Therefore it is highly desirable to be able to represent them in modelling languages for biology BioPEPA is a language for the modelling and analysis of biochemical networks in its present version compartments can be de64257ned but they are only use ID: 69368
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CiocchettaandGuerriero additiontoallowingmoleculestopassacrossmembranes,membraneproteinsarealsoim-portantforthetransmissionofsignalsbetweencompartments.Indeed,signallingpathwaysinvolvespecialmembrane-boundproteins,calledreceptors,whichrespondtotheinputofsignallingmoleculesononesideofthemembranebytriggeringacascadeofeventsontheotherside.Manydi erentmoleculesresideonmembranes.Thetwomaintypesofmembranepro-teinsareintegralandperipheralproteins.Integralproteinsarealwaysattachedtomem-branes;theycaneitherspantheentiremembrane(transmembraneproteins)orbeattachedtoonesideofthemembrane(monotopicproteins).Theycanbindandinteractwithclosemolecules.Peripheralproteinsaretemporarilyattachedtomembranes:theycanbindandinteractwithclosemolecules,andalsodetachfromthemembranes.Non-membranepro-teinsarefreetomovewithinthecompartmentvolume:theycanbindandinteractwithclosemoleculesand,insomecases,passacrossmembranes.Bio-PEPAisalanguagedenedrecentlyforthemodellingandanalysisofbiochemicalnetworks[ 8 , 7 ].Abiochemicalnetworkiscomposedofasetofmolecularspecies,suchasproteins,smallmolecules,andgenes,thatinteractwitheachotherthroughsomereactions.Themolecularspeciesarelocatedincompartments,suchasthenucleusandthecytosol,oronthemembraneswhichenclosethem.Bio-PEPAsupportsthedenitionofstaticcom-partmentsasnames:compartmentsarecontainersforthemolecularspeciesandarenotinvolvedinanyreactionswhichchangetheirsizeorstructure.Aconstantvolume(orsize)canbeassociatedwiththem,andthisinformationisjustusedinthederivationoftheratesforstochasticsimulation[ 10 ]fromthefunctionalratesgivenintheBio-PEPAsystem.Fur-thermore,forthederivationofthetransitionratesandthestepsizeforthetransitionsystemitisimplicitlyassumedthateitherallthespeciesareinthesamecompartmentorallthecompartmentshavethesamesize.Theapproachisnotappropriateinthegeneralcaseofmultiplecompartmentswithdi erentsizes,aswewilldiscussinSections 3.2 and 3.3 .TheaimofthisworkistoinvestigatetheuseofBio-PEPAforrepresentingmulti-compartmentmodelsandextendthelanguagewithnotionstoexpressmoredetailsaboutlocationsofspeciesandreactions.Weintroducethegenericnotionoflocations,whichindicatebothmembranesandcompartments,andwedeneacompactnotationfortherep-resentationofspecieswhichcanbeindi erentcompartments.Thestructureofthesystemisdescribedintermsofahierarchyoflocations,thatalsoallowsustodenerelationswhichclassifyreactionsbasedontheirlocation.Thepossiblekindsofanalysisofmulti-compartmentsystemsarediscussedand,inparticular,thederivationofthetransitionratesfortheCTMCisshown.Herewefocusonstaticlocations(i.e.compartmentscannotmerge,split,orundergoanystructuralchange)whosesizecanvarywithrespecttotime.Thisassumptionismoti-vatedbythefactthatthesekindsofcompartmentaretheonesconsideredinmodelspresentindatabasesandintheliterature(seeforinstancemodelsintheBioModelsdatabase[ 13 ]).Staticlocationscanbedescribedatdi erentlevelsofdetail.Inthemodelsintheliteratureorindatabasessomesimplicationsareoftenmade.Forinstance,receptorsareassumedtobeinaspeciccompartment(insteadofbeingonamembrane),volumesandmembranesarenotconsideredexplicitly.Ontheotherhand,inthemodelsderivedfromexperimentaldata,moredetailscouldbegiven:membranescanbeconsideredexplicitlyandcompart-mentshavedi erentsizes.Wekeepthelanguageexibleinordertoallowthespecicationofbothlevelsofdetail. 2 CiocchettaandGuerriero Thestructureofthepaperisasfollows.Section 2 reportsabriefdescriptionofBio-PEPA,whileSection 3 isdevotedtothedenitionofourextension.InSection 4 wepresentthepossiblekindsofanalysis.Asanexample,amodelfordescribingintracellularCa2+oscillationsispresentedinSection 5 .Section 6 isanoverviewoftherelatedworks,whereasthelastsectionreportssomeconclusiveremarks.2Background:Bio-PEPAInthefollowingwepresentabriefdescriptionofBio-PEPA[ 8 , 7 ].ThesyntaxofBio-PEPAisdenedas:S::=( ;)opSjS+SjCP::=PBCIPjS(x)whereop=#j"jj j.ThecomponentS(speciescomponent)abstractsamolecularspeciesandthecompo-nentP(modelcomponent)describesthesystemandtheinteractionsamongcomponents.Theprexterm( ;)opScontainsinformationabouttheroleofthespeciesinthereactionassociatedwiththeactiontype :isthestoichiometrycoecientofthespeciesandtheprexcombinator“op”representsitsroleinthereaction.Specically,#indicatesareac-tant,"aproduct,anactivator, aninhibitor,andagenericmodier.Theoperator“+”expressesachoicebetweenpossibleactionsandtheconstantCisdenedbyanequationCdef=S.Theparameterx2R+inS(x)representsthe(current)concentration.Finally,theprocessPBCIQdenotesthecooperationbetweencomponents:thesetIdeterminesthoseactivitiesonwhichtheoperandsareforcedtosynchronize.InBio-PEPA,reactionratesarenotexpressedinthesyntaxofcomponentsbutaredenedasfunctionalratesassociatedwithactions,whichallowustoexpressanykindofkineticlaw.ApossiblemodellingstylesupportedbyBio-PEPAisintermsofconcentrationlevels.Theamountofeachmolecularspeciescanbediscretisedintoanumberoflevels,fromlevel0(i.e.speciesnotpresent)toamaximumlevelN.ThelevelNdependsonthemaximumconcentrationofthespecies.EachlevelrepresentsanintervalofconcentrationandthegranularityofthesystemisexpressedintermsofthestepsizeH,i.e.thelengthoftheconcentrationinterval.TheinformationaboutthestepsizesandthenumberoflevelsforeachspeciesiscollectedinasetN.Theviewintermsoflevelsisconsideredforsomekindsofanalysis(seebelow)andthesetNisoptional.TheBio-PEPAsystemisdenedinthefollowingway: Denition2.1 ABio-PEPAsystemPisa6-tuplehV;N;K;FR;Components;Pi,where:Visthesetofcompartments,Nisthesetofquantitiesdescribingspecies,Kisthesetofparameterdenitions,FRisthesetoffunctionalrates,Componentsisthesetofdenitionsofspeciescomponents,Pisthemodelcomponentdescribingthesystem.Thebehaviourofthesystemisdenedintermsofanoperationalsemantics,whichreferstothelevel-basedmodellingstyle.TherulesarereportedintheAppendix.Twore-lationsaredened.Therstone,calledcapabilityrelation,isindicatedby�!candischar-acterisedbythelabel.Thislabelisoftheform( ;w),wherew:=[S:op(l;)]jw::w,withSaspeciescomponent,lthelevelandthestoichiometrycoecient.Thesecondrelation,calledstochasticrelation,is�!s.Ithasalabel ,denedas :=( ;r ),wherer 2R+istherateassociatedwiththeaction.Theratesareobtainedfromthefunctionalrates,rescaledaccordingtothestepsizeofthereactants.Inthisdenition,r represents 3 CiocchettaandGuerriero (a)Compartmentshierarchy (b)LT (c)LTwithoutmembranesFig.1.Locationtreewithexplicit(b)andimplicit(c)membranesforthehierarchyshownin(a)(membranenamesaredenotedbylowercasecharactersforthesakeofreadability). Example3.2 ThecompartmenthierarchyrepresentedinFigure 1 (a)canbemodelledinBio-PEPAbythefollowinglistoflocations:L=[S:volS;C;A:volA;C;B:volB;C;C:volC;C;D:volD;C;e:vole;M;f:volf;M;g:volg;M;h:volh;M;i:voli;M].ThelocationtreeassociatedwiththiscompartmenthierarchyisrepresentedinFig-ure 1 (b),whileFigure 1 (c)referstothesamemodelwithnoexplicitdenitionofmem-branes.3.2SpeciesThedenitionofspeciescomponentscontainssomedetailsabouttheirlocalisation.Wedenethelocationofthespeciesconsideringtheelementspecies location:species location::=LjL1=L2whereL;L1;L22Landkind(L1),kind(L2).Thisdenitionallowsustospecifyifaspeciesisinsideacompartment(species location=L,withkind(L)=C),isacrossamembrane,suchastransmembraneproteins(species location=L,withkind(L)=M)oritisontheborderbetweenacompartmentandanadjacentmembrane(species location=L1=L2,withkind(L1)=Candkind(L2)=M).Inordertomakethelocationofspeciesexplicit,thelocationnameisaddedtoeachspeciescomponentwiththenotationS@L,indicatingthatthespeciesrepresentedbytheprocessSisinthelocationL.GiventhelocationsL1;L2,thecomponentsS@L1andS@L2representthespeciesSinthetwolocations.Byusingthisapproach,reactionsinvolvingspecieslocatedindi erentcompartmentscanbemodelledanalogouslytostandardreac-tions.AtransportofamoleculeSfromL1toL2,forinstance,issimplyareactioninwhichS@L1isareactantandS@L2isaproduct.Representingthesamemoleculeindi erentcompartmentsasdi erentspeciescompo-nentsseemsareasonablechoicefromabiologicalpointofview,sincethesamemoleculeisgenerallyinvolvedindi erentreactionsaccordingtothecompartmentinwhichitislocated.Inordertoavoidthepossibleduplicationofactionsinthemodel(e.g.analo-gousreactionsinvolvingthesamemoleculesoccurringindi erentcompartmentshavetobeduplicatedforeachspecies),weproposeanotationtorepresentaspeciesinmultiple 5 CiocchettaandGuerriero 3.3SemanticsBio-PEPAisgivenanoperationalsemanticsintermsofconcentrationlevels.Thesameapproachisusedinthisextension.Twomainchangesarerequired: thetransitionratesfortheSLTSmusttakeintoaccountthelocationofthespecies; theinformationaboutthelocationisaddedtothetransitionlabels,andisusedtorecordthelocationofthespeciesinvolvedinthereaction(usingthecapabilityrelation)andtoderivethelocationofthereactionitself(usingthestochasticrelation).Inthiscontextwelimitourattentiontocompartmentswithxedsize.Indeed,thepossibilitytoconsidercompartmentswhosesizechangeswithrespecttotimeisconsideredintheotherkindsofanalysis,asdiscussedinSection 4 .3.3.1CapabilityrelationThelabelofthecapabilityrelationisdenedas( ;w),wherewnowcontainsfurtherdetailsconcerningthelocation:w::=[S:op(l;;species location)]jw::wwhereS2C,listhelevel,isthestoichiometrycoecientandspecies locationisthelocationofS.Allthelabelsintherulesaremodiedaccordingtothisnewdenition.3.3.2StochasticrelationThelabel ofthestochasticrelationisredenedas( ;r;reaction location),where istheactiontype,ristherateandreaction locationexpressesthelocationwherethereactionassociatedwith occurs.Thedenitionofreaction locationisderivedfromthelistwofthecapabilityrelation.Thefollowingsetsoflocationnamesaredened: LR::=fLi2Lj(S:op(;l;Li))2w^op=#g LP::=fLi2Lj(S:op(;l;Li))2w^op="g LM::=fLi2Lj(S:op(;l;Li))2w^op2f; ;gg LRM::=LR[LMandLPM::=LP[LM.IfLRM=LPM=fLgthenreaction location::=Lelsereaction location::=fLi2LRMg=)fLj2LPMg.Ifallthereagentsofagivenreactionareinasinglelocationthenthereactionoc-cursinthatlocation.Otherwise,ifreagentsarenotinthesamelocation,thenotationfLi2LRMg=)fLj2LPMgallowsustocollecttheinformationaboutthelocationofthere-actants/modiers(ontheleft)andabouttheproduct/modifers(ontheright).Forinstance,ifwehavethatthelocationofareactionisfL1g=)fL2g,wecandeducethatthereactionisatransportreactionbetweenthelocationsL1andL2.3.3.3DenitionoftheSLTS/CTMCtransitionratesThetransitionratesfortheSLTSandtheCTMCassociatedwithaBio-PEPAmodelarederivedfromthefunctionalrates.Eachfunctionalratecorrespondstoakineticlawandisexpressedasthereactionrateequation(RRE)fortheassociatedreaction.Whenmultiple 7 CiocchettaandGuerriero inhigher-orderreactionsthestochasticratesmustberescaledaccordingtothevolumewherethereactantsare.Thisisstraightforwardwhenonlyonecompartmentisconsid-eredorallthereactantsareinthesamecompartment.Howeveritisnotobvioushowtodenethestochasticrateswhenareactionhasreactantsindi erentcompartments.Wecandistinguishthefollowingsituations. (i) Onesinglewell-mixedlocation.Thenumberofmoleculesarederivedfromtheconcentrationsandtheratesarederivedfromthefunctionalratesbysimplecalcula-tions(see[ 7 ]fordetails).Thisapproachisvalidevenifthecompartmentvolumescanchangeovertime:wecanconsidertheextensionofGillespie'salgorithmgivenin[ 14 ]. (ii) Multiplewell-mixedlocations,withnointeractionsbetweendi erentlocations(i.e.locationsareisolatedfromeachother).Wecanapplythesameapproachof(i)toeachlocation. (iii) Multiplewell-mixedlocations,withreactionsthatcaninvolvespeciesindi erentlocationsbutallthereactantsareinthesamelocation.Undertheassumptionthatthesystemasawholeiswell-mixed,thederivationofthestochasticratesistheusualone,andastochasticsimulationalgorithmcanbeapplied. (iv) Multiplewell-mixedlocations,withreactionsthatcaninvolvespeciesindi erentlocationsandreactantsthatcanbeindi erentlocations.Here,inadditiontotheas-sumptionofawell-mixedsystemwehavethefurthercomplicationofthederivationoftheappropriatestochasticrates.Thisremainsanopenquestionandmoreinvesti-gationsarenecessary.Inthefollowingweassumethatthemodellerisabletogivetheserates. MappingtoODEs. ThemappingintoODEsisstraightforwardandidenticaltoBio-PEPA(fordetailssee[ 7 ]).ThemappingisbasedonthederivationofthestoichiometricmatrixfromtheBio-PEPAsystem,thedenitionofthespeciesvariables(representingconcen-trationsormoles)andonthedenitionofakineticvectorcontainingthekineticlaws.Weconsidertwocases: (i) Onecompartmentormultiplecompartmentswiththesamesize.Thevariablesex-pressthespeciesconcentrationsandthekineticsisobtaineddividingthefunctionalratebythevolume.Thesearethestandardkineticlaws. (ii) Multiplecompartmentswithdi erentsizes.ThevariablesexpressmolesandweusethekineticlawsasexpressedintheBio-PEPAmodel.Locationswithsizedependingontimeareallowed.Inthissituationweneedtoaddsomereactionrateequationsforlocationsizes.5AmodelforintracellularcalciumoscillationsIntracellularCa2+oscillationsareobservedinalargevarietyofcelltypesandplayanim-portantroleinthecontrolofmanycellularprocesses.TherearevariousmodelsintheliteraturethatrepresentsimpleperiodicoscillationsforCa2+.Inadditiontotheseoscilla-tionssomeexperimentsshowcomplexperiodicbehaviourresemblingbursting,inwhichphasesofhighfrequencyoscillationsareseparatedbyphasesofquiescence,inapatternthatoccursatregularintervals.Suchcomplexoscillatingbehaviourcanbeduetodi erentfactors,suchas,forinstance,thepresenceofafeedbackloopwherethereleaseofCa2+directlyinhibitsaCa2+channel(throughtheIP3receptor).Severalmodelsinvestigating 10 CiocchettaandGuerriero Fig.2.Schemaofthebasicmodelforintracellularcalciumoscillation(CICRmodel). thesebehaviourshavebeenpresentedin[ 3 ].Figure 2 describesthebasicoscillatingmodel,referredtoasCICR(Ca2+-inducedCa2+release).Compartmentsandtransportofmoleculesamongthemplayamajorroleinthismodel.Threemainspeciesareconsidered:extracellularCa2+,cytosolicCa2+,andCa2+inthesarcoplasmicreticulum(SR).TheconcentrationofcytosolicCa2+changesduetotheinuxofextracellularCa2+(reaction1f),thepassiveeuxofCa2+fromthecytosoltotheextracellularmedium(reaction1b)andfromtheSRintothecytosol(reaction3).Moreover,Ca2+ispumpedinto(reaction2f)andreleasedfrom(reaction2b)theSR.TheconcentrationofextracellularCa2+isconsideredconstant.OnepossibilitytoexplainthepresenceofcomplexoscillationsistotakeintoaccounttheinhibitionoftheIP3receptorchannelactivity.Thischannelisbothactivatedandinhib-itedbycytosolicCa2+.Inparticular,itisassumedthattheIP3receptorhastwotypesofCa2+bindingsitesatthecytosolicsideofthechannel,oneforpositiveandonefornegativeregulation.ThechannelcanonlytransportCa2+ifitisintheactivestate.InadditiontothereactionsshowninFigure 2 weconsidertheactivationandinhibitionoftheIP3receptor.Weindicatethesetworeactionsas4fand4bandweusethenamesR AcandR Intorefertothetwostatesofthereceptor.InthefollowingwepresentthespecicationofthemodelinBio-PEPA,inordertoillustratehowlocationsandreactionsinvolvingmultiplecompartmentscanbedescribedinourlanguage.Inthemodelproposedin[ 3 ]compartmentsareassumedtohaveallthesamesize,andtheparametersandconcentrationsaredenedaccordingly.Thisassumptionisgenerallyintroducedtosimplifythemodelandtheanalysis,andisparticularlyuse-fulwhenprecisedataoncompartmentvolumesarenotavailable.HereweconsiderthesameassumptioninthedenitionoftheBio-PEPAsystem,inordertoobtainacomparablemodel.However,sinceBio-PEPAsupportsmultiplecompartmentswithdi erentsizes,amoreaccuraterepresentationofthebiochemicalsystemcouldbeobtainedifquantitativeinformationaboutcompartmentvolumesareavailable.5.1Bio-PEPAsystemDenitionofcompartments:V=[Ext:1l;C;Cyt:1l;C;SR:1l;C] 11 CiocchettaandGuerriero Fig.3.ODEsimulation formula(see[ 2 ])veriesthatthespeciesCa@SRandCa@Cytoscillatepermanently:thisconrmstheevolutionexhibitedbytheODEssimulation,andalsoguaranteesthatsuchbehaviourisnotlimitedtothechosentimebound,butisperpetual.6RelatedworksSeverallanguageshavebeenproposedtomodelbiologicalcompartmentsandmem-branes[ 18 , 4 , 16 , 19 , 15 ].Allofthemhavesomedi erencesintheconsiderednotionofcompartment,inthekindsofoperationsallowed,andintheunderlyingassumptions.TheBioAmbientscalculus[ 18 ]wastherstprocesscalculusformodellingbiologicalsystemswithanexplicitnotionofcompartments.Asystemisrepresentedasahierarchyofnestedambients,whichrepresenttheboundariesofcompartmentscontainingcommuni-catingprocesseswhoseactionsspecifytheevolutionofthesystem.Operationsinvolvingcompartments,complexformationandtransportofsmallmoleculesacrosscompartmentscanbeeasilyrepresentedinBioAmbients.InBranecalculi[ 4 ],membranesarenotjustcontainers,butactiveentitiesthatarere-sponsibleforcoordinatingspecicactivities.Asystemisrepresentedasasetofnestedmembranes,andamembraneisrepresentedasasetofactions.Operationssuchasthetransportofsmallmoleculesacrossmembranescanbeeasilyrepresented;moreover,mem-branescanmove,merge,split,enterintoandexitfromothermembranes.InBeta-binders[ 16 ]systemsaremodelledasacompositionofboxesrepresentingbi-ologicalentities.Althoughthenestingofboxesisforbidden,thetypingforsitesprovidesavirtualformofnesting,whichmakestherepresentationofhierarchiesofcompartmentspossible.ExplicitstaticcompartmentsandtransportofobjectsacrossthemhavebeenaddedtoBeta-bindersin[ 11 ].ThepossibilitytodealwithcompartmentswhosevolumedependsontimeisconsideredinBlenX[ 9 ],alanguagebasedonBeta-binders.Thestochastic-calculushasalsobeenequippedwithnotionsoflocationsinsomeofitsvariants.Inparticular,wementiontheS@calculus[ 19 ],anextensionofthestochastic-calculusinwhichcompartmentsareexplicitlyaddedtothesyntax.Thislanguagehasbeenprimarilydesignedtobeacorelanguageforencodingdi erentcompartment-basedformalisms,andithandlesvaryingvolumesanddynamicalcompartmentsbydeningthe 13 CiocchettaandGuerriero ofthesystem,noteasilyobservablefromsimulationresults.Thepossibilityofdi erentanalysesisalsosupportedbymembranesystems,whereasmostoftheotherlanguageslistedaboveessentiallylimitthepossibleanalysestostochasticsimulation.7DiscussionandconclusionsCompartmentsandmembranesplayakeyroleinbiochemicalsystemsand,consequently,itisessentialforamodellinglanguagetoallowacorrectandintuitiverepresentationofthosenotions.WehaveenrichedtheBio-PEPAprocessalgebrawithspecicfeaturesusefultomodelbiologicalcompartments.Anotionoflocationhasbeenintroducedand,inadditiontothree-dimensionalcompartments,theirenclosingmembranescanalsobeexplicitlydened.Transportsandotherreactionsinvolvingmolecularspeciesinmultiplecompartmentscanbeeasilymodelledinthisextension.Di erentkindsofanalysiscanbeperformed(basedonODEs,CTMCs,andstochasticsimulation).Severalassumptionsarerequiredinordertoapplytheseanalysestomulti-compartmentmodels(e.g.mostanalysismethodsassumesystemstobeawell-stirredmix-tureofmolecules).Thoughtheseassumptionsaregenerallyverystrong,suchmethodshavebeenshowntoprovidegoodapproximationsofthebehaviourofmulti-compartmentsystems.InthisextensionofBio-PEPAwehavereliedonthesestandardassumptionsandwehaveshownhowtocorrectlyderivemodelsforthesekindsofanalysisfromBio-PEPAmodelswithmultiplecompartments.Inthisworkwehavefocusedonstaticlocations(i.e.whosestructuredoesnotchange,butwhosesizecanchangeovertime);themanagementofdynamicallocations(i.e.whichcansplit,merge,etc.)wouldrequireevenstrongerassumptionsandwouldmakethelan-guagemuchmoreheavy.Moreover,currently,neithertheexistingmathematicalmodelsnortheexperimentaldataprovidesuchalevelofdetail.AcknowledgementTheauthorsthankJaneHillston,StephenGilmore,CristianVersariandLaurenceLoewefortheirhelpfulcomments.FedericaCiocchettaissupportedbytheU.K.EngineeringandPhysicalSciencesResearchCouncil(EPSRC)researchgrantEP/C543696/1“ProcessAlgebraApproachestoCollectiveDynamics”.MariaLuisaGuerrieroissupportedbytheEPSRCgrantEP/E031439/1“StochasticProcessAlgebraforBiochemicalSignallingPath-wayAnalysis”.References 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[2] P.Ballarini,R.Mardare,andI.Mura.AnalysingBiochemicalOscillationthroughProbabilisticModelChecking.InProc.ofFBTC2008,ToappearinENTCS,2008. [3] J.A.M.Borghans,G.Dupont,andA.Goldbeter.Complexintracellularcalciumoscillations:Atheoreticalexplorationofpossiblemechanisms.BiophysicalChemistry,66(1):25–41,1997. [4] L.Cardelli.BraneCalculi-InteractionsofBiologicalMembranes.InProceedingsofComputationalMethodsinSystemsBiology(CMSB'04),volume3082ofLNCS,pages257–278,2005. 15 CiocchettaandGuerriero TableA.1AxiomsandrulesforBio-PEPA. prefixReac(( ;)#S)(l)( ;[S:#(l;)])���������!cS(l�)lNprefixProd(( ;)"S)(l)( ;[S:"(l;)])���������!cS(l+)0l(N�)prefixMod(( ;)opS)(l)( ;[S:op(l;)])����������!cS(l)withop2f;; gand0lNifop=;0lNotherwisechoice1S1(l)( ;w)����!cS01(l0) (S1+S2)(l)( ;w)����!cS01(l0)choice2S2(l)( ;w)����!cS02(l0) (S1+S2)(l)( ;w)����!cS02(l0)constantS(l)( ;S:[op(l;)])����������!cS0(l0) C(l)( ;C:[op(l;)])����������!cS0(l0)withCdef=Scoop1P1( ;w)����!cP01 P1BCIP2( ;w)����!cP01BCIP2with Icoop2P2( ;w)����!cP02 P1BCIP2( ;w)����!cP1BCIP02with Icoop3P1( ;w1)�����!cP01P2( ;w2)�����!cP02 P1BCIP2( ;w1::w2)�������!cP01BCIP02with 2I 17