From Petri nets to differential equations - an int

FromPetriNetstoDifferentialEquations-anIntegrativeApproachforBiochemicalNetworkAnalysisDavidGilbert1andMonikaHeiner21BioinformaticsResearchCentre,UniversityofGlasgowGlasgowG128QQ,Scotland,[e-mail]2DepartmentofComputerScience,BrandenburgUniversityofTechnologyPostbox101344,03013Cottbus,[e-mail]investigationintotheinte-grationofPetrinetsandordinarydifferentialequations(ODEs)forthemodellingandanalysisofbiochemicalnetworks,andtheapplicationofourapproachtothemodeloftheinfluenceoftheRafKinaseInhibitorProtein(RKIP)ontheExtracellularsignalRegulatedKinase(ERK)signallingpathway.WeshowthatanalysisbasedonadiscretePetrinetmodelofthesystemcanbeusedtoderivethesetsofinitialconcen-trationsrequiredbythecorrespondingcontinuousordinarydifferentialequationmodel,andnootherinitialconcentrationsproducemeaningfulsteadystates.Altogether,thispaperrepresentsatutorialinstep-wisemodellingandanalysisoflargermodelsaswellasinstructureddesignofODEs.1MotivationClassical,i.e.time-lessdiscretePetrinetscombineanintuitivemodellingstylewithwell-foundedanalysistechniques.Itisforthisreasonthattheyarewidelyusedinvariousapplicationareas,wheretheyhavebeenproventobeusefulforaqualitativeverificationoftechnicalaswellas“natural”systems,i.e.biochemicalnetworkslikemetabolicnetworks,signaltransductionnetworks,orgeneregula-torynetworks.However,anyrealsystembehaviourhappensintime.Thusthenextstepfollowingonfromaqualitativeanalysistypicallyconsistsinquantitativeanaly-sestakingintoaccounttiminginformation.Inthecaseofbiochemicalsystems,allatomicactionstakeplacecontinuously.Moreover,theratesofalltheatomicactionstypicallydependonthecontinuousconcentrationsoftheinvolvedsub-stances.Hencesystemsofordinarydifferentialequations(ODEs)appeartobeanaturalchoiceforquantitativemodellingofbiochemicalnetworks.Inthispaperwebridgethegapbetweenthesetwoworlds,i.e.the(time-less)discreteandthe(timed)continuousone,anddemonstratebymeansofoneofthestandardexamplesusedinthesystemsbiologycommunity–thecoremodeloftheinfluenceoftheRaf-1KinaseInhibitorProtein(RKIP)ontheERKsignallingpathway–howbothsidescanplaytogetherbyprovidingdifferent,butcomplementaryviewpointsonthesamesubject.Thispapercanbeconsideredasatutorialinthestep-wisemodellingandanalysisoflargermodelsaswellasinthestructureddesignofODEs.Thediscretemodelisintroducedasasupplementaryintermediatestep,atleastfromtheviewpointofthebiochemistaccustomedtoODEmodellingonly,andservesmainlyformodelvalidationsincethiscannotbeperformedonthecontinuouslevel.Havingsuccessfullyvalidatedthediscretemodel,thecontinuousmodelisderivedfromthediscreteonebyassigningrateequationstoalloftheatomicactionsinthenetwork.Thusthecontinuousmodelpreservesthestructureofthediscreteone,andthecontinuousPetrinetisnothingelsethanastructureddescriptionofODEs.Theapproachispresentedbyasmallexample,whichishoweversophisti-catedenoughtohighlightthemainideas—itiscommonsensetopracticenewtechniquesonsmallexamplesatfirst,beforeattemptinglargerones,wheretheoutcometobeexpectedtendstobelesswell-defined.Moreoverwedemonstratehowthediscretemodelcanbeusedtodrivethecontinuousmodelbyautomaticallygeneratingsetsofbiochemicallyplausiblevaluesfortheinitialconcentrationsofproteinspecies.Thispaperisorganizedasfollows.Thenextsectionprovidesanoverviewonthebiochemicalcontextonhandandintroducestherunningexample.Af-terwards,wedemonstratethestep-wisemodellingandanalysis,wheresection3dealswiththecontributionsbythediscreteviewpoint,whilesection4isdevotedtothecontinuousviewpoint.Havingpresentedourownapproach,wediscusssomerelatedworkinsection5.Weconcludewithasummaryandoutlookonintendedfurtherresearchdirections.2BiochemicalContextTherearemanynetworksofinteractingcomponentsknowntoexistaspartofthemachineryoflivingorganisms.Biochemicalnetworkscanbemetabolic,regulatoryorsignaltransductionnetworks.Theroleofmetabolicnetworksistosynthesizeessentialbiochemicalcompoundsfrombasiccomponents,ortodegradecompounds.RegulatorynetworksareusedtocontrolthewaysinwhichgenesareexpressedasRNAsorproteins,whereassignaltransductionnetworkstransmitbiochemicalsignalsbetweenorwithincells.Thetwoterms“pathway”and“network”tendtobeusedinterchangeablyintheliterature,with“pathway”being(implicitly)takentobeapartofamoregeneralnetwork.Inthispaperwefollowthegenerallyaccepteduseoftheterm“pathway”torefertothecoreofabiochemicalnetwork,comprisingasequenceofactivities,forexampleakinasecascade.Thus,forexample,wewilldescribetheERKpathwayasbeingembeddedinamoregeneralsignaltransductionnetwork,andthattheERKpathwayisamemberofalargefamilyofMAPKinasepathways.Inthispaperwefocusonsignaltransduction,whichisthemechanismwhichenablesacelltosensechangesinitsenvironmentandtomakeappropriatere-sponses.Thebasisofthismechanismistheconversionofonekindofsignalintoanother.Extracellularsignalingmoleculesaredetectedatthecellmembranebybeingboundtospecifictrans-membranereceptorsthatfaceoutwardsfromthemembraneandtriggerintracellularevents,whichmayeventuallyeffecttran-scriptionalactivitiesinthenucleus.Theeventualou