorrelationcoefficientsinreactionsystems:(a)Metabolitesinequilibriumduetoreversiblereactionscanreadilyberecovered,whereasirreversiblereactionsposeasubstantialproblemtoasso-ciation-basedreconstructionattempts(inconcordancewith[6]).(b)Inputandoutputreactionsforintermedi-atemetabolites,however,improvethereconstructionaccuracy.Suchexchangereactionsarelikelytobepre-sentformostnaturallyoccurringmetabolitesduetohighlyinterconnectedmetabolicpathways.(c)Withanincreasingamountoffluctuationsontheinputreaction,thepartialcorrelationdifferencebetweendirectandindirectinteractionsincreasesforcertainnetworktopol-ogies(e.g.fortheirreversiblelinearmetabolitechains).ThisindicatesthatahighheterogeneityofmetabolicstatesinapopulationdatasetliketheKORAcohortmightbebeneficialratherthanproblematicforourapproach.(d)Metaboliteconnectivityincofactor-drivennetworkscanbeaccuratelyreconstructed.Thepresenceofexchangereactionsforcofactors,astheyarelikelytobepresentinrealsystems,hassubstantialimpactonthereconstructionquality.Theconnectivityofthecofactorsthemselves,however,remainsspurious.(e)Saturationeffectsinenzyme-catalyzedreactionsdonotposeapro-blemforthereconstructionprocess.However,inhibitoryinfluencesinmetabolicmodulesthatincludeexchangereactionsmightdecouplecertainmetabolitesandleadtofalsenegativeresults.(f)Non-linearratelawsandantag-onistic,correlation-generatingmechanismsmightimpairreconstructionquality.InthenextstepweinferredbothaGGMandaregu-larcorrelationnetworkfromalarge-scalemetabolomicsdatasetwith1020strictlystandardizedsamplesfromovernightfastingindividualsmeasuredbystate-of-theartmetabolomicstechnologies[19].Weinvestigatedtheinfluenceofthe15genome-wide-significantSNPsfromthisstudyonourGGManddemonstratedthatgeneticvariationinthegeneralpopulationisneglectableforpartialcorrelationcalculation.WefoundthattheGGMdisplaysamuchsparserstructurethanregularcorrela-tionnetworks.Onlyaround400partialcorrelationvalueswereabovesignificancelevel(~3.6%),whereashalfofallPearsoncorrelationvaluesweresignificantafterBonferronicorrection.Thisdepictedthenatureofpartialcorrelationcoefficientstoneglectindirectasso-ciationsbetweendistantlyrelatedmetabolites.WedetectedastronglymodularstructureintheGGMwithrespecttothedifferentmetaboliteclasses,exceptforthefourtypesofphospholipidswhichappearslightlyinter-woven.Thisprovidesauniquepictureoftheseparationofmetabolicpathways(synthesis,degradationandaminoacidmetabolism),butalsotheinteractionbetweendifferentlipidclassesdependentonasingleintracellularfattyacidpool.Finally,GGMswerestablewithrespecttobothchoiceandnumberofsamplesinthedataset.Evenasmallerdatasetwithonlyafewhundredsampleswouldhavebeensufficienttoachievetheresultsfromthisstudy.TheestimationofGGMsfordatasetswithlesssamplesthanmetabolitesispossible[26],butnotabledeviationsfromthetruepartialcorre-lationcoefficientshavetobeexpected.Manualinvestigationofhigh-scoringsubstructuresintheGGMrevealedgroupsofmetabolitesthatcouldbedirectlyattributedtoreactionstepsfromthehumanfattyacidbiosynthesisanddegradationpathways.WeetalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page11of16 detectedeffectsofELOVL-mediatedelongationsandFADS-mediateddesaturationsoffattyacidsaswellassignaturesofthecatabolic-oxidationpathway.Forinstance,ourmethodsuccessfullyrecoveredadirectelongationfromC16:1toC18:1,whichhasbeenexperi-mentallyshownby[27]butisnotpresentinthepublicreactiondatabases.Furthermore,weidentifiedhighlynegativepartialcorrelationsasanindicationforapath-waydistanceoftwo,servingasafurtherhintinthereconstructionofmetabolicnetworktopology.Inordertosystematicallyevaluatewhetherhighpartialcorrela-tionsrepresentdirectinteractions,wegeneratedacon-sensusmodeloffattyacidbiosynthesisreactionsfromthreepublicallyavailablereactiondatabases.Bymappingpartialcorrelationcoefficientstothenumberofreactionstepsbetweentwometabolitesweobservedastatisticallysignificantenrichmentofhighvaluesforapathwaydis-tanceofone.Wecalculatedahighaccuracyforpartialcorrelationstodiscriminatebetweendirectlyandindir-ectlyassociatedmetabolites,asmeasuredbysensitivity,specificityandthemeasure.Interestingly,wecouldshowthatthediscriminationqualityoflow-orderpartialcorrelations[12],especiallythefirst-ordervariants,isclosetothefull-orderGGM.Eventhoughthismightbeafeaturespecifictothemetabolitepanelusedinthisstudy,low-orderpartialcorrelationsrepresentasuitablealternativeespeciallyforstudieswithonlyfewsamples.Ifmoresamplesthanvariablesareavailable,however,werecommendGGMsasanunbiasedapproachcondi-tioningagainstasmanyparametersaspossible.Takentogether,ourresultsdemonstratethatGGMsinferredfrommetabolomicsmeasurementsinbloodplasmasamplesrevealstrongsignaturesofintracellularandeveninner-mitochondrialprocesses.Previousstudiesonbloodplasmasamplesdetectedsimilarrelationshipswithcellularprocessesbasedongeneticassociations[19]andcase/controldrugtrials[34].Inthisworkwecouldnowshowthatmetaboliteprofilesalonearesufficienttocapturethedynamicsofmetabolicpathways.However,GGMscanneverprovideaperfectrecon-structionoftheunderlyingsystem.Thereareseveralfactorsthatleadtotheabsenceofhighpartialcorrela-tionsbetweeninteractingmetabolites,thatisfalsenega-tiveedgesintheGGM:(a)Counterantagonisticcorrelation-generatingprocessesandbimolecularreac-tions(seeabove)mightleadtotheeliminationofpair-wiseassociation;cf.[6].(b)Therespectiveenzymemightnotbeactiveinthecurrentmetabolicstate,oritseffectsontherespectivemetabolitepoolsareneglect-able.(c)Contrarytoourgeneralfindingthatevenbloodplasmametabolitescarrystrongsignaturesofmetabolicpathways,thesignalmightbediminishedforcertaintypesofmetabolites.Furthermore,theactualoriginsofbloodplasmametabolites,e.g.intermsofmeasuredcelltypesorcausaltissueactivity,stillremaintobeunra-veled.Theabove-mentionedmechanismsarepossibleexplanationsforthenon-perfectsensitivityvaluesobservedinFigure5C.FalsepositiveGGMedges,ontheotherhand,provideinterestingnewmetabolicpath-wayhypothesis.Thepresenceofstrongpartialcorrela-tionsintheabsenceofknownmetabolicconnectionscouldpointoutmissingpathwayinformationorregula-toryeffectsnotcapturedinasimplestoichiometricrepresentationofthepathway.Conclusively,thisstudypresentedGaussiangraphicalmodelsasavaluabletoolfortherecoveryofbiochem-icalreactionsfromhigh-throughputtargetedmetabolo-micsdata.Thepresentworkcouldbeextendedbycomparinghighpartialcorrelationcoefficientswithenzymeactivityorexpressiondata,orbytheexperimen-talvalidationofpromisinginteractioncandidates.WesuggestusingGGMsasastandardtoolofinvestigationinfuturemetabolomicsstudies,utilizingtheupcomingwealthofmetabolicprofilingdatatoformamorecom-prehensivepictureofcellularmetabolism.MethodsInsilicosimulationofartificialreactionnetworks)beavectorofmetaboliteconcentrationsthestoichiometrymatrixofadynamicalsys-temwithmetabolitesandreactions.Eachcolumninrepresentsthecompoundstoichiometryofasinglereaction,withnegativevaluesfortheeductsofareactionandpositivevaluesforitsproducts(cf.[35]).Further-more,wedefineaneductstoichiometrymatrixS,whichonlycontainsthenegativevaluesfrom.Thereactionratelawscanbewrittenas)=diag(),where:=(,...,)representsavectorofelementaryratecon-stantsand cxxjr=\b=:,,..., containstheproductsofsubstrateconcentrationsaccordingtothelawofmassaction[36].Forexample,forthereactionweobtain,and2.Forenzyme-catalyzedreactions,thecorre-spondingentriesinareformulatedusingreversibleMichaelis-Menten-typekinetics[37,38]insteadofthemass-actiontermabove: \tŠ\tmaxmax[][][][] Where Vmax+ and VmaxŠ aretheproductandsubstrateformationconstants,respectively, KMs and KMp senttheMichaelisconstantsforsubstrateandproduct,etalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page12of16 S]representsthesubstrateconcentrationand[]repre-sentstheproductconcentration.Notethatweomittedreaction-specificparameterindicesforsimplicityhere.Allostericregulationwasmodeledusingamixedinhibi-tionmechanism,whichextendstheratelawfromequa-tion(1)asfollows: Š\t[]+++    [][][]maxmax ++ with[]beingtheinhibitorconcentration,thebind-ingrateoftheinhibitortotheenzymeandthebind-ingrateoftheinhibitortothesubstrate-enzyme(orproduct-enzyme)complex.Inasimplemixed(non-com-)inhibitionscenario,weassumeTheordinarydifferentialequationsdescribingthetemporalevolutionofthesystemarenowgivenas Svxk Tointroducevariabilityeachparameterissubjecttofluctuationsaccordingtoalog-normaldistributionwithmean1andchangingvariances: LogN1 .Finally,forfixed,Pearsonandpartialcorrelations(seebelow)arecalculatedbydrawingthevectormultipletimesfromthepara-meterdistribution,calculatingthecorrespondingmeta-bolitesteadystateconcentrationsandlogarithmizingtheobtainedvalues.Ifthesystemcontainsonlyzer-oth-orderandfirst-orderreactions(i.e.inputreactionsandreactionswithonlyonesubstrate),thesteadystateconcentrationsforagivencanbereadilycomputedbyequating(2)tozeroandsolvingforusinglinearalgebratechniques.Ontheotherhand,ifhigherorderreactionsarepresent,theODEsareintegratednumeri-callyandsimulateduntilequilibriumtogetcorre-spondingsteadystates.Forthispurpose,avariable-ordersolverforstiffdifferentialequations(ode15s)fromMATLABwasused[39].Thepresenceofaunique,singlepositivesteadystatewasshownforeachnetworkindividuallyusingtheERNESTtoolbox[21],orbyempiricalevaluation(parameterandinitialvaluesampling).ForadetailedanalysiswereferthereadertoAdditionalfile1,section1.ComputationofcorrelationnetworkandGaussiangraphicalmodelLet)bethematrixoflogarithmizedmeta-boliteconcentrations(eithermeasureddatasamplesorcomputer-simulatedsteadystates),whereisthenumberofsamples,andagainrepresentsthenumberofmetabolites.ThenthestandardPearsonproduct-momentcorrelationcoefficients)betweenmeta-bolitesarecalculatedas kiikjjkiikjjxxxxxxxx where xi representsthemeanvalueofmetaboliteSinceweuseaGaussiangraphicalmodel,thecondi-tionaldistributionsarealsoGaussian.Theirwidthandthecorrespondingpartialcorrelationcoefficientscanbecalculatedas ijijiijj]ZZZ with ZijP()=Š1 Apartialcorrelationvaluedenotesthepairwisecor-relationofmetabolitescorrectedfortheeffectsofallremainingmetabolites.Sinceourstudydesigncon-tainsmoresamplesthanmeasuredvariables,thecorrela-tionmatrixhasfullrankanditsinversecanbestraightforwardlydetermined.First-,second-,andthird-orderpartialcorrelationswerecalculatedusingthesoft-warepublishedin[12].Toassessthesignificanceofpar-tialcorrelations,p-values)werecalculatedusing-transform[40]: pnmzijij]I]=ŠŠŠ123)) wherestandsforthecumulativedistributionfunc-tionofthestandardnormaldistribution.Inordertoaccountformultipletesting,Bonferronicorrectionwasappliedtoobtainanestimateofthesignificancelevel.NotethatBonferronicorrectionisthemostconservativeapproachformultipletesting;itassumesindependenceofalltestedvalues,whichiscertainlynotthecaseforpartialcorrelationcoefficients.Basedonanominalsig-nificancelevelof=0.01,weretrieveanadjustedlevel ==\t0011132588310./. afterBonferronicor-rection.Solvingequation(3)foryieldsaminimumabsolutepartialcorrelationcoefficientof0.1619forthegivensignificancelevel.Thatis,allpartialcorrelationssmallerthan-0.1619orlargerthan0.1619areconsid-eredsignificant.Bootstrappingwasperformedbyrandomlydrawing1020sampleswithreplacementfromtheoriginaldataset.Forthesecondstabilityanalysis,theinvestigationofdifferentdatasetsizes,therespectivenumberofsam-pleswasrandomlydrawnfromtheoriginaldataset.etalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page13of16 Thewholeprocedurewasrepeated100timestogetastableestimateofthedeviation.NetworkmodularitycalculationWedefinetheadjacencymatrixofanewunweighted,undirectedgraphinducedbyallsignificantlypositivepartialcorrelationsin ]Dijij:,,=\f\r10ifelse where D representsthesignificancelevelaftermulti-pletestingcorrection.Nowlet(,...,)betheparti-tioningofthemetabolitesintothesixmetaboliteclasses:acyl-carnitines,diacyl-PCs,lyso-PCs,acyl-alkyl-PCs,sphingomyelinsandaminoacids(thehexoseisleftoutasonlyasinglemetabolitebelongstothatclass).Wecalculatedtherelativeout-degreeR6×6eachclasstotheotherclasses,(i.e.theproportionofitsedgeseachclassshareswiththeotherclasses)as: (,) where (,")’,"iVjV representsthetotalnumberofedgesbetween,andcontainsallmetabolitesinthenetwork.Thetotalnet-workmodularityofthenetworkcanbequantifiedaccordingto[41]as: iii(,)(,)(,) Intuitively,thismeasurecomparesthewithin-classedgeswiththeedgestotherestofthenetwork.Themoreedgestherearewithineachclassincomparisontotheotherclasses,thehigherwillbe.Notethatequa-tion(4)canbeappliedtobothweightedandunweightedgraphs.Toassessthesignificanceoftheobservedvalue,weperformedgraphrandomizationbyedgerewiring[42,43]andsubsequentcalculationofDuringtherewiringprocesswerandomlypicktwoedgesfromthenetworkandexchangethetargetnodesofeachedge.Inordertoachievesufficientrandomiza-tion,thisoperationisrepeated5·times,whererepresentsthenumberofedgesinthegraph.Toper-formedgereshufflingonweightedgraphs,wedecidedonaneighbor-preservingvariantasdescribedin[44].StudycohortandmetabolitepanelKORA(KooperativeGesundheitsforschunginderRegionAugsburg)isaresearchplatforminsouthernGermanywithaprimaryfocusoncardiovasculardis-eases,Diabetesmellitustype2,andgeneticepidemiol-ogy[18].Fastingserumconcentrationsfrom=1020individualsintheKORAF4weredeterminedbyelectro-sprayionizationtandemmassspectrometry(ESI-MS/MS)usingtheBiocratesAbsoluteIDQtargetedmetabo-lomicskittechnology.Thesesamplesrepresentasubsetofthedatasetpreviouslyevaluatedinagenome-wideassociationstudyin[19].Atotalof=151metabolitesweremeasuredintheexperiments:14aminoacidsincluding13proteinogenicaminoacidsandornithine;hexose(sugarswith6carbonatoms,e.g.glucoseandfructose);23acylcarnitines[Cx:y-carn](withcarbonatomsanddoublebonds),7hydroxy-acylcarnitines[Cx:y-OH-carn],6dicarboxy-acylcarnitines[Cx:y-DC-carn],and2methylateddicar-boxy-acylcarnitinesvariants[Cx:y-M-DC-carn];9sphin-gomyelins[SMCx:y]and5hydroxy-sphingomyelins[SMCx:y-OH];and87phosphatidylcholines(PC).Theseglycerophospholipidsarefurthersubdividedwithrespecttothepresenceofesterandetherbondsoffattyacidresidueswiththeglycerolmoiety.Thesetcontains36diacyl-PCswithtwoesterifiedfattyacidresidues[PCaaCx:y],38acyl-alkyl-PCswithoneether-bondatthesn-2position[PCaeCx:y]and13lyso-PCswithonlyoneesterifiedfattyacidresidueatthesn-1orsn-2position[lysoPCaCx:y].Ourmassspectometrytechnologycan-notdistinguishbetweenthesidechainsofdiacyl-phos-pholipids.Themeasuredcompoundsarethusassociatedwiththesumofcarbonatomsanddoubleboundsforbothfattyacidresidues.Toensurelog-normality,wecomparedQQ-plotsagainstnormaldistributions[45]forbothnon-logarithmizedandlogarithmizedmetabo-liteconcentrations.Alldistributionswereclosertolog-normalitythantoregularnormality(notshown),sowelogarithmizedthemetaboliteconcentrationsforthefol-lowinganalysissteps.SensitivityandspecificityInordertoobjectivelyevaluatethediscriminationbetweendirectlyandindirectlyconnectedmetabolites,wecalculatedsensitivityandspecificityas: sensTPFNspecTNFP:,: withTPtruepositives,FPfalsepositives,TNtruenegatives,FNfalsenegatives[46].Ametabolitepairisconsideredtruepositiveifitexhi-bitsapartialcorrelationabovethethresholdandhasadirectpathwayconnection;afalsepositiverepresentsametabolitepairalsoabovethethresholdbutwithnodirectpathwayconnection;afalsenegativepairliesbelowthethresholdbutdoeshaveadirectpathwayetalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page14of16 connection;andfinallyatruenegativepairliesbelowthethresholdandalsohasnodirectpathwayconnec-tion.Thescorewascalculatedastheharmonicmeanofbothquantities: 2:sensspecsensspec PathwaymodelPathwayreactionsinthehumanfattyacidmetabolismweredrawnfromthreeindependentdatabases:(1)sapiensRecon1fromtheBiGGdatabases(confidencescoreofatleast4)[7],(2)theEdinburghHumanMeta-bolicNetworkreconstruction[31]and(3)theKEGGPATHWAYdatabase[29]asofJuly2010.Acompletelistofallcuratedreactionsandthecorrespondingdata-baseidentifierscanbefoundinAdditionalfile6.Thereactionsetwassubdividedintotwogroups:(1)Fattyacidbiosynthesisreactionswhichapplytothemetabo-liteclasseslyso-PC,diacyl-PC,acyl-alkyl-PCandsphin-gomyelins.(2)-oxidationreactionsrepresentingfattyaciddegradationtomodelreactionsbetweentheacyl-carnitines.The-oxidationmodelconsistsofalinearchainofC2degradationsteps(C10C6etc.).Fattyacidresidueswithidenticalmasses,thatcannotbedistinguishedbyourmass-spectrometrytechnology,aremergedintoasinglemetaboliteinthereactionset.Forinstance,thepolyunsaturatedfattyacidsC20:48,11,14,17fromtheomega-3pathwayandC20:45,8,11,14fromtheomega-6pathwayhaveidenticalnumbersofcarbonatomsanddoublebondsandarethusmergedintoasinglemetaboliteC20:4.AdditionalmaterialAdditionalfile1:Furtherresultsoncomputer-simulatednetworksAdditionalfile2:EffectsofgeneticvariationonGGMcalculationAdditionalfile3:Modularity:OptimizedpartitioningandweightedcalculationAdditionalfile4:StabilityoftheGGMwithrespecttochangesintheunderlyingdatasetAdditionalfile5:comparisonwithlow-orderpartialcorrelationapproachesAdditionalfile6:Literature-curatedpathwaymodelofhumanfattyacidbiosynthesisanddegradationAcknowledgementsTheauthorsthanktheanonymousreviewersforvaluablecommentsandsuggestionstoimprovetheoriginalmanuscript.ThisresearchwaspartiallysupportedbytheInitiativeandNetworkingFundoftheHelmholtzAssociationwithintheHelmholtzAllianceonSystemsBiology(projectCoReNe),byagrantfromtheGermanFederalMinistryofEducationandResearch(BMBF)totheGermanCenterDiabetesResearch(DZDe.V.),andbytheBMBF-fundedMedizinischeSystembiologie-MedSys(subprojectSysMBo,projectlabel0315494A).JanKrumsiekissupportedbyaPhDstudentfellowshipfromtheStudienstiftungdesDeutschenVolkesThankstoHaroldGutchforcriticallyproofreadingandcorrectingthisAuthordetailsInstituteofBioinformaticsandSystemsBiology,HelmholtzZentrumMünchen,Germany.FacultyofBiology,Ludwig-Maximilians-Universität,Planegg-Martinsried,Germany.InstituteofEpidemiology,HelmholtzZentrumMünchen,Germany.InstituteofExperimentalGenetics,GenomeAnalysisCenter,HelmholtzZentrumMünchen,Germany.DepartmentofMathematics,TechnischeUniversitätMünchen,Germany.JK,KSandFJTconceivedthisdataanalysisproject.TIandJAperformedthesamplepreparationanddataacquirement.JKperformedtheanalysisandwrotetheprimarymanuscript.Allauthorsapprovedthefinalmanuscript.Received:1October2010Accepted:31January2011Published:31January20111.TweeddaleH,Notley-McRobbL,FerenciT:EffectofslowgrowthonmetabolismofEscherichiacoli,asrevealedbyglobalmetabolitepool)analysis.JBacteriol2.WenkMR:Theemergingfieldoflipidomics.NatRevDrugDiscovDiscov3.GriffinJL:TheCinderellastoryofmetabolicprofiling:doesmetabolomicsgettogotothefunctionalgenomicsball?PhilosTransRSocLondBBiolBiol4.FiehnO:Metabolomics-thelinkbetweengenotypesandphenotypes.PlantMolBiol5.SteuerR,KurthsJ,FiehnO,WeckwerthW:Observingandinterpretingcorrelationsinmetabolomicnetworks.6.CamachoD,delaFuenteA,MendesP:Theoriginofcorrelationsinmetabolomicsdata.data.s11306-005-1107-3].7.DuarteNC,BeckerSA,JamshidiN,ThieleI,MoML,VoTD,SrivasR,PalssonBO:Globalreconstructionofthehumanmetabolicnetworkbasedongenomicandbibliomicdata.ProcNatlAcadSciUSAUSA8.ArkinA,ShenP,RossJ:ATestCaseofCorrelationMetricConstructionofaReactionPathwayfromMeasurements.Measurements.5330/1275].9.VanceW,ArkinA,RossJ:Determinationofcausalconnectivitiesofspeciesinreactionnetworks.ProcNatlAcadSciUSAUSA10.SchäferJ,StrimmerK:LearningLarge-ScaleGraphicalGaussianModelsfromGenomicData.InProcNatlAcadSciUSA,Volume776,AIP263-276[http://link.aip.org/link/?APC/776/263/1].11.LeeJM,LeeJM,GianchandaniEP,EddyJA,PapinJA:Dynamicanalysisofintegratedsignaling,metabolic,andregulatorynetworks.PLoSComputComput12.delaFuenteA,BingN,HoescheleI,MendesP:Discoveryofmeaningfulassociationsingenomicdatausingpartialcorrelationcoefficients.coefficients.bioinformatics/bth445].13.MagwenePM,KimJ:Estimatinggenomiccoexpressionnetworksusingfirst-orderconditionalindependence.GenomeBiolBioldx.doi.org/10.1186/gb-2004-5-12-r100].14.SchäferJ,StrimmerK:AnempiricalBayesapproachtoinferringlarge-scalegeneassociationnetworks.networks.dx.doi.org/10.1093/bioinformatics/bti062].15.WilleA,ZimmermannP,VranováE,FürholzA,LauleO,BleulerS,HennigL,PrelicA,vonRohrP,ThieleL,ZitzlerE,GruissemW,BühlmannP:graphicalGaussianmodelingoftheisoprenoidgenenetworkinArabidopsisthaliana.GenomeBiolBiol10.1186/gb-2004-5-11-r92].KrumsieketalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page15of16 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Thorough peer review No space constraints or color Þgure chargesResearch which is freely available for redistribution www.biomedcentral.com/submit etalBMCSystemsBiologyhttp://www.biomedcentral.com/1752-0509/5/21Page16of16