Rule based modelling of cellular signalling
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Rule-based modelling of cellular signalling Vincent Danos1,3,4, Jerome Feret2, Walter Fontana3, Russell Harmer3,4, and Jean Krivine5 1 Plectix Biosystems 2 Ecole Normale Superieure 3 Harvard Medical School 4 CNRS, Universite Denis Diderot 5 Ecole Polytechnique Abstract. Modelling is becoming a necessity in studying biological sig- nalling pathways, because the combinatorial complexity of such systems rapidly overwhelms intuitive and qualitative forms of reasoning. Yet, this same combinatorial explosion makes the traditional modelling paradigm based on systems of differential equations impractical. In contrast, agent- based or concurrent languages, such as ? [1–3] or the closely related BioNetGen language [4–10], describe biological interactions in terms of rules, thereby avoiding the combinatorial explosion besetting differential equations. Rules are expressed in an intuitive graphical form that trans- parently represents biological knowledge. In this way, rules become a nat- ural unit of model building, modification, and discussion. We illustrate this with a sizeable example obtained from refactoring two models of EGF receptor signalling that are based on differential equations [11, 12]. An exciting aspect of the agent-based approach is that it naturally lends it- self to the identification and analysis of the causal structures that deeply shape the dynamical, and perhaps even evolutionary, characteristics of complex distributed biological systems. In particular, one can adapt the notions of causality and conflict, familiar from concurrency theory, to ?, our representation language of choice.
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Rule-based modelling of cellular signalling Vincent Danos1,3,4, Jerome Feret2, Walter Fontana3, Russell Harmer3,4, and Jean Krivine5 1 Plectix Biosystems 2 Ecole Normale Superieure 3 Harvard Medical School 4 CNRS, Universite Denis Diderot 5 Ecole Polytechnique Abstract. Modelling is becoming a necessity in studying biological sig- nalling pathways, because the combinatorial complexity of such systems rapidly overwhelms intuitive and qualitative forms of reasoning. Yet, this same combinatorial explosion makes the traditional modelling paradigm based on systems of differential equations impractical. In contrast, agent- based or concurrent languages, such as ? [1–3] or the closely related BioNetGen language [4–10], describe biological interactions in terms of rules, thereby avoiding the combinatorial explosion besetting differential equations. Rules are expressed in an intuitive graphical form that trans- parently represents biological knowledge. In this way, rules become a nat- ural unit of model building, modification, and discussion. We illustrate this with a sizeable example obtained from refactoring two models of EGF receptor signalling that are based on differential equations [11, 12]. An exciting aspect of the agent-based approach is that it naturally lends it- self to the identification and analysis of the causal structures that deeply shape the dynamical, and perhaps even evolutionary, characteristics of complex distributed biological systems. In particular, one can adapt the notions of causality and conflict, familiar from concurrency theory, to ?, our representation language of choice.
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Rule-basedmodellingofcellularsignallingVincentDanos1,3,4,Je´roˆmeFeret2,WalterFontana3,RussellHarmer3,4,andJeanKrivine51PlectixBiosystems2E´coleNormaleSupe´rieure3HarvardMedicalSchool4CNRS,Universite´DenisDiderot5E´colePolytechniqueAbstract.Modellingisbecominganecessityinstudyingbiologicalsig-nallingpathways,becausethecombinatorialcomplexityofsuchsystemsrapidlyoverwhelmsintuitiveandqualitativeformsofreasoning.Yet,thissamecombinatorialexplosionmakesthetraditionalmodellingparadigmbasedonsystemsofdifferentialequationsimpractical.Incontrast,agent-basedorconcurrentlanguages,suchasκ[1–3]orthecloselyrelatedBioNetGenlanguage[4–10],describebiologicalinteractionsintermsofrules,therebyavoidingthecombinatorialexplosionbesettingdifferentialequations.Rulesareexpressedinanintuitivegraphicalformthattrans-parentlyrepresentsbiologicalknowledge.Inthisway,rulesbecomeanat-uralunitofmodelbuilding,modification,anddiscussion.WeillustratethiswithasizeableexampleobtainedfromrefactoringtwomodelsofEGFreceptorsignallingthatarebasedondifferentialequations[11,12].Anexcitingaspectoftheagent-basedapproachisthatitnaturallylendsit-selftotheidentificationandanalysisofthecausalstructuresthatdeeplyshapethedynamical,andperhapsevenevolutionary,characteristicsofcomplexdistributedbiologicalsystems.Inparticular,onecanadaptthenotionsofcausalityandconflict,familiarfromconcurrencytheory,toκ,ourrepresentationlanguageofchoice.UsingtheEGFreceptormodelasanexample,weshowhowcausalityenablestheformalizationofthecolloquialconceptofpathwayand,perhapsmoresurprisingly,howcon-flictcanbeusedtodissectthesignallingdynamicstoobtainaqualitativehandleontherangeofsystembehaviours.Bytamingthecombinatorialexplosion,andexposingthecausalstructuresandkeykineticjuncturesinamodel,agent-andrule-basedrepresentationsholdpromiseformak-ingmodellingmorepowerful,moreperspicuous,andofappealtoawideraudience.1BackgroundAlargemajorityofmodelsaimedatinvestigatingthebehaviorofbiologicalpath-waysarecastintermsofsystemsofdifferentialequations[11–16].Thechoiceseemsnatural.Thetheoryofdynamicalsystemsoffersanextensiverepertoireofmathematicaltechniquesforreasoningaboutsuchnetworks.Itprovides,atleastinthelimitoflongtimes,awell-understoodontologyofbehaviors,likesteady
states,oscillations,andchaos,alongwiththeir(linear)stabilityproperties.Thereadyavailabilityofnumericalproceduresforintegratingsystemsofequations,whilevaryingoverparametersandinitialconditions,completesapowerfulwork-benchthathassuccessfullycarriedmuchofphysicsandchemicalkinetics.Yet,thisworkbenchisshowingclearsignsofcrackingundertheponderouscombina-torialcomplexityofmolecularsignallingprocesses,whichinvolveproteinsthatinteractthroughmultiplepost-translationalmodificationsandphysicalassocia-tionswithinanintricatetopologyoflocales[17].Representationsofchemicalreactionnetworksintermsofdifferentialequa-tionsareaboutchemicalkinetics,nottheunfoldingofchemistry.Infact,allmolecularspeciesmadepossiblebyasetofchemicaltransformationsmustbeexplicitlyknowninadvanceforsettingupthecorrespondingsystemofkineticequations.Everymolecularspecieshasitsownconcentrationvariableandanequationdescribingitsrateofchangeasimpartedbyallreactionsthatpro-duceorconsumethatspecies.Thesereactions,too,mustbeknowninadvance.Manyionchannels,kinases,phosphatases,andreceptors–tomentionjustafew–areproteinsthatpossessmultiplesitesatwhichtheycanbemodifiedbyphosphorylation,ubiquitination,methylation,glycosidilation,andaplethoraofotherchemicaltaggingprocesses.Aboutoneinahundredproteinshaveatleast8modifiablesites,whichmeans256states.Asimpleheterodimeroftwodistinctproteins,eachwiththatmuchstate,wouldweighinatmorethan65,000equations.Itiseasilyseenthatthiscombinatoricscanrapidlyamounttomorepossiblechemicalspeciesthancanberealizedbytheactualnumberofmoleculesinvolvedinacellularprocessofthiskind.Theproblemisnotsomuchthatadeterministicdescriptionisnolongerwarranted,butratherthattheequations,whetherdeterministicorstochastic,cannolongerbewrittendown—andiftheycould,whatwouldonelearnfromthem?Thisdifficultyiswellrecognized.Onewayoutistouseaggregatevariablesdescribingsetsofmodificationforms.Forexample,onemightbundletogetherallphosphoformsofareceptor,regardlessofwhichsitesarephosphorylated.This,however,isproblematic.First,thechoiceofwhattoaggregateandnotisunprin-cipled.Second,theappropriatelevelofaggregationmaychangeovertimeasthesystemdynamicsunfolds.Third,theaggregationiserrorprone,sinceithastobedonewithoutapriormicroscopicdescription.Afurther,moresubtle,difficultyisthatanextensionalsystemofdifferentialequationsdescribestheconstituentmoleculesonlyintermsofinteractionsthatarerelevantinagivencontextofothermolecules.Itdoesnotcharacterizemolecularcomponentsintermsoftheirpotentialinteractionsthatcouldbecomerelevantifthecompositionofthesys-temweretochange.Asaconsequence,“compositionalperturbations”,suchasaddinganovelmolecularcomponent(adrug)ormodifyinganextantone(tomodeltheeffectsofknockingoutasiteoraddinganewdomain)arevirtu-allyimpossibletocarryoutbyhand,sincetheyrequire,again,enumeratingallchemicalconsequencesinadvanceandthenrewritingallaffectedequations.Theseproblemshaveledtorecentattemptsatdescribingmolecularreactionnetworksintermsofmoleculesas“agents”,whosepossibleinteractionsarede-2
