Ossareh Department of Electrical Engineering and Computer Science University of Michigan Ann Arbor MI Alejandra C Ventura Department of Biology University of Buenos Aires Buenos Aires Argentina Department of Internal Medicine University of Michigan ID: 58871
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LongSignalingCascadesTendtoAttenuateRetroactivityHamidR.OssarehDepartmentofElectricalEngineeringandComputerScienceUniversityofMichigan,AnnArbor,MIAlejandraC.VenturaDepartmentofBiologyUniversityofBuenosAires,BuenosAires,ArgentinaDepartmentofInternalMedicineUniversityofMichigan,AnnArbor,MISoaD.MerajverDepartmentofInternalMedicineUniversityofMichigan,AnnArbor,MIDomitillaDelVecchioDepartmentofMechanicalEngineeringMIT,Cambridge,MA11Address:DepartmentofMechanicalEngineering,MIT,Cambridge,MA02139,ddv@mit.edu AbstractSignalingpathwaysconsistingofphosphorylation/dephosphorylation(PD)cycleswithnoexplicitfeedbackallowsignalstopropagatenotonlyfromupstreamtodownstreambutalsofromdownstreamtoupstreamduetoretroactivityattheinter-connectionbetweenPDcycles.However,theextenttowhichadownstreampertur-bationcanpropagateupstreaminasignalingcascadeandtheparametersthata ectthispropagationarepresentlyunknown.Here,wedeterminethedownstream-to-upstreamsteadystategainateachstageofthesignalingcascadeasafunctionofthecascadeparameters.Thisgaincanbemadesmallerthan1(attenuation)bysucientlyfastkinaseratescomparedtothephosphataseratesand/orbysuf-cientlylargeMichaelis-Mentenconstantsandsucientlylowamountsoftotalstageprotein.Numericalstudiesperformedonsetsofbiologicallyrelevantparam-etersindicatedthatabout50%oftheseparameterscouldgiverisetoamplicationofthedownstreamperturbationatsomestageina3-stagecascade.Inann-stagecascade,thepercentageofparametersthatleadtoanoverallattenuationfromthelaststagetotherststagemonotonicallyincreaseswiththecascadelengthnandreaches100%forcascadesoflengthatleast6.Keywords:Signaling,mathematicalmodel,numericalsimulation,analyticalderivations CascadesAttenuateRetroactivity2IntroductionSignalingpathwaysareubiquitousinlivingsystemsandcoveracentralroleinacell'sabilitytosenseandrespondtobothexternalandinternalinputstimuli(1,2).Numeroussignalingpathwaysconsistofcyclesofreversibleproteinmodication,suchasphosphorylation/dephosphorylation(PD)cycles,whereinaproteinisre-versiblyconvertedbetweentwoforms(3).MultiplePDcyclesoftenappearcon-nectedinacascadefashion,suchasintheMAPKcascades(4,5),andthelengthofthecascadehasbeenshowntohaveimportante ects,forexample,onsignalamplication,signalduration,andsignalingtime(68).Inparticular,awealthofworkhasbeenemployingmetaboliccontrolanalysis(MCA)approachestoanalyt-icallydeterminetheamplicationgainsacrossthecascadeasasmallperturbationappliedatthetopofthecascadepropagatestowardthebottomstages(810).Nostudyhasbeenperformedonhowperturbationsatthebottomofacascadepropa-gatetowardthetopofthecascade.Sincecascadesoftenintersecteachotherbysharingcommoncomponents,suchasproteinsubstratesorkinases(11,12),perturbationsatbottomorintermediatestagesinacascadecanoftenoccur.Theseintersectionsarealreadyknowntocauseunwantedcrosstalkbetweenthesignalingstagesdownstreamoftheintersectionpoint(1316).However,noattentionwasgiventocrosstalkbetweenthestagesupstreamoftheintersectionpoint.Severaloftheseworks,infact,viewedasignal-ingcascadeasthemodularcompositionofPDcycles,resultinginasystemwherethesignaltravelsonlyfromupstreamtodownstream.Theoreticalwork,however,hasshownthatPDcycles(asseveralotherbiomolecularsystems)cannotbemodu-larlyconnectedwitheachotherbecauseofretroactivitye ectsatinterconnections(1722).Initialexperimentalvalidationofthesee ectsonthesteadystateresponseofaPDcyclehavealsoappeared(2325).Thesee ectschangethebehaviorofanupstreamsystemwhenitisconnectedtoitsdownstreamclientsandarerelevantespeciallyinsignalingcascades,inwhicheachPDcyclehasseveraldownstreamtargets.Asaresultofretroactivity,signalingcascadesallowsignalstoalsotravelfromdownstreamtoupstream,thatis,theyallowbidirectionalsignalpropagation(22,26).Asaconsequence,aperturbationatthebottomofthecascadecanpropa-gatetotheupstreamstagesandhaverepercussionsontheoverallsignaling.Aperturbationatthebottomofacascadecanbeduetoanumberoffactors.Forexample,whenadownstreamtargetorasubstrateissharedwithothersignal-ingpathways,itsfreeconcentrationisperturbedbytheseotherpathways.Hence,theamountoftarget/substrateavailabletothecascadeunderstudycansuddenlychange.Similarly,theintroductionofaninhibitorofanactiveenzyme,asper-formedintargeteddrugdesign,createsaperturbationatthetargetedstageofthecascade.Howlargeisthee ectofsuchperturbationsontheupstreamstages?How CascadesAttenuateRetroactivity3doesthelengthofacascadeimpactbackwardsignaltransfer?Ontheonehand,answeringthesequestionswillrevealtheextentbywhichaberrantsignalingintheupstreamstagesofacascadecanbecausedbyretroactivityfromsharingdown-streamtargets/substrates.Ontheotherhand,itwillprovidetoolsfortargeteddrugdesignbyquantifyingtheo -targete ectsofinhibitorsontheupstreamstages.Inthispaper,weaddressthesequestionsincascadeswithasinglephospho-rylationcycleperstagebyexplicitlyincorporatingretroactivityinthePDcyclemodel.Specically,weconsidersmallperturbationsatthebottomofthecascadeandexplicitlyquantifyforthersttimehowsuchperturbationspropagatefromdownstreamtoupstream.Ourmainresultsareasfollows.Weprovideanalyticalexpressionsforthedownstream-to-upstreamtransmissiongains.Theseestablishtheextenttowhichaperturbationatthebottomofthecascadecanpropagateup-streamandprovidesucientconditionsforattenuation.Throughextensivenu-mericalsimulation,wediscoveredthat,surprisingly,naturalcascadescanamplifyaperturbationasitpropagatesupstream,buttheprobabilityofattenuationissub-stantiallyhigherthanthatofamplication.Also,theprobabilityofattenuationincreaseswiththenumberofstagesinthecascade.MethodsWeconsiderasignalingcascadecomposedofnphosphorylation/dephosphorylation(PD)cyclesasdepictedinFigure1.Thesensitivityofresponsetoperturbationsoccurringatthetopofthecascade,forexampleinW0,hasbeenextensivelystudiedemployingMCAapproaches(810).Bycontrast,hereweinvestigatethesensitiv-ityofresponseofeachcycletoaperturbationatthebottomofthecascade.Thisperturbationcanbedue,forexample,toaninhibitoroftheactiveenzymeWn,asitisemployedintargeteddrugdesign(27)ortothesignalingfromanotherpathwaysharingasubstratewithWn.Ourmethodisbasedonassumingasmallperturbation,onlinearizingthesystemdynamicsaboutthesteadystate,andonde-terminingthecorrespondingchangeofeachcyclephosphorylatedprotein.Sinceourapproachisbasedonlinearization,itissimilarinspirittoMCAapproaches,whichalsoassumesmallperturbationsandlinearizethesystemdynamics.Here,weareinterestedindetermininghowe ectivelytheperturbationpropagatesup-stream.Wethusexplicitlycomputethesensitivitygainfromonestagetothenextupstreamasafunctionofthecascadeparameters. CascadesAttenuateRetroactivity4CascadeModelAteachstagei,fori2f1;:::;ng,wedenotebyWi 1thekinase,byEithephos-phatase,byWitheproteinsubstrate,andbyWithephosphorylatedformofWi.ThekinaseWi 1bindstoWitoformthesubstrate-kinasecomplexXi.Thiscom-plexthenturnsintoWi.ThephosphorylatedproteinWiisinturnakinaseforthenextcycleandbindstodownstreamsubstrates,formingthecomplexXi+1.ThephosphataseEiactivatesthedephosphorylationoftheproteinWibybindingtoWiandformingthecomplexYi.ThiscomplexisinturnconvertedtoWi.Weemploythefollowingtwo-stepreactionmodelforeachphosphorylationanddephosphory-lationreaction(28,29)atstagei2f1;:::;ngofthecascade:Wi+Wi 1ai *) aiXiki !Wi+Wi 1Wi+Eibi *) biYi ki !Wi+Ei:WeassumethatproteinWiandphosphataseEiareconservedateverystage,andareintotalamountsWiTandEiT,respectively.Therefore,wehavetheconservationrelationsWi+Wi+Xi+Yi+Xi+1=WiTEi+Yi=EiT;(1)inwhichforaspeciesXwehavedenotedbyX(italics)itsconcentration.Weassumethattheinputkinasetotherststage,W0,isproducedatratek(t)anddecaysatrate,thatis,W0 *) k(t);:Finally,weassumethattheoutputproteinofthelaststage,Wn,reactswithspeciesDdownstreamofthecascade.ThesespeciesDcanmodel,forexample,asignalingmoleculeoraninhibitoroftheactiveenzymeWn(adrug)suchasconsideredintargeteddrugdesign(27),inwhichthetotalconcentrationofDcanbeperturbed,forexample,byaddingmoredrug.SpeciesDcanalsomodelasubstratethatissharedwithothersignalingpathways.Inthiscase,Disasubstrateforanotheractiveenzyme,sayS,whoseconcentrationiscontrolledbyanothersignalingcas-cade.Hence,theamountoffreeDplustheamountofDboundtoWn,whichwecallDT,canbeperturbed(itcanincreaseordecrease)byachangeintheconcen-trationoftheactiveenzymeS.DenotingbyXn+1thecomplexformedbyWnandD,wehavethatWn+Dan+1 *) an+1Xn+1withDT,D+Xn+1: CascadesAttenuateRetroactivity5Inthisstudy,weconsiderDTastheparametertobeperturbedandcalculatethesensitivityofthesteadystateresponseofeachcycleactiveproteintosmallpertur-bationsinDT.Thedi erentialequationsthatdescribethedynamicsofthecascadearegiven,fori2f1;:::;ng,byW0= W0+k(t) (a1W0W1 ( a1+k1)X1) Xi=aiWi 1Wi ( ai+ki)Xi;Wi=kiXi biWiEi+ biYi (ai+1WiWi+1 ( ai+1+ki+1)Xi+1) Yi=biWiEi ( bi+ ki)YiWn=knXn bnWnEn+ bnYn (an+1DWn an+1Xn+1) Xn+1=an+1DWn an+1Xn+1:RecognizingthatthetermsintheboxescorrespondtoX1,Xi+1,andXn+1,respec-tively,andemployingtheconservationlaw(1),weobtainfori2f1;:::;ngthatW0= W0+k(t) X1Xi=aiWi 1(WiT Wi Xi Yi Xi+1) ( ai+ki)XiWi=kiXi biWi(EiT Yi)+ biYi Xi+1(2)Yi=biWi(EiT Yi) ( bi+ ki)YiXn+1=an+1(DT Xn+1)Wn an+1Xn+1:PerturbationAnalysisInthissection,weconsiderthecascadetobeatthesteadystateandinvestigatehowasmallperturbationintheconcentrationDTperturbsthesteadystateconcentra-tionsateverystageofthecascade.Wedenotethesteadystatevalueoftheupstreaminputk(t)by kandthatofDTby DT.ThecorrespondingequilibriumvaluesoftheproteinconcentrationsW0,Wi,Xi,Yi,Wi,fori2f1;:::;ng,andXn+1aredenotedby W0, Wi, Xi, Yi, Wi,fori2f1;:::;ng,and Xn+1,respectively.WerepresenttheperturbationofDTwithrespecttoitssteadystatevaluebydT:=DT DT.NotethatifdT�0,thedownstreamperturbationispositive,thatis,theconcentrationDTincreases.IfinsteaddT0,thedownstreamperturbationisnegative,thatis,theconcentrationDTdecreases.Hence,bothpositiveandnegativeperturbationsareconsidered.Thecorrespondingperturbationsofthestatesofthecascadeabouttheequilibriumvalues W0, Wi, Xi, Yi,fori2f1;:::;ng,and Xn+1aredenotedbyw0,wi,xi,yi,fori2f1;:::;ng,andxn+1,respectively.Similarly,denotebyZifor CascadesAttenuateRetroactivity6i2f1;:::;ngtheconcentrationofthetotalphosphorylatedproteinatstagei,thatis,Zi:=Wi+Yi+Xi+1.Denotethecorrespondingperturbationaboutthesteadystate Zi= Wi+ Yi+ Xi+1byzi,whichcanbewrittenaszi=wi+yi+xi+1foralli2f1;:::;ng.Thelinearizationofsystem(2)abouttheequilibrium W0, Wi, Xi, Yi,and Xn+1,fori2f1;:::;ngisgivenbyw0= w0 x1xi=ai Wiwi 1+ai Wi 1( wi xi yi xi+1) ( ai+ki)xiwi=kixi+bi Wiyi bi Eiwi+ biyi xi+1(3)yi= bi Wiyi+bi Eiwi ( bi+ ki)yixn+1=an+1 Dwn an+1 Wnxn+1 an+1xn+1+an+1 WndT;inwhichwehavefori2f1;:::;ngthat(fromsettingthetimederivativesinequations(2)equaltozero) W0= k (4) Wi=Ki ki ki Ei Wi+ Ki0BBBBB@1+ Wi Ki1CCCCCA Wi Wi 1(5) Yi=EiT 1+ Ki= Wi(6) Xi= ki ki Yi;(7)inwhich Ki:=¯bi+ ki biistheMichaelis-Mentenconstantofthedephosphorylationreaction,whileKi:=¯ai+ki aiistheMichaelis-Mentenconstantofthephosphorylationreaction.Sinceweareinterestedinthesteadystatevaluesofwi,wesetthetimederiva-tivestozeroinsystem(3)toobtainxi= ki kieEiwi(8)wi=Ti( Wiwi 1 Wi 1xi+1);(9) CascadesAttenuateRetroactivity7fori2f1;:::;ng,inwhicheEi, KiEiT (¯Wi+ Ki)2(10)Ti,1 Wi 1+eEi( Wi 1+ ki ki( Wi 1+Ki)):(11)Figure2representsrelations(8)and(9)inablockdiagramform,whichhighlightsthedirectionalityofsignalpropagationthroughthestagesinthecascade.Basically,theperturbationdTpropagatesupstreaminthecascadethroughperturbationsinthecomplexesXi.Hence,inthissteadystateresponsemodel,retroactivityisduetothecomplexXioftheactiveproteinwithitsdownstreamsubstrate.ResultsAnalyticalResultsReferringtoFigure2,theperturbationdTpropagatesupstreamthroughperturba-tionsxiandcausesperturbationsziandwiinthetotalandfreephosphorylatedproteinconcentrations,respectively,ateverystage.Howdotheseperturbationstransferfromonestageofthecascadetothenextoneupstream?Inordertoanswerthisquestion,wecalculatethegainsi:=jzij jzi+1jand i:=jwij jwi+1jateverystagei.Againgreaterthanonemeansthatsmallperturbationsareampli-edastheytransferfromdownstreamtoupstream,whileagainsmallerthanonemeansthatsmallperturbationsareattenuatedastheytransferfromdownstreamtoupstream.Sincejzij=ijzi+1j,wehavethatjz1j=0BBBBBB@n 1Yi=1i1CCCCCCAjznjwhereQdenotesmulti-plication.Wethusdenethetotalgaintotfromstagentostage1astot:=n 1Yi=1i:Similarly,thetotalgain totfromstagentostage1isdenedas tot:=n 1Yi=1 i: CascadesAttenuateRetroactivity8Havingatotalgainsmallerthanonemeansthatoverallthecascadeattenuatesdownstreamperturbations,evenifsomestagesmayamplifytheperturbation.Werstfocusonthegainsioftotalactiveproteinconcentration.Thetotalactiveproteinconcentrationcanbeexperimentallydeterminedbymeasuringpro-teinactivitythroughphospho-specicantibodies(30).Bycontrast,thefreeactiveproteinmaybemorediculttomeasure.Whenitisanactivetranscriptionfactor,itcanbemeasuredindirectly,forexample,byplacingareportergeneunderthecontrolofthepromoterthatitregulates.TheexpressionofthegainiateachstageicanbeexplicitlycalculatedasafunctionofthecascadeparametersfromtherelationsintheblockdiagramofFigure2(seeSI).Thisexpressionisgivenbyi=0BBBBBBBB@eEi ki ki+Fi 1+eEi+eEi ki ki+Fi1CCCCCCCCA0BBBBBBBB@ ki+1 ki+1eEi+1 ki+1 ki+1eEi+1+Fi+11CCCCCCCCAforalli2f1;:::;n 1g;inwhichFiandFi+1arepositivequantities.SinceeEi ki ki+Fi 1+eEi+eEi ki ki+Fi1and ki+1 ki+1eEi+1 eEi+1 ki+1 ki+1+Fi+11,wehavethati1;foralli2f1;:::;n 1g;Furthermore,wehavethat(seeSI)sign(zi)= sign(zi+1)foralli2f1;:::;n 1g;thatis,anincreaseofZi+1impliesadecreaseofZi.Therefore,thereisasignrever-saloftheperturbationonthetotalphosphorylatedproteinconcentrationacrossthestagesandthemagnitudeoftheperturbationateverystageisalwaysattenuatedasitpropagatesupstreaminthecascade.Thatis,jz1jjz2j:::jzn 1jjznjforallparametervalues.Furthermore,thisimpliesalsothatwehaveoverallattenuationfromdownstreamtoupstreaminthecascade,thatis,tot1.Sincethesefactsdonotdependonthespecicparametervaluesorthelengthofthecascade,theyhighlightanewstructuralpropertyofsignalingcascades.Fortheperturbationonthefreeactiveproteinconcentration,wealsohavethat(seeSI)sign(wi)= sign(wi+1)foralli2f1;:::;n 1g;thatis,whentheperturbationwi+1ispositivethenextupstreamstagehasaper-turbationwiwithnegativesign.Hence,ifthedownstreamperturbationcausesadecreaseoftheactiveproteinconcentrationatonestage,itcausesanincreaseoftheactiveproteinconcentrationinthenextupstreamstage.Anexpressionofthestagegain icanbecalculatedasafunctionofthecascadeparametersstartingfromthe CascadesAttenuateRetroactivity9relationsoftheblockdiagramofFigure2.TheexactexpressioniscalculatedintheSIanditissuchthat i ki+1 ki+1E(i+1)T Ki+1 1+EiT Ki+WiT1+WiT Ki1+ ki ki1+Ki W(i 1)T:(12)Therefore,onecancontroltheamountofattenuation/amplicationthroughthecas-cadeparametersasfollows.ThesmallertheW(i 1)T,themoretheattenuationfromstagei+1toi(i.e.,thesmallertheupperboundon iinequation(12)).Moreover,sucientlylargevaluesofKiand Kiforallileadtoanincreasedattenuationateverystage.Inturn,largeKiand KiandsmallWiTareresponsibleforadecreasedsensitivityoftheresponseofstageitoupstreamstimuli(29).Asaconsequence,amoregradedupstreamtodownstreamresponseatallstagesisassociatedwithanincreasedattenuationofdownstreamperturbations.Fromexpression(12),italsofollowsthatasucientconditionforhavingat-tenuationatstageiofthedownstreamperturbationisthat¯ki+1 ki+1E(i+1)T Ki+11:ThisconditionisvalidforgeneralPDcascades.However,ithasaparticularlysimplemeaninginthecaseinwhichthesignalingpathwayisweaklyactivatedasexplainedinwhatfollows.In(6),itwasfoundthatarequirementforupstreamtodownstreamsignalamplicationisthatthephosphorylationrateconstantshouldbelargerthanthedephosphorylationrateconstant.ForaweaklyactivatedpathwaywithKi W(i 1)T,thephosphorylationrateconstantiswellapproximatedby i:=kiWiT=Ki(seeSI).Inthecaseinwhich Ki WiT,thedephosphorylationrateconstantiswellapproximatedby i:= kiEiT= Ki(seeSI).Asaconsequence,tohaveupstream-to-downstreamsignalamplication,itisrequiredthat i� i,which,whenKiWiT,impliesthat¯ki kiEiT Ki1.This,inturn,impliesthat i 11andhencethatthedownstreamperturbationisattenuatedasittransfersfromstageitostagei 1.Hence,inweaklyactivatedpathwaysinwhichKiWiT, Ki WiT,andKi W(i 1)T,upstreamtodownstreamsignalamplicationisassociatedwithattenuationofdownstreamperturbationsastheytransferupstream.This,inturn,impliesunidirectionalsignalpropagationfromupstreamtodownstream.Fromexpression(12),italsofollowsthatanecessaryconditionforhaving i�1,thatis,foramplifyingadownstreamperturbationasittransfersfromstagei+1tostagei,isthat¯ki+1 ki+1E(i+1)T Ki+1�1:Thiscondition,inturn,inthecaseinwhich Ki+1 W(i+1)T,W(i+1)TKi+1,andKi+1 WiTimpliesthatthephosphorylation CascadesAttenuateRetroactivity10rateconstant i+1issmallerthanthedephosphorylationrateconstant i+1.Asaconsequence,thereisnoamplicationatstagei+1ofthesignaltravelingfromupstreamtodownstreamastherequiredconditionforamplicationasdeterminedby(6)isviolated.Hence,inweaklyactivatedpathwaysinwhichKi+1W(i+1)T, Ki+1 W(i+1)T,andKi+1 WiTifadownstreamperturbationisampliedasitpropagatesfromstagei+1tostagei,thenthereisnoamplicationfromstageitostagei+1forthesignaltravelingfromupstreamtodownstreaminresponsetoastimulusatthetopofthecascade.Fromtheexpressionsof i,wecanalsoderiveanecessaryconditionforatten-uation(seeSI).Specically,tohave i1atstageiitisnecessarythat ki+1 ki+1 Ki+1E(i+1)T (¯Wi+1+ Ki+1)2 1+ KiEiT (¯Wi+ Ki)21+ ki ki1+Ki Wi 11+1+ Wi Ki Wi Wi 11:(13)Ifthenecessaryconditionisviolatedatstagei,theneitherstagei 1orstageiamplifythedownstreamperturbation.Thisexpressioncanbeemployedtodeter-mineparametervaluesforwhichamplicationofthedownstreamperturbationcanresultatanygivenstageandcanbeusefultodeterminetheecacyoftheo -targete ectsofaninhibitor.Toconcludetheanalyticalstudy,weinvestigatehowdTa ectswnandzn.Itcanbeshown(seeSI)thatjwnjjdTjandthatjznjjdTj.Thatis,theperturbationdTinduceschangeswnandznabout Wnand Zn,respectively,thatarelessthandTinmagnitude,regardlessoftheparameters.Also,wehavethatsign(dT)= sign(wn)andsign(dT)= sign(zn).NumericalResultsInthissection,werstillustratetheresultsonathree-stagecascadeexample.Wethenemploytheanalyticallycomputedexpressions itodeterminetheprobabilitythatnaturalcascadesattenuateadownstreamperturbationasittransfersupstreaminthecascade.Wenallystudythee ectofthelengthofthecascadeontheoverallgain tot.Allsimulationsareperformedonthefullnonlinearmodelofequations(2)inMATLABusingthebuilt-inODE23ssolver.Figure3showshowtheperturbationpropagatesupstreaminathree-stagecas-cadefortheparametervaluesof(28).ThisFigureillustratesthat,surprisingly,therelationshipbetweenwianddTisapproximatelylinearevenforlargeperturba-tionsdT(upto400nM).Hence,thetheoreticalresultsmusthold.Inparticular,thevaluesofw1andw3arenegativewhilethevalueofw2ispositive.Thatis,theper-turbationonWiswitchessignfromonestagetothenextupstream.Thegains i CascadesAttenuateRetroactivity11calculatedfromtheexpressionintheSIfortheparametervaluesof(28)aregivenby 1=2:4510 5and 2=2:1410 2.Since 1and 2arebothsmallerthan1,thecascadeshouldattenuatethedownstreamperturbationateverystage.ThisisconrmedbyFigure3inwhichforthesamevalueofdT,wehavethatjwijbecomessmallerandsmallerasthestageidecreases(i.e.,astheperturbationpropagatesupstream).Sincethevaluesof iaremuchsmallerthan1,thisthree-stagecascadepracticallyenforcesunidirectionalsignalpropagationfromupstreamtodownstream.NotethataslongastheappliedperturbationdTissmallenough,therelationshipbetweendTandwiislinearandhenceallourresultsholdindepen-dentlyoftheparametervalues.Additionalexamplesfordi erentparametervaluesareprovidedintheSI.Tovalidatethenecessaryconditionforattenuationatstagei,weconstructedaparametersetthatviolatesthenecessaryconditionforattenuation(13).Inthiscase,weshouldexpectthatatthestageiforwhich i�1,thedownstreamperturbationisamplied,thatis,jwij�jwi+1j.Thenecessarycondition(13)canbeviolatedbychoosingphosphataseamountsthatincreasewiththestagenumber,thatis,E1T E2T E3Tandsubstrateamountsthatdecreasewiththestagenumber,thatis,W1T W2T W3T.Weutilizedtheseconditionsandconstructedacascadethatampliesdownstreamperturbations.TheresultisshowninFigure4.Theresultingparametervaluesarestillbiologicallymeaningfulastheyarecontainedintheparameterintervalsestimatedin(28).Therefore,thesecascadesarecapableofalsotransmittingaperturbationfromdownstreamtoupstreambyamplifyingitsamplitude.Donaturalsignalingcascadesattenuatedownstreamperturbations?Inordertodeterminetheprobabilitythatanaturalsignalingcascadeattenuatesorampliesdownstreamperturbations,weevaluatedtheexpressionofthegains ionparametersextractedwithuniformprobabilitydistributionfromintervalstakenfromtheliterature(28,3133).Wepresenttheresultsrstforathree-stagecascadestartingfromconservativeintervalsandweprogressivelyreducethesizeoftheintervals.Inallcases,eachparameterhasarangeandauniformprobabilitydistributionisusedtosampleparametersforeachrange.Also,eventhoughtherangeofparametersforeachcycleisthesame,inthesimulationseachcyclehasdi erentparameters(randomlypickedfromthegivenrange).Conservativeintervals.Inthiscase,werandomlychoseparametersthroughauniformprobabilitydistributionfromtheintervalsgiveninTable1.Themaximumandminimumvaluesoftheintervalswerechosentobethemaximumandminimumoftheunionoftheintervalsdenedin(28)and(31).Thisisaconservativewayofchoosingtheintervalsastheparametersof(28)and(31)aretakenfromdi erent CascadesAttenuateRetroactivity12organisms.InselectingtherangeforDT,weassumedthatDisadownstreamproteinsubstrateandthusitsintervalofvariationwaschosentobethesameasthatforWiT.Wesimulatedthethree-stagecascade10000timesandtheresultsarereportedinTable2.Thistableshowsthepercentageofsimulationsthatresultedin i1foreveryi2f1;2g,thatis,thatresultedinattenuationatstagei.Theprobabilityofstage1attenuatingthedownstreamperturbationis71:34%andtheprobabilityofstage2attenuatingitis55%.Moreover,sincetheprobabilitythat tot1is79:4%theprobabilityofsuchcascadesprovidinganoverallattenuationofadownstreamperturbationisquitehigh.Toexplorewhether10000simulationswereenoughtoobtainmeaningfulprobabilitygures,wecalculatedateachnewsimulationthepercentageofallperformedsimulationsthatresultedinattenuation.TheprobabilitiesconvergeforeverystagetothevaluesgiveninTable2,henceperformingmoresimulationswillnotsignicantlychangetheresults(seeSI).IntervalsbasedonBhallaetal.(31).Weconsideredthenominalparame-tervaluesgivenin(31)andthenconstructedintervalsbyvaryingthesevaluesby20%,50%,and80%.Specically,foreveryparameterwithnominalvaluep,weconsideredacondenceintervaloftheform[(1 0:x)p;(1+0:x)p]forthethreedi erentcasesinwhichx=2,x=5,andx=8.Theresultsforthesethreedif-ferentcasesareshowninTable3.Evenwhentheparametersareallowedtovaryby80%fromthenominalvalues,theprobabilitythatanygivenstageattenuatestheperturbationisveryhighandtheprobabilitythatthecascadeprovidesoverallattenuation(i.e., tot1)is1.Asperformedinthepreviouscase,theresultsofTable3areobtainedperforming10000numericalsimulations.IntheSI,weshowthatthisnumberislargeenoughtoattainconvergenceoftheprobabilities.IntervalsbasedonLevchenkoetal.(32).Wenextconsideredthenominalparametervaluesgivenin(32)andconstructedintervalsbyvaryingthesevaluesby20%,50%,and80%.Specically,foreveryparameterwithnominalvaluep,weconsideredacondenceintervaloftheform[(1 0:x)p;(1+0:x)p]forthethreedi erentcasesinwhichx=2,x=5,andx=8.Theresultsforthesethreedi erentcasesareshowninTable4.Whentheparametersareallowedtochangeby50%withrespecttothenominalvalues,theprobabilityofattenuationateachstageislowerthanthevaluesobtainedfortheparametersof(31)(Table3).With80%parametervariation,thereisasignicantpercentageofthepossibleparameters(10%)thatallowstooverallamplifythedownstreamperturbationfromstage3tostage1.Moreover,50%oftheparametersledtohaving 1&#x-200;1or 2&#x-200;1andonly2:2%oftheparametersledtohavingboth 1&#x-200;1and 2&#x-200;1.Therefore,50%ofthepossibleparametervaluesleadtoamplicationinatleastonestageinthecascade.TheresultsofTable4areobtainedperforming10000numericalsimulations.TheSIshowsthatbythetimethe10000thsimulationisperformedtheprobabilityhasconvergedtoitsnalvalue. CascadesAttenuateRetroactivity13Wethenanalyzedhowthelengthnofthecascadea ectstheoverallattenua-tionfromstagentostage1,thatis,howita ectsthegain tot.Toperformthisstudy,werstsimulatedaten-stagecascade10000timeswiththesameparameterrangesasgiveninTable1.TheresultisshowninTable5.Theprobabilityofthelasttwostages(i=8;9)attenuatingtheperturbationhassignicantlyincreasedcomparedtothethree-stagecase(Table2).Furthermore,theprobabilityofoverallattenuation,thatis,that tot1,is100%.Hence,evenwhensomestagesam-plifythedownstreamperturbation,therestofthestagesprovideattenuationsothattheoverallattenuationinthecascadeismuchmorethantheoverallamplication.Toconrmthat10000simulationswereenoughtoprovidemeaningfulprobabilitygures,weanalyzedtheconvergenceoftheprobabilityaftereachsimulationrunintheSI.Finally,tostudyhowthenumberofstagesinacascadeimpactstheprobabilityofoverallattenuation,thatis,theprobabilitythat tot1,weperformedanum-berofnumericalsimulationsextractingparametersfromtheintervalsofTable1forcascadeswithincreasingnumberofstages.Theprobabilityofoverallatten-uationmonotonicallyincreasesasthenumberofstagesinthecascadeincreasesanditreaches100%forcascadesoflengthatleast6(Figure5).Foreachnumberofstages,n,weperformedasucientlylargenumberofsimulationsfordi erentvaluesoftheparameterssampledintheintervalsofTable1(seeSI).Thisresultimpliesthatforaxedrangeofparameters,addingmorestagescontributessigni-cantlytotheprobabilityofoverallattenuationfromstagentostage1.Forexample,theprobabilityofathree-stagecascadeprovidingoverallattenuationwasfoundtobe79:4%while,forthesamerangeofparameters,theprobabilityofaten-stagecascadeprovidingoverallattenuationwasfoundtobe100%.DiscussionUpstreamtodownstreamsignaltransferinsignalingcascadesdetermineshowex-ternalstimuliatthetopofthecascade,suchasgrowthfactors,hormones,andneu-rotransmitters,a ectdownstreamtargets,suchasgeneexpression.Severalworksfocusedondeterminingthesensitivityofeachstageofacascadetosmallpertur-bationsatthetopofthecascade.Inthesestudies,itwasfoundthatmultiplestagesinthecascadecanboosttheoverallcascadesensitivitytoupstreaminputstimuli(810).Downstreamtoupstreamsignaltransferdetermineshowaperturbationatthebottomofthecascadedue,forexample,toadrugortosharingasubstratewithanothersignalingpathway,a ectstheupstreamstagesofthecascade.Thishasnotbeenstudiedbefore.Here,wehavestudiedforthersttimetheresponseofeachstageofacascadetosmallperturbationsinasubstrateorinhibitoratthebottomof CascadesAttenuateRetroactivity14thecascade.Oneofourresultsisthatlargernumbersofstagesinthecascadeleadtohigheroverallattenuationofthesignaltransferfromdownstreamtoupstream.Thisprovidesanotherreasonwhynaturalsignalingcascadesareusuallycomposedofmultiplestages:morestagesenforceunidirectionalsignalpropagation,whichiscertainlydesirableinanynaturalorhuman-madesignaltransmissionsystem.Wehavecomputedanalyticalexpressionsofthedownstream-to-upstreamgainsateachstageofthecascadeasafunctionofthecascadeparameters.Theseex-pressionsuncovertwomainstructuralpropertiesofsignalingcascades,whichareindependentofthespecicparametervalues.First,theperturbationonthetotalorfreeactiveproteinconcentrationswitchessignateachstageofthecascadeasitpropagatesupstream.Thatis,ifatonestagetheamountofactiveproteinincreasesbecauseoftheperturbation,itmustdecreaseatthenextupstreamstage.Second,theperturbationonthetotalamountofactiveproteinisattenuatedasitpropagatesfromonestagetothenextoneupstream.Bycontrast,thewaytheperturbationpropagatesonthefreeamountofactiveproteindependsonthespecicparametervalues.Wehaveprovidedasucientconditionforattenuation,whichappliestogeneralPDcascadesandhasaparticularlysimplemeaninginthespecialcaseofweaklyactivatedpathways.Thatis,forweaklyactivatedpathwaysinwhicheachcycleoperatesinthehyperbolicregime,amplicationofaperturbationatthetopofthecascadeasitpropagatesdownstreamimpliesattenuationofaperturbationatthebottomofthecascadeasitpropagatesupstream.Whilesimulationstudiesperformedin(22)suggestedthataperturbationisat-tenuatedasitpropagatesupstreaminthecascade,theanalyticalexpressionsofthegainsfoundinthispaperclearlyshowthatamplicationoftheperturbationonthefreeproteinconcentrationisalsopossible.Inordertounderstandwhethernaturalsignalingcascadesaremorelikelytoattenuateortoamplifyadownstreampertur-bationonthefreeactiveproteinconcentration,weperformedanumericalstudy.Inthisstudy,thegain iateachstagewascomputedwithparametervaluesrandomlyextractedfrombiologicallymeaningfulsetsobtainedfromtheliterature(28,3133).Thisnumericalstudyrevealsthatsignalingcascadesaresubstantiallymorelikelytoattenuateadownstreamperturbationthantoamplifyitandthatlongersignalingcascadeshaveahigherprobabilityofoverallattenuation.However,insignalingcascadesoflength3,whichisthemostcommonlengthfoundinprac-tice,about50%ofthebiologicallymeaningfulparameterstakenfrom(32)leadtoamplicationatleastatonestageandabout10%ofthemresultedinoverallamplication(fromstage3tostage1).Insummary,ourndingssuggestthatthee ectsofcrosstalkbetweensignalingpathwayssharingcommoncomponentscanbefeltevenupstreamofthecommoncomponentasopposedtoonlydownstreamofitaspreviouslybelieved.Thispro-videsanewmechanismbywhichapathwaycanbecomeover-activatedasfound CascadesAttenuateRetroactivity15inseveralpathologicalconditionssuchascancer(1316).Atthesametime,ourstudyprovidestoolstounderstandhowthee ectsofatargeteddrug(26,27)maypropagatetoobtaino -targete ectsandhowthesee ectsdependonthecascadeparameters.Thispaperaddressescascadesinwhichateachstagethereisasinglephos-phorylationcycle.However,severalnaturalcascades,suchastheMAPKcascade,displaydoublephosphorylationandexperimentalworkperformedinDrosophilaembryoshasdemonstratedthataperturbationinoneofthesubstratesatthebottomofthecascadea ectsthephosphorylationlevelatthelastcycleofthecascade(24).Whethersuchaperturbationcanpropagateonthehigherlevelsofthecascadewasnotaddressed.Infuturework,wethusplantoextendourgaincalculationstocas-cadeswithdoublephosphorylationinordertoestablishtheextenttowhichsuchperturbationspropagateonthehigherlevelsoftheMAPKcascade.Itwasshowninpreviousworkthatthepresenceofdoublephosphorylationcanleadtosustainedos-cillationsevenintheabsenceofexplicitnegativefeedback(34).Insuchinstances,ouranalysiswillhavetoextendtodynamicperturbationsasopposedtostaticper-turbationsinordertounderstandhowtheseoscillationspropagateupstreaminthecascade.Recentlypublishedexperimentalpapersclearlyshowthatperturbationsinthedownstreamtargetsofasignalingcascadecauseaperturbationintheimmediateupstreamsignalingstage.Specically,(24)showed,throughinvivoexperimentsintheDrosophilaEmbryo,thatchangingthelevelofoneofthesubstratesoftheMAPKcascadeinuencesthelevelofMAPKphosphorylation.Additionally,(23)showed,throughexperimentsonareconstitutedcovalentmodicationcycle,thattheadditionofadownstreamtargetchangesthesteadystatevalueofthemodiedproteinoftheupstreamcycle.Theseresultsarepromising;however,additionalexperimentsarerequiredtovalidatetheattenuation/amplicationpredictionsofthispaperonthehigherlevelsofacascade.Specically,validatingthepredictionthattheperturbationonthetotalproteinconcentrationisattenuatedasitpropagatesupstreamisparticularlyappealingasitdoesnotdependonthespecicparametervalues.Furthermore,itrequirestomeasurethetotalphosphorylatedprotein,whichisamucheasiertasktoaccomplishthanmeasuringthefreephosphorylatedprotein.Weplantoexperimentallyvalidatethispredictioninourfuturework.AcknowledgmentsDomitillaDelVecchioandHamidR.OssarehwereinpartsupportedbyAFOSRGrantnumberFA9550-09-1-0211.AlejandraC.VenturaandSoaD.MerajverweresupportedbygrantsfromtheDepartmentofDefenseBreastCancerResearch 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CascadesAttenuateRetroactivity19Tables Parameter Intervalforsimulation Intervalfrom(28) Intervalfrom(31) k, k [6.3,600] [150,150] [6.3,600] a,b [18.018,4545.45] [2500,2500] [18.018,4545.45] a, b [25.2,2400] [600,600] [25.2,2400] EiT [0.3,224] [0.3,120] [3.2,224] WiT [3,1200] [3,1200] [180,360] W0 [0.3,100] [0.3,0.3] [100,100] DT [0,1200] - - Table1:ConservativeIntervals.Foreachoftheparametersofthecascade,weindicatetheintervalconsideredforsimulationandtheintervalsgivenin(28)and(31).Forsimu-lation,auniformprobabilitydistributionovereachintervalischosentosampleparametervalues.Also,eachstagehasdi erentparameterseventhoughallextractedfromauniformprobabilitydistribution. 1 2 tot %of i1 71.34 55 79.4 Table2:Three-stagecascadeattenuationpercentage.TheparametersaretakenrandomlyfromTable1. 1 2 tot %of i1with20%variation 100 100 100 %of i1with50%variation 99.98 100 100 %of i1with80%variation 96.895 99.91 100 Table3:Three-stagecascadeattenuationpercentagefordi erentintervalsaboutthenom-inalparametervaluesofBhallaetal.(31). 1 2 tot %of i1with20%variation 77.49 100 100 %of i1with50%variation 65.85 93.32 97.07 %of i1with80%variation 64.69 82.68 90.91 Table4:Three-stagecascadeattenuationpercentagefordi erentintervalsaboutthenom-inalparametervaluesofLevchenkoetal.(32). CascadesAttenuateRetroactivity20 i 1 2 3 4 5 6 7 8 9 tot %of i1 67.3 71.8 72.9 73.3 73.7 74.5 72.9 76.2 59.8 100 Table5:Ten-stagecascadeattenuationpercentagefortheparametervaluesinTable1.FigureLegendsFigure1.AsignalingcascadewithnstagesofPDcycles.ThephosphorylatedproteinWi 1ofstagei 1functionsasakinaseforproteinWiofthenextstagedownstream.DephosphorylationisbroughtaboutbythephosphataseEi.Adownstreampertur-bationintheconcentrationofD,inwhichDcanbeasubstratesharedwithothersignalingpathwaysoraninhibitoroftheactiveenzymeWn,resultsinaperturba-tionofproteinconcentrationinallupstreamstages.Figure2.AblockdiagramrepresentationofthesteadystateresponseofstageitoasmalldownstreamperturbationinDT.Thedownstreamperturbationpropagatesup-streamthroughperturbationsxiinthecomplexesofactiveproteinswiththeirdown-streamsubstrates.Figure3.Attenuationandsign-reversalinathree-stagecascade.Thex-axisshowsthevalueoftheperturbationdTandthey-axisshowsthesteadystatevalueofthere-sultingperturbationsw1,w2,andw3.SimulationisperformedonthefullnonlinearODEmodelgivenbyequation(2).Theparametersofeachstageiaretakenfrom(28)andaregivenbyki=150(min) 1, ki=150(min) 1,ai=2:5(nMmin) 1, ai=600(min) 1,bi=2:5(nMmin) 1, bi=600(min) 1,E3T=120nM,E2T=0:3nM,E1T=0:3nM,W3T=1200nM,W2T=1200nM,W1T=3nM, W0=0:3nM,and DT=0nM.Asaresult,Ki=300nMand Ki=300nM.Figure4.Amplicationinathree-stagecascade.Numericalsimulationofsystem(2):valueofjwijfori2f1;2;3ginresponsetoaunitperturbationdT=1.Thisplotshowsthatviolationofthenecessaryconditionleadstoamplicationofthedownstreamperturbationasittransfersupstreaminthecascade.Parametersofstageiaregivenby:ki=150(min) 1, ki=150(min) 1,ai=2500(nMmin) 1, CascadesAttenuateRetroactivity21 ai=600(min) 1,bi=2500(nMmin) 1, bi=600(min) 1,E3T=120nM,E2T=30nM,E1T=0:3nM,W3T=3nM,W2T=30nM,W1T=1200nM, W0=0:3nM,and DT=0:9nM.Figure5.Percentageofoverallattenuation( tot1)asafunctionofthenumberofstagesinacascadewithparametersrandomlyselectedfromtheintervalsofTable1. CascadesAttenuateRetroactivity22 Figure1 CascadesAttenuateRetroactivity23 Figure2 CascadesAttenuateRetroactivity24 0 200 400 -2 -1.5 -1 -0.5 0x 10-4 (nM)(nM) 0 200 400 0 2 4 6 8 (nM)(nM) 0 200 400 -300 -250 -200 -150 -100 -50 0 (nM)(nM) Figure3 CascadesAttenuateRetroactivity25 1 2 3 -115 -110 -105 -100 -95 -90 -85 -80 -75 -70 -65 stage20log10 Figure4 CascadesAttenuateRetroactivity26 2 3 4 5 6 7 8 9 10 40 50 60 70 80 90 100 number of stages in cascade% Ytot Figure5