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Behaviorsofindividualmicrotubulesandmicrotubulepopulationsrelativetocriticalconcentrations:Dynamicinstabilityoccurswhencriticalconcentrationsaredrivenapartbynucleotidehydrolysis

ErinM.Jonassona,d,1†,AvaJ.Mauroa,b,e,2†,ChunleiLib,3,EllenC.Norbya,4,ShantM.Mahserejianb,5,JaredP.Scripturea,IvanV.Gregorettia,6,MarkS.Alberb,f,

andHollyV.Goodsona,c*

aDepartmentofChemistryandBiochemistry;bDepartmentofAppliedandComputationalMathematicsandStatistics;cDepartmentofBiologicalSciences

UniversityofNotreDame,NotreDame,IN46556;dDepartmentofNaturalSciences,SaintMartin’sUniversity,Lacey,WA98503

eDepartmentofMathematicsandStatistics,UniversityofMassachusettsAmherst,AmherstMA,01003

fDepartmentofMathematics,UniversityofCalifornia,Riverside,CA92521

Keywords:microtubule,dynamicinstability,criticalconcentration,steady-statepolymerShortTitle:Microtubulecriticalconcentrations

PresentAffiliations:1DepartmentofNaturalSciences,SaintMartin’sUniversity,Lacey,WA985032DepartmentofChemistryandBiochemistry,UniversityofNotreDame,NotreDame,IN465563AML,Apple,Sunnyvale,CA940854BiophysicsProgram,StanfordUniversity,Stanford,CA943055PacificNorthwestNationalLaboratory,Richland,WA993526CellSignalingTechnologies,Danvers,MA01923

*Authorforcorrespondence:HollyGoodsonDepartmentofChemistryandBiochemistry251NieuwlandScienceHallNotreDame,IN46556hgoodson@nd.edu(574) 631-7744

†Co-firstauthors

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ARTICLEFILE

TableofContents

Abstract...................3

Articlemaintext........4–31

Acknowledgements........31

References...........32–35

Tables...............36–39

FigureswithLegends...40–56

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ABSTRACT

Theconceptofcriticalconcentration(CC)iscentraltounderstandingbehaviorsofmicrotubulesandothercytoskeletalpolymers.Traditionally,thesepolymersareunderstoodtohaveoneCC,measuredmultiplewaysandassumedtobethesubunitconcentrationnecessaryforpolymerassembly.However,thisframeworkdoesnotincorporatedynamicinstability(DI),andthereisworkindicatingthatmicrotubuleshavetwoCCs.Weuseourpreviouslyestablishedsimulationstoconfirmthatmicrotubuleshave(atleast)twoexperimentallyrelevantCCsandtoclarifythebehaviorsofindividualsandpopulationsrelativetotheCCs.AtfreesubunitconcentrationsabovethelowerCC(CCIndGrow),growthphasesofindividualfilamentscanoccurtransiently;abovethehigherCC(CCPopGrow),thepopulation’spolymermasswillincreasepersistently.OurresultsdemonstratethatmostexperimentalCCmeasurementscorrespondtoCCPopGrow,meaning“typical”DIoccursbelowtheconcentrationtraditionallyconsiderednecessaryforpolymerassembly.Wereportthat[freetubulin]atsteadystatedoesnotequalCCPopGrow,butinsteadapproachesCCPopGrowasymptoticallyas[totaltubulin]increasesanddependsonthenumberofstablemicrotubuleseeds.WeshowthatthedegreeofseparationbetweenCCIndGrowandCCPopGrowdependsontherateofnucleotidehydrolysis.Thisclarifiedframeworkhelpsexplainandunifymanyexperimentalobservations.

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INTRODUCTION

Theconceptofcriticalconcentration(CC)isfundamentaltoexperimentalstudiesofbiologicalpolymers,includingmicrotubules(MTs)andactin,becauseitisusedtodeterminetheamountofsubunitneededtoobtainpolymerandtointerprettheeffectsofpolymerassemblyregulators.Inthestandardframeworkforpredictingthebehaviorofbiologicalpolymers,thereisonecriticalconcentrationofsubunitsatwhichpolymerassemblycommences(e.g.,(Albertsetal.,2015;Mirigianetal.,2013)).However,asindicatedbyotherwork(HillandChen,1984;Walkeretal.,1988),thisframeworkfailstoaccountforthedynamicinstability(DI)displayedbymicrotubulesandotherDIpolymers(e.g.,PhuZ,ParM)(MitchisonandKirschner,1984a;Garneretal.,2004;Erbetal.,2014).OnepurposeoftheworkpresentedhereistoexaminethemanyexperimentalandtheoreticaldefinitionsofCCinordertoshowhowthedefinitionsrelatetoeachother.AnotherpurposeistoclarifyhowthebehaviorsofindividualdynamicallyunstablefilamentsandtheirpopulationsrelatetoeachotherandtotheexperimentalmeasurementsofCC.Toaddresstheseproblems,wecomputationallymodeledsystemsofdynamicmicrotubuleswithoneofthetwoendsofeachMTfixed(e.g.,asoccursforMTsgrowingfromcentrosomes)andperformedanalysesthataredirectlycomparabletothoseusedinexperiments.Asignificantadvantageofcomputationalmodelingforthisworkisthatitallowssimultaneousexaminationofthebehaviorsofindividualsubunits,individualmicrotubules,andthepopulation’sbulkpolymermass.

TraditionalunderstandingofCriticalConcentration(CC)basedonequilibriumpolymersTraditionally,“thecriticalconcentration”isunderstoodtobetheconcentrationofsubunitsneededforpolymerassemblytooccur(CCPolAssem,measuredbyQ1inFigure1A,D);equivalently,theCCisdefinedastheconcentrationoffreesubunitsleftinsolutiononcepolymerassemblyhasreachedasteady-statelevel(CCSubSoln,measuredbyQ2inFigure1A,D).Thissetofideasisbasedonearlyempiricalobservationswithactin(Oosawaetal.,1959).TheseobservationswereinitiallygivenatheoreticalframeworkbyOosawaandcolleagues,whoexplainedthebehaviorofactinbydevelopingatheoryfortheequilibriumassemblybehaviorofhelicalpolymers(OosawaandKasai,1962;Oosawa,1970).ThisequilibriumtheorywasextendedtotubulinbyJohnsonandBorisy(JohnsonandBorisy,1975).

Forequilibriumpolymers,theCCiscommonlydefinedaskoff/kon=KD,wherekonandkoffaretherateconstantsforattachment/detachmentofasubunitto/fromafilamenttip;polymerwillthenundergonetassemblywhenkon*[freesubunit]isgreaterthankoff(Table1).Theideathatpolymerassemblycommencesatthecriticalconcentrationisnowusedroutinelytodesignandinterpretexperimentsinvolvingcytoskeletalpolymers(e.g.(Amayedetal.,2002;Bueyetal.,2005;Wieczoreketal.,2015;Díaz-Celisetal.,2017;Schummeletal.,2017;Concha-Marambioetal.,2017)),anditisastandardtopicincellbiologytextbooks(e.g.,(Albertsetal.,2015;Lodishetal.,2016)).Overtime,asetofexperimentalmeasurementsanddefinitionsofcriticalconcentrationhaveemerged(Table1,Figure1),allofwhichwouldbeequivalentforanequilibriumpolymer.

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NucleotidehydrolysisallowsmicrotubulestoexhibitdynamicinstabilityMicrotubules(composedofsubunitscalledtubulindimers)aresteady-statepolymers,notequilibriumpolymers,becausetheyrequireaconstantinputofenergyintheformofGTP(guanosinetriphosphate)nucleotidestomaintaina(highly)polymerizedstate.Microtubulesexhibitabehaviorknownasdynamicinstability(DI),inwhichtheystochasticallyswitchbetweenphasesofgrowthandshorteningviatransitionsknownascatastropheandrescue(Figure1E)(MitchisonandKirschner,1984a;Walkeretal.,1988).TheDIbehaviorofMTsisdrivenbyGTPhydrolysis(conversionofGTP-tubulintoGDP-tubulin):tubulinsubunitscontainingGTPassembleintoMTs,whiletubulinsubunitscontainingGDPdonot(thekonandkoffvaluesforGTP-tubulindifferfromthoseforGDP-tubulin).Incontrast,tubulinsubunitscontainingnon-orslowly-hydrolyzableGTPanalogs(e.g.,GMPCPP)assembleintostableMTsthatdonotdisplayDI(Hymanetal.,1992).ThoughsomedetailsaboutthemechanismofDIremainunclear,theconsensusexplanationforDIbehavioristhatgrowingMTshaveacapofGTP-tubulinsubunits(the“GTPcap”)thatstabilizestheunderlyingGDP-tubulinlattice.TheMTsswitchtorapiddisassembly(i.e.,undergocatastrophe)whentheylosetheirstabilizingcap,exposingtheunstableGDP-tubulinlatticebelow.WhenMTsregaintheircap,theyundergorescue(transitionfromshorteningtogrowth)(reviewedin(GoodsonandJonasson,2018)).Onthesurface,itmayseemreasonabletoapplythetraditionalcriticalconcentrationframeworkasoutlinedabove(seealsoTable1)tounderstandingDIpolymerslikemicrotubulesbecausethisframeworkisfoundedontheory(albeitequilibriumpolymertheory)andappearstobeconsistentwithmanyexperimentalresults(Howard,2001).AproblemwiththisapproachisthatitleavesopenvariousquestionsregardinghowdynamicinstabilityandenergyutilizationfitintothetraditionalCCframework.Forexample,howdoestheDIbehaviorofanindividualfilamentinFigure1B,Erelatetothepopulation-levelbehaviorinFigure1A,C?Isthereoneexperimentallyrelevantcriticalconcentration(asassumedfromequilibriumpolymertheory)ormorethanone?Ifmorethanone,howmany?Morebroadly,whydosomesteady-statepolymers(e.g.,microtubules)displaydynamicinstability,whileothers(e.g.,actin)donot?Asonemightimagine,thesequestionshavebeenstudiedpreviously,butambiguityinunderstandingcriticalconcentrationstillexists.AbriefsummaryofsomekeypreviouseffortsonCCforMTsisasfollows:•Inthe1980s,Hillandcolleaguesinvestigatedsomeofthequestionsoutlinedaboveandworkedtodevelopatheoryofsteady-statepolymerassembly.Theirconclusionsincludedtheideathatgrowthofmicrotubulesisgovernedbytwodistinctcriticalconcentrations:alowerCCatwhich“themeansubunitfluxperpolymer”during“phase1”(growthphase)equalszeroandanupperCCatwhich“themeannetsubunitfluxperpolymer”iszero(similartoFigure1C)(e.g.,(HillandChen,1984),elaboratedonin(Hill,1987)).However,thepublishedworkdidnotclarifyforreadersthebiologicalsignificanceofthesetwoCCsnorhowtheyrelatetothebehaviorsofindividualfilamentsandtheirpopulations.•Laterinthe1980s,Walkeretal.usedvideomicroscopytoanalyzeindetailthebehaviorofindividualMTsundergoingdynamicinstability.Theydemonstratedthatmicrotubulesobservedinvitrohavea“criticalconcentrationforelongation”(CCelongation),whichtheydescribedastheconcentrationatwhichtherateoftubulinassociation(!ongrowth[freetubulin])isequaltotherate

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ofdissociation(!offgrowth)duringtheelongationphase(Walkeretal.,1988)(Figure1B;Table1anditsfootnotes).Consequently,attubulinconcentrationsbelowCCelongation,thereisnoelongation.Laterinthissamepaper,theauthorsdiscussedtheexistenceofahighercriticalconcentrationabovewhichapopulationofpolymerswillundergo“netassembly”.Thus,theanalysisinthismanuscriptclearlyindicatesthatmicrotubuleshavetwocriticalconcentrations.However,thisconclusionisnotstatedexplicitly,andthemanuscriptdoesnotaddressthequestionofhoweitherofthetwoWalkeretal.CCsrelatestothetwoCCspredictedbyHill.•Inthe1990s,Dogterometal.andFygensonetal.usedacombinationofmodeling(DogteromandLeibler,1993)andexperiments(Fygensonetal.,1994)toshowthatthereisa“criticalvalueofmonomerdensity,c=ccr”,abovewhichmicrotubulegrowthis“unbounded”(i.e.,theaveragelengthincreasesindefinitelyanddoesnotleveloffwithtime)(DogteromandLeibler,1993;Fygensonetal.,1994;Dogterometal.,1995).Hereafter,werefertothisccrasCCunbounded.Dogterometal.alsoprovidedequations(similartothoseproposedinitiallyby(HillandChen,1984)and(Walkeretal.,1988))thatcanbeusedtorelateCCunbounded,whichisapopulation-levelcharacteristic,tothedynamicinstabilityparameters(Figure1E-F),whichdescribeindividual-levelbehaviors.OneofthemanysignificantoutcomesofthesepaperswasthattheyencouragedreaderstothinkabouthowsmallchangestoDIparameters(e.g.,ascausedbyregulatorychangestoMTbindingproteins)couldchangethebehaviorofasystemofMTs,especiallyinacellularcontext.However,theimplicationsofthesearticlesforunderstandingcriticalconcentrationsmorebroadlyremainedpoorlyappreciatedbecausetheydidnotexplicitlyrelateCCunboundedtothemoreclassicalCCdefinitionsandmeasurementsinTable1ortothosediscussedby(HillandChen,1984)and(Walkeretal.,1988).Thus,althoughdynamicinstabilityhasbeenstudiedformorethan30years,confusionremainsabouthowthetraditionallyequivalentdefinitionsofcriticalconcentrationandtheinterpretationofCCmeasurementsshouldbeadjustedtoaccountfordynamicinstability.Remarkably,theliteratureasyetstilllacksacleardiscussionofhowtheCCelongationandCCunboundedmentionedaboverelatetoeachother,totheCCspredictedbyHill,ortotheclassicalexperimentalmeasurementsofCCdepictedinFigure1A.HowmanydistinctCCsareproducedbythedifferentexperimentallymeasurablequantities(Qvalues,Figure1andTable1),whatisthepracticalsignificanceofeach,andwhichmeasurementsyieldwhichCC?Howdoanyofthesevaluesrelatetobehaviorsatthescalesofsubunits,individualmicrotubules,andthebulkpolymermassofpopulationsofmicrotubules?HowdoesdynamicinstabilitybehaviorrelatetotheseparationbetweendistinctCCs?Undoubtedly,manyresearchershaveanintuitiveunderstandingoftheanswerstoatleastsomeofthesequestions.However,theobservationthatevenrecentliteraturecontainsmanyreferencesto“the”CCformicrotubuleassembly(e.g.,(Wieczoreketal.,2015;Alfaro-AcoandPetry,2015;Hussmannetal.,2016;Schummeletal.,2017)indicatesthatthisproblemdeservesattention.Whiletheseissuesareinterestingfromabasicscienceperspective,theyalsohavesignificantpracticalrelevance:properdesignandinterpretationofexperimentsthatinvolveperturbingmicrotubuledynamics(e.g.,characterizationofMT-directeddrugsorproteins)requiresanunambiguousunderstandingofcriticalconcentrationsandhowtheyaremeasured(e.g.,(Verdier-Pinardetal.,2000;Bonfilsetal.,2007;Hussmannetal.,2016;CytoskeletonInc.).

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ComputersimulationsasanapproachtoaddressingthesequestionsToinvestigatetheconceptofcriticalconcentrationasitappliestodynamicallyunstablepolymers,wehaveusedcomputationalmodeling.Computationalmodelsareidealforaddressingthistypeofproblembecausethebiochemistryofthereactionscanbeexplicitlycontrolledand"experiments"canbeperformedquicklyandeasily.Furthermore,itispossibletosimultaneouslyfollowthebehaviorofthesystematallrelevantscales:addition/lossofindividualsubunitsto/fromthefreeend,dynamicinstabilityofindividualfilaments,andanychangesinpolymermassofthepopulationoffilaments.Incomparison,itischallengingtoaddressthesequestionsusingphysicalsystemsbecauseexperimentshavethusfarbeenlimitedtechnicallytomeasurementsatone(oratmosttwo)ofthesescalesatatime.Asdescribedmorebelow(Resultssubsection“ComputationalModels”andFigure2),therulesandconditionscontrollingoursimulationscorrespondtothosethatwouldbesetbytheintrinsicpropertiesofthebiologicalsystem(e.g.,kineticrateconstants)orbytheexperimenter(e.g.,totaltubulinconcentration).Typicalexperimentalresults(DIparameters,concentrationsoffreeandpolymerizedtubulin)areemergentpropertiesofthesystemofbiochemicalreactions,justastheywouldbeinaphysicalexperiment.SummaryofConclusionsUsingthesesystemsofsimulatedmicrotubules,weshowthatclassicalinterpretationsofexperimentssuchasthoseinFigure1canbemisleadingintermsofunderstandingthebehaviorofindividualMTs.Inparticular,weusethesimulationstoillustratethefactthatdynamicallyunstablepolymerslikemicrotubulesdohave(atleast)twomajorexperimentallydistinguishablecriticalconcentrations,asoriginallyproposedbyHillandcolleagues(summarizedin(Hill,1987)).WeclarifyhowtheCCsrelatetobehaviorsofindividualMTsandpopulationsofMTs.At[freetubulin]abovethelowerCC,extendedgrowthphasesofindividualfilamentscanoccurtransiently.At[freetubulin]abovethehigherCC,thepolymermassofalargepopulationwillincreasesteadily,evenwhileindividualfilamentsinthepopulationpotentiallystillexhibitdynamicinstability.WeshowthatthelowercriticalconcentrationcorrespondstoCCelongationasmeasuredbyWalkeretal.(Table1;(Walkeretal.,1988)),whichcanbedescribedasthefreetubulinconcentrationabovewhichindividualMTsareabletoelongateduringthegrowthphase.ThisCCcanbemeasuredbyexperimentalquantityQ3inFigure1B.ThehigherCCcorrespondstoCCunboundedasidentifiedbyDogterometal.,i.e.,theconcentrationoffreetubulinabovewhich“unboundedgrowth”occurs(Table1;(DogteromandLeibler,1993;Fygensonetal.,1994;Dogterometal.,1995).ThisupperCCcanbemeasuredbyQ1,Q2,andQ4inFigure1A,C.ToclearlydistinguishthesetwoCCsandavoidconfusingeitherwithasituationwhereaphysicalboundaryisinvolved,wesuggestcallingthemCCIndGrowandCCPopGrow,respectively.1InadditiontothesetwoexperimentallyaccessibleCCs,therearetwomoreCCs(perhapsnotexperimentallyaccessible)thatcorrespondtotheKDfor

1ThismanuscriptfocusesonsystemscomposedofMTswithoneendfreeandtheotherendanchored,suchaswouldexistforMTsgrowingfromcentrosomes.Inothercases,microtubulescanhavetwofreeends(plusandminus).ForeachofCCIndGrowandCCPopGrow,thenumericalvalueattheplusendcouldpotentiallydifferfromthevalueattheminusend.

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theGTPandGDPformsoftubulinsubunits.WesuggestcallingtheseCCKD_GTPandCCKD_GDP,respectively.Whileourstudiesfocusonmicrotubules,wesuggestthatthesecriticalconcentrationdefinitionsandinterpretationscanapplytosteady-statepolymersmoregenerallybutareespeciallysignificantforthosethatexhibitdynamicinstability.Weshowthatmostexperimentsintendedtomeasure“theCC”actuallymeasureCCPopGrow(i.e.,thehigherCC).Thisconclusionmeansthat“typical”microtubuledynamicinstability(whereMTsgrowanddepolymerizebacktotheseed)islimitedtoconcentrationsbelowwhathastraditionallybeenconsidered"the"CCneededforpolymerassembly.Furthermore,weshowthatincompetingsystems(i.e.,closedsystemswhereMTscompeteforalimitedtotalnumberoftubulinsubunits),theconcentrationoffreetubulinatsteadystate([freetubulin]SteadyState)doesnotequalCCPopGrowaswouldbeexpectedfromtraditionalinterpretationsofclassicCCexperiments(Figure1A).Instead,[freetubulin]SteadyStateasymptoticallyapproachesCCPopGrowas[totaltubulin]increases.Inaddition,wedemonstratethatthedegreeofseparationbetweenCCIndGrowandCCPopGrowdependsontheGTPhydrolysisrateconstant(kH).WealsoshowthatCCIndGrowcandifferfromCCKD_GTP,contrarytopreviousassumptionsthatgrowingMTsalwayshaveGTP-tubulinattheirtips(topmostsubunits)(e.g,(Bowne-Andersonetal.,2015)).Finally,wedemonstratethatdynamicinstabilityitselfcanproduceresults(e.g.,sigmoidalseedoccupancyplots)previouslyinterpretedasevidencethatgrowthfromstableseedsrequiresanucleationstep. RESULTSComputationalModelsInthiswork,weusedbotha“simplified”modelofMTdynamics,inwhichMTsaremodeledassimplelinearpolymers(Gregorettietal.,2006),anda“detailed”model,wheremicrotubulesarecomposedof13protofilaments,withlateralandlongitudinalbondsbetweensubunits(tubulindimers)modeledexplicitly(Margolinetal.,2011;Margolinetal.,2012)(Figure2).Thesimulationsweredesignedtobeintuitivelyunderstandabletoresearchersfamiliarwithbiochemicalaspectsofcytoskeletalpolymers.Consequently,therulesgoverningthesimulationscorresponddirectlytobiochemicalreactionkinetics.KeyelementsofthesemodelsaredescribedinBox1.Weutilizeboththesimplifiedanddetailedcomputationalmodelsbecauseeachhasparticularstrengthsforaddressingproblemsrelatedtomicrotubuledynamics.Thesimplifiedsimulationhasfewerkineticparameters,allofwhicharedirectlycomparabletoparametersintypicalanalyticalmodels(i.e.,mathematicalequations).Thus,thesimplifiedsimulationisusefulfortestinganalyticalmodelpredictionsrelatingbiochemicalpropertiestoindividualfilamentlevelandbulkpopulationlevelbehaviors.Incontrast,theincreasedresolutionofthedetailedmodelisimportantfortestingthegeneralityandrelevanceofconclusionsderivedfromthesimplifiedmodel.

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Inaddition,theinputtedkineticrateconstantsinthetwomodelsweretunedtoproducedynamicinstabilitybehaviorthatisquantitativelydifferentbetweenthetwomodels,soitfollowsthatthespecificnumericalvaluesforcriticalconcentrationsextractedfromthesetwosimulationswillbedifferent.However,asdiscussedmorebelow,thebehavioralchangesthatoccurateachCCarequalitativelysimilarinthetwomodels.Thus,thesetwomodelsenableustodeterminewhichconclusionsaregeneralandtoavoidmakingconclusionsthatarespecifictoparticularparametersetsorpolymertypes.

Box1:Keyelementsofthetwocomputationalmodels(simplifiedanddetailed)usedinthisstudy

•Subunitaddition/lossandGTPhydrolysis(bothmodels)andlateralbondformation/breaking(detailedmodelonly)aremodeledasstochasticeventsthatoccuraccordingtokineticrateequationsbasedonthebiochemistryoftheseprocesses(Figure2)(Gregorettietal.,2006;Margolinetal.,2012).

•Theuser-defined(adjustable)parameterscorrespondtothefollowing:(a)thebiochemistryoftheproteinsbeingstudied(i.e.,kineticrateconstantsforthereactionslistedabove);and(b)attributesoftheenvironmentthatwouldbesetbyeithertheexperimenterorthecell(e.g.,theconcentrationoftubulininthesystem,whetherthesystemiscompeting(closed)ornon-competing(open),thenumberofstableseeds,andthesystemvolume).

•Asinphysicalexperiments,emergentpropertiesofthesimulatedsystemsincludethedynamicinstabilityparameters(Vg,Vs,Fcat,Fres,seeFigure1E-F)andtheconcentrationsoffreeandpolymerizedtubulinatsteadystate.Inparticular,transitionsbetweengrowthandshortening(catastropheandrescue)arespontaneousprocessesthatoccurwhenthestabilizingGTPcaphappenstobelostorregainedasaresultofthebiochemicalreactionsdescribedabove.

•Becausemicrotubulesincellsandinmanyinvitroexperimentsgrowfromstableseeds(nucleationsitessuchascentrosomes,axonemes,orGMPCPPseeds),oursimulationsassumethatoneendofeachMTisfixed(aswouldbethecaseforgrowthfromcentrosomes),andthatalladditionandlossoccuratthefreeend.Inoursimulations,theseedsarecomposedofnon-hydrolyzableGTP-tubulin.Exceptwhereotherwisenoted,thenumberofstableseedswassetto100inthesimplifiedmodeland40inthedetailedmodel.

•Bothsimulationsspontaneouslyundergothefullrangeofdynamicinstabilitybehaviors(includingrescue),andtheycansimulatesystemsofdynamicmicrotubulesforhoursofsimulatedtime(Gregorettietal.,2006;Margolinetal.,2012).

•Thebehaviorsoftheevolvingsystemsofdynamicmicrotubulescanbefollowedatthescalesofsubunits,individualfilaments,orpopulationsoffilaments.

•ThekineticrateconstantsusedasinputparametersforthedetailedmodelwerepreviouslytunedtoapproximatetheDIparametersofmammalianbrainMTsinvitro(Margolinetal.,2012).Thesimplifiedmodelparametersusedherearemodifiedfromthoseof(Gregorettietal.,2006)andwerechosenforuseherebecausetheyproduceDIbehaviorthatisquantitativelydifferentfromthatofthedetailedmodel.

Thesumoftheseattributesmakethesesimulationsidealforstudyingtherelationshipsbetweentheconcentrationoftubulin,thebehaviorsofindividualMTs,andbehaviorsofsystemsofdynamicMTs.SeetheMethods,SupplementaryInformation,and(Gregorettietal.,2006;Margolinetal.,2012)foradditionaldetailsincludinginputparameters.

ApproachtounderstandingtherelationshipbetweenmicrotubulebehaviorsandcriticalconcentrationsToclarifytheconceptofcriticalconcentrationasitappliestomicrotubules,weexaminedwhichofthecommonlyusedcriticalconcentrationdefinitions(outlinedinTable1)aremeaningfulwhenstudyingmicrotubules,andforthesetthataremeaningful,whichareequivalent.The

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termcriticalconcentrationcanhaveaspecificthermodynamicmeaningasthesoluteconcentrationatwhichaphasechangeoccurs.Herewedefinethetermoperationally,astheconcentrationatwhichabehavioralchangeoccurs.TodeterminehowthevariousCCdefinitionsrelatetoeachotherandtodynamicinstability,weusedthesimulationstosimultaneouslyexaminethebehaviorsofindividualMTsandpopulationsofMTs.Morespecifically,weransetsofsimulationsforboththesimplifiedanddetailedmodelsatvarioustubulinconcentrationsinbothcompetingsystems(closedsystemswithconstant[totaltubulin],asmighthappeninatesttube)andnon-competingsystems(opensystemswithconstant[freetubulin],similartowhatmighthappeninamicroscopeflowcell).Thisapproachmimicsvariousexperiments(Table2)thatareclassicallyusedtomeasuremicrotubulecriticalconcentration(Table1).Wethenassessedandcomparedthebehaviorsoftheindividualmicrotubules(e.g.,DIparameters),population-levelproperties(e.g.,[freetubulin]atsteadystate),andcriticalconcentrationsasdeterminedbythetraditionaldefinitions(Table1).Fortheworkpresentedhere,itisimportanttorecognizethattherelevantobservationsarethebehaviorsofthesystemsatdifferentscalesandtheconcurrence(ordisagreement)betweenthevaluesofCCthatresultfromvariousdefinitionsormeasurementapproaches;thespecificnumericalCCvaluesobservedaresimplyoutcomesoftheparticularinputkineticrateconstantsusedandsoarenotbythemselvessignificant.ThissituationisanalogoustophysicalMTs,whereDIparametersandCCvaluesdependontheproteinsequences,temperatures,andbufferconditionsused(e.g.,(Williamsetal.,1985;Gildersleeveetal.,1992;Fygensonetal.,1994;Hussmannetal.,2016;Schummeletal.,2017)).WeusethetermsQ1,Q2,etc.torefertospecificexperimentallymeasurablequantities(i.e.,valuesobtainedthroughexperimentalapproachesasindicatedinthefigures),andthetermsCCKD,CCPolAssem,CCSubSoln,etc.torefertotheoreticalvalues(concepts)thatmayormaynotcorrespondtoparticularexperimentallymeasurablequantitiesandmayormaynotbeequivalent.Table1summarizestraditionalcriticalconcentrationdefinitionsandmeasurementsusedintheliterature.Table3summarizesourclarificationsofcriticalconcentrationdefinitionsandadditionalQvaluemeasurementsbasedontheresultsthatwillbepresentedinthiswork.Addressingtheideathat“the”criticalconcentrationiskoff/kon(CCKD)Theideathat“theCC”istheKDforadditionofsubunitstopolymer(i.e.,CC=koff/kon=KD;CC=CCKD;Table1)isaseriousoversimplificationwhenappliedtomicrotubules.Thoughthisformulaisfrequentlystatedintextbooks,itiswell-recognizedthatthisrelationshipcannotbeappliedinastraightforwardwaytopopulationsofdynamicmicrotubules,ortosteady-statepolymersmoregenerally(Albertsetal.,2015).Morespecifically,experimentallyobservedcriticalconcentrationsforsystemsofdynamicMTs(howevermeasured)cannotbeequatedtosimplekoff/kon=KDvaluesbecausetheGTPandGDPformsoftubulinhavesignificantlydifferentvaluesofkoff/kon.Forexample,thecriticalconcentrationforGMPCPP(GTP-like)tubulinhasbeenreportedtobelessthan1µM(Hymanetal.,1992),whilethatforGDP-tubulinisveryhigh,perhapsimmeasurablyso(Howard,2001).

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ExactlyhowthemeasuredCCvalue(s)forasystemofdynamicmicrotubulesrelatetotheKDvaluesforGTP-andGDP-tubulinhasnotpreviouslybeenestablished.However,intuitionsuggeststhatanyCCsmustliebetweentherespectiveKDvaluesforGTP-andGDP-tubulin(Howard,2001).Consistentwiththisidea,experimentallyreportedvaluesformammalianbraintubulinCCtypicallyliebetween~1and~20µM(Verdier-Pinardetal.,2000;Bonfilsetal.,2007;Mirigianetal.,2013;Wieczoreketal.,2015).NotethatwhiletheideathatCC=KDcannotapplyinasimplewaytoasystemofdynamicmicrotubules,itcanapplytotubulinpolymersintheabsenceofhydrolysis,whereassemblyisanequilibriumphenomenon.ExamplesincludesystemscontainingonlyGDP-tubulin(whenpolymerizedwithcertaindrugs)ortubulinboundtonon-/slowly-hydrolyzableGTPanalogs(e.g.,GTP-γS,GMPCPP)(Hymanetal.,1992;Díazetal.,1993;Bueyetal.,2005).

DIpolymersgrowatconcentrationsbelowstandardexperimentalquantitiescommonlythoughttomeasurethecriticalconcentrationforpolymerassemblyAtypicalwaytomeasure“thecriticalconcentration”formicrotubuleassemblyistodeterminethe[totaltubulin]atwhichpolymerassemblesinacompeting(closed)experimentsuchasthatportrayedinFigure1A,whereQ1measurestheCCforpolymerassembly(CCPolAssem)(seee.g.,(Mirigianetal.,2013)).Analternativeapproachtreatedasequivalentistomeasuretheconcentrationoffreetubulinleftinsolutiononcesteady-statepolymerassemblyhasoccurred(Figure1A,Q2),traditionallyconsideredtoyieldCCSubSoln(Mirigianetal.,2013).Inotherwords,theexpectationisthatQ1≈Q2,andthattheseexperimentallyobtainedquantitiesprovideequivalentwaystomeasurethecriticalconcentrationforpolymerassembly,whereCCPolAssem=CCSubSoln(Table1).WetestedthesepredictionsbyperformingsimulationsofcompetingsystemswhereindividualMTsgrowingfromstableseedscompeteforalimitedpooloftubulin(i.e.,[totaltubulin]isconstant).Thissituationisanalogoustoatest-tubeexperimentinwhichmicrotubulesgrowfrompre-formedMTseeds,andboth[polymerizedtubulin]and[freetubulin]aremeasuredafterthesystemhasreachedpolymer-masssteadystate(FigureS1A-D).2Atfirstglance,thebehaviorofthesystemsofsimulatedmicrotubulesmightseemconsistentwiththatexpectedfromcommonunderstanding(Figure1A):significantpolymerassemblywasfirstobservedat[totaltubulin]≈Q1,andQ1≈Q2(Figure3A-B).However,closerexaminationofthesedatashowsthatthereisnosharptransitionateitherQ1orQ2(Figure3A-B),astraditionallyexpected(Figure1A).Significantly,smallbutnon-zeroamountsofpolymerexistat[totaltubulin]belowreasonableestimatesforQ1(Figure3A-B,S1E-F).Inaddition,thesteady-stateconcentrationoffreetubulin([freetubulin]SteadyState)isnotconstantwithrespectto[totaltubulin]for[totaltubulin]>Q1asisoftenassumed.Instead, 2Polymer-masssteadystatedescribesasituationwherethepolymermasshasreachedaplateauandnolongerchangeswithtime(otherthansmallfluctuationsaroundthesteady-statevalue)(FigureS1A-D).Systemsofdynamicmicrotubulescanalsohaveothersteadystates(e.g.,polymer-lengthsteadystate)(seealso(Mourãoetal.,2011)).

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[freetubulin]SteadyStateapproachesanasymptoterepresentedbyQ2(Figure3A-B).Nonetheless,Q1isstillapproximatelyequaltoQ2.3Consistentwiththeseobservations,examinationofindividualMTsinthesesimulationsshowsMTsgrowingandexhibitingdynamicinstabilityat[totaltubulin]belowQ1≈Q2(Figure3C-D;comparetoFigure3A-B).Thesedata(Figure3)suggestthatoneofthemostcommonlyacceptedpredictionsoftraditionalcriticalconcentrationunderstandingisinvalidwhenappliedtosystemsofdynamicmicrotubules:insteadofbothQ1andQ2providinganexperimentalmeasureoftheminimumconcentrationoftubulinneededforpolymerassembly(CCPolAssem),neitherdoes,sinceMTsexhibitingdynamicinstabilityappearatconcentrationsbelowQ1≈Q2.Correspondingly,theresultsinFigure3A-BindicatethatthecriticalconcentrationcalledCCSubSolnwouldbemoreaccuratelydefinedastheasymptoteapproachedbythe[freetubulin]SteadyStateas[totaltubulin]increases,notthevalueof[freetubulin]SteadyStateitself(Figure1A).ThenumberofstableMTseedsimpactsthesharpnessofthetransitionatQ1andQ2.WhyisthetransitionatQ1andQ2inFigure3A-BmoregradualthanthetheoreticaltransitionasdepictedinFigure1A?Previousresultsofoursimplifiedmodel(Gregorettietal.,2006)andothermodels(e.g.,(VorobjevandMaly,2008;Mourãoetal.,2011))indicatethat[freetubulin]SteadyStatedependsonthenumberofstableMTseeds.Therefore,weinvestigatedhowchangingthenumberofstableMTseedsaffectstheshapeofthecurvesinclassicalCCplots.Examinationoftheresults(Figure4A-B,zoom-insinFigure4C-D)showsthatchangingthenumberofMTseedsdoeschangethesharpnessofthetransitionatQ1andQ2.Morespecifically,whenthenumberofMTseedsissmall,arelativelysharptransitionisseenatbothQ1andQ2ingraphsofsteady-state[freetubulin]and[polymerizedtubulin];littleifanybulkpolymerisobservedat[totaltubulin]belowQ1(Figure4,fewerseeds,darkestcurves;similartoFigure1A).Incontrast,whenthenumberofMTseedsishigh,measurableamountsofpolymerappearatconcentrationswellbelowQ1,andconsequently[freetubulin]SteadyStateapproachestheQ2asymptotemoregradually(Figure4,moreseeds,lightestcurves).Moreover,thedataforvariousnumbersofseedsallapproachthesameasymptotes(greydashedlines,Figure4).TheseobservationsindicatethatthenumberofMTseedsdoesnotimpactthevalueofQ1≈Q2,butdoesaffecthowsharplysteady-state[freetubulin]approachestheQ2asymptote.

3Sincethetransitionsarenotsharp,itcanbedifficulttodeterminetheexactvaluesofQ1andQ2.Dependingonhowthemeasurementsareperformed,thevaluesofQ1andQ2mightappeardifferentfromeachother.However,Q1=Q2doesholdifthemeasurementsareperformedasfollows:Q2isthevalueofthehorizontalasymptotethat[freetubulin]SteadyStateapproachesas[totaltubulin]increases;Q1isthe[polymerizedtubulin]=0interceptofthelinewithslope1that[polymerizedtubulin]approachesas[totaltubulin]increases(Figure3A-B).NotethatQ1wouldbeexactlyequaltoQ2inasystemwithnomeasurementerror,nonoise,andnonon-functionaltubulin,butforaphysicalexperimentthesefactorscaninterferewiththemeasurements.

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Theobservationsthusfarraiseaquestion:SinceCCSubSolnisnottheminimumtubulinconcentrationneededforpolymerassembly(CCPolAssem),whatisthesignificanceofQ1≈Q2≈CCSubSolnformicrotubulebehavior?AcriticalconcentrationforpersistentgrowthofMTpopulations(CCPopGrow)ToinvestigatethesignificanceofQ2,i.e.,theasymptoteapproachedby[freetubulin]SteadyStateas[totaltubulin]isincreased(Figures3A-B,4),weexaminedthedependenceofMTbehaviorontheconcentrationoffreetubulininnon-competingsimulations.Forthesestudies,wefixed[freetubulin]atvariousvaluesinsteadofallowingpolymergrowthtodepletethefreetubulinovertime.ThissetofconditionsisanalogoustoalaboratoryexperimentinvolvingMTspolymerizingfromstableseedsinaconstantlyreplenishingpooloffreetubulinataknownconcentration,suchasmightexistinaflowcell.Asdescribedabove,Q1andQ2fromcompetingsystemsdonotyieldthecriticalconcentrationforpolymerassembly(CCPolAssem)asexpectedfromtraditionalunderstanding.Instead,comparisonwiththenon-competingsimulations(Figure5)showsthatQ1andQ2correspondtoadifferentCC,whichcanbedescribedasthe[freetubulin]abovewhichindividualMTswillexhibitnetgrowthoverlongperiodsoftime(Figure5A-B).Equivalently,thisCCcanbedescribedasthe[freetubulin]abovewhichthepolymermassofalargepopulationofMTswillgrowpersistently(weusethisterminologybasedontheexperimentally-observed“persistentgrowth”in(Komarovaetal.,2002)).Asdiscussedmorebelow,thisCCisthesameasthatpreviouslyidentifiedbyDogterometal.astheCCatwhichthetransitionfrom“boundedgrowth”to“unboundedgrowth”4occurs(DogteromandLeibler,1993;Fygensonetal.,1994;Dogterometal.,1995),byWalkeretal.astheCCfor“netassembly”(Walkeretal.,1988),andbyHilletal.astheCCwherenetsubunitfluxequalszero(HillandChen,1984).Toavoidimplyingthataphysicalboundaryisinvolved,wesuggestidentifyingthisCCasthecriticalconcentrationforpersistentmicrotubulepopulationgrowth(CCPopGrow).CCPopGrowcanbemeasuredbyQ5a,the[freetubulin]atwhichthenetrateofchangeinaverageMTlength(Figure5C-D,leftaxes)orinpolymermass(Figure5C-D,rightaxes)transitionsfromequalingzerotobeingpositive.AdditionalapproachestomeasuringCCPopGrowarediscussedlater.HowMTbehaviorsrelatetoCCPopGrow.ExaminationofFigure5showsthatMTpolymerizationbehaviorinnon-competingconditions(i.e.,where[freetubulin]isconstant)canbedividedintotworegimes:Polymer-masssteadystate:AtconcentrationsoffreetubulinbelowCCPopGrow(measuredby

Q5a),boththeaverageMTlengthand[polymerizedtubulin]withinapopulationreachsteady-statevaluesthatincreasewith[freetubulin]butareconstantwithtime(Figures5C-D,S3A-B).Individualmicrotubulesinthesesystemsexhibitwhatmightbecalled“typical”

4Note:Herea“bounded”systemreferstoonethathasaconstantsteady-statepolymermassoraverageMTlength;“unbounded”referstoasystemwherethepolymermassoraverageMTlengthexhibitsnetgrowthovertime(DogteromandLeibler,1993;Dogterometal.,1995).ThissituationshouldnotbeconfusedwithoneinwhichthesystemofMTsexperiencesaphysicalboundary(e.g.,MTsincells).

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dynamicinstability:theyundergoperiodsofgrowthandshortening,buttheyeventuallyandrepeatedlydepolymerizebacktothestableMTseed(Figure5A-B).

Polymer-growthsteadystate:AtCCPopGrow,thepopulationsofdynamicMTsundergoamajor

changeinbehavior:theybegintogrowpersistently.Morespecifically,when[freetubulin]isabovelabelQ5ainFigure5C-D,thereisnopolymer-masssteadystatewhere[polymerizedtubulin]isconstantovertime(FigureS3A-B).Instead,thesystemofMTsarrivesatadifferenttypeofsteadystatewhere[polymerizedtubulin]increasesataconstantrate(Figures5C-D,S3A-B).IndividualMTswithinthesepopulationsstillexhibitdynamicinstability(exceptperhapsatveryhigh[freetubulin]),buttheyexhibitunboundedgrowth(DogteromandLeibler,1993)(alsodescribedasnetassembly(Walkeretal.,1988))iftheirbehaviorisassessedoversufficienttime(Figure5A-B).

Significantly,forbothsimulations,Q5a(Figure5C-D)liesatapproximatelythevalueofQ1≈Q2(Figures3A-B).Thisobservationindicatesthatsteady-state[freetubulin]incompetingsystemsasymptoticallyapproachesthesame[freetubulin]atwhichmicrotubulesbegintoexhibitnetgrowth(i.e.,unboundedgrowth)innon-competingsystems.Inotherwords,thesedatashowthatCCSubSoln≈CCPopGrow.Thisconclusionmeansthatclassicalmethodsformeasuring“theCCforpolymerassembly”donotyieldtheCCatwhichindividualDIpolymersappear(astraditionallyexpected),butinsteadyieldtheCCatwhichpopulationsofpolymersgrowpersistently.OtherexperimentalmethodsformeasuringCCPopGrowAsnotedabove,DogteromandcolleaguespreviouslypredictedtheexistenceofacriticalfreetubulinconcentrationCCunbounded,atwhichMTswilltransitionfromexhibiting“boundedgrowth”toexhibiting“unboundedgrowth”4(DogteromandLeibler,1993;Dogterometal.,1995).ThispredictionwasexperimentallyverifiedbyFygensonetal.(Fygensonetal.,1994).Dogteromandcolleagues(Verdeetal.,1992;DogteromandLeibler,1993)usedanequationfortherateofchangeinaverageMTlengthasafunctionoftheDIparameterstocharacterizeboundedandunboundedgrowth(notethatJisatypicalabbreviationforflux):

JDI=steady-staterateof

changeinaverageMTlength=

0duringboundedgrowthVg Fres – Vs Fcat

Fcat+ Fres >0duringunboundedgrowth (Equation1)

Dogterometal.identifiedCCunboundedasthe[freetubulin]atwhichVgFres=|Vs|Fcat (seelabelQ5binFigure5C-D).Significantly,CCunboundedaspredictedbyQ5bfromthisequationevaluatedwithourDIparametermeasurementsmatchesQ5a(compare+symbolstoosymbolsinFigure5C-D).Hence,CCPopGrowcorrespondstoCCunbounded,andpolymer-masssteadystateandpolymer-growthsteadystatecorrespondto“boundedgrowth”and“unboundedgrowth”respectively.

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Thus,CCPopGrowcanbemeasuredclassically,bydeterminingQ1orQ2,butitcanalsobedeterminedbymeasuringDIparametersforindividualMTsatdifferent[freetubulin]andinputtingthemintotheequationforrateofchangeinaverageMTlength(Equation1).Havingsaidthis,determiningDIparametersacrossarangeofconcentrationsrequiresextended(>tensofminutes)analysisofmanyindividualMTsandsoislaboriousandtimeconsuming.

AnalternativeapproachtomeasuringCCPopGrowthatmaybemoretractableexperimentallyistousevideomicroscopytosimultaneouslyanalyzethebehaviorofmanyindividualMTswithinapopulationaccordingtothedriftparadigmofBorisyandcolleagues(Vorobjevetal.,1997;Vorobjevetal.,1999;Komarovaetal.,2002).5ThedriftcoefficientisthemeanrateofchangeinpositionoftheMTends(forplusorminusendsseparately),alsodescribedasthemeanvelocityofdisplacementoftheMTends.Incaseswhereoneendisfixed,asinoursimulations,thedriftcoefficientisequivalenttotherateofchangeinaverageMTlength.Hereweusedamethodbasedon(Komarovaetal.,2002),whichcalculatesthedriftcoefficientfromthedisplacementsofMTendsoversmalltimesteps,e.g.,betweenconsecutiveframesofamovie(seeSupplementalMethodsforadditionalinformation).AscanbeseeninFigures5E-F(xsymbols)andS3G-H(allsymbols),apopulationofMTsatsteady-stateexhibitszerodriftat[freetubulin]belowQ5c(i.e.,inthisstate,theaveragelengthofMTsinthepopulationisconstantwithtime)butexhibitspositivedriftat[freetubulin]aboveQ5c(i.e.,theaverageMTlengthincreaseswithtime;thepopulationgrowspersistently)(Komarovaetal.,2002).Asonemightintuitivelypredict,Q5a≈Q5b≈Q5c(Figure5C-F).TheevidentsimilaritybetweenthedifferentmeasurementsinFigure5C-FsuggeststhatDogterom’sequationusingDIparameters(+symbolsinFigure5C-D)(Verdeetal.,1992;DogteromandLeibler,1993)andtheequationofKomarovaetal.usingshort-termdisplacements(xsymbolsinFigure5E-F)(Komarovaetal.,2002)equationsaresimplytwodifferentrepresentationsofthesamerelationship.Indeed,bothyieldtherateofchangeinaverageMTlengthasfunctionsofexperimentallyobservedgrowthanddepolymerizationbehaviors.Additionally,variousformsofthisequationwerepresentedearlierbyHillandcolleagues(HillandChen,1984;Hill,1987)andWalkeretal.(Walkeretal.,1988),andhavesincebeenusedinotherwork(e.g.,(Bicout,1997;Gliksmanetal.,1992;Maly,2002;VorobjevandMaly,2008;Mourãoetal.,2011;Mahrooghyetal.,2015;Aparnaetal.,2017)).Thus,experimentalistsshouldbeabletousewhicheveranalysismethodistractableandappropriatefortheirexperimentalsystem.MeasuringCCPopGrowusingpopulationdilutionexperiments.NextwetestedifCCPopGrowisthesameastheCCobtainedfromthepopulationdilutionexperimentsusedinearlystudiesofsteady-statepolymers(e.g.,(Carlieretal.,1984b;Carlieretal.,1984a);seeQ4inTable1andFigure1C).Theseexperimentsmeasuretherateofchangein[polymerizedtubulin],whichisalsodescribedtheflux(typicallyabbreviatedasJ)oftubulinintooroutofpolymer.ThismeasurementisperformedafterapopulationofMTsatsteadystateisdilutedintoalargepool 5Foramathematicalexplanationofhowmicrotubulebehaviorcanbeapproximatedbyadrift-diffusionprocess,see(Maly,2002;VorobjevandMaly,2008).

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offreetubulinatanewconcentration.6ThemeasureddatafromthedilutionexperimentsarethenusetoproduceJ(c)plots,whereJisplottedasafunctionofsubunitconcentration“c”(Figure6A-B).Intheseplots,“theCC”isidentifiedasthedilution[freetubulin]atwhichJ=0(i.e.,wheretheplottedcurvecrossesthehorizontalaxis,Q4).Atthisconcentration,individualMTsundergoperiodsofgrowthandshortening,butthepopulation-levelfluxesintoandoutofpolymerarebalanced(i.e.,netgrowthiszero).WerefertotheCCmeasuredviaJ(c)plotsasCCflux(Table1).CCfluxcorrespondstooneoftheCCsthatwasidentifiedbyHillandcolleagues,variouslynamedcoin(HillandChen,1984;ChenandHill,1985b)anda!in(Hill,1987).Significantly,thevalueofCCfluxasmeasuredbyQ4inthedilutionsimulationscorrespondstoCCPopGrow(greydashedline,Figure6A-B)asmeasuredbyQ1≈Q2inthecompetingsimulations(Figures3A-B)andbyQ5abcinthenon-competingsimulations(Figure5C-F).Notealsothatfordilution[freetubulin]aboveCCPopGrow,theJ(c)curveobtainedfromthedilutionsimulationsissuperimposablewiththenetrateofchangeinaverageMTlengthobtainedfromtheconstant[freetubulin]simulations(Figure6C-D).Thisobservationmightseemsurprisinggiventhedifferencesintheexperimentalapproaches;however,itmakessensebecauseineachcasethemeasurementisperformedduringatimeperiodwhen[freetubulin]isconstantandtherateofchangehasreacheditssteady-statevalueforeach[freetubulin](FiguresS3A-BandS4E-F).Thus,alloftheexperimentalapproachesformeasuringcriticalconcentrationdiscussedthusfaryieldthecriticalconcentrationforpersistentpopulationgrowth(CCPopGrow).Thisconclusionleavesuswithanunresolvedquestion:WhatisthesignificanceoftheremainingcommonexperimentalCCmeasurementQ3(obtainedfromexperimentsmeasuringgrowthvelocityduringgrowthphasesforindividualMTsasafunctionof[freetubulin],seeFigure1B,Table1)?Acriticalconcentrationfortransientelongation(growth)ofindividualfilaments(CCelongation=CCIndGrow)Q3(Figure1B)haspreviouslybeenusedasameasureofthe“criticalconcentrationforelongation”(CCelongation)(Walkeretal.,1988).Accordingtostandardmodels,CCelongationisthefreesubunitconcentrationwheretherateofsubunitadditiontoanindividualfilamentinthegrowthphaseexactlymatchestherateofsubunitlossfromthatindividualfilament,meaningthatindividualfilamentswouldbeexpectedtogrowatsubunitconcentrationsaboveQ3≈CCelongation(seeTable1anditsfootnotes).TodeterminethevalueofQ3inoursimulations,weusedthestandardapproachforMTsasoutlinedinTable1(experimentsin(Walkeretal.,1988);seealsotheoryin(HillandChen,1984;Hill,1987)).Weplottedthegrowthvelocity(Vg)ofindividualfilamentsobservedduringthe

6Inthephysicalexperiments,therewasnormallyadelayofafewsecondsafterthedilutionandbeforethedatawererecorded(Carlieretal.,1984a).AnalysisofoursimulatedJ(c)experimentsincorporatesasimilardelay.Thisdelaymayhavebeennecessaryfortechnicalreasonsinthephysicalexperiments,butitalsoservesapurposeinallowingthestabilizingGTPcaptoredistributetoitssteady-statesize(Duellbergetal.,2016).SeeFigureS4forplotsof[freetubulin]and[polymerizedtubulin]asfunctionsoftimeinthedilutionsimulations.

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growthphaseofdynamicinstabilityasafunctionof[freetubulin],andextrapolatedalinearfitbacktothe[freetubulin]atwhichtoVgiszero.7Inadditiontoperformingthesemeasurementsontheconstant[freetubulin]simulations(Figure7A-B),wealsousedthegrowthphasesthatoccurredinthedilutionexperimentstoobtainameasurementofCCelongation(Q6inFigure7C-D).ComparingthesemeasurementsofCCelongationinFigure7A-DtothedatainFigures3-6showsthatinbothsimulationsCCelongation(asdeterminedbyQ3≈Q6)iswellbelowCCPopGrowasmeasuredbyanyoftheotherapproaches(Q1≈Q2≈Q4≈Q5abc).8

ThisobservationdemonstratesthatQ3≈Q6providesinformationaboutMTbehaviornotprovidedbytheothermeasurements.Specifically,sinceQ3andQ6aredeterminedfrommeasurementsofthegrowthvelocityofindividualsduringthegrowthphaseofdynamicinstability,Q3andQ6provideestimatesofthe[freetubulin]abovewhichindividualfilamentscangrowtransiently(i.e.,toextendduringthegrowthphaseofdynamicinstability).Whethergrowthphaseswilloccuralsodependsonavarietyofotherfactors,suchasrescuefrequencyandfrequencyofinitiatinggrowthfromseeds.ConsistentwiththeidentificationoftheupperCCasCCPopGrowforpopulationgrowth,wesuggestreferringtoCCelongationasCCIndGrowforindividualfilamentgrowth.ComparisonofFigure7withofFigures5-6leadstoadditionalconclusionswithpracticalsignificanceformeasuringtheCCs.Figure8A-BshowsthattheVgdatafromindividualgrowthphasesinFigure7A-BandthenetrateofchangeinaverageMTlengthdatafrompopulationsinFigure5C-D(orequivalentlythepopulationdriftcoefficientinFigure5E-F)overlayeachotherwhen[freetubulin]issufficientlyhigh(i.e.,farenoughaboveCCPopGrowthatcatastropheisrare).Thismakessensebecausewhencatastropheisunlikely,almostallMTswillbegrowing,someasurementsofindividualsandpopulationsshouldgiveapproximatelythesameresults.Thus,linearextrapolationfromthenetrateofchangeinaverageMTlengthdataathigh[freetubulin]toobtainQ7asshowninFigure8C-DyieldsapproximatelythesamevalueforCCIndGrowasQ3≈Q6.Additionally,sincethenetrateofchangeinaverageMTlengthdatafromtheconstant[freetubulin]experimentsandJ(c)fromthedilutionexperimentsmatcheachotherathigh[freetubulin](Figure6C-D),theQ7extrapolationcanalsobeperformedontheJ(c)datatomeasureCCIndGrow.Thus,bothconstant[freetubulin]experimentsanddilutionexperimentscanbeusedtoobtainnotonlyCCPopGrow(viaQ4≈Q5abc)butalsoCCIndGrow(viaQ3≈Q6≈Q7).

7ThisVgversus[freetubulin]relationshipisexpectedtobelinearonthebasisoftheassumptionthatgrowthoccursaccordingtotheequationVg=kTonT[freetubulin]–kToffT,wherethefirsttermcorrespondstotherateatwhichGTP-tubulinattachestoaGTPtip,andthesecondtermcorrespondstotherateatwhichGTP-tubulindetachesfromaGTPtip(Bowne-Andersonetal.,2015).Wereturntothisrelationshiplaterinthemaintext.8WhilethenumericalvaluesoftheCCsinthesimplifiedmodelshouldberegardedasarbitrary,thedetailedmodelCCIndGrowandCCPopGrownumericalvaluescloselymatchthoseobtainedbyWalkeretal.(Walkeretal.,1988).ThisisnotablebecausewetunedthedetailedmodelparameterstomatchWalker’sDIparametersat[freetubulin]=10µMbutdidnottunetotheCCvalues.

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CCIndGrowisnotCCPolAssemTheinformationaboveleadstothestraightforwardconclusionthatCCIndGrowrepresentsalowerlimitformicrotubulegrowth.OnemightbetemptedtousethisideatopredictthatCCIndGrowistheconcentrationoffreetubulinatwhichpolymerappears(i.e.,thatCCIndGrow=CCPolAssem).However,thispredictionfails.Contrarytotraditionalexpectation,thereisnototalorfreetubulinconcentrationatwhichpolymerassemblycommencesabruptly.Instead,theamountofpolymerinitiallyincreasesinaslowandnonlinearwaywithrespectto[freetubulin],increasingmorerapidlyonlyas[freetubulin]approachesCCPopGrow(FigureS3A-F).Thesameconclusionisreachedwhetherexaminingpolymermass(FigureS3A-B),averageMTlength(FigureS3A-F),ormaximalMTlength(FigureS3C-F).9Theseobservationsindicatethatmicrotubules(andDIpolymersmorebroadly)donothaveacriticalconcentrationforpolymerappearance(CCPolAssem)astraditionallyunderstood.CCIndGrowisthetubulinconcentrationabovewhichextendedgrowthphasescanoccur,butsignificantamountsofpolymerdonotaccumulateinexperimentswithbulkpolymeruntil[freetubulin]nearsorexceedsCCPopGrow(Figures3,S1-S3).Thesebehaviorsmightseemcounterintuitive,buttheycanbeexplainedbythefollowingreasoning.First,when[freetubulin]isjustaboveCCIndGrow,thegrowthvelocityduringthegrowthphaseislow(Vg=0atQ3)andthefrequencyofcatastrophe(Fcat)ishigh.Undertheseconditions,individualmicrotubuleswillbebothshort(Figure5A-B,S3A-F)andshort-lived(Figure5A-B),andthusdifficulttodetect.As[freetubulin]rises,MTswillexperiencegrowthphasesthatlastlonger(becauseFcatdrops)andalsohavefastergrowthvelocity(Figure7).Thecombinedimpactofthesetwoeffectscreatesanonlinearrelationshipbetween[freetubulin]and[polymerizedtubulin]orequivalentlytheaverageMTlengthobservedatsteadystate;itsimilarlycreatesanonlinearrelationshipbetween[freetubulin]andmaximalMTlengthasobservedwithinaperiodoftime(FigureS3C-F).MeasurementofCCIndGrowbyQ3,Q6,orQ7isapproximateCCIndGrowandCCPopGrowareintrinsicpropertiesofasystem(i.e.,aparticularproteinsequenceinaparticularenvironment),whereastheexperimentalmeasurements(Qvalues)aresubjecttomeasurementerrorsandarethereforeapproximate.ThemeasurementsofCCIndGrowbyQ3,Q6,orQ7canbeparticularlysensitivetomeasurementerrorsandnoisebecausetheyarebasedonextrapolations.Morespecifically,sinceQ3andQ6aredeterminedbyextrapolationsfromregressionlinesfittedtoplotsofVgversus[freetubulin],smallchangesintheVgdata(e.g.,fromnoise)canbeamplifiedwhenextrapolatingtotheVg=0intercept.Additionally,inthesimulationresults,nonlinearitiesareobservedintheVgversus[freetubulin]plotsinbothsimulations.Inthepresenceofnoiseand/ornonlinearities,thevaluesofQ3andQ6willdependonthe[freetubulin]rangewheretheregressionlinesarefittedtotheVgplots. 9NotethatHillandChenconcludedthatevenequilibriumpolymershavesomeassemblybelowtheCC,butthattheaveragelengthisverysmalluntil[freesubunit]is“extremelyclose”totheCC(HillandChen,1984).

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Thedeviationfromlinearityinthesimulationplotsisexplainedinpartbymeasurementbias:atthelowest[freetubulin],therearefewgrowingMTs,allofwhichareshort(Figures5A-B,S3C-F).ThemeasuredVgdataisbiasedtowardsthoseMTsthathappenedtogrowfastenoughandlongenoughtobedetected(seemaximumMTlengthdatainFigureS3C-F).ThisindicatesthatthelowestconcentrationsshouldnotbeusedinthelinearextrapolationtoidentifyQ3orQ6.Toourknowledge,suchdeviationsfromlinearityatlowconcentrationshavenotbeendetectedexperimentally.However,thesimulationsgeneratemuchmoredataandatsmallerlengththresholdsthanispossiblewithtypicalexperiments.Becausemeasurementbiascouldalsobeaprobleminphysicalsystems,wespeculatethatsimilareffectsmayeventuallybeseenexperimentally.Giventhenonlinearitiesandthemeasurementbiasdescribedabove,onemightbeconcernedthatdetectionthresholdswouldaffectthemeasuredvalueofCCIndGrow.WethereforecomparedtwodifferentanalysismethodsfordeterminingVg(Figure7).Specifically,fortheDIanalysismethod(Figure7,+symbols),wesetathresholdof25subunits(200nm)oflengthchangetodetectgrowthorshorteningphases(wesetthisthresholdtobecomparabletotypicallengthdetectionlimitsinlightmicroscopyexperiments).Incontrast,forthetime-stepmethod(Figure7,squaresymbols),wedidnotimposeathresholdonthelengthchangeduringeachtimestep(seeSupplementalMethods).TheVgresultsfromthetwomethodsagreewellwitheachotherinthe[freetubulin]rangeusedtodetermineCCIndGrow(i.e.,therangewhereVgisapproximatelylinear).Thus,whenimplementingVganalysistoestimateCCIndGrow,theregressionlinesshouldbefittedtothelinearregiontoavoidtheeffectofdetectionthresholds.IftheregressionlinesarenotfittedinthetubulinrangewhereVgislinear,thenQ3andQ6willbelessaccurateapproximationsofCCIndGrow.Dependingonthespecificsystem,Q7maybealessaccurateapproximationthanQ3orQ6.Q7isobtainedfromtherateofchangeinaverageMTlengthatfreetubulinconcentrationsthataresufficientlyhighthat(almost)allMTsaregrowing(i.e.,whereVgandtherateofchangeinaverageMTlengthoverlapwitheachother,Figure8).SincetheQ7extrapolationisperformedfromhigherconcentrationsthantheQ3orQ6extrapolations,measurementerrorsornoiseinthedatacanbefurtheramplified.Moreover,VgandtherateofchangeinaverageMTlengthmaynotoverlapuntiltubulinconcentrationsaresohighthatexperimentalmeasurementsmaynolongerbefeasible(e.g.,becauseofproblemssuchasfreenucleation).InboththedetailedmodelandinphysicalMTs,anadditionalfactorcancauseVgtohavenonlinearitiesasafunctionof[freetubulin]andthereforelikelyinterfereswiththeaccuracyofidentifyingCCIndGrowviaQ3,Q6,orQ7.PreviousworkhasprovidedexperimentalandtheoreticalevidencethattheGTP-tubulindetachmentratedependsonthetubulinconcentration(Gardneretal.,2011),contrarytotheassumptionsclassicallyusedtodetermineCCelongation.Thisobservationhasbeenexplainedbytheoccurrenceofconcentration-dependentchangesintheMTtipstructure(Coombesetal.,2013).

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Thus,bothdetectionissuesandactualstructuralfeaturescanpotentiallymakeobservedVgmeasurementsnonlinearwithrespectto[freetubulin].Asaresult,thevalueobtainedforCCIndGrowfromQ3≈Q6≈Q7maydependonwhat[freetubulin]rangeisusedforthelinearfit.Theseobservationsmeanthatthesevalues(Q3,Q6,Q7)provideatbestapproximatemeasurementsofCCIndGrow.Effectofhydrolysisrateconstant(kH)onCCIndGrowandCCPopGrow.TheresultsaboveshowthatCCIndGrowisobtainedfrommeasurementsofindividualMTsthatareinthegrowthphase,whileCCPopGrowisobtainedfrommeasurementsperformedonpopulations(oronindividualsoversufficienttime)thatincludebothgrowthandshorteningphases(seealso(Hill,1987;Walkeretal.,1988)).Thus,theco-existenceofbothgrowthandshorteningphases(i.e.,dynamicinstabilityitself)occursinconjunctionwiththeseparationbetweenCCIndGrowandCCPopGrow.Dynamicinstabilityinturndependsonnucleotidehydrolysis,sinceGTP-tubulinispronetopolymerizationandGDP-tubulinispronetodepolymerization.Therefore,todevelopanimprovedunderstandingoftheseparationbetweenCCIndGrowandCCPopGrowinDIpolymers,wenextexaminedtheeffectofthehydrolysisrateconstantkHonCCIndGrowandCCPopGrow.Toallowastraightforwardcomparisonbetweentheobservedbehaviorsandtheinputkineticparameters,weutilizedthesimplifiedmodel.Morespecifically,weransimulationsinthesimplifiedmodelunderconstant[freetubulin]conditionsacrossarangeofkHvalues,whileholdingtheotherbiochemicalkineticparametersconstant.FromthesedatawedeterminedCCIndGrowasmeasuredbyQ3andCCPopGrowasmeasuredbyQ5a(Figures9,S5).WhenthehydrolysisrateconstantkHequalszero,onlyGTP-tubulinsubunitsarepresent.Aswouldbeexpected,thebehavioristhatofanequilibriumpolymer:noDIoccurs(seelengthhistoriesinFigureS6A),andallobservedCCvaluescorrespondtotheKDforGTP-tubulinasdefinedbytheinputrateconstants.Inotherwords,whenkHiszero,CCIndGrow=CCKD_GTP=kToffT/kTonT=CCPopGrow(Figure9A).WhenkHisgreaterthanzerointhesesimulations,bothGTP-andGDP-tubulinsubunitscontributetopolymerdynamics,concurrentwiththeappearanceofdynamicinstability(FigureS6B-F).AskHincreases,CCIndGrow(Q3)andCCPopGrow(Q5a)bothincreaseanddivergefromeachother(Figures9,S5),anddynamicinstabilityoccursoverawiderrangeof[freetubulin](FigureS6).CCIndGrowcandifferfromCCKD

InadditiontoshowingthatnucleotidehydrolysisdrivesCCIndGrow(Q3)andCCPopGrow(Q5a)apartfromeachother,theresultsinFigure9alsoshowthathydrolysisdrivesbothawayfromCCKD_GTP(x-interceptofgreydashedlineinFigure9A-F;greydashedlineinFigure9G).Inparticular,whiletherelationshipCCKD_GTP=kToffT/kTonTisindependentofkH,weobservethatCCIndGrowchangeswithkH.ThiscouldbeviewedassurprisingbecauseonemightexpectCCIndGrowtoequalCCKD_GTPeveninthepresenceofDI.Thereasoningbehindthisexpectationisasfollows.First,therateofgrowthofanindividualMTduringthegrowthstateisassumedtochangelinearlywith[freetubulin]accordingtotherelationship,

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Vg=kongrowth[freetubulin]–koff

growth, (Equation2)10 (Walkeretal.,1988)wherekoff

growthandkongrowth(calledk-1e+andk2e+in(Walkeretal.,1988))areeffective(observed)

rateconstantsforadditionandlossofGTP-tubulinsubunitsonagrowingtip.By“effective”wemeanthattheyareemergentquantitiesextractedfromtheVgdata,asopposedtodirectlymeasuredkineticrateconstants.Morespecifically,thevaluesofkon

growthandkoff growthare

measuredfromtheslopeandthey-interceptrespectivelyofaregressionlinefittedtoVgdata,giventheequationabove.SinceCCIndGrowismeasuredasthevalueof[freetubulin]atwhichVgiszero,settingEquation2equaltozeroandsolvingfor[freetubulin]leadstotheconclusionthatCCIndGrow=koff

growth/kongrowth.Thisratiokoff

growth/kongrowthismeasuredasthex-interceptofthe

regressionline(Equation2)(Walkeretal.,1988).Second,itiscommonlyassumedthatrapidlygrowingtipshaveonlyGTP-subunitsattheend(e.g.,(Howard,2001;Bowne-Andersonetal.,2015)).Underthisassumption,andalsoassumingthatallunpolymerizedtubulinisboundtoGTP,Equation2becomes Vg=kTonT[freetubulin]–kToffT, (Equation3)whichleadstothepredictionthatCCIndGrow=kToffT/kTonT=CCKD_GTP.Instead,theresults(Figures9A-F,S5A)showthatEquation3fitsthedatawellonlywhenkHisclosetozero.AskHincreases,theVgregressionlineandCCIndGrowdivergeawayfromvaluesthatwouldbepredictedfromEquation3.11Morespecifically,whenkHisgreaterthanzero,theeffective!ongrowthand!offgrowth(slopeandinterceptofVginEquation2)inthesimulationsdivergefromkTonTandkToffT;inthiscase,VgdoesnotsatisfyEquation3,andCCIndGrowdivergesfromCCKD_GTP.TheseobservationsindicatethatGDP-subunitscaninfluencebehaviorduringgrowthphases.PossiblemechanismsforexposureofGDP-tubulinatgrowingMTtips.TherearestrongreasonstoexpectthatGDP-subunitswillinfluencegrowthbehaviorinphysicalMTs.TheideathatgrowingMTtipscouldhaveGDP-tubulinsubunitsmightseemsurprising,butGDP-tubulinsubunitscouldbecomeexposedonthesurfaceofagrowingtipeitherbydetachmentofasurfaceGTP-subunitfromaGDP-subunitbelowitorbydirecthydrolysis.ThefirstmechanismconflictswithearlierideasthatGTP-subunitsrarelydetach,butisconsistentwithrecentexperimentaldataindicatingrapidexchange(attachmentanddetachment)ofGTP-subunitsonMTtips(Gardneretal.,2011;Coombesetal.,2013);seealso(Margolinetal.,2012)).

10Thesymbol≈maybemoreappropriatethan=becausethisequationassumes(i)thatVgincreaseslinearlywith[freetubulin]and(ii)thatthedetachmentrateisindependentof[freetubulin].OurVgresultspresentedaboveindicatethatassumption(i)maybeinaccurate.See(Gardneretal.,2011)forevidenceagainstassumption(ii).11Recallthatthekineticrateconstants(e.g.,kTonT,kH)inoursimulationsareinputtedbytheuser.Incontrast,thevaluesofVgandCCIndGrowareemergentpropertiesofthesystem.

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TheideathatGDP-tubulincannotbeexposedatMTtipsduringgrowthphasesmaybearemnantofvectorialhydrolysismodels,12whereGDP-tubulinwouldbecomeexposedonlywhentheGTPcapisentirelylost(atleastforsingleprotofilamentmodels).However,variousauthorshaveshownthatvectorialhydrolysisisneithersufficient(Flyvbjergetal.,1994;Flyvbjergetal.,1996; Padinhateerietal.,2012)nornecessary(Margolinetal.,2012;Padinhateerietal.,2012)toexplainMTdynamicinstabilitybehavior.Additionally,Hillandcolleaguesexaminedbothvectorialandrandomhydrolysismodels.Inthevectorialhydrolysismodel,thegrowthvelocitysatisfiedanequationequivalenttoEquation2above,whichassumesonlyGTPtipsduringgrowth(Hill,1987).Intheirrandomhydrolysismodel,theobserved(emergent)slopeandinterceptofVgdidnotequaltheinputrateconstantsforadditionandlossofGTP-subunits,asexplicitlypointedoutin(Hill,1987;HillandChen,1984).ThisconclusionfromHill’srandomhydrolysismodelisconsistentwiththeresultsofourmodel,whichalsohasrandomhydrolysis.TheconclusionabovethatCCIndGrow≠CCKDalsohelpsexplaintheobservationfromearlierinthepaperthatthereisnoconcentrationatwhichpolymerassemblyabruptlycommences(i.e.,thereisnoCCPolAssem).Instead,theamountofpolymerincreasesslowlywithincreasing[freetubulin](FigureS3A-F).Morespecifically,althoughtheMTstypicallyreachexperimentallydetectablelengths(e.g.,>200nm,dependingonthemethodused)atsomeconcentrationaboveCCIndGrow(FigureS3A-F),smallamountsofgrowthcanoccurevenbelowCCIndGrow(FigureS3E-F;squaresymbolsinFigure7).When[freetubulin]isaboveCCKD_GTP,attachmenttoaGTP-subunitwillbemorefavorablethandetachment;thus,smallamountsofgrowthcanoccur.Incontrast,asnotedabove,CCIndGrowisthe[freetubulin]necessaryforamicrotubuletoexhibitextendedgrowthphases.ThedependenceofCCIndGrowonkHindicatesthatattachmentmustinsomesenseoutweighbothdetachmentandhydrolysisofGTP-subunitsinorderforextendedgrowthphasestooccur.DynamicinstabilitycanproducerelationshipspreviouslyinterpretedasevidenceofanucleationprocessforgrowthfromstableseedsPreviously,twoexperimentalobservationshavebeeninterpretedasevidencethatgrowthofMTsfromstabletemplates(e.g.,centrosomes,axonemes,GMPCPPseeds)involvesanucleationprocess(e.g.,conformationalmaturationorsheetclosure)(Wieczoreketal.,2015;RoostaluandSurrey,2017).First,MTsaregenerallynotobservedgrowingat[freetubulin]nearCCIndGrow.Second,whenthefractionofseedsoccupiedisplottedasfunctionof[freetubulin],theshapeoftheresultingcurveissigmoidal,suggestingacooperativeprocessand/orathermodynamicbarrier.Inthissectionweshowthatthesetwonucleation-associatedbehaviorsareobservedinoursimulations,whichisnotablebecauseneithersimulationincorporatesanexplicitnucleation

12Invectorialhydrolysismodels,hydrolysisoccursonlyattheinterfacebetweentheGDP-tubulinlatticeandtheGTP-tubulincap(e.g.,(Carlieretal.,1987;Hill,1987)).Incontrast,inrandomhydrolysismodels,anyinternalGTP-subunitcanhydrolyze(terminalGTP-subunitsmayalsohydrolyzedependingonthemodel).

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step(ourseedsarecomposedofnon-hydrolyzableGTP-tubulin).Weshowthatbothexperimentallyobservedrelationshipscanresultfromdynamicinabilityincombinationwithlengthdetectionthresholds.ThebehaviorsofDIpolymersrelativetoCCIndGrowandCCPopGrow,asdescribedabove(e.g.,Figures5A-B,S3A-F),canthereforebehelpfulinunderstandingtheserelationships.FailuretodetectMTgrowtheventsinexperimentsat[freetubulin]nearCCIndGrowcanresultfromphysicaldetectionlimitationscoupledwithDI.Asdescribedabove,when[freetubulin]isnearCCIndGrow,VgissmallandFcatishigh,meaningthatMTsareshort(FigureS3A-F)andshort-lived(Figure5A-B);theaverageMTlengthremainssmalluntil[freetubulin]isclosertoCCPopGrow(FigureS3A-F).Thisbehaviorcoupledwithlengthdetectionthresholds(suchaswouldbeimposedbyphysicalexperiments)couldmakeitdifficulttodetectMTsat[freetubulin]nearCCIndGrow.Totestthishypothesis,weusedthesimulations(whichoutputtheMTlengthwithoutanydetectionthreshold)toexaminetheeffectofimposinglengthdetectionthresholdssimilartothosepresentinphysicalexperiments.Indeed,whenweimposeda200nmdetectionthreshold(comparabletolightmicroscopy)onthelengthchangeneededforagrowthphasetoberecognized(Figure7,+symbols),wesawthatMTgrowththatwasdetectedintheabsenceofthisthreshold(Figure7,squaresymbols)isnolongerdetected.TheseresultsindicatethatfailuretoobserveMTsgrowingfromstableseedsat[freetubulin]nearCCIndGrowcanresultfromusingexperimentalmethodsthathavelengthdetectionlimitations,providingevidencethatsuchbehaviorcanresultfromprocessesotherthannucleation.AsigmoidalPocccurveispredictablefromdetectionthresholdsandMTpopulationlengthdistributionsresultingfromDI.PoccistheproportionofstableMTtemplates/seedsthatareoccupiedbya(detectable)MT(Figure10A-B).PreviousexperimentalworkhasshownthatPocchasasigmoidalshapewhenplottedasafunctionof[freetubulin](e.g.,(MitchisonandKirschner,1984b;Walkeretal.,1988;Wieczoreketal.,2015)).ThisshapehasbeeninterpretedasevidencethatstartinganewMTfromaseedisharderthanextendinganexistingMTandthusthatgrowthfromseedsinvolvesanucleationprocess(Walkeretal.,1988;Fygensonetal.,1994;Wieczoreketal.,2015)(compareFigure11Ato11B).However,theVganalysisdescribedaboveledustohypothesizethatthissigmoidalPoccshapecanalsoresultfromthecombinationoflengthdetectionthresholdsandDI.Totestthishypothesis,weexaminedPoccasafunctionof[freetubulin]withvaryingdetectionthresholds(Figures10C-D,S7).Theresultsshowthatateach[freetubulin](belowCCPopGrow),asthedetectionthresholdisincreased,thedetectedPoccdecreases(i.e.,fewerMTsarelongerthanthethreshold).Thisresultsinasigmoidalshapeemergingintheplotsforbothsimulationsasthelengthdetectionthresholdisincreased.Thesteepnessofthesigmoiddependsstronglyonthedetectionthreshold.Theseobservationsindicatethatthesigmoidalshapecanresultsimplyfromimposingalengthdetectionthresholdonasystem(suchasdynamicMTs)wheresomeofthefilamentsareshorterthanthedetectionthreshold.InthepresenceofDIwith

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completedepolymerizationsbacktotheseeds(asoccursbelowCCPopGrow),MTswillnecessarilybebelowanynon-zerodetectionthresholdforatleastsomeamountoftime.ThePocccurvereaches1at[freetubulin]nearCCPopGrow.TheresultsinFigures10andS7provideanotherobservationrelevanttounderstandingcriticalconcentrations:inbothsimulations,Poccapproaches1as[freetubulin]approachesCCPopGrow(exceptpossiblyatverysmallthresholds,wherePoccnears1atlower[freetubulin]).Thisresultispredictable,withorwithoutanucleationprocess,becauseonlyat[freetubulin]aboveCCPopGrow(wherethepopulationundergoesnetgrowth)wouldallactiveseedsbeoccupiedbyMTslongerthananarbitrarilychosenlengththreshold.Thisfulloccupancywouldoccurifsufficienttimeisallowed,becauseat[freetubulin]aboveCCPopGrow,MTswilleventuallybecomelongenoughtoescapedepolymerizingbacktotheseed.Thus,theideathatPocc=1at[freetubulin]aboveCCPopGrowaftersufficienttimemayprovideapracticalwaytoidentifyCCPopGrowexperimentally(seealso(ChenandHill,1985a;Fygensonetal.,1994;Dogterometal.,1995)).Takingallthisinformationtogether,weproposethatacombinationofdynamicinstabilityitselfandtheexistenceofdetectionthresholdscontributestophenomena(failuretoobservegrowingMTsat[freetubulin]nearCCIndGrow,Figure7;andsigmoidalPoccplots,Figures10,S7)thathavepreviouslybeeninterpretedasevidencethatgrowthofMTsfromstableseedsinvolvesanucleationprocess(Fygensonetal.,1994;Wieczoreketal.,2015).Infact,anyprocessthatmakesgrowthfromaseedmoredifficultthanextensionofagrowingtip(i.e.,anucleationprocesssuchassheetclosure)wouldmakethePocccurvemorestep-like,notlessso(Figure11,comparepanelsBandC).Whilewecannotexcludetheexistenceofnucleationprocessessuchasconformationalmaturationorsheetclosureinphysicalmicrotubules,ourworksuggeststhatneithersigmoidalPocccurvesnorabsenceofdetectableMTsonseedsat[freetubulin]nearCCIndGrowaresufficientevidencetoconcludethatgrowthfromtemplates(e.g.,centrosomes,stableseeds)involvesaphysicalnucleationprocess.DISCUSSIONThebehaviorofMTsisgovernedbytwomajorcriticalconcentrationsUsingthedynamicmicrotubulesinourcomputationalsimulations,weexaminedtherelationshipsbetweensubunitconcentrationandpolymerassemblybehaviorsfordynamicinstability(DI)polymers.OurresultsshowthatthereisnotrueCCPolAssemastraditionallydefined,meaningthatthereisnoconcentrationwhereMTsabruptlycomeintoexistence.Instead,thereareatleasttwomajorcriticalconcentrations.ThereisalowerCC(CCIndGrow),abovewhichindividualfilamentscangrowtransiently,andanupperCC(CCPopGrow),abovewhichapopulationoffilamentswillgrowpersistently(Figure12B,D).For[freetubulin]aboveCCPopGrow,individualMTsmaystillundergodynamicinstability(Figure12D,bluelengthhistory),butwillexhibitnetgrowthovertime(Figure12D,blueandgreenlengthhistories).Whatmightbeconsidered“typical”or“bounded”dynamicinstability(whereindividualMTsrepeatedly

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depolymerizebacktotheseeds)occursat[freetubulin]betweenCCIndGrowandCCPopGrow(Figure12D,purplelengthhistory;Figure12C).CCIndGrowisestimatedbyQ3,Q6,andQ7,andCCPopGrowisestimatedbyQ1,Q2,Q4,andQ5abc(Figure12A-B,Table3).Classicalcriticalconcentrationmeasurements(e.g.,Figure1AQ1andQ2)donotyieldthetraditionallyexpectedCCPolAssem,butinsteadyieldCCPopGrow(Figure12AQ1andQ2).Importantly,[freetubulin]SteadyStateinacompetingsystemdoesnotequalCCPopGrow,butapproachesCCPopGrowasymptoticallyas[totaltubulin]increasesanddependsonthenumberofstableseeds(Figure12A,comparedarkandlightgreenlines).BulkpolymerexperimentscancreatetheillusionthatCCPopGrowcorrespondstoCCPolAssem.TheaboveconclusionthatMTsgrowtransientlyat[tubulin]betweenCCIndGrowandCCPopGrowmightappeartoconflictwithexperimentalobservationsreportingthatbulkpolymerisdetectableonlyaboveQ1(Figure1A,seee.g.,(JohnsonandBorisy,1975;Mirigianetal.,2013)).Asdiscussedabove,Q1providesameasureofCCPopGrow,butistraditionallyexpectedtoprovidethecriticalconcentrationforpolymerassembly,CCPolAssem.ThisapparentconflictbetweentheseobservationsandtheconclusionsabovecanberesolvedbyrecognizingthatthefractionoftotalsubunitsconvertedtopolymerwillbelowuntilthefreetubulinconcentrationnearsCCPopGrow.Thus,for[totaltubulin]<CCPopGrow,[freetubulin]willbeapproximatelyequalto[totaltubulin](Figure12A,darkgreenline),unlesstherearemanystableseeds(Figure12A,lightgreenline).Incontrast,for[totaltubulin]>CCPopGrow,allfreetubulininexcessofCCPopGrowwillbeconvertedfromfreetopolymerizedformifsufficienttimeisallowed(FigureS1A-D).13ThisconversionwillhappenbecausetheaverageMTfilamentwillexperiencenetgrowthuntil[freetubulin]fallsbelowCCPopGrow(Figure12C,compareearlyintimetolaterintime).Theoutcomeoftheserelationshipsisthatinbulkpolymerexperiments,littleifanyMTpolymermasswillbedetected14untilthetotaltubulinconcentrationisaboveCCPopGrow(Figure12A,darkblueline),eventhoughdynamicindividualMTfilamentscantransientlyexistattubulinconcentrationsbelowCCPopGrow(Figure3C-D).Thus,theexperimentalquantitiesQ1andQ2maylookliketheexpectedcriticalconcentrationforpolymerassembly(CCPolAssem),buttheyactuallyrepresentthecriticalconcentrationforpersistentpopulationgrowth(CCPopGrow).Poccplotscancreatetheillusionthatthereisa[freetubulin]atwhichMTassemblycommencesabruptly,i.e.,thatCCPolAssemexists.Poccplotswithlengthdetectionthresholds(suchasthresholdsintrinsictomicroscope-basedexperiments)(Figure10A-B)mayhaveledtotheconclusionthatthereisaCCPolAssem,atwhichPoccfirstbecomespositive.However,atlow

13Moreprecisely,asindicatedbytheearlierdiscussionofFigure3A-B,allsubunitsinexcessofthesteady-state[freetubulin]willbeconvertedtopolymer;thesteady-state[freetubulin]isnecessarilybelowbutperhapsclosetoCCPopGrow.14Theamountofpolymerpresentdependsonthekineticrateconstantsoftheparticularsystemandthenumberofstableseeds(Figure4).Theamountofpolymerdetecteddependsontheamountofpolymeractuallypresentandonwhattheexperimentalsetupcandetect.

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[freetubulin],MTsareshortandshort-livedasaresultoflowVgandhighFcat,asdescribedabove,andthereforecanbeundetectablebystandardmicroscopy.Byvaryingthelengthdetectionthresholdimposedonsimulationdata(Figure10C-D),itcanbeseenthatthe[freetubulin]atwhichPoccfirstbecomespositivedependsonthethreshold.Theseresults,togetherwiththepolymermass,averagelength,andmaximallengthdata(FiguresS1C-F,S3A-F)indicatethatthereisnoconcentrationatwhichassemblyofDIpolymerscommencesabruptly.TwoadditionalCCshelpdefinepolymerbehaviors.InadditiontothemajorCCs(CCIndGrowandCCPopGrow),thereareatleasttwoadditionalCCsthatimpactMTassembly.ThefirstoftheseisCCKD_GTP=kToffT/kTonT,whichcorrespondstotheKDforbindingofafreeGTP-tubulinsubunittoaGTP-tubulinataMTtip.ThesecondadditionalCCistheKDforbindingofafreeGDP-tubulinsubunittoaGDP-tubulinataMTtip,CCKD_GDP=kDoffD/kDonD.SinceCCKD_GTPandCCKD_GDPprovidebiochemicallimitsonthebehaviorsofGTP-tubulinandGDP-tubulin,anyCCsmustliebetweenthesetwonucleotide-specificCCs(CCKD_GTP≤CCIndGrow≤CCPopGrow≤CCKD_GDP).CCKD_GTPisthe[freetubulin]abovewhichGTP-tubulinpolymerswillgrowintheabsenceofhydrolysisandprovidesthelowerlimitforshort-termassemblyofpolymersinthepresenceofhydrolysis.Asthehydrolysisrateconstantincreases,CCIndGrow(the[freetubulin]abovewhichextendedgrowthphasescanoccur)candivergefromCCKD_GTP(Figure9G,compareyellowCCIndGrowlinetogreyCCKD_GTPline).UnlikeCCKD_GTP,CCKD_GDPisnotstraightforwardlymeasurableforMTs,becauseGDP-tubulinsubunitsalonedonotpolymerizeintomicrotubules(Howard,2001),butcouldberelevanttoothersteady-statepolymers(e.g.,actin).SeparationbetweentheCCsiscreatedbyGTPhydrolysis.ByrunningsimulationsinthesimplifiedmodelatdifferentkHvalues,weshowthatincreasingkHcausesCCIndGrowandCCPopGrowtodivergefromeachotherandfromCCKD_GTP(Figure12E).WeexpectthatthemagnitudeoftheseparationbetweenthevariousCCswilldependnotonthevalueofkHperse,noronanyindividualrateconstants,butratherontherelativerelationshipsbetweenthevariousrateconstants.Thisisatopicofongoinginvestigation.WespeculatethattheseparationbetweentheCCshassignificanceforunderstandingthedifferencebetweenactinandtubulin,asdiscussedmorebelow.Relationshiptopreviouswork.Asdiscussedabove,theideathatMTsandothersteady-state(energy-using)polymershavetwomajorcriticalconcentrationswasfirstinvestigatedindepthbyHillandcolleagues,whostudiedthebehaviorofthesesystemsusingacombinationoftheory,computationalsimulations,andexperiments(HillandChen,1984;Carlieretal.,1984a;Hill,1987).Theirc1(alsoreferredtobyothernamesincludinga1)correspondstoourCCIndGrow;theirc0(alsocalledaα)correspondstoourCCPopGrow(HillandChen,1984;Hill,1987).Moreover,HillandChenconcludedthatMTsgrowatconcentrationsbelowwhattheyreferredtoasthe“real”CC(correspondingtoourQ4inFigure1C)(HillandChen,1984).However,thesignificanceofthisworkforMTDIbehaviorwasnotfullyincorporatedintotheCCliterature,perhapsbecauseitwasnotclearhowtheirtwoCCsrelatedtoclassicalCCmeasurements(e.g.,Q1andQ2inFigure1A).Walkeretal.’sseminal1988manuscriptondynamicinstabilityparametersincludedmeasurementsoftwodifferentcriticalconcentrationsthattheytermedtheCCforelongation(CCIndGrowinournotation)andtheCCfornetassembly(ourCCPopGrow)

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(Walkeretal.,1988).TheycalculatedtheirvalueoftheCCfornetassemblyfromtheirmeasuredDIparametersusingaversionoftheJDIequation(seeequationonpage1445of(Walkeretal.,1988)).However,perhapsbecausethemanuscriptfocusedonCCelongationanddidnotdirectlyrelateeitheroftheseCCstothosepredictedbyHillandcolleagues,theideathatMTshavetwoCCsstilldidnotbecomewidelyacknowledged.Soonthereafter,themanuscriptsofDogterometal.andFygensonetal.wereimportantinshowingclearlyandintuitivelyhowthebehaviorofMTschangesattheCCforunboundedgrowth(ourCCPopGrow),whichtheydescribedusingtheJDIequationshowninEquation1(DogteromandLeibler,1993;Fygensonetal.,1994;Dogterometal.,1995).However,theseauthorsdidnotrelatetheirCCforunboundedgrowthtotheCCsdiscussedbyHillorWalkeretal.ortomoreclassicalCCs(Table1,Figure1).Someofthecontinuedconfusionaboutcriticalconcentrationmayhaveresultedfromthefactthatthepublishedexperimentalworktypicallyinvolvedeithercompetingconditionsornon-competingconditionsbutnotboth.Morespecifically,classicalexperimentsfordetermining“thecriticalconcentration”(e.g.,Figure1A)involvedcompetingconditions,butmuchofthepreviousworkdescribedabovewasperformedunderconditionsofconstant[freetubulin](e.g.,Figure1B-C).Walkeretal.(Walkeretal.,1988)didnoteintheirDiscussionsectionthattheconcentrationoffreetubulinatsteadystateintheircompetingsystemwasbelowtheircalculatedCCfornetassembly(i.e.,CCPopGrow),contrarytotheexpectationthat[freetubulin]SteadyStatewouldequaltheCCfornetassembly.Theyattributedthisdifferenceto“uncertaintiesinherentin[their]assumptionsandmeasurements”(Walkeretal.,1988).Instead,asshownabove,theobservationthat[freetubulin]SteadyStateapproachesCCPopGrowwithoutactuallyreachingitisapredictableaspectofdynamicinstability.Morespecifically,[freetubulin]SteadyStatewillbemeasurablybelowCCPopGrowif[totaltubulin]isnothighenoughrelativetothevalueofCCPopGrowand/orifthenumberofstableseedsislarge(Figures3A-B,4).Morerecently,Mourãoetal.focusedonsystemsofMTsgrowingundercompetingconditions(Mourãoetal.,2011).UsingstochasticsimulationsandmathematicalanalysistostudyMTgrowthfromstableseeds,theyexaminedaquantitythattheycalled“abaselinesteadystatefreesubunitconcentration(MDSS)”,whichisconceptuallysimilartoourCCSubSoln(measuredbyQ2).Theyconcludedthat[freetubulin]SteadyStateisnotequaltoMDSSbutbelowit;ourresultsareconsistentwiththisconclusion.Inparticular,theydemonstratedhowtheseparationbetween[freetubulin]SteadyStateandMDSSdependsonvariousfactorsincludingthenumberofstableMTseeds.ThedependenceofMTbehavioronsubunitconcentrationwasnottheirprimaryfocus,sotheydidnotexplicitlyshowthat[freetubulin]SteadyStateasymptoticallyapproachesMDSS=CCPopGrowas[totaltubulin]increases(Figures3A-B,4);however,theydidperformsimulationsatthreedifferentvaluesof[totaltubulin]andtheirresultsareconsistentwithourconclusions.Additionally,thecriterionthattheyusedtodeterminethevalueofMDSSisthatMDSSisthefreetubulinconcentrationatwhichVg/|Vs|=Fcat/Fres.Wenotethatthisequationisalgebraicallyequivalentto|Vs|Fcat=VgFres,whichwasthecriteriongivenbyDogterometal.(DogteromandLeibler,1993)foridentifyingtheCCforunboundedgrowth(equivalenttoourCCPopGrow).Thus,therehasbeenaneedforaunifiedunderstandingofhowcriticalconcentrationsrelatetoeachotherandtoMTbehaviorsatdifferentscales.Ourworkfillsthisgapbyclearlyshowing

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howthebehaviorsofindividualMTsandpopulationsofMTsrelatetoeachother,to[freesubunit]and[totalsubunit],andtoarangeofdifferentexperimentalmeasurementsinbothcompetingandnon-competingsystems(conclusionssummarizedinFigure12andTable3).Takentogether,oursimulationsandanalysesshouldprovideamoresolidfoundationforunderstandingthebehaviorofMTsandotherDIpolymersundervariedconcentrationsandexperimentalconditions.ConcurrencebetweendifferentapproachesformeasuringMTbehaviorhaspracticalsignificanceAsshowninFigure5C-F,thereisremarkableconcurrencebetweenthreeseeminglydisparatewaysofmeasuringandanalyzingMTbehavior:(i)thenetrateofchangein[polymerizedtubulin](Figure5C-F,osymbols),whichisabulkpropertyobtainedbyassessingthemassofthepopulationofpolymersatdifferentpointsintime(e.g.,across15minutes);(ii)theJDIequation(Figure5C-D,+symbols),whichusesDIparametersextractedfromindividualfilamentlengthhistoryplotsobtainedovertensofminutes;(iii)thedriftcoefficient(Figures5E-F,xsymbols;S3G-H,allsymbols)asmeasuredfromobservingindividualMTsinapopulationofMTsforshortperiodsoftime(e.g.,2-secondtimestepsacrossaslittleasoneminute).Theseapproachesdifferinattributesincludingphysicalscale,temporalscale,andexperimentaldesign.Whilethesimilarityofthedataproducedbythesedifferentapproachesmayinitiallybesurprising,itcanbeshownthatthesemeasurementsshouldyieldthesamevaluesbecausetheequationsunderlyingthemarealsoalgebraicallyequivalentifcertainassumptionsaremet(reviewarticleinpreparation).InadditiontoyieldingmeasurementsofCCPopGrow(Q5abc,Figure5),thesethreeexperimentalapproachescanalsoprovideapproximatemeasurementsofCCIndGrow(Q7,Figure8).Theagreementbetweentheresultsofthesemeasurementsindicatesthattheexperimentallymoretractabletime-stepapproach(Komarovaetal.,2002)(seeSupplementalMethods)canbeusedtomeasurebothCCIndGrowandCCPopGrowandshouldbeusedmorefrequentlytoquantitativelyassessMTassemblybehaviorinthefuture.BiologicalsignificanceofhavingtwomajorcriticalconcentrationsTheunderstandingofcriticalconcentrationaspresentedaboveshouldhelpresolveapparentlycontradictoryresultsinthemicrotubuleliterature.Inparticular,ourresultsindicatethatreportedmeasurementsof“the”criticalconcentrationforMTpolymerizationvaryatleastinpartbecausesomeexperimentsmeasureCCIndGrow(e.g.(Walkeretal.,1988;Wieczoreketal.,2015)),whileothersmeasureCCPopGrow(e.g.,(Carlieretal.,1984a;Dogterometal.,1995;Mirigianetal.,2013)).Thisclarificationshouldhelpindesignandinterpretationofexperimentsinvolvingcriticalconcentration,especiallythoseinvestigatingtheeffectsofMTbindingproteins(e.g.(Amayedetal.,2002;Wieczoreketal.,2015;Hussmannetal.,2016)),osmolytes(e.g,(Schummeletal.,2017))ordrugs(e.g.(Bueyetal.,2005;Vermaetal.,2016)).Additionally,theseideascanbeappliedtohelpclarifythebehaviorofMTsinvivo.MTsinmanyinterphasecelltypesgrowpersistently(perhapswithcatastropheandrescue,butwithnetpositivedrift)untiltheyreachthecelledge,wheretheyundergorepeatedcyclesofcatastropheandrescuewithrarecompletedepolymerizations(Komarovaetal.,2002).Weshowed

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previouslythatthispersistentgrowthisapredictableoutcomeofhavingenoughtubulininaconfinedspace:ifsufficienttubulinispresent,theMTsgrowlongenoughtocontactthecellboundary,whichcausescatastrophe;thisdrivesthe[freetubulin]aboveitsnaturalsteady-statevalue,whichreducescatastrophe,enhancesrescue,andinducesthepersistentgrowthbehavior(Gregorettietal.,2006).Inlightofthecurrentresults,wecannowphrasethispreviousworkmoresuccinctly:persistentgrowthofMTsininterphasecellsoccurswhencatastrophesinducedbythecellboundarydrive[freetubulin]aboveCCPopGrow.Incontrast,atmitosis,whentheMTsaremorenumerousandthusshorter,[freetubulin]remainsbelowCCPopGrow.Seealso(Dogterometal.,1995;Gregorettietal.,2006;VorobjevandMaly,2008;Mourãoetal.,2011)forrelevantdataanddiscussions.Furthermore,itisimportanttoemphasizethatCCIndGrowandCCPopGrowarefundamentalattributesofaspecifictypeoftubulininaparticularenvironment,similartothewayaKDcharacterizesaprotein-proteininteractionoraKMcharacterizesanenzyme-substratereaction.Thus,wesuggestusingCCIndGrow(asmeasuredbyQ3,Q6,orQ7)andCCPopGrow(especiallyasmeasuredbyQ5cfromthetime-stepdriftcoefficientapproach)inadditiontousingdynamicinstabilityparametersasawaytocharacterizetubulin(orotherproteinsthatformpolymers)andtheactivitiesofproteinsthatalterpolymerassembly(seealsothediscussionin(Komarovaetal.,2002)).Relevanceforothersteady-statepolymersThoughthestudiespresentedherewereformulatedspecificallyforMTs,wesuggestthattheycanbeappliedtoanynucleated,steady-statepolymersthatdisplaydynamicinstability,andperhapstosteady-statepolymersmorebroadly.Inparticular,weproposethatthekeycharacteristicthatdistinguishesdynamicallyunstablesteady-statepolymers(e.g.mammalianMTs)fromothersteady-statepolymers(e.g.,mammalianactin)isasfollows:forDIpolymers,CCIndGrowandCCPopGrowareseparablevaluesdrivenapartbyhydrolysis,butforotherpolymers,theyareeitheridentical(asistrueforequilibriumpolymers)orsocloseastobenearlysuperimposed(e.g.,mammalianactin).ThevaluesofCCIndGrowandCCPopGrowareemergentpropertiesofthekineticrateconstants,whichinturnareintrinsicpropertiesoftheproteinsequenceofthesubunits(whichcomesfromthegenesequence)andpost-translationmodifications(whichcomefromthecelltypeandcellularsignaling).WhetherornotdynamicinstabilityisphysiologicallyrelevantforagivenpolymertypeinaspecificcellularenvironmentwilldependonhowthevaluesofCCIndGrowandCCPopGrowrelatetothecellularsubunitconcentration.METHODSSimulations SimplifiedModel(Figure2A):Asdiscussedinthemaintext,thesimplifiedmodelofstochasticmicrotubuledynamicswasdescribedpreviously(Gregorettietal.,2006),buttheimplementationusedherewasupdatedsignificantly.First,thecodewasrewritteninJavaso

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thatitcouldbemoreeasilyimplementedonpersonalcomputers.Second,thetimebetweeneventsisnowsampledusinganexactversionoftheGillespiealgorithm(Gillespie,1976),insteadofanapproximateversionwithafixedtimestep.Thischangeimprovestheaccuracywithwhichthesimulationcarriesouttheunderlyingbiochemicalmodelwithuser-inputtedrateconstants.Third,thesimulationwasadjustedsothateachsimulatedsubunitnowcorrespondstoan8nmMTring(1x13dimers)insteadofa20nmMTbrick(2.5x10dimers)asin(Gregorettietal.,2006).Also,thesimulationsin(Gregorettietal.,2006)hadacelledge,whichlimitedtheMTlengths;thesimulationspresentedherehavenophysicalconstraintsontheMTlengths.ThechangeinsubunitsizeandthelackofphysicalboundaryinthepresentsimulationmeanthatthenumericalvaluesoftheDIparametersandQmeasurements(Figures3-8,leftpanels)arenotdirectlycomparablebetweenthisimplementationandourearlierpublication(Gregorettietal.,2006).However,thegeneralbehaviorofthesimulationisthesame.Theinputparametersusedhereareasfollows:kTonT 2.0µM/sec kineticrateconstantforadditionofGTP-tubulinontoGTP-MTendkTonD 0.1µM/sec kineticrateconstantforadditionofGTP-tubulinontoGDP-MTendkToffT,kToffD 0.0/seckineticrateconstantforlossofGTP-tubulinfromGTP-orGDP-MTendkDoffT,kDoffD 48/seckineticrateconstantforlossofGDP-tubulinfromGTP-orGDP-MTendkh 1/seckineticrateconstantfornucleotidehydrolysis(GTP-tubulin-->GDP-tubulin)Vol 500fLvolumeofsimulationUnlessotherwiseindicated,eachofthesimplifiedmodelsimulationswasrunwithMTsgrowingfrom100stableseedscomposedofnon-hydrolyzableGTP-tubulin.DetailedModel(Figure2B):ThedetailedmodelofstochasticmicrotubuledynamicswithparameterstunedtoapproximateinvitrodynamicinstabilityofmammalianbrainMTswasfirstdevelopedin(Margolinetal.,2011;Margolinetal.,2012)andlaterutilizedin(Guptaetal.,2013;Lietal.2014;Duanetal.,2017).Thecoresimulationisthesameasthatinthesepriorpublications,butthisversionhasminormodificationsincludingtheadditionofadilutionfunctiontoenableproductionofJ(c)plotssuchasthoseinFigure6.Pleasereferto(Margolinetal.,2012)fordetailedinformationonthemodel,itsparametersetC,andhowitsbehaviorcomparestothatofinvitrodynamicinstability.Unlessotherwiseindicated,eachofthedetailedmodelsimulationswasrunwithMTsgrowingfrom40stableseedscomposedofnon-hydrolyzableGTP-tubulininavolumeof500fL.ThenumericalvaluesoftheDIparametersforbothmodelsasmeasuredbyourautomatedDIanalysistool(describedintheSupplementalMethods)areprovidedintheSupplementalExcelfiles.Thevaluesforthedetailedmodelaresimilartothosethatwepublishedpreviouslyforthismodel(Margolinetal.,2012;Duanetal.,2017).AnalysisCCPopGrowisestimatedbydeterminingQ1,Q2,Q4,orQ5(Figures3-6).CCIndGrowisestimated(perhapspoorly)bydeterminingQ3,Q6,orQ7(Figures7-8).SeeTable3BforinformationonhowtoperformeachoftheQmeasurements.Thefigurelegendsprovidedetailsaboutapplyingthemeasurementstothesimulationdata,includinginformationaboutthetimeperiodsduring

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whichmeasurementswereperformed.Thetimeperiodswerechosentoensurethatthevariablebeingmeasured(e.g.,rateofchangeinaveragelength)hasreacheditssteady-statevalue.Formostofthemeasurements,thisoccurswhenthesimulatedsystemhasreachedeitherpolymer-masssteadystate(non-competingsystemswith[freetubulin]<CCPopGrow,FigureS3A-B;andcompetingsystems,FiguresS1A-D)orpolymer-growthsteadystate(non-competingsystemswith[freetubulin]>CCPopGrow,FigureS3A-B).IntheSupplementalMethods,wedescribethetime-stepanalysismethod(basedon(Komarovaetal.,2002))usedtomeasuredriftandVg,aswellasourDIanalysismethodusedtomeasureVg,Vs,Fcat,andFres.CodeAvailabilityThesimulationcodes(writteninJava)andanalysiscodes(writteninMATLAB)areavailableuponrequest.ACKNOWLEDGMENTSThisworkwassupportedbyNSFgrantsMCB-1244593toHVGandMSA,MCB-1817966toHVG,andMCB-1817632toEMJ.PortionsoftheworkwerealsosupportedbyfundingfromtheUniversityofMassachusettsAmherst(AJM)andbyafellowshipfromtheDoloresZohrabLiebmannFund(SMM).WethankthemembersoftheChicagoCytoskeletoncommunityfortheirinsightfuldiscussions,andmembersoftheGoodsonlaboratoryforassistanceineditingthemanuscript.

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Table1:Traditionalcriticalconcentration(CC)definitionsusedintheliterature.Thesedefinitionsofcriticalconcentration(CC)areinterchangeableforequilibriumpolymers,buthavenotallbeencomparedinasingleanalysisforDIpolymers.ForeachCCdefinition,wehaveassignedaspecificabbreviationandprovideanexampleofanearlypublicationwherethatdefinitionwasused.ThetermsCCPolAssem,CCSubSoln,etc.refertotheoreticalvalues(concepts),andQ1,Q2,etc.refertoexperimentallymeasurablequantities(i.e.,valuesobtainedthroughexperimentalapproachesasindicatedinthefigures).AlldefinitionsexceptCCKDcanbeappliedtobothequilibriumandsteady-statepolymers(CCKDassumesthesystemisatequilibriumandthereforecanbeappliedtoonlyequilibriumpolymers).ThetraditionalframeworkinTable1willberevisedintheResultsSection(seeTable3forasummary).Classicalcriticalconcentrationdefinition Abbreviation ExperimentalmeasurementofCCasappliedtoMTsystems

Minimalconcentrationoftotalsubunits(e.g.,tubulindimers)necessaryforpolymerassembly(Oosawa,1970;JohnsonandBorisy,1975).

CCPolAssem CCPolAssemisdeterminedbymeasuringsteady-state[polymerizedtubulin]atdifferent[totaltubulin]inacompetingsystemandextrapolatingbackto[polymerizedtubulin]=0.SeeQ1inFigure1A;alsoFigures3A-B,4.

Concentrationoffreesubunitsleftinsolutiononceequilibriumorsteady-stateassembly15hasbeenachieved(Oosawa,1970;JohnsonandBorisy,1975).

CCSubSoln CCSubSolnisdeterminedbymeasuring[freetubulin]leftinsolutionatsteadystatefordifferent[totaltubulin]inacompetingsystemanddeterminingthepositionoftheplateaureachedby[freetubulin].SeeQ2inFigure1A;alsoFigures3A-B,4.

Dissociationequilibriumconstantforthebindingofsubunittopolymer,i.e.,CC=KD=koff/kon

16(OosawaandAsakura,1975).

CCKD CCKDcanbedeterminedbyseparateexperimentalmeasurementsofkonandkoffforaddition/lossoftubulinsubunitsto/fromMTpolymer,respectively,andcalculatingtheratiokoff/kon.

Concentrationoffreesubunitatwhichtherateofassociationequalstherateofdissociationduringtheelongationphase17(calledSc

ein(Walkeretal.,1988);similartoc1in(HillandChen,1984).

CCelongation CCelongationisdeterminedbymeasuringthegrowthrateduringthegrowthstate(Vg)atavariousvaluesof[freetubulin]andextrapolatingbacktothe[freetubulin]atwhichVg=0.SeeQ3inFigure1B;alsoFigure7A-B.

Concentrationoffreesubunitatwhichthefluxesofsubunitsintoandoutofpolymerarebalanced,i.e.,thenetfluxiszero(e.g.,(Carlieretal.,1984a;HillandChen,1984).

CCflux CCfluxisdeterminedbygrowingMTstosteady-stateatveryhigh[totaltubulin],thenrapidlydilutingtoanew[freetubulin]andmeasuringtheinitialrateofchangein[polymerizedtubulin](i.e.,[polymerizedtubulin]flux).CCfluxisthevalueof[freetubulin]where[polymerizedtubulin]flux=0.SeeQ4inFigure1C;alsoFigure6.

Concentrationoffreesubunitatwhichpolymerstransitionfrom“boundedgrowth”to“unboundedgrowth”(calledccrin(DogteromandLeibler,1993)).

CCunbounded CCunboundedisthe[freetubulin]atwhichtherateofchangeinaverageMTlengthtransitionsfromequalingzerotobeingpositive.SeeQ5inFigure5.CCunboundedcanbeidentifiedbymeasuringDIparametersfromMTlengthhistories(Figure1E-F)acrossarangeofdifferent[freetubulin]anddeterminingthe[freetubulin]atwhichVgFres=|Vs|Fcat.

15Assumingthatassemblystartsfromastatewithnopolymer,maximalpolymerassemblywilloccuratequilibriumforequilibriumpolymers,andatpolymer-masssteadystateforsteady-statepolymers.Steady-statepolymerswillbe(mostly)disassembledatthermodynamicequilibriumbecausethenucleotidesinthesystemwillbe(effectively)entirelyhydrolyzed. 16TheideathatCC=KDforsimpleequilibriumpolymersisderivedasfollows.Thenetrateofpolymerlengthchangeatasinglefilamenttip=rateofaddition–rateofloss.Therateofadditionisassumedtobekon[freesubunit],andtherateoflossisassumedtobekoff.Therefore,therateatwhichnewsubunitsaddtoapopulationofnpolymersisn*kon[freesubunit],andtherateatwhichsubunitsdetachfromapopulationofnpolymersisn*koff.Atequilibrium,rateofpolymerization=rateofdepolymerization,son*kon[freesubunit]=n*koff.Therefore,atequilibrium,[freesubunit]=koff/kon=KD=CCKD.17CCelongationhasbeeninterpretedastheminimalconcentrationoffreesubunitrequiredtoelongatefromagrowingpolymer.ThederivationofCCelongationissimilartothatforCCKD,butconsidersthebehaviorofasinglefilament,notapopulation,andcanapplytosteady-statepolymersbecauseitdoesnotrequireequilibrium.Forpolymersdisplayingdynamicinstability,measurementsofCCelongationareperformedduringthegrowthstateofdynamicinstability.ThederivationofCCelongationassumesthatVgisalinearfunctionof[freesubunit],i.e.,Vg = kongrowth freesubunit -koff growth,where kongrowthandkoff growthareobservedrateconstantsduringgrowth.Then,the[freesubunit]atwhichVg=0iskoff

growth kongrowth =CCelongation.

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Table2:TypesofExperiments/Simulations.TypeofExperiment/Simulation

Description

Competing Closedsystemwhere[totaltubulin]isheldconstantandMTscompetefortubulin(e.g.,intesttube)

Non-Competing Opensystemwhere[freetubulin]isheldconstant(e.g.,inaflowcell)

Dilution SystemwhereMTsaregrowntopolymer-masssteadystateundercompetingconditionsatveryhigh[totaltubulin]andthenmovedintonon-competingconditionsatvariousvaluesof[freetubulin]

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Table3A:Revisedunderstandingofcriticalconcentrationfordynamicinstabilitypolymers.Notethatforsteady-statepolymers(includingDIpolymers),CCKD_GTP��CCIndGrow�CCPopGrow��CCKD_GDP,butforequilibriumpolymers,CCKD��CCIndGrow�CCPopGrow.CriticalConcentration

RepresentativeFigures

CriticalConcentrationDescription

Equivalentto(seeTable1)18

Measuredby(seeTable3B)

CCPopGrow 1A,C,3-6 CCabovewhichthepolymermassofapopulationwillincreasepersistently,andindividualfilamentswillundergonetgrowthovertime

CCSubSoln,19CCflux,20CCunbounded

Q1,Q2,Q4,Q5

CCIndGrow 1B,7,8 CCabovewhichindividualfilamentscanexhibittransient,butextended,growthphases

CCelongation Q3,Q6,Q7

CCKD_GTP 9 EquilibriumdissociationconstantforbindingofafreeGTP-subunittoaGTP-subunitatapolymertip

AnyoftheQvaluesaboveunderconditionswhereGTPisnothydrolyzed

CCKD_GDP EquilibriumdissociationconstantforbindingofafreeGDP-subunittoaGDP-subunitatapolymertip

GDP-tubulinalonedoesnotformMTs,soCCKD_GDPisnotstraightforwardlymeasured

18CCPolAssemisnotlistedherebecausethereisnothresholdconcentrationatwhichpolymersabruptlyappear.Instead,themeasurementclassicallyexpectedtoyieldCCPolAssem(seeQ1inTable3B)actuallyyieldsCCPopGrow.19NotethatCCSubSolnisclassicallydefinedasthevalueof[freetubulin]SteadyStateinacompetingsystemwhenever[totaltubulin]isabove“CCPolAssem”(Table1,Figure1A).However,CCSubSolnismoreaccuratelydefinedastheasymptoteapproachedby[freetubulin]SteadyStateas[totaltubulin]isincreased(Q2inFigures3A-B,4).20ItshouldbestressedthatCCfluxisthe[freetubulin]atwhichthepopulation-levelfluxesoftubulinintoandoutofpolymerarebalanced,whileindividualsmaygrowandshortenwhen[freetubulin]=CCflux.

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Table3B:Summaryofexperimentallymeasureablequantities(Qvalues)usedtoestimateCCs.SeeTable3AfordescriptionsoftheCCs.Qvalue Representative

FiguresDescriptionofExperimentallyMeasureableQuantity CCestimated

byQQ1 1A,3A-B,4 Q1isthex-interceptoftheline(withslope=1)approachedbysteady-

state[polymerizedtubulin]as[totaltubulin]isincreasedinacompetingsystem.

CCPopGrow

Q2 1A,3A-B,4 Q2isthehorizontalasymptoteapproachedby[freetubulin]SteadyStateas[totaltubulin]isincreasedinacompetingsystem.

CCPopGrow(=CCSubSoln)

Q3 1B,7A-B Q3isthe[freetubulin]atwhichVg=0.Q3isestimatedbyplottingVgasafunctionof[freetubulin],fittingaregressionlinetotheapproximatelylinearpartoftheVgdata,andextrapolatingbacktothe[freetubulin]atwhichVg=0.

CCIndGrow(=CCelongation)

Q4 1C,6 Q4isthe[freetubulin]atwhichtherateofchangein[polymerizedtubulin]equalszeroinadilutionexperiment(J<0whendilution[freetubulin]<Q4;J>0whendilution[freetubulin]>Q4)21.Q4isdeterminedbygrowingMTstopolymer-masssteadystateathigh[totaltubulin],thenrapidlydilutingtoanew[freetubulin]andmeasuringtherateofchangein[polymerizedtubulin]afterashortdelay.22

CCPopGrow(=CCflux)

Q5(a,b,andc)

5C-F Q5isthe[freetubulin]abovewhichtherateofchangeinaverageMTlengthispositiveinanexperimentwhere[freetubulin]isheldconstantandthepopulationhasreachedpolymer-massorpolymer-growthsteadystate(J=0when[freetubulin]<Q5;J>0when[freetubulin]>Q5)23.Q5canalsobedescribedastheconcentrationabovewhichthepopulationdriftcoefficientispositive.WeusethenamesQ5a,Q5b,orQ5cdependingonhowJismeasured.

CCPopGrow(=CCunbounded)

Q5a 5C-F,6C-D Q5aisQ5withJcalculatedfromthenetrateofchangeinapopulation’saverageMTlengthbetweentwotimepoints,i.e.,J=(averagelengthattimeB–averagelengthattimeA)/(timeB–timeA).

Q5b 5C-D Q5bisQ5withJcalculatedfrommeasuredDIparametersusingtheJDIequation(Equation1ofmaintext).Q5bisthe[freetubulin]atwhichVgFres=|Vs|Fcat.

Q5c 5E-F Q5cisQ5withJcalculatedbysummingdisplacementsmeasuredovershorttimesteps(seeSupplementalMethodssubsectiononmeasuringdriftcoefficient).

Q6 7C-D Q6ismeasuredthesamewayasQ3,butusinggrowthphasesfromadilutionexperimentafterthesystemhasbeendilutedintoconstant[freetubulin]conditions(insteadof[freetubulin]beingconstantfortheentireexperimentaswithQ3).

CCIndGrow

Q7 8C-D Q7isthex-interceptofthelineapproachedbyJas[freetubulin]isincreased(note,Japproachesthelinewhen[freetubulin]>>CCPopGrow).

CCIndGrow

21JcanbedefinedintermsofpolymermassoraverageMTlength:J=rateofchangein[polymerizedtubulin]=fluxoftubulinintoandoutofpolymer(e.g.,inµM/s);orJ=rateofchangeinaverageMTlength=driftcoefficient(e.g.,inµm/s).22ThedelayallowstheGTPcapsizetoadjustinresponsetothenew[freetubulin].23Note,thecloser[freetubulin]istoCCPopGrow,thelongeritwilltakeforthesystemtoreachsteadystate.IfJismeasuredbeforepolymer-masssteadystatehasbeenreachedfor[freetubulin]<CCPopGrow,thenJwillappeartobepositivefor[freetubulin]nearbutbelowCCPopGrow;thiswouldmakeitdifficulttoidentitytheprecisevalueofQ5.ThetransitionfromJ=0toJ>0atQ5willbesharperthelongerthesystemisallowedtorun.

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FIGURESANDLEGENDS

Figure1:Classicalunderstandingofmicrotubule(MT)polymerassemblybehavior.SeeTable1foradditionaldescriptionofthecriticalconcentrationmeasurementsdepictedhere.[Freetubulin]istheconcentrationoftubulindimersinsolution,[polymerizedtubulin]istheconcentrationoftubulindimersinpolymerizedform,and[totaltubulin]=[freetubulin]+[polymerizedtubulin].(A)Inacompeting(closed)system,[totaltubulin]isheldconstantovertimeandMTscompetefortubulin.Astypicallypresentedintextbooks,thecriticalconcentration(CC)canbe

Figure 1

[Free Tubulin]

C

Ste

ady-

Sta

te [T

ubul

in]

A B

D

Net

Flu

x of

Tub

ulin

in

to a

nd o

ut o

f Pol

ymer

Dilution [Free Tubulin]

Q3 +

Velo

city

of G

row

th

Dur

ing

Gro

wth

Pha

ses

Q4

Experimentally Measureable

Quantity

Traditional Measurement Method (square brackets [ ] represent concentration)

Theoretical Critical Concentration

corresponding to Q

Q1 Determined by measuring steady-state [polymerized tubulin] at different [total tubulin] in a competing (closed) system and extrapolating back to [polymerized tubulin] = 0.

CCPolAssem

Q2 Plateau reached by steady-state [free tubulin] as [total tubulin] is increased in a competing system. CCSubSoln

Q3 [Free tubulin] at which Vg = 0 (where Vg is the growth velocity during growth phases of individuals), measured by plotting Vg as a function of [free tubulin] and extrapolating back to Vg = 0.

CCelongation

Q4 [Free tubulin] at which the flux (measured as the rate of change in [polymerized tubulin]) equals zero in a dilution experiment. CCflux

[Total Tubulin]

Q1

Q2

Free Tubulin Polymerized Tubulin

Competing Non-Competing Dilution

Time

Catastrophe

Rescue

Gro

wth

Shortening

MT

Leng

th

E Length History Abbreviation Definition

DI Dynamic instability (stochastic switching between phases of growth and shortening)

DI parameters

Four measurements commonly used to quantify DI behavior: Vg, Vs, Fcat, and Fres as defined below

Vg Growth velocity during growth phases

Vs Shortening velocity during shortening phases Note: We use Vs to mean shortening velocity (negative number). Some papers use Vs to mean shortening speed (positive number).

Fcat Catastrophe frequency = # of catastrophes / time in growth

Fres Rescue frequency = # of rescues / time in shortening

F

+ +

+

Traditional Measurements of Critical Concentration

41

measuredinacompetingsystembyobservingeithertheconcentrationoftotaltubulinatwhichMTpolymerappears(Q1)ortheconcentrationoffreetubulinleftinsolutiononcetheamountofpolymerhasreachedsteadystate(Q2).(B)Inanon-competing(open)system,[freetubulin]isheldconstantovertime.Insuchasystem,criticalconcentrationisconsideredtobetheminimumconcentrationoftubulinnecessaryforMTpolymerstogrow,whichisestimatedbymeasuringthegrowthrateofindividualfilaments(Vg)andextrapolatingbacktoVg=0(Q3).(C)Indilutionexperiments,MTsaregrownundercompetingconditionsuntilthesystemreachespolymer-masssteadystate,andthendilutedintovarious[freetubulin].Theinitialrateofchangein[polymerizedtubulin]ismeasured.Here,criticalconcentrationistheconcentrationofdilution[freetubulin]atwhichtherateofchangein[polymerizedtubulin]iszero,(i.e.,thedilution[freetubulin]atwhichthenetfluxoftubulinintoandoutofMTpolymeriszero)(Q4).(D)SummarytableofthedefinitionsoftheexperimentallymeasureablequantitiesQ1-4depictedinpanelsA-C.(E)IndividualMTsexhibitabehaviorcalleddynamicinstability(DI),inwhichtheindividualsundergophasesofgrowthandshorteningseparatedbyapproximatelyrandomtransitionstermedcatastropheandrescue.(F)TableofdefinitionsofDIparameters(fourmeasurementscommonlyusedtoquantifyDIbehavior).

42

Figure2:Processesthatoccurinthecomputationalmodels.(A)Inthesimplifiedmodel,microtubulesareapproximatedassimplelinearfilamentsthatcanundergothreeprocesses:subunitaddition,loss,andhydrolysis.Additionandlosscanoccuronlyatthetip.HydrolysiscanoccuranywhereinthefilamentwherethereisaGTP-subunit.(B)Inthedetailedmodel,thereare13protofilaments,whicheachundergothesameprocessesasinthesimplifiedmodelbutalsoundergolateralbondingandbreakingbetweenadjacentprotofilaments.(C)Informationaboutthesubunitsinthemodels.Inbothmodels,thekineticrateconstants(panelD)controllingtheseprocessesareinputtedbytheuser,andtheMTsgrowoffofauser-definedconstantnumberofstableMTseeds(composedofnon-hydrolyzableGTP-tubulin).Thestandarddynamicinstabilityparameters(Vg,Vs,Fcat,Fres;seeFigure1E-F)areemergentpropertiesoftheinputrateconstants,[freetubulin],andotheraspectsoftheenvironmentsuchasthenumberofstableseeds.Formoreinformationaboutthemodelsandtheirparametersets,seeBox1,Methods,SupplementalMethods,and(Gregorettietal.,2006;Margolinetal.,2011;Margolinetal.,2012).

Figure 2

B

detachment attachment

breakage

lateral bond

formation

hydrolysis T D

A

attachment

non-hydrolyzable stable GTP-tubulin seed

Simplified Model Detailed Model

detachment

C Symbol Definition of model subunit

Simplified model subunit, represents a 1 x 13 ring of tubulin dimers

Detailed model subunit, represents one tubulin dimer

Abbreviation Definition of biochemical kinetic rate constant (values inputted by user)

kTonT, kTonD, kDonT, kDonD Kinetic rate constants for attachment (Fig. 2A-B) of a free subunit to a filament tip.

kToffT, kToffD, kDoffT, kDoffD Kinetic rate constants for detachment (Fig. 2A-B) of a subunit from a filament tip.

kH Kinetic rate constant for hydrolysis (Fig. 2A-B) of nucleotide bound to tubulin (conversion of GTP-tubulin to GDP-tubulin).

In the detailed model, there are additional inputs such as kinetic rate constants for lateral bond formation and breakage (Fig. 2B) between adjacent protofilaments (please see (Margolin et al., 2011; Margolin et al., 2012)).

D

x 13 protofilaments

D T

T

Abbreviation Nucleotide state of subunit

T GTP-tubulin subunit (purple), GTP = guanosine triphosphate

D GDP-tubulin subunit (teal), GDP = guanosine diphosphate

hydrolysis

43

Figure3:Behaviorofmicrotubules(populationsandindividuals)underconditionsofconstanttotaltubulin.Leftpanels:simplifiedmodel;rightpanels:detailedmodel;colorsofdatapointsreflecttheconcentrationsoftotaltubulin.(A,B)Classicalcriticalconcentrationmeasurements(comparetoFigure1A).SystemsofcompetingMTsattotaltubulinconcentrationsasindicatedonthehorizontalaxiswereeachallowedtoreachpolymer-masssteadystate(showninFigureS1A-D).Thenthesteady-stateconcentrationsoffree(squares)andpolymerized(circles)tubulinwereplottedasfunctionsof[totaltubulin].(C,D)RepresentativelengthhistoryplotsforindividualMTsfromthesimulationsusedinpanelsA-B.Thevalueof[totaltubulin]foreachlengthhistoryisindicatedinthecolorkeysatthetopofpanelsC-D.Interpretation:Classically,Q1estimatesCCPolAssem,andQ2estimatesCCSubSoln.However,ascanbeseeninpanelC-D,MTsgrowinbothsimulationsat[totaltubulin]belowQ1≈Q2(~2.85µMinthesimplifiedmodeland~11.8µMinthedetailedmodel).Consistentwiththisobservation,themaintextprovidesjustificationfortheideathatCCasestimatedbyQ1≈Q2insteadmeasuresCCPopGrow,theCCforpersistentpopulationgrowth.NotethatthedifferenceinthevaluesofQ1≈Q2betweenthetwosimulationsisexpectedfromthefactthattheinputtedkineticparametersforthesimulationswerechosentoproducequantitativelydifferentDImeasurementsinordertoprovideatestofthegeneralityofconclusionsaboutqualitativebehaviors;theresultsshowthatthebehaviorsareindeedqualitativelysimilarbetweenthetwosimulations.Foradditionaldatarelatedtothesesimulations(e.g.,plotsof[freetubulin]and[polymerizedtubulin]asfunctionsoftime),seeFigureS1.Methods:DatapointsinpanelsA-Brepresentthemean+/-onestandarddeviationofthevaluesobtainedinthreeindependentrunsofthesimulations.Thevaluesfromeachofthreerunsareaveragesover15to30minutesforthesimplifiedmodel(panel3A)andover30to60minutesforthedetailedmodel(panel3B).Thesetimeperiodswerechosensothat[freetubulin]and[polymerizedtubulin]havereachedtheirsteady-statevalues(FigureS1A-D).

A

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Figure 3

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Q2

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30 60 0 0

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8

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Free Tubulin

Polymerized Tubulin

Competing Simulations

9 10 11 < Q1 ≈ Q2 < 12 15 20 µM

44

Figure4:Impactofchangingthenumberofmicrotubuleseeds.Steady-stateconcentrationsoffree(squares)andpolymerized(circles)tubulininacompetingsystemasinFigure3A-B.(A,C)SimplifiedmodelwithMTsgrowingfrom5,100,or500stableMTseeds(datafor100seedsre-plottedfromFigure3A).(B,D)DetailedmodelwithMTsgrowingfrom5,40,or100stableMTseeds(datafor40seedsre-plottedfromFigure3B).Panels4C-Dshowzoom-insofthedataplottedinpanels4A-B,respectively.Thedarkercurveswithsmallersymbolscorrespondtofewerseedsandthelightercurveswithlargersymbolscorrespondtomoreseeds.Interpretation:ThesedatashowthatchangingthenumberofstableMTseedsalterstheapproachtotheasymptotesdeterminingQ1andQ2(dashedgreylinesre-plottedherefromFigure3A-B),butdoesnotchangethevalueofQ1≈Q2.Methods:Datapointsrepresentthemean+/-onestandarddeviationofthevaluesobtainedinthreeindependentrunsofthesimulations.SimilartoFigure3,[freetubulin]and[polymerizedtubulin]fromeachrunwereaveragedoveraperiodoftimeafterpolymer-masssteadystatewasreached.ThetimetoreachthissteadystatedependsonthenumberofstableMTseeds(seeFigureS2).Forthesimplifiedmodel,theaveragesof[freetubulin]and[polymerizedtubulin]weretakenfrom120to150minutesfor5MTseedsandfrom15to30minutesfor100and500MTseeds.Forthedetailedmodel,theaveragesweretakenfrom100to150minutesfor5MTseedsandfrom30to60minutesfor40and100MTseeds.Wewereabletouseahighernumberofseedsinthesimplifiedmodelthaninthedetailedmodelbecauseitismorecomputationallyefficient.

A

9.0

6.0

3.0

0.0 Ste

ady-

Sta

te [T

ubul

in] (µM

)

[Total Tubulin] (µM) 0 10 15 20

Simplified Model

Figure 4

Detailed Model

30.0

20.0

10.0

0.0 Ste

ady-

Sta

te [T

ubul

in] (µM

)

[Total Tubulin] (µM) 0 10 30 20 50 40

B 12.0

Q1 Q1

Q2

Ste

ady-

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te [T

ubul

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)

2.0

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)

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Q2

Q1

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Q2

Q1

6.0

Free Tubulin: 5 seeds 100 seeds 500 seeds Polymerized

Tubulin: 5 seeds 100 seeds 500 seeds

Polymerized Tubulin: 5 seeds 40 seeds 100 seeds

Free Tubulin: 5 seeds 40 seeds 100 seeds

Polymerized Tubulin: 5 seeds 100 seeds 500 seeds

Free Tubulin: 5 seeds 100 seeds 500 seeds

Free Tubulin: 5 seeds 40 seeds 100 seeds

Polymerized Tubulin: 5 seeds 40 seeds 100 seeds

Competing Simulations

45

Figure5:Behaviorofmicrotubules(individualsandpopulations)underconditionsofconstantfreetubulin.Leftpanels:simplifiedmodel;rightpanels:detailedmodel;colorsofdatapointsreflecttheconcentrationsoffreetubulin.(A,B)RepresentativelengthhistoryplotsforoneindividualMTateachindicatedconstantfreetubulinconcentration.(C,D)Steady-statenetrateofchange(osymbols)inaverageMTlength(leftaxis)orinconcentrationofpolymerizedtubulin(rightaxis)forthefreetubulinconcentrationsshown.Q5aindicatestheconcentrationatwhichthisratebecomespositive.ThispanelalsoshowsthetheoreticalrateofchangeinaverageMTlength(+symbols)ascalculatedfromtheextractedDImeasurementsusingtheequationJDI=(VgFres–|Vs|Fcat)/(Fcat+Fres)inthe[freetubulin]rangewhereJDI>0(Equation1inthe“unboundedgrowth”regime)(HillandChen,1984;Walkeretal.,1988;Verdeetal.,1992;DogteromandLeibler,1993).Q5bistheconcentrationatwhichJDIbecomespositive.(E,F)Driftcoefficient(Komarovaetal.,2002)ofMTpopulationsasafunctionof[freetubulin](xsymbols).Q5cistheconcentrationabovewhichdriftispositive.Foreaseofcomparison,therateofchangeinaverageMTlength(osymbols)frompanelsCandDisre-plottedinpanelsEandFrespectively.Foradditionaldatarelatedtothesesimulations,seeFigureS3.Interpretation:TheresultsshowthatQ5a≈Q5b≈Q5c,

µM 25.0 20.0 15.0 13.0 12.0 11.8 11.4 11.0 10.8 10.0 9.0

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A Simplified Model Detailed Model

C

Figure 5

20.0

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E F

[Free Tubulin] (µM) 0 5 10 25 20

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[Free Tubulin] (µM) 0 2 4 6

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50.0

Non-Competing Simulations

Q1≈Q2 Q1≈Q2 Q

1 ≈

Q2

Q1 ≈

Q2

JDI equation

Net rate of change

0.03

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Net rate of change Drift coefficient

JDI equation

Net rate of change 0.030

0.015

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Q1 ≈

Q2

Q1 ≈

Q2

46

hereafterreferredtoasQ5.AtconcentrationsbelowQ5,populationsofMTsreachapolymer-masssteadystatewheretheaverageMTlengthisconstantovertime(therateofchangeinaverageMTlengthorpolymermassisapproximatelyzero;panelsC-D),andthesystemofMTsexhibitszerodrift(panelsE-F).AtfreetubulinconcentrationsaboveQ5,populationsofMTsreachapolymer-growthsteadystatewheretheaverageMTlengthandpolymermassincreaseovertimeatconstantaverageratesthatdependon[freetubulin](panelsC-D),andsystemofMTsexhibitspositivedrift(panelsE-F).TheaverageMTlengthasafunctionoftimeisshowninFigureS3A-B.NotethattheconcentrationrangebelowQ5correspondstothe“bounded”regimeasdiscussedbyDogterometal.,whilethataboveQ5correspondstothe“unbounded”regime(DogteromandLeibler,1993).Theoverallconclusionsofthedatainthisfigurearethat(i)MTsexhibitnetgrowth(asaveragedovertimeoroverindividualsinapopulation)at[freetubulin]abovethevalueQ5(Q5a≈Q5b≈Q5c);(ii)Q5issimilartothevalueQ1≈Q2(greydashedline)asdeterminedinFigure3A-B.Thus,Q1,Q2,andQ5allprovidemeasurementsofthesamecriticalconcentration,definedasCCPopGrowinthemaintext.Methods:Allpopulationdatapoints(panelsC-F)representthemean+/-onestandarddeviationofthevaluesobtainedinthreeindependentrunsofthesimulations.InpanelsC-D,thenetrateofchangewascalculatedfrom15to30minutes.InpanelsE-F,thedriftcoefficientwascalculatedusingamethodbasedonKomarovaetal.(Komarovaetal.,2002)(SupplementalMethods).

47

Figure6:FluxoftubulinsubunitsintoandoutofMTpolymerasafunctionofdilution[freetubulin](i.e.,aJ(c)plotasin(Carlieretal.,1984)andFigure1C).Leftpanels:simplifiedmodel;rightpanels:detailedmodel.(A,B)Inthedilutionsimulations,competingsystemsofMTsathigh[totaltubulin]wereallowedtopolymerizeuntiltheyreachedpolymer-masssteadystate.TheMTswerethentransferredinto(“dilutedinto”)thefreetubulinconcentrationsshownonthehorizontalaxis.Afterabriefdelay,theinitialflux(rateofchangein[polymerizedtubulin](leftaxis)orinaverageMTlength(rightaxis))wasmeasured,similarto(Carlieretal.,1984).(C,D)Datare-plottedtoshowthattheJ(c)curvesfromthedilutionsimulationsinpanelsA-B(trianglesymbols)andthenetrateofchangeinaverageMTlengthfromtheconstant[freetubulin]simulationsinFigure5C-D(circlesymbols)overlaywitheachotherfor[freetubulin]aboveCCPopGrow.Interpretation:ThesedatashowthatCCasdeterminedbyQ4fromJ(c)plotsisapproximatelythesamevalueasQ1≈Q2(greydashedline),andthusQ4alsoprovidesameasurementofCCPopGrow.Methods:CompetingsystemsofMTsat22µMtotaltubulinwereallowedtoreachpolymer-masssteadystate.Then,atminute10ofthesimulationinthesimplifiedmodelandatminute20ofthesimulationinthedetailedmodel,theMTsweretransferredintothefreetubulinconcentrationsshownonthehorizontalaxis.Aftera5seconddelay,thefluxwasmeasuredovera10secondperiod(seeFigureS4forplotsof[freetubulin]and[polymerizedtubulin]asfunctionsoftime).Notethatthedelayafterdilutionwasnecessaryintheoriginalexperimentsbecauseofinstrumentdeadtime,butitisimportantforobtainingaccurateJ(c)measurementsbecauseitallowsthecaplengthtorespondtothenew[freetubulin](Duellbergetal.,2016;Bowne-Andersonetal.,2013).Foraccuratemeasurements,thepre-dilutionMTsmustbesufficientlylongthatnonecompletelydepolymerizeduringthe15-secondperiodafterthedilution.Datapointsfordifferentconcentrationsofdilution[freetubulin](seecolorkey)representthemean+/-onestandarddeviationofthevaluesobtainedinthreeindependentrunsofthesimulations.

Figure 6

B

D

A Simplified Model

Dilution [Free Tubulin] (µM)

2 4 6

Q4

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8.00 9.00 10.00 10.25 10.50 10.75 11.00 11.25 11.50 11.75 12.00 12.25 12.50 12.75 13.00 15.00 20.00 25.00

µM

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µM

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CC

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48

Figure7:GrowthvelocityofindividualMTsduringthegrowthstateasafunctionof[freetubulin].Leftpanels:simplifiedmodel;rightpanels:detailedmodel;colorsofdatapointsreflecttheconcentrationsoffreetubulin.(A-D)Growthvelocity(Vg)measuredusinggrowthphasesfromeithertheconstant[freetubulin]simulationsofFigure5(panels7A-B)orthedilutionsimulationsofFigure6A-B(panels7C-D).EachofpanelsA-DshowsVgasmeasuredbyastandardDI-basedanalysismethod(+symbols)andatime-stepbasedmethod(squaresymbols).Regressionlines(solidblack)werefittedtothelinearrangeofthesedata,andextrapolatedbacktoVg=0toobtainQ3fortheconstant[freetubulin]simulations(panelsA-B)andQ6forthedilutionsimulations(panelsC-D).Interpretation:ThesedatashowthattheCCasmeasuredbyQ3isapproximatelyequaltothatmeasuredbyQ6andisdifferentfromCCPopGrow(greydashedline)asmeasuredbyQ1≈Q2(≈Q4≈Q5)fromFigures3-6.ThemaintextprovidesjustificationfortheideathatQ3andQ6estimateCCIndGrow,theCCforextended,buttransient,growthphasesofindividualfilaments.Methods:InpanelsA-B(constant[freetubulin]),theVgmeasurementsweretakenduringthetimeperiodfromminute15tominute30ofthesimulations(chosensothatthesystemhasreachedeitherpolymer-massorpolymer-growthsteadystate).InpanelsC-D(dilutionsimulations),theVgdatawereacquiredfrom5to15secondsafterthedilution,i.e.,theJ(c)measurementperioddescribedinFigure6.FortheDI-basedanalyses(panelsA-D,+symbols),weusedacustomMATLABprogramtoidentifyandquantifygrowthphasesbyfindingpeaksinthelengthhistorydata.Thetime-stepbasedmethod(panelsA-D,squaresymbols)divideseachlengthhistoryinto2-secondintervalsandidentifiesintervalsduringwhichthereisapositivechangeintheMTlength.SeeSupplementalMethodsformoreinformationaboutbothmethods.Regressionlineswerefittedtothetime-stepmeasurementsofVgfor[freetubulin]inrangeswheretheVgdataareapproximatelylinearasafunctionof[freetubulin]:from3to7µMforthesimplifiedmodel(panelsA,C)andfrom7to15µMforthedetailedmodel(panelsB,D).Alldatapointsrepresentthemean+/-onestandarddeviationofthevaluesobtainedinthreeindependentrunsofthesimulations.

0.00

Figure 7

Non-Competing Simulations Non-Competing Simulations B 0.10

0.08

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[Free Tubulin] (µM) 5 15 20 0 0 10

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DI analysis Time-step analysis Extrapolation

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49

Figure8:AnalternativemethodformeasuringCCIndGrow.Leftpanels:simplifiedmodel;rightpanels:detailedmodel.Inallpanels,thegreydashedlinesrepresentCCIndGrowasmeasuredbyQ3(Figure7A-B)andCCPopGrowasmeasuredbyQ1≈Q2(Figure3A-B).(A,B)OverlayofVgfromthetime-stepanalysisofgrowingindividualMTs(squaresymbols;re-plottedfromFigure7A-B)andthenetrateofchangeinaverageMTlengthoftheMTpopulation(circlesymbols;re-plottedfromFigure5C-D),bothfromtheconstant[freetubulin]simulations.InterpretationofpanelsA,B:Thesedatashowthatathigh[freetubulin],thenetrateofchangeinaverageMTlengthapproachestheVgofindividualMTs.Thesetwodatasetsconvergebecauseatsufficientlyhigh[freetubulin]individualMTsaregrowing(nearly)allthetime,asseeninthelengthhistories(Figure5A-B).Thus,CCIndGrow,whichwasobtainedfromVginFigure7,shouldalsobeobtainablebyextrapolatingfromthenetrateofchangedata.(C,D)ExtrapolationtoobtainQ7fromthenetrateofchangeinaverageMTlength.InterpretationofpanelsC,D:Ineachofthemodels,thevalueofQ7isapproximatelyequaltoQ3≈Q6(Figure7).Thus,Q7providesanotherwayofanestimatingofCCIndGrow.Methods:RegressionlineswerefittedtothenetrateofchangeinaverageMTlengthfor[freetubulin]inrangeswherethenetrateofchangedataisapproximatelylinearasafunctionof[freetubulin]:from6to7µMforthesimplifiedmodel(panelC)andfrom14to20µMforthedetailedmodel(panelD).Q7isthex-interceptoftheregressionline.Notethatthe[freetubulin]rangesusedfordeterminationofQ7arehigherthanthoseusedforQ3andQ6becausetheQ7extrapolationrequiresconditionswhereallMTsinthepopulationaregrowing(nodepolymerizationphases).

Figure 8 Non-Competing Simulations

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50

Figure9(legendonnextpage)

Figure 9

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51

Figure9:Effectofvaryingtherateconstantfornucleotidehydrolysis(kH)inthesimplifiedmodel.Non-competingsimulationsofthesimplifiedmodelwereperformedforvariousvaluesofkH(allotherinputkineticrateconstantsarethesameasintheotherfigures).EachofpanelsA-FcorrespondstoadifferentvalueofkH,rangingfrom0to10sec-1,asindicatedinthepaneltitles.(A-F)Thegrowthvelocityduringgrowthphases(Vg)(+symbols;colorcodedbykHvalue)andtherateofchangeinaverageMTlength(colorandsymbolvarybykHvalue)asfunctionsof[freetubulin].WealsoplotthetheoreticalequationforVgthatassumesthatgrowingendshaveonlyGTP-tubulinatthetips(greydashedline).NotethatthescalesoftheaxesvaryamongpanelsA-F;fordatare-plottedatthesamescale,seeFigureS5.(G,H)CCIndGrowandCCPopGrowasfunctionsofkH,withCCIndGrowandCCPopGrowmeasuredrespectivelybyQ3andQ5afrompanelsA-F.TheaxeshavelinearscalesinpanelGandlogscalesinpanelH.TheverticalseparationbetweenCCIndGrowandCCPopGrowateachkHinthelog-logplot(panelH)representstheirratioCCPopGrow/CCIndGrow.Interpretation:WhenkHiszero(panelG;seealsopanelA),CCIndGrowandCCPopGrowareequaltoeachotherandtoCCKD_GTP.AskHincreases(panelsGandH;seealsopanelsB-F),thevaluesofCCIndGrowandCCPopGrowincrease,anddivergefromeachotherandfromCCKD_GTP.Thus,theseparationsbetweenCCKD_GTP,CCIndGrow,andCCPopGrowdependonkH.ToseehowDIbehaviorsrelatetotheCCs,seeFigureS6forrepresentativelengthhistoryplotsofindividualMTsateachkHvaluepresentedhere.Methods:Thesimulationswereperformedusingthesimplifiedmodelwith50stableMTseeds.VgwasmeasuredusingtheDIanalysismethod(SupplementalMethods).Thesteady-staterateofchangeinaverageMTlengthwasmeasuredfromthenetchangemethod(seeQ5a,Table3).Allmeasurementsweretakenfrom40to60minutes.Regressionlines(blacksolidline)werefittedtotheVgdatapointsinthe[freetubulin]rangeaboveCCPopGrowandthenextrapolatedbacktoVg=0.

52

Figure10:RelationshipbetweenPocc(proportionofstableMTseedsthatareoccupied)and[freetubulin].SimplifiedmodelinpanelsA,C;detailedmodelinpanelsB,D.Therawdataanalyzedinthisfigurearefromthesamenon-competing(constant[freetubulin])simulationsusedinFigures5,6C-D,7A-B,and8.Inallpanels,thegreydashedlinesrepresentCCIndGrow(Q3fromFigure7A-B)andCCPopGrow(Q1,Q2fromFigure3A-B).(A,B)

Figure 10

B A

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1 subunit = 8 nm 2 subunits = 16 nm 3 subunits = 24 nm 4 subunits = 32 nm 5 subunits = 40 nm 10 subunits = 80 nm 25 subunits = 200 nm 50 subunits = 400 nm 75 subunits = 600 nm 100 subunits = 800 nm 125 subunits = 1 µm

Length Threshold:

18 20 4 8 12 14 10

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4 2

Simplified Model Detailed Model

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0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 6.00 7.00

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53

Proportionofstableseedsbearing“experimentally-detectable”MTs(Pocc)asafunctionof[freetubulin].HeredetectableMTsarethosewithlength≥25subunits=200nm(chosenbecausetheAbbediffractionlimitfor540nm(green)lightina1.4NAobjectiveis~200nm).(C,D)Poccwithdetectionthresholdsvariedfrom1subunit(8nm)to125subunits(1000nm).Thedatawiththe25subunitthresholdisre-plottedfrompanelsA-B.Interpretation:ThedatainpanelsA-Bshowthatwithadetectionthresholdsimilartothatintypicalfluorescencemicroscopyexperiments,littlepolymerisobservedgrowingoffoftheGTP-tubulinseedsineithersimulationuntil[freetubulin]iswellaboveCCIndGrow.Morespecifically,withthis200nmthreshold,Poccdoesnotreach0.5until[freetubulin]ismorethanhalfwayfromCCIndGrowtoCCPopGrow.Notethatthelowestvalueof[freetubulin]atwhich100percentoftheseedshaveadetectableMTcorrespondsto~CCPopGrow(seealso(Fygensonetal.,1994;Dogterometal.,1995)).ThedatainpanelsC-DindicatethatshortMTs(withlengthsbelowthe200nmdetectionthresholdfrompanelsA-B)arepresentatfreetubulinconcentrationsnearCCIndGrow.Additionally,wenotethatthePocccurveofthedetailedmodelissteeperthanthatofthesimplifiedmodelwhenthesamethresholdiscompared.Wesuggestthatthisresultsfromthemorecooperativenatureofgrowthinthedetailed(13-protofilament)model,whichisanoutcomeofinteractionsbetweenprotofilaments.Methods:Alldatapointsrepresentthemean+/-onestandarddeviationofthePoccvaluesobtainedinthreeindependentrunsofthesimulations.Thevaluesfromeachrunareaveragesfrom25to30minutes,chosensothatPocchasreacheditssteady-statevalue.MTlengthismeasuredasthenumberofsubunitsofabovetheseed.Notethatinthedetailedmodel,theMTlengthistheaverageofthe13protofilamentlengthsandcanthereforehavenon-integervalues;seesupplementalFigureS7forfractionalthresholdsbelow2subunits,whichfillinthelargegapbetween1and2subunits.

54

Figure11:HypotheticalPoccvs.[freetubulin]curves,wherePoccistheproportionofseedsthatareoccupiedbyMTs.ItmighthavebeenexpectedthatGTP-likeseedsshouldstartgrowingonce[freetubulin]isaboveCCIndGrow,andthatPoccwouldthereforeincreaseabruptlyfrom0to1when[freetubulin]isatorjustaboveCCIndGrow,similartothestepfunctioninpanelA.Incontrast,sigmoidalPocccurves,similartopanelB,havebeenobservedexperimentally(MitchisonandKirschner,1984b;Walkeretal.,1988;Dogterometal.,1995;Wieczoreketal.,2015).Obtainingasigmoidalshape(B)insteadofastepfunction(A)hasbeeninterpretedasevidenceofanucleationprocessthatmakesgrowthofMTsfromstableseedsmoredifficultthanextensionfromagrowingend(Fygensonetal.,1994;Wieczoreketal.,2015).However,asdiscussedinthemaintext,thissigmoidalshapecanbeaconsequenceofDIincombinationwithexperimentallengthdetectionlimitations,andthereforeisnotnecessarilyevidenceofanucleationprocess.NotethatanucleationprocessthatmakesgrowthfromseedsmoredifficultwouldleadtoaPocccurvethatincreasesmorerapidlyfrom0to1anddoessoat[freetubulin]nearCCPopGrow,similartothestepfunctioninpanelC.Thisbehaviorcanbeexplainedinthefollowingway.When[freetubulin]isbelowCCPopGrow,MTswillrepeatedlydepolymerizebacktotheseed.Whennucleationfromseedsisdifficult,itwilltakelongerforanewgrowthphasetoinitiateaftereachcompletedepolymerization;seedswillthereforeremainunoccupiedforlongertimesandtheproportionofseedsinthepopulationthatareoccupiedatanyparticulartimebewilllower.Thus,asthedifficultyofnucleationincreases,theshapeofthePocccurvewouldchangefromasigmoid(asinpanelB)toastepfunctionatCCPopGrow(asinpanelC).

CC

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Figure 12

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steady-state DI range steady-state DI range steady-state DI range

Summary of Conclusions

56

Figure12:Schematicsummaryoftherelationshipsbetweendynamicinstability(DI)behaviorandcriticalconcentrationsforDIpolymers.(A)Relationshipsbetween[totalsubunit]and[freesubunit](green)or[polymerizedsubunit](blue)forapopulationoffilamentscompetingforafixedpoolofsubunits(constant[totalsubunit])atpolymer-masssteadystate,similartoFigures3A-B,4).Noticethatthesteady-state[freesubunit]insuchcompetingsystemsapproachesCCPopGrowandthatthesharpnessoftheapproachdependsonthenumberofseeds.Inparticular,formanyseeds,steady-state[freesubunit]isnoticeablybelowCCPopGrowevenatveryhigh[totalsubunit].(B)Relationshipsbetween[freesubunit]andtherateofpolymerization/depolymerizationundernon-competingconditions(constant[freesubunit])foreitherindividualfilamentsduringgrowthphasesorpopulationsoffilaments.Morespecifically,thepanelshows:(i)thegrowthvelocity(Vg)ofindividualfilamentsduringthegrowthphase(purpledashedline;similartoFigure7);(ii)thenetrateofchangeinaveragefilamentlengthinapopulationoffilamentsasassessedfromexperimentsperformedwith[freesubunit]heldconstantfortheentiretimeoftheexperiment(lightturquoisedashedcurve;similartoFigure5C-F);and(iii)thenetrateofchangeinaveragefilamentlengthinapopulationoffilamentsasassessedfromdilutionexperiments(darkturquoisesolidcurve;similartoFigure6).Noticethatcurves(ii)and(iii)aresuperimposedfor[freesubunit]>CCPopGrow,andthatcurves(ii)and(iii)approachcurve(i)for[freesubunit]>>>CCPopGrow.(C-D)Lengthhistoriesofindividualfilamentsincompetingsystems(panelC)andnon-competingsystems(panelD).Notethatwhen[freesubunit]isbelowCCPopGrow(purplelengthhistoryinpanelDandalllengthhistoriesafterpolymer-masssteadystateinpanelC),individualfilamentsdisplaysteady-statedynamicinstabilityinwhichtheyeventuallyandrepeatedlydepolymerizebacktotheseed;furthermore,theaveragefilamentlengthandthepolymermassreachfinitesteady-statevaluesgivensufficienttime(seepolymer-masssteadystateinFiguresS1C-DandS3A-B).When[freesubunit]isaboveCCPopGrow(panelD),individualfilamentsdisplaynetgrowthovertime,whilestillundergoingdynamicinstability(skyblue,panelD)exceptperhapsatveryhighconcentrations(seagreen,panelD).ThetextunderneaththehorizontalaxisinpanelsA-BrelatesthedynamicinstabilitybehaviorofindividualfilamentsinpanelsC-Dtotheindicatedrangesof[freesubunit]andthepopulationbehaviorsinpanelsA-B.(E)EffectofchangingkH,therateconstantfornucleotidehydrolysis(similartoFigure9).WhenkH=0(equilibriumpolymer),CCKD_GTP=CCIndGrow=CCPopGrow.WhenkH>0(steady-statepolymer),CCIndGrowandCCPopGrowaredistinctfromeachotherandfromCCKD_GTP.AskHincreases,CCIndGrowandCCPopGrowbothincreaseandtheseparationbetweenthemincreases(atlowenoughkH,CCIndGrowandCCPopGrowwouldbeexperimentallyindistinguishable);CCKD_GTPdoesnotchangewithkH.The[freesubunit]rangewheresteady-stateDIoccurs,i.e.,therangebetweenCCPopGrowandCCIndGrow(yellowbracketsinpanelE;comparetopanelsBandD),increaseswithkH.Notethatthisfigureisaschematicrepresentationofbehaviorsoverawiderangeofconcentrationsandisnotdrawntoscale.