Marwan,Rohr,HeinerPetrinetsinSnoopy - PDF document

Marwan,Rohr,HeinerPetrinetsinSnoopy ways.Experimentaliststhinkinmoleculesandmolecularmechanisms,whichtheyillustratewithpictures,qualitativeschemes,andbiochemicalreactionpathways.Theoreticianscommunicatebyusingequationsandmathematicalsymbols,whicharediculttoreadforthemajorityofexperimentalists,whofrequentlydonothaveanysignicantmathematicalbackground.Atrueun-derstandingofeachotherwouldbegreatlyfacilitatedbyestablishingacommu-nicationplatform(andlanguage)whichisequallyeasytouseforbothexperi-mentalistsandtheoreticians.Suchcommonlanguageshouldbeofunequivocalexpressivenessanddirectlyrefertothetraditionallanguagesofboth.WithSnoopy,weprovideatoolthatsupportstheuseofPetrinetsasacommoncommunicationplatform(modellinglanguage)forexperimen-talistsandtheoreticians,togetherwithaunifyingframeworkforthegraphicaldisplay,computationalmodelling,simulation,andbioinformaticannotationofbiochemicalnetworks,suchasbacterialregulatorynetworks.Petrinetsasexecutablemodels.APetrinetisamathematicalgraphwithastrictlydenedsyntax.APetrinetisdirectlyexecutable,ifitisrep-resentedwiththehelpofanappropriatetoollikeSnoopy.Snoopytranslatesautomaticallyandthusreproduciblythegraphicalschemeintoasetofequationsusedbytheprogramtorunsimulations.Inotherwords,agraphicalrepresen-tationofaPetrinetdrawninSnoopycanbeexecuted,i.e.simulationscanberunwithamouseclick;nospecialadditionalencodingisrequired.Petrinetsrepresentmolecularandothermechanismswithastrictlydenedsyntax.ThesyntaxofthePetrinetlanguageissimpleandthereforeveryeasytolearn(seebelow).Becausethesyntaxisstrictlydened,thereisnoambiguityinwhatagraphicalrepresentationofthePetrinetmodelmeansintermsofreactionmechanism.ThePetrinetformalismisidealtonaturallyrepresentchemicalreactionsandtheirmechanisms,andanytypeofbiochem-icalinteractions.Asdenedbytheuser,Petrinetcomponentsmayrepresentmoleculesandreactions,orevenmorecomplexentitiesandprocesses.Thisallowstodescribeandrepresentbiologicalprocessesatarbitrarylevelofab-straction(i.e.witharbitraryresolutionindetail),butalwaysmechanisticallyunambiguous,andtojointhoseprocessesintoacoherent,executablemodel.Thisoptionisextremelyusefulforsignaltransductionnetworksorgeneregula-torynetworksasusuallynotallprocessesinvolvedinaphenomenonofinterestareandwillbeknownwiththesameresolutionindetail.Petrinetsprovideaunifyingframeworkformodellingandsim-ulationinsystemsbiology.Petrinetscanbeusedtoperformallmajormodellingandsimulationapproachescentraltosystemsbiology.Brie y,thereareseveralreasonssuggestingthedeploymentofPetrinetstoinvestigatebio-chemicalnetworks.1.Petrinets[1]enjoyanintuitivebipartitegraphicalrepresentation,whichmakesthemeasilycomprehensibleandfacilitatesthecommunicationbe-tweenwet-labexperimentalistsandcomputationaltheoreticians.HumanaPress,MethodsinMolecularBiology,Chapter21-preprint2 Marwan,Rohr,HeinerPetrinetsinSnoopy Figure2:Sometypicalbasicstructuresofbiochemicalnetworks.(a)formationanddecayofamacro-molecularcomplex;(b)reversiblereactions;(c)sequentialreactions;(d)alternativereactions;(e)concurrent,i.e.independentreactions.asiteAtoasiteB.Reversiblechemicalreactionsaremodelledbytwooppositetransitions(seeFigure2-b).Duetotheirstrongcorrespondence,weoftenusethetwoterms'transition'and'reaction'interchangeably.2.Thedirectedarcs(edges)connectalwaysnodesofdierenttype.Theygofromprecursorstoreactions,andfromreactionstoproducts.Inotherwords,thepre-placesofatransitioncorrespondtothereactionsprecur-sors,anditspost-placestothereactionsproducts.3.Arcsareweightedbynaturalnumbers.Thearcweightmaybereadasthemultiplicityofthearc,re ectingknownstoichiometriesorothersemi-quantitativeassumptions.Thearcweight1isthedefaultvalueandisusuallynotgivenexplicitly.4.Aplacecarriesanarbitraryamountoftokens,representedasblackdotsoranaturalnumber.Thenumberzeroisthedefaultvalueandusuallynotgivenexplicitly.Tokenscanbeinterpretedastheavailableamountofagivenspeciesinnumberofmoleculesoranyabstract,i.e.discretecon-centrationlevel.Thetokensonallplacesestablishtogetherthemarkingofthenet,whichrepresentsthecurrentstateofthesystem.HumanaPress,MethodsinMolecularBiology,Chapter21-preprint6 Marwan,Rohr,HeinerPetrinetsinSnoopy 1.sequentialreactionsre ectingcausality(e.g.,reactionr5cannothappenbeforer4happened);2.alternativereactionscompetingfortokensonsharedpre-placesandthere-forebranchingintoalternativebehaviour;inPetrinetterminology,thetransitionsaresaidtobeincon ict(e.g.,reactionsr6andr7sharethepre-placeL;atokenonLcaneitherbeconsumedbyr6orbyr7);3.concurrentreactions,whichareneitherincausalnorcon ictrelation;thus,theyareindependentandcanreinanyorderorevenconcurrently(e.g.,reactionsr8andr9).InadditiontoabasictoolkitfordrawingPetrinets,SnoopysupportstwodistinguishedfeaturesforthedesignandsystematicconstructionoflargerPetrinets(seeFigures1and5).1.Placesandtransitionscanbespeciedaslogicalnodes(alsocalledfusionnodes).Theyareautomaticallycolouredingrey.Logicalnodeswiththesamenameareidentical,i.e.,graphicalcopiesofasinglenode(placeortransition).Theyareoftenusedforcompoundsinvolvedinseveralreactionsorreactionsinvolvingdistributedcompounds.2.Therearetwotypesofmacronodes.Macrotransitions(drawnastwocentricsquares)helptohidetransition-borderedsubnets(i.e.subnetshavingonlytransitionsasinterfacetothesuper-net).Likewise,macroplaces(drawnastwocentriccircles)canbeusedtohideplace-boundedsubnets(i.e.subnetshavingonlyplacesasinterfacetothesuper-net).BothtypesofmacronodesallowahierarchicalstructuringofPetrinets.Logicalnodesandmacronodeshelptodealwithlargernetworks.Theymaycontributetoanet'sreadabilityand,therefore,arecrucialfornon-trivialnetexamples.However,theydonotextendtheexpressivenessofthemodellinglanguage.Contrary,thefeaturesintroducedinthenextsectiondoextendtheexpressiveness.HowtodrawanhierarchicalPetrinetwithSnoopy.TohierarchicallystructureaPetrinet,repeatthefollowingsteps.1.Drawyour atnet(orportionofit)asyouwishtohaveit.2.Selectthesub-graphwhichyouwanttobeabstractedbyamacronode,andselectHierarchy!Coarse.ChosetheappropriatecoarseelementandhittheOKbutton.3.Adoubleclickonthemacronodeopensitsattributewindowandallowstoassignasuitablename,aclickontheentrywiththisnameinthehierarchypanelontheleft-handsideopensthesub-graphinaseparatewindow.Thebluenetpartshavebeenautomaticallygeneratedandrepresenttheconnectionofthesub-graph(macronode)withtheneighbouringnodesonthenexthigherhierarchylevel.HumanaPress,MethodsinMolecularBiology,Chapter21-preprint9 Marwan,Rohr,HeinerPetrinetsinSnoopy 4.Finally,donotforgettosaveyourworkunderanewname(File!Saveas). Figure5:Theuseofmacrotransitionsandlogicalnodes.DierentgraphicalrepresentationsofoneandthesamemodelrepresentingtheenzymaticreactionA+E$AjE!B+E,whereAjEistheenzyme-substratecomplex.(a)Macrotransitionsyieldhierarchicallystructuredmodels;(b)reaction-centricrepresentationbytheuseoflogicalplaces;(c)species-centricrepresentationbytheuseoflogicaltransitions.3.2ExtendedPetrinets(xPN)ExtendedPetrinetsbuildonPetrinetsandadditionallyprovidespecialarctypes,whichgoalwaysfromaplacetoatransition,seeFigure1and6.Theycanalsocarrymultiplicities.Thetwomostpopularspecialarcsare:1.Readarcs(alsocalledtestarcs),graphicallyrepresentedbyablackdotasarchead,queryfortokensinaplacerepresentingpositiveside-conditions,e.g.,theconformationofaproteincomplexthatmaydeterminewhetherareactioncanoccur,oraspecicphysiologicalstateofacellthatmaydeterminewhetheracellisresponsivetoacertainstimulus.Thetestedplaceneedsatleastasmanytokensasgivenbythereadarc'smultiplicitytoenableatransition.Theringofthetransitiondoesnotchangethenumberoftokensonthetestedplace.Areadarcmaybesimulatedbytwooppositearcs.Althoughareadarcandtwooppositearcshavethesametotaleect,thereisasubtleseman-ticdierence,whichissometimesusefultopreciselyrepresentmolecu-larmechanismswithinabiochemicalnetwork.AssumethatanenzymeHumanaPress,MethodsinMolecularBiology,Chapter21-preprint10 Marwan,Rohr,HeinerPetrinetsinSnoopy EcatalysesthereactionofsubstrateXtoproductY.ThenEistran-sientlyconsumedbyformingtheenzyme-substratecomplexandreformedastheenzyme-productcomplexdecaystoformY.Suchanenzymaticre-actionisrepresentedinamechanism-orientedstylebytwooppositearcsifformationanddecayoftheenzyme-substratecomplexarenotexplicitlymodelled.Ontheotherhand,usingthereadarcisappropriateif,forexample,anunphosphorylatedreceptor(Y)autophosphorylates(Y-P)astheconsequenceofaconformationalchangetoR(thetokeninRrepre-sentstheactiveconformationofthereceptor).Uponautophosphorylation,theactiveconformationofthereceptorisnottransientlyconsumedandaccordingly,thetokenstaysinR(comparethetwopanelsofFigure7;seealsoNote2).2.Inhibitoryarcs,graphicallyrepresentedbyanhollowdotasarchead,indicatenegativeside-conditionsinanabstractway,e.g.,ifthepresenceofagivenprotein(inhibitorysubunitofaproteincomplex)orconditioninhibitsaspecicreaction.Theinhibitingplacemusthavelesstokensthangivenbytheinhibitoryarc'smultiplicitytoenableatransition.Theringofthetransitiondoesnotchangethenumberoftokensontheinhibitingplace.InhibitoryarcscanonlybesimulatedbystandardPetrinets,iftheinhibitingplaceisbounded(thenumberoftokensneverexceedsagivennitenumber).Sotheystrictlyincreasetheexpressiveness.Therearetwootherspecialarctypes,notusedinthischapter:equalarcsandresetarcs(formoredetails,see[11]). Figure7:Semanticdierencesbetweentwooppositearcsandareadarc.Ecatalysesthereactionr1andr2.Thus,r1andr2competeforthetokensonEandcanonlyusethemalternatively.Pstandsforanactiveconrmation,allowingr3andr4tohappenconcurrently,thussharingthetokensonP.Thetwoarctypesarenotdistinguishableininterleavingsemantics,butinpartialordersemantics(seealsoNote2).3.3AnimationofqualitativePetrinets(PN,xPN)HavingobtainedanewPetrinet,thenextstepoftenaimsatabetterunder-standingofthenetbehaviour.Aswehavealreadyseen,wecanexecutetheHumanaPress,MethodsinMolecularBiology,Chapter21-preprint12 Marwan,Rohr,HeinerPetrinetsinSnoopy Petrinetbyplayingthetokengametoexperiencethenetbehaviour.Eachexe-cutionexempliessomepossiblenetbehaviour.Theanimationcanbetriggeredmanuallybyclickingonanenabledtransition.Itcanalsobedoneinautomaticmodebyclickingonthecontrolpanel.Ineachstepoftheautomaticmodethesetofallenabledtransitionsisdetermined.Therearethreestrategiestochoosethetransition(s)toberedinthenextexecutionstepamongallenabledtransitions.1.Singlestep{onesingletransitionisrandomlychosen.2.Intermediatestep{anarbitrarysubsetofconcurrenttransitionsisrandomlychosen.3.Maximalstep{amaximalsetofconcurrenttransitionsisrandomlychosen.Stepsjustdonecanbeplayedbackwardsuptoadepthof10;thisdefaultvaluecanbechangedintheGlobalPreferencesdialogue.Byplayingthetokengame,wecanproduceandobserveanyreachablestate.Allstates,whichcanbereachedfromagivenstatebyanyringsequenceofarbitrarylength,constitutethesetofreachablestates.EachexecutionrunofanxPNcorrespondstoawalkthroughthestatespacedenedbythisxPN.Thesetofstatesreachablefromtheinitialstateissaidtobethestatespaceofagivensystem.Often,thisqualitativestatespaceisinnite,causedbyconsideringallpossiblebehaviourofastructurallyunboundedPetrinetunderanytimingconstraints.WecallaPetrinetbounded,ifthenumberoftokensonallplacesisboundedbyaconstantindependentlyofwhathappens.Thenwegetanitestatespace,whichhowevercanstillbetoohugetobeexploredexhaustively.ExecutingaPetrinetgenerallyneedstomakedecisionsbetweenalternativebehaviour.Encounteredalternatives(con icts)aretakennon-deterministically(automaticmode)oruser-guided(manualmode).Togetanexhaustivepicture,wehavetoconsiderallpossibleexecutionsequences.Obviously,that'snotpossibleforsystemswithinnitebehaviour,whichcanbecausedbycyclicbehaviourand/orinnitestatespace.Thuswehavetoconneourselvestoasubsetofexecutionsequences,whichareconsideredtoberepresentative.Inthenextsectionwewillintroducetimeandwewillseehowspecicki-neticassumptionswilltypicallyrestrictthequalitativelyinnitestatespacetoaquantitativelynitestatespace,ifweareonlyinterestedinstateswithaprobabilityaboveacertainthreshold.HowtochangethenetclassinSnoopy.A(qualitative)PetrinetcanbeconvertedintoastochasticPetrinet,whichallowstore-usethestructure.Onlytheadditionalattributeshavetobeset.1.OpenthequalitativePetrinet,andgototheexportWindow(File!Export),chosetheappropriatetarget(s)andhittheOKbutton.2.OpenthestochasticPetrinet,justgenerated.Itlooksexactlythesameasthequalitativenet.HumanaPress,MethodsinMolecularBiology,Chapter21-preprint13 Marwan,Rohr,HeinerPetrinetsinSnoopy 3.5ExtendedStochasticPetrinets(xSPN)ExtendedstochasticPetrinetscombinestochasticPetrinetswiththespecialarctypesofextendedPetrinets.Additionally,therearedeterministicallytimedtransitions,whichcomeinthree avours.1.Deterministictransitionshavecontrarytostochastictransitions{adeterministicringdelaywhichisspeciedbyanintegerconstant.Thedelayisalwaysrelativetothetimepointwherethetransitiongetsenabled.Thetransitionmayloseitsenablednesswhilewaitingforthedelaytoexpire.Then,thetransitionwillnotre.Thisbehaviourisalsoknownastheso-calledpre-emptiveringrule.Asforstochastictransitions,theringitselfofadeterministictransitiondoesnotconsumetimeandfollowsthestandardringruleofqualitativePetrinets.Deterministictransitionsmaybeusefultoreducenetworks,andthusspeed-upsimulations,e.g.byreplacingalinearsequenceofstochastictransitionsbyonedeterministictransitionwiththedelaysettotheexpectationvalueofthesumofthesequence.2.Immediatetransitionsareapopularspecialcaseofdeterministictran-sitions.Theyhaveazerodelayandalwayshighestpriority.Thelattercreatesasubtledierencebetweenanimmediatetransitionandadeter-ministictransitionwithzeroringdelay:ifthereisacon ictbetweenthetwo,.i.e.asituation,wheretwotransitionscompetefortokens,theimmediatetransitiongetspriority.Immediatetransitionsmayhelptoavoidstisystemsbyusingthemfortransitionswithextremelyhighrates(non-signicantdelay),comparedtotheothertransitionsinthesystem.3.Scheduledtransitionsareanotherspecialcaseofdeterministictransi-tions.Thedeterministicringoccursaccordingtoaschedulespecifyingabsolutepointsofthesimulationtime.Aschedulecanspecifyjustasingletimepoint,orequidistanttimepointswithinagiveninterval,trig-geringtheringonceorperiodically.However,transitionsonlyreattheirscheduledtimepointsiftheyareenabledatthistime.Scheduledtransitionscandramaticallyrestrictthe(qualitative)netbehaviour.Thecrucialpointisthattheyallowtodisturbthecoremodelatwell-denedtimepointsasitisdoneexperimentallywiththeactualbiologicalsystemunderinvestigationinthewet-lab.AnunrestricteduseofdeterministictransitionsdestroytheMarkovprop-erty,whichprecludesananalyticalevaluationbyconstructingandanalysingtheCTMC.IfweconsiderstochasticPetrinetswithoutdeterministictransitions,theprobabilityoftwotransitionstoreatthesametimeispracticallyzero.Contrary,instochasticPetrinetswithdeterministictransitions,itispossiblethattwotransitionswanttoresimultaneously.Toanalysesuchasystem,allpossiblechoiceshavetobeconsidered,whileinthesimulationarandomchoicetakesplace,seenextsection(formoredetailsandrelatedexamplessee[12]).HumanaPress,MethodsinMolecularBiology,Chapter21-preprint16 Marwan,Rohr,HeinerPetrinetsinSnoopy Inmostpracticalcases,extendedstochasticPetrinetsaretheobviouschoice.Therefore,SnoopydoesnotdistinguishbetweenstochasticPetrinets(SPN)andextendedstochasticPetrinets(xSPN).3.6SimulationofquantitativePetrinets(SPN,xSPN)ThesimulationofstochasticPetrinetscanbereadasatimedversionofthequal-itativetokengame,takingintoaccountthestochasticanddeterministicdelaysofenabledtransitions.SnoopybuildsupontheGillespiealgorithm[13].Thisexactmethoddoesastep-by-stepsimulationofpossiblestatesofthestochasticPetrinet.Consequently,theresultrepresentsalwaysavalidstateoftheun-derlyingstochasticprocessatanytimepointofthesimulation.Thestandardstochasticsimulationalgorithmworksinthefollowingway:1.InitialisetheSPNnetworkwiththechoseninitialmarking.2.Calculatetheringratesofallenabledtransitionsusingtheirratefunc-tions.3.Calculatethecombinedratebysummingupalltransitionrates.4.Determinethetimeintervaluntilthenextstatechangetakesplace.Thisisdonebycomputinganexponentiallydistributedrandomnumberde-pendingonthecurrentcombinedtransitionrateofthenet.5.Increasethesimulationtimebythistimeinterval.6.Determinethenextsystemstatechange.Forthispurpose,aweightedrandomselectionofthetransitionismadewhichgetsthelicensetore.7.LettheselectedtransitionreandupdatethemarkingoftheSPN.8.Gobacktostep2,ifthesimulationtimehasnotyetreacheditsendpoint.ThesimulationofxSPNrequiressomestraightforwardmodicationsofthestandardsimulationalgorithm.Deterministictransitionsrequireahigherpri-oritythanstochastictransitionstoensuretheircorrecthandling.Immediatetransitionsneedtohavethehighestpriority(overdeterministicandscheduledtransitions)becausetheyhavetoreinstantaneouslywhentheygetenabled.Consequently,thesimulationalgorithmhastocheckforenabledimmediatetransitionsafteranychangeofthemarking.Inthecaseofmorethanoneenabledimmediatetransitions,theselectionisdonerandomly,butuniformlydistributed.Thesystemmayrunintoatimedeadlock,ifalwaysanimmediatetransitionisenabled.Then,timewillnotprogressanymore.Timedeadlocksareanindicationofinconsistenttimeassignments.Theuserhastotakecareofavoidingthem.Deterministicandscheduledtransitionsaretreateddierently.Ifadeter-ministictransitiongetsenabled,itstimerstartsrunninguntilitexpires.IfHumanaPress,MethodsinMolecularBiology,Chapter21-preprint17 Marwan,Rohr,HeinerPetrinetsinSnoopy Dierenttablesandplotscanbecreatedtoswitchconvenientlybetweendierentviewsonthesimulationresults.Eachtableischaracterisedbyasetofselectedplaces(transitions).Ifanodewascoloured,itscorrespondingcurvegetsthesamecolourintheplot.Simulationplotscanalsobesavedinepsformat.4Casestudy4.1Acasestudy:thephosphateregulationnetworkinentericbacteriaLetusnowconsiderthephosphateutilisationgeneregulatorynetworkinEs-cherichiacoli(seeFigure9discussedin[14]).Notethatthiscasestudyisneithermeantasascienticcontributiontotheunderstandingofthegeneregulatorynetworkofphosphateutilisationnordoesitprovideanycomprehensiverepre-sentationoftheknowledgeonthissubject;forarecentsurveysee[15].Instead,thecasestudyisintendedtoshowhowatypicalbiochemicalmodel,agene Figure9:Biochemicalmodelofthephosphateregulatorynetworkinentericbacteria.TheschemeisadaptedfromNeidhardtetal.(1990).HumanaPress,MethodsinMolecularBiology,Chapter21-preprint19 Marwan,Rohr,HeinerPetrinetsinSnoopy regulatorynetwork,maybeformallytranslatedintoaPetrinet.Aswewillshow,thePetrinetcanserveasaqualitativeschemerepresentingthemolecularreactionmechanisms,butitcanalsobeusedfordynamicsimulations.Fordierentreasons,inorganicphosphatemaybecomeagrowth-limitingfactorforabacterialcellpopulation.Undertheseconditionscellssynthesisealkalinephosphatase(PhoA),anenzymewhichissecretedintotheperiplasm(i.e.intothespacebetweencytoplasmicmembraneandoutermembraneofaGram-negativebacterium),whereitdegradesorganicphosphateestersintoinorganicphosphatetobetakenupandrecycledbythecell.Transportofin-organicphosphateismediatedbyanuptakesystemcomposedoffourproteins,PstS,PstC,PstAandPstBthatformatransmembraneproteincomplex.Ex-perimentalevidencesuggeststhatphosphatetransportissensedbythePhoUprotein.Ifthephosphatetransportsystemisactive,thePhoUproteinresidesinitsinactivestate.Phosphatelimitationrendersthetransportsysteminactive,whichactivatesPhoUandcausesphosphorylationofthePhoRproteinwhichinturnphosphorylatesPhoB.PhosphorylatedPhoB(PhoB-P)isapositivereg-ulatorwhichbindstothepromotorregionofcertainoperons.UponPhoB-Pbinding,amongothers,thephoAgeneisexpressedandalkalinephosphatase(PhoAprotein)issynthesisedandexportedintotheperiplasm.WhileactivePhoUcausesthephosphorylationofPhoR,inactivePhoUisthoughttoactasaPhoB-PphosphatasewhichswitchesotheDNAbindingactivityofthePhoBprotein.AsummaryofthemolecularcomponentsinvolvedintheregulationofthephosphateutilisationnetworkisshowninTable1.Table1:Molecularcomponentsinvolvedinphosphateregulation. AbbreviationMolecularComponent PhoAAlkalinephosphataseenzymedegradingorganicphos-phatecompoundstoinorganicphosphatePiInorganicphosphatePoOrganicphosphatePstSCABTransmembraneproteincomplex,transporterofinor-ganicphosphatePhoUSignaltransducerrelayingthePstSCABcomplexactiv-ityPhoRPhosphorylatableregulatoryproteinPhoBPhosphorylatableregulatoryprotein 4.2ApplicationofxSPNtostudycase:thephosphatenetworkinentericbacteriaQualitativemodelling(xPN).TheregulatorymechanismillustratedinFig-ure9isimplementedinthePetrinetshowninFigure10.Themolecularcompo-HumanaPress,MethodsinMolecularBiology,Chapter21-preprint20 Marwan,Rohr,HeinerPetrinetsinSnoopy nentsintheirdierentstates(e.g.active,inactive,phosphorylated,unphospho-rylated)arerepresentedasplaces(Table2)andtheirbiochemicalreactionsorfunctionalinteractionsarerepresentedasstochastictransitions(Table3).Addi-tionally,therearedeterministicallytimedtransitions(immediateandscheduledtransitions)tomodeltheexperimentaladditionofinorganic/organicphosphate.Therearedierentarctypes.Standardarcsrepresentthemass ow.Readarcsandinhibitoryarcsareusedtomodelregulatoryinteractionsbetweenproteinsthroughphysicalprotein-proteininteraction.Doublearcs(asort-handnotationfortwooppositearcs)representcatalyticreactionswhichmaybecomplex.Thegraphicalrepresentationofthecytoplasmicmembraneandofsomepro-teinsofinterestisputunderneaththePetrinetinordertoexplainthemodularstructureofthenetandtohighlightwhichreactionsoccurinsideoroutsidethecell,respectively.NotethatthesegraphicsarenotafunctionalelementofthePetrinetasgeneratedinSnoopy.ThePstSCABtransmembraneproteincom-plexhasadualfunctionastransporterandasasensorofinorganicphosphate.Wheninorganicphosphateispresentintheperiplasm,thePstSCABcomplexisphosphorylatedthroughreactionr7.Thephosphorylatedform,PstSCAB-Pisactiveintransportinginorganicphosphatethroughthecytoplasmicmem-braneintothecytoplasm(r5)whereitisusedforbiosyntheticreactions(r6).ThePstSCABproteinallostericallycontrolstheactivityofthePhoUprotein,modelledbyreadarcsthatcontrolthetransitionsr9andr10representingtheTable2:Petrinetplacesrepresentmolecularcomponentsintheirdierentstates. PlaceMolecularcomponent Pi PeriPlasmInorganicphosphateintheperiplasmPi CytoplasmInorganicphosphateinthecytoplasmPo PeriPlasmOrganicphosphateintheperiplasmPhoA PeriplasmPhoAproteinintheperiplasmPhoAPhoAproteininthecytoplasmPhoAmRNAPhoAgenemRNAPstSCABTransporterofinorganicphosphate,inactiveformPstSCAB-PTransporterofinorganicphosphate,phosphorylated,ac-tiveformPhoU inactiveSignaltransducer,inactiveformPhoU activeSignaltransducer,activeformPhoRPhoRprotein,dephosphorylatedformPhoR-PPhoRprotein,phosphorylatedformPhoBPhoBprotein,dephosphorylatedformPhoB-PPhoBprotein,phosphorylatedformswitch on,switch oAuxiliaryplacestomodeltheexperiments HumanaPress,MethodsinMolecularBiology,Chapter21-preprint21 Marwan,Rohr,HeinerPetrinetsinSnoopy degradesorganicphosphatetosupplythecellwithinorganicphosphatewhichinturnswitchesothesignalingcascadeandsubsequentlythebiosynthesisofthePhoAprotein,aslongassucientinorganicphosphateentersthecell.WehavealreadyseenhowtodrawasimplePetrinet,soyouwillnothaveanytroubleindrawingthePetrinetforyourrstcasestudy.Coarsetransitionsandcoarseplacesmaybeusedtostructurelargenetworksintomodulesinordertoimprovethereadabilityofthenet.SincethePetrinetpresentedforthecasestudyisrathersmall,wedidnotusecoarsenodeshere.AnimationofthexPN.Toexplorethenetbehaviour,dothefollowingsteps:1.loadthenet(File!Open),2.starttheanimation(View!StartAnimationMode),3.runthenetinmanualorautomaticmode.Specicallyyoushouldcheckthefollowingscenarios:1.Whichreactionsarenecessaryinwhich(partial)ordertorethetransi-tionsBiosynthesis(r6),Decay(r15),orDenaturation&Decay(r18)?2.Whichnetbehaviourispossibleiftheinorganicsupplyisswitchedon(placeswitch onismarked),whichnetbehaviouristriggerediftheinor-ganicsupplyisswitchedo?3.Trytogureoutthemaximaltokennumbersyoucangetoneachplace!4.Isitpossibletoreachamarkingwherenoneofthetransitionsisenabled?5.Havingplayedwiththenetforawhile,isitalwayspossibletocomebacktothegiveninitialmarking?Quantitativemodelling(xSPN).Allstochasticreactionsfollowmass-actionkineticswithallparametersinitiallysetto0.1.Afterwards,theparame-terswereadjustedsothatthesteadystateconcentrationofinorganicphosphateintheperiplasmisapproximatelythesamenomatterwhethertheexternalsourceisinorganicororganicphosphate,respectively:theparameterofr4issetto0.2,andtheparameterofr15to0.075.Thetransitiont oisscheduledtoreattimepoint100,andthetransitiont onisscheduledtoreattimepoint1000.StochasticanimationofthexSPN.Followthestochastictoken owintheautomaticanimationmode.Whichreactionsequencesdoyouobservewhiletheinorganicsupplyisswitchedon?Whatisthedierencetothereactionandstatesequences,whichyouhaveobservedintheanimationofthecorrespondingxPN?Theanswerliesintheimmediatetransitions(herer1),whichhavenowalwayshighestpriority.HumanaPress,MethodsinMolecularBiology,Chapter21-preprint23 Marwan,Rohr,HeinerPetrinetsinSnoopy Figure10:Petrinetmodelofthephosphateregulatorynetwork,alongwithaschematicrepresentationofthemembraneandofsomerelevantproteins.Biomolecularcomponentsandtheirfunctionalstatesarerepresentedasplaces,(bio-)chemicalreactionsarerepresentedastransitions.Stoichiometricreactionsarerepresentedasstandardarcs,allostericinteractionsofproteinsbyreadarcsandinhibitoryarcs.Organicandinorganicphosphateissuppliedbytheenvironmentofthecellatconstantconcentration.Inthesimulations,theconstantconcentrationofthesephosphatecompoundsisobtainedbyusingtransitionswithimme-diateringbehaviour(r1,r3)deliveringtokenstothecorrespondingplaces(Pi PeriPlasm;Po PeriPlasm).Theringofr1andr3iscontrolledbyinhibitoryarcs.Theseinhibitoryarcsshutthetransitionsoifthepre-denednumberoftokensisinthepre-placeoftherespectivetransition.Experimentaladditionandremovalofinorganicphosphatetothecellismodelledwiththehelpoftheswitch onplacewhichisconnectedbyareadarctor1.Thereforer1deliversinorganicphosphateaslongas(1)thereisatokenintheswitch onplaceand(2)thenumberoftokensinthePi PeriPlasmplaceislessthanten.Thesupplywithinorganicphosphatemaybeswitchedoandonatdenedtimepointsbyswitchingthetokenintheswitch onplacewiththedeterministicallytimed(scheduled)transitionst onandt o.Inthesimulations,organicphosphateintheperiplasmremainsconstantatalltimes.ThePetrinetrepresentstheprocessesinvolvedinphosphateregulationwithdierentresolutionofde-tails.Onlypartofthenetrepresentsmolecularprocessesintermsofinteractionsofindividualmolecules.Complexreactionmechanisms,likebiosynthesesusinginorganicphosphate(r6),transcriptionofthephoAgene(r14)ortranslationofthephoAmRNA(r16)arerepresentedasindividualtransitionsthatcondensemultipleindividualstepsintoonesinglereaction,as-sumingrstorderrateconstantsintheexampleprovided.Notehowever,thatthequantitativebehaviourofatransitioninthexSPNversionofSnoopycanbeprogrammedindividuallysothatevencomplexandnon-linearkineticmechanismscanbemodelled.HumanaPress,MethodsinMolecularBiology,Chapter21-preprint24 Marwan,Rohr,HeinerPetrinetsinSnoopy StochasticsimulationofthexSPN.Inthestochasticsimulation,there-actionofeachindividualmoleculeisconsideredintheformofindividualtokensthatmovethroughthenet.Whenthenumberofmoleculesofeachbiochem-icalcomponentpercellisknown,onecanobtainrealistictracesofhowthenumberofmoleculesdevelopsovertime.Theresultmaybeatime-dependentchangeintheconcentrationofthecompoundifthenumberofmoleculespercellissucientlyhighorthenumberofmoleculesmaybesubjectedtostochastic uctuationsovertimeifitislow.Whenastochasticsimulationisrunmanytimesandthesimulationresultsareaveraged,onecanapproachtheresultofadeterministicsimulationasobtained,forexample,bysolvingasetofordinarydierentialequations.Eachwayofsimulation,stochasticasperformedhere,ordeterministic,e.g.bysolvingordinarydierentialequations,maybeofpartic-ularadvantagedependentonthequestionunderconsideration.Forstudyingandsimulatingthebehaviourofindividualcells,stochasticsimulationsmaybeessential.Inordertoprovideanexampleofasimulation,wehaverunthemodel50,000timestoshowhowthephosphateregulationnetworkmaybehavedynamically.Theconcentrationofthesources,inorganicandorganic,wereassumedtobeconstantovertime.Thesystemwasrstallowedtoequilibrateinthepresenceofconstantexternalinorganicphosphate(Figure11).Afterthesteadystatewasreached,externalinorganicphosphatewasremoved(switchedo)byastep-downtozeroconcentrationandthesystemwasallowedtoapproachthenewsteadystate.Theoscillationsintheconcentrationofthecomponentsobtainedinthesimulationareduetothefeed-backloopsinthesystem.Notethattheprecisedynamicbehaviourofthesystemdependsontheratiooftherateconstantswhichwedonotknow.Finally,theexternalsourceofinorganicphosphateisswitchedonagainandthesystemequilibratesintoitspre-stimulusstate.TheresultsofthestochasticsimulationareshownasapanelobtainedbyimportingthesimulationresultsintotheplotprogramKaleidagraph(Figure11)andasscreenshotsofthesimulationresultswindowofSnoopy(Figure12-13).Youcancontinuebytryingthefollowingsimulationscenarios:1.Whichratesin uenceamplitudes,frequenciesanddampingoftheoscilla-tionscausedbyfeedbackregulation?2.Checkhowthedynamicsystembehaviourisin uencedbythedecayratesofmRNAandPhoAprotein.HumanaPress,MethodsinMolecularBiology,Chapter21-preprint25 Marwan,Rohr,HeinerPetrinetsinSnoopy Figure11:Responseofthephosphateregulatorynetworktostep-wisechangesinthesupplyofexternalinorganicphosphate.Traceswereobtainedbyaverag-ing50,000simulationruns.Therateconstantsofthestochastictransitionsare:r40.2,r150.075,else0.1.InitialmarkingasgiveninFigure9.SimulationresultswereexportedfromSnoopyascsvleandimportedintoKaleidagraph,seehttp://www.kaleidagraph.com/,fordisplay.HumanaPress,MethodsinMolecularBiology,Chapter21-preprint26 Marwan,Rohr,HeinerPetrinetsinSnoopy Figure12:ScreenshotofthesimulationdialoguewindowofSnoopydisplay-ingsimulationresultsshowninFigure10.Simulationrunparameters:Intervalstart:0;Intervalend:1500;Outputstepcount:500;Simulationruncount:50,000.Snoopyisamodellingandsimulationtool.Petrinetscanbegraphi-callyedited,parameterlistseditedandsimulationsrunandresultsgraphicallydisplayedwithoutleavingtheprogram. Figure13:Simulationdialogueshowingtransitionringtimes(amountofringinthelastgridinterval).HumanaPress,MethodsinMolecularBiology,Chapter21-preprint27 Marwan,Rohr,HeinerPetrinetsinSnoopy ousinthegivencase:(switch on,switch o),(PstSCAB,PstSCAB-P),(PhoU inactive,PhoU active),(PhoR,PhoR-P),and(PhoB,PhoB-P).Thefollowingveplacesarenotcoveredbyplaceinvariants:Pi PeriPlasm,Po PeriPlasm,Pi CytoPlasm,PhoA Periplasm,PhoA,phoAmRNA;theirmaximaltokennumbersaredeterminedbythetimingconstraints.4.Anestablishedanalysistechniqueforspecialbehaviouralpropertiesismodelchecking.ItchecksthePetrinetbehaviouragainstpropertiesformallyspeciedintemporallogics,see(chapterBattetal.,thisvol-ume).Charlieprovidessomestandardexplicitmodelcheckers,basicallyforteachingpurposes.SnoopyallowsalsotoexportthedesignedPetrinetstoquiteanumberofmodelcheckers,amongthemthesymbolicmodelcheckersBDD-CTL,IDD-CTL[20]andIDD-CSL[21].SimulationtracesgeneratedbySnoopy's(stochasticorcontinuous)simulationenginescanbecheckedagaintsPLTLcpropertieswiththeMonteCarloModelCheckerMC2[22].See[4,6]forcasestudiesdemonstratingasystematicandseam-lessmodelcheckingapproachinthequalitative,stochasticandcontinuousmodellingparadigms.5.AgivenPetrinetmayalsobereadasacontinuousPetrinet,ifitisamenabletocontinuizationandthepopulationsemanticsallowstocon-siderjusttheaveragedcase.ContinuousPetrinetsdeneuniquelyasys-temofordinarydierentialequations(ODEs)[4],butnotviceversa.Thepaper[23]demonstratesthestructureddesignofODEsbythestep-wisecompositionofhierarchicallystructuredcontinuousPetrinets.AfamilyofrelatedPetrinetmodelsbuildsthecoreofanintegrativeapproachcom-prisingqualitative,stochasticandcontinuousPetrinets,whichisdemon-stratedbyarunningexampleeachin[4,6].Italsoprovidesanadequateframeworktoreasonaboutthebehaviouralrelationofmodels,sharingstructure,buthavingdierentkinetics;see,e.g.,[24].Forageneralout-lineofBioModelEngineeringsee[25].AcknowledgementsChristianRohrisfundedbytheInternationalMaxPlanckResearchSchoolforAnalysis,DesignandOptimizationinChemicalandBiochemicalProcessEngineeringMagdeburg.References1.Petri,C.A.,Reisig,W.(2008)Petrinet.Scholarpedia3(4):6477http://www.scholarpedia.org/article/Petri_net.2.Marwan,W.,Wagler,A.,andWeismantel,R.(2009)Petrinetsasaframe-workforthereconstructionandanalysisofsignaltransductionpathwaysandregulatorynetworks.J.Nat.Comput.,inpress[DOI10.1007/s11047-009-9152-x].HumanaPress,MethodsinMolecularBiology,Chapter21-preprint29 Marwan,Rohr,HeinerPetrinetsinSnoopy 3.Baldan,P.,Cocco,N.,Marin,A.andSimeoni,M.(2010)Petrinetsformodellingmetabolicpathways:asurvey.J.Nat.Comput.,inpress[DOI10.1007/s11047-010-9180-6].4.Heiner,M.,Gilbert,D.,andDonaldson,R.(2008)Petrinetsinsystemsandsyntheticbiology.Lect.NotesComput.Sci.5016:215-264.5.Rohr,C.,Marwan,W.,andHeiner,M.(2010)Snoopy-aunifyingPetrinetframeworktoinvestigatebiomolecularnetworks;Bioinformatics26,7:974-975.6.Heiner,M.,Donaldson,R.,andGilbert,D.(2010)PetriNetsforSystemsBiology.InMSIyengar(ed.):SymbolicSystemsBiology:TheoryandMethods,Chapter3,Jones&BartlettPublishers,LLC.7.PetriNetsWorld:OnlineServicesfortheInternationalPetriNetsCom-munity,http://www.informatik.uni-hamburg.de/TGI/PetriNets/.8.SBMLSoftwareSummary:http://sbml.org/SBML_Software_Guide/SBML_Software_Summary.9.BitesizeBioFreeOnlineBioinformaticstools,http://bitesizebio.com/2009/01/06/free-online-bioinformatics-tools/.10.SystemsBiologySoftware,http://systems-biology.org/software/.11.Tovchigrechko,A.(2008)EcientsymbolicanalysisofboundedPetrinetsusingIntervalDecisionDiagrams;Ph.D.Thesis,BTUCottbus,Germany.12.Heiner,M.,Lehrack,S.,Gilbert,D.,andMarwan,W.(2009)ExtendedStochasticPetriNetsforModel-BasedDesignofWet-labExperiments.Lect.NotesBioinformatics5750:138-163.13.Gillespie,D.T.(1977)Exactstochasticsimulationofcoupledchemicalreactions;J/Phys.Chem.81(25):2340-2361.14.Neidhardt,F.C.,Ingraham,J.L.etal.(1990).PhysiologyoftheBac-terialCell-AMolecularApproach.Sunderland,Massachusetts,SinauerAssociates.p.370.15.Yi-JuHsieh,Y.-J.andWanner,B.L.(2010)Globalregulationbytheseven-componentPisignalingsystem.Curr.Opin.Microbiol.13:198-203.16.Franzke,A.(2009)Charlie2.0-amulti-threadedPetrinetanalyser.DiplomaThesis,BTUCottbus,Germany.17.Heiner,M.,andKoch,I.(2004)PetriNetBasedSystemValidationinSystemsBiology.Lect.NotesComput.Sci.3099:216-237.HumanaPress,MethodsinMolecularBiology,Chapter21-preprint30