View
4
Download
0
Category
Preview:
Citation preview
1
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
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
2
ARTICLEFILE
TableofContents
Abstract...................3
Articlemaintext........4–31
Acknowledgements........31
References...........32–35
Tables...............36–39
FigureswithLegends...40–56
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
3
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
4
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
5
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
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
6
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.).
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
7
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
8
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
9
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
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
10
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).
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
11
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)).
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
12
[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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
13
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).
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
14
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
15
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).
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
16
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
17
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
18
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).
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
19
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).
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
20
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,
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
21
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
22
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).
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
23
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
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
24
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
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
25
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
26
[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)
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
27
(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
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
28
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
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
29
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
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
30
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
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
31
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
32
REFERENCESCITED Alberts,B.,Johnson,A.,Lewis,J.,Morgan,D.,Raff,M.,Roberts,K.,andWalter,P.(2015).MolecularBiologyoftheCell,SixthEdition(NewYork,NewYork:GarlandScience).Alfaro-Aco,R.,andPetry,S.(2015).BuildingtheMicrotubuleCytoskeletonPiecebyPiece.JBiolChem290,17154-17162.Amayed,P.,Pantaloni,D.,andCarlier,M.F.(2002).Theeffectofstathminphosphorylationonmicrotubuleassemblydependsontubulincriticalconcentration.JBiolChem277,22718-22724.Aparna,J.S.,Padinhateeri,R.,andDas,D.(2017).Signaturesofamacroscopicswitchingtransitionforadynamicmicrotubule.SciRep7,45747.Bicout,D.J.(1997).Green’sfunctionsandfirstpassagetimedistributionsfordynamicinstabilityofmicrotubules.PhysicalReviewE56,6656-6667.Bonfils,C.,Bec,N.,Lacroix,B.,Harricane,M.C.,andLarroque,C.(2007).Kineticanalysisoftubulinassemblyinthepresenceofthemicrotubule-associatedproteinTOGp.JBiolChem282,5570-5581.Bowne-Anderson,H.,Hibbel,A.,andHoward,J.(2015).RegulationofMicrotubuleGrowthandCatastrophe:UnifyingTheoryandExperiment.TrendsCellBiol25,769-779.Bowne-Anderson,H.,Zanic,M.,Kauer,M.,andHoward,J.(2013).Microtubuledynamicinstability:anewmodelwithcoupledGTPhydrolysisandmultistepcatastrophe.Bioessays35,452-461.Buey,R.M.,Barasoain,I.,Jackson,E.,Meyer,A.,Giannakakou,P.,Paterson,I.,Mooberry,S.,Andreu,J.M.,andDíaz,J.F.(2005).Microtubuleinteractionswithchemicallydiversestabilizingagents:thermodynamicsofbindingtothepaclitaxelsitepredictscytotoxicity.ChemBiol12,1269-1279.Carlier,M.F.,Didry,D.,andPantaloni,D.(1987).Microtubuleelongationandguanosine5’-triphosphatehydrolysis.Roleofguaninenucleotidesinmicrotubuledynamics.Biochemistry26,4428-4437.Carlier,M.F.,Hill,T.L.,andChen,Y.(1984a).InterferenceofGTPhydrolysisinthemechanismofmicrotubuleassembly:anexperimentalstudy.ProcNatlAcadSciUSA81,771-775.Carlier,M.F.,Pantaloni,D.,andKorn,E.D.(1984b).EvidenceforanATPcapattheendsofactinfilamentsanditsregulationoftheF-actinsteadystate.JBiolChem259,9983-9986.Chen,Y.,andHill,T.L.(1985a).Theoreticaltreatmentofmicrotubulesdisappearinginsolution.ProcNatlAcadSciUSA82,4127-4131.Chen,Y.D.,andHill,T.L.(1985b).MonteCarlostudyoftheGTPcapinafive-starthelixmodelofamicrotubule.ProcNatlAcadSciUSA82,1131-1135.Concha-Marambio,L.,Maldonado,P.,Lagos,R.,Monasterio,O.,andMontecinos-Franjola,F.(2017).ThermaladaptationofmesophilicandthermophilicFtsZassemblybymodulationofthecriticalconcentration.PLoSOne12,e0185707.Coombes,C.E.,Yamamoto,A.,Kenzie,M.R.,Odde,D.J.,andGardner,M.K.(2013).Evolvingtipstructurescanexplainage-dependentmicrotubulecatastrophe.CurrBiol23,1342-1348.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
33
Cytoskeleton,Inc.Tubulinandmicrotubulebasedassays,compoundscreening,drugscreening,pre-clinicaldrugscreenfortubulininhibitorsandmicrotubuleinhibitors.https://www.cytoskeleton.com/custom-services3-compound-screening-tubulin-microtubulesDíaz-Celis,C.,Risca,V.I.,Hurtado,F.,Polka,J.K.,Hansen,S.D.,Maturana,D.,Lagos,R.,Mullins,R.D.,andMonasterio,O.(2017).BacterialTubulinsAandBExhibitPolarizedGrowth,Mixed-PolarityBundling,andDestabilizationbyGTPHydrolysis.JBacteriol199, e00211-17.Díaz,J.F.,Menéndez,M.,andAndreu,J.M.(1993).Thermodynamicsofligand-inducedassemblyoftubulin.Biochemistry32,10067-10077.Dogterom,andLeibler(1993).Physicalaspectsofthegrowthandregulationofmicrotubulestructures.PhysRevLett70,1347-1350.Dogterom,M.,Maggs,A.C.,andLeibler,S.(1995).Diffusionandformationofmicrotubuleasters:physicalprocessesversusbiochemicalregulation.ProcNatlAcadSciUSA92,6683-6688.Duan,A.R.,Jonasson,E.M.,Alberico,E.O.,Li,C.,Scripture,J.P.,Miller,R.A.,Alber,M.S.,andGoodson,H.V.(2017).InteractionsbetweenTauandDifferentConformationsofTubulin:ImplicationsforTauFunctionandMechanism.JMolBiol429,1424-1438.Duellberg,C.,Cade,N.I.,Holmes,D.,andSurrey,T.(2016).ThesizeoftheEBcapdeterminesinstantaneousmicrotubulestability.Elife5,e13470.Erb,M.L.,Kraemer,J.A.,Coker,J.K.,Chaikeeratisak,V.,Nonejuie,P.,Agard,D.A.,andPogliano,J.(2014).AbacteriophagetubulinharnessesdynamicinstabilitytocenterDNAininfectedcells.Elife3,e03197.Flyvbjerg,Holy,andLeibler(1994).Stochasticdynamicsofmicrotubules:Amodelforcapsandcatastrophes.PhysRevLett73,2372-2375.Flyvbjerg,Holy,andLeibler(1996).Microtubuledynamics:Caps,catastrophes,andcoupledhydrolysis.PhysRevEStatPhysPlasmasFluidsRelatInterdiscipTopics54,5538-5560.Fygenson,Braun,andLibchaber(1994).Phasediagramofmicrotubules.PhysRevEStatPhysPlasmasFluidsRelatInterdiscipTopics50,1579-1588.Gardner,M.K.,Charlebois,B.D.,Jánosi,I.M.,Howard,J.,Hunt,A.J.,andOdde,D.J.(2011).Rapidmicrotubuleself-assemblykinetics.Cell146,582-592.Garner,E.C.,Campbell,C.S.,andMullins,R.D.(2004).DynamicinstabilityinaDNA-segregatingprokaryoticactinhomolog.Science306,1021-1025.Gildersleeve,R.F.,Cross,A.R.,Cullen,K.E.,Fagen,A.P.,andWilliams,R.C.(1992).Microtubulesgrowandshortenatintrinsicallyvariablerates.JBiolChem267,7995-8006.Gillespie,D.T.(1976).Ageneralmethodfornumericallysimulatingthestochastictimeevolutionofcoupledchemicalreactions.JCompphys22,403-434.Gliksman,N.R.,Parsons,S.F.,andSalmon,E.D.(1992).Okadaicacidinducesinterphasetomitotic-likemicrotubuledynamicinstabilitybyinactivatingrescue.JCellBiol119,1271-1276.Goodson,H.V.,andJonasson,E.M.(2018).MicrotubulesandMicrotubule-AssociatedProteins.ColdSpringHarbPerspectBiol10,a022608.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
34
Gregoretti,I.V.,Margolin,G.,Alber,M.S.,andGoodson,H.V.(2006).Insightsintocytoskeletalbehaviorfromcomputationalmodelingofdynamicmicrotubulesinacell-likeenvironment.JCellSci119,4781-4788.Gupta,K.K.,Li,C.,Duan,A.,Alberico,E.O.,Kim,O.V.,Alber,M.S.,andGoodson,H.V.(2013).Mechanismforthecatastrophe-promotingactivityofthemicrotubuledestabilizerOp18/stathmin.ProcNatlAcadSciUSA110,20449-20454.Hill,T.L.(1987).LinearAggregationTheoryinCellBiology(NewYork:Springer-Verlag).Hill,T.L.,andChen,Y.(1984).PhasechangesattheendofamicrotubulewithaGTPcap.ProcNatlAcadSciUSA81,5772-5776.Howard,J.(2001).MechanicsofMotorProteinsandtheCytoskeleton(SinauerAssociates).Hussmann,F.,Drummond,D.R.,Peet,D.R.,Martin,D.S.,andCross,R.A.(2016).Alp7/TACC-Alp14/TOGgenerateslong-lived,fast-growingMTsbyanunconventionalmechanism.SciRep6,20653.Hyman,A.A.,Salser,S.,Drechsel,D.N.,Unwin,N.,andMitchison,T.J.(1992).RoleofGTPhydrolysisinmicrotubuledynamics:informationfromaslowlyhydrolyzableanalogue,GMPCPP.MolBiolCell3,1155-1167.Johnson,K.A.,andBorisy,G.G.(1975).Theequilibriumassemblyofmicrotubulesinvitro.SocGenPhysiolSer30,119-141.Komarova,Y.A.,Vorobjev,I.A.,andBorisy,G.G.(2002).LifecycleofMTs:persistentgrowthinthecellinterior,asymmetrictransitionfrequenciesandeffectsofthecellboundary.JCellSci115,3527-3539.Lodish,H.,Berk,A.,Kaiser,C.A.,andKrieger,M.(2016).MolecularCellBiology(NewYork,NewYork:W.H.Freeman).Mahrooghy,M.,Yarahmadian,S.,Menon,V.,Rezania,V.,andTuszynski,J.A.(2015).Theuseofcompressivesensingandpeakdetectioninthereconstructionofmicrotubuleslengthtimeseriesintheprocessofdynamicinstability.ComputBiolMed65,25-33.Maly,I.V.(2002).Diffusionapproximationofthestochasticprocessofmicrotubuleassembly.BullMathBiol64,213-238.Margolin,G.,Goodson,H.V.,andAlber,M.S.(2011).Mean-fieldstudyoftheroleoflateralcracksinmicrotubuledynamics.PhysRevEStatNonlinSoftMatterPhys83,041905.Margolin,G.,Gregoretti,I.V.,Cickovski,T.M.,Li,C.,Shi,W.,Alber,M.S.,andGoodson,H.V.(2012).Themechanismsofmicrotubulecatastropheandrescue:implicationsfromanalysisofadimer-scalecomputationalmodel.MolBiolCell23,642-656.Mirigian,M.,Mukherjee,K.,Bane,S.L.,andSackett,D.L.(2013).Measurementofinvitromicrotubulepolymerizationbyturbidityandfluorescence.MethodsCellBiol115,215-229.Mitchison,T.,andKirschner,M.(1984a).Dynamicinstabilityofmicrotubulegrowth.Nature312,237-242.Mitchison,T.,andKirschner,M.(1984b).Microtubuleassemblynucleatedbyisolatedcentrosomes.Nature312,232-237.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
35
Mourão,M.,Schnell,S.,andShaw,S.L.(2011).Macroscopicsimulationsofmicrotubuledynamicspredicttwosteady-stateprocessesgoverningarraymorphology.ComputBiolChem35,269-281.Oosawa,F.(1970).Sizedistributionofproteinpolymers.JTheorBiol27,69-86.Oosawa,F.,andAsakura,S.(1975).ThermodynamicsofthePolymerizationofProteins(Horecker,B.,Kaplan,NO,Matmur,J.,andScheraga,HA,eds)(NewYork,NewYork,AcademicPress).Oosawa,F.,andKasai,M.(1962).Atheoryoflinearandhelicalaggregationsofmacromolecules.JMolBiol4,10-21.Oosawa,F.,Asakura,S.,Hotta,K.,Imai,N.,andOoi,T.(1959).G-Ftransformationofactinasafibrouscondensation.JournalofPolymerSciencePartA:PolymerChemistry37,323-336.Padinhateeri,R.,Kolomeisky,A.B.,andLacoste,D.(2012).Randomhydrolysiscontrolsthedynamicinstabilityofmicrotubules.BiophysJ102,1274-1283.Roostalu,J.,andSurrey,T.(2017).Microtubulenucleation:beyondthetemplate.NatRevMolCellBiol18,702-710.Schummel,P.H.,Gao,M.,andWinter,R.(2017).ModulationofthePolymerizationKineticsofα/β-TubulinbyOsmolytesandMacromolecularCrowding.Chemphyschem18,189-197.Verde,F.,Dogterom,M.,Stelzer,E.,Karsenti,E.,andLeibler,S.(1992).ControlofmicrotubuledynamicsandlengthbycyclinA-andcyclinB-dependentkinasesinXenopuseggextracts.JCellBiol118,1097-1108.Verdier-Pinard,P.,Wang,Z.,Mohanakrishnan,A.K.,Cushman,M.,andHamel,E.(2000).Asteroidderivativewithpaclitaxel-likeeffectsontubulinpolymerization.MolPharmacol57,568-575.Verma,S.,Kumar,N.,andVerma,V.(2016).Roleofpaclitaxeloncriticalnucleationconcentrationoftubulinanditseffectsthereof.BiochemBiophysResCommun478,1350-1354.Vorobjev,I.A.,andMaly,I.V.(2008).Microtubulelengthanddynamics:Boundaryeffectandpropertiesofextendedradialarray.CellandTissueBiology2,272–281.Vorobjev,I.A.,Rodionov,V.I.,Maly,I.V.,andBorisy,G.G.(1999).Contributionofplusandminusendpathwaystomicrotubuleturnover.JCellSci112,2277-2289.Vorobjev,I.A.,Svitkina,T.M.,andBorisy,G.G.(1997).Cytoplasmicassemblyofmicrotubulesinculturedcells.JCellSci110,2635-2645.Walker,R.A.,O’Brien,E.T.,Pryer,N.K.,Soboeiro,M.F.,Voter,W.A.,Erickson,H.P.,andSalmon,E.D.(1988).Dynamicinstabilityofindividualmicrotubulesanalyzedbyvideolightmicroscopy:rateconstantsandtransitionfrequencies.JCellBiol107,1437-1448.Wieczorek,M.,Bechstedt,S.,Chaaban,S.,andBrouhard,G.J.(2015).Microtubule-associatedproteinscontrolthekineticsofmicrotubulenucleation.NatCellBiol17,907-916.Williams,R.C.,Correia,J.J.,andDeVries,A.L.(1985).Formationofmicrotubulesatlowtemperaturebytubulinfromantarcticfish.Biochemistry24,2790-2798.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
36
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
37
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]
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
38
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
39
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.
.CC-BY-NC-ND 4.0 International licenseacertified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under
The copyright holder for this preprint (which was notthis version posted April 15, 2019. ; https://doi.org/10.1101/260646doi: bioRxiv preprint
40
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
3.0
2.0
1.0
0.0 Ste
ady-
Sta
te [T
ubul
in] (µM
)
[Total Tubulin] (µM) 0 2 4 6
Simplified Model
Figure 3
Detailed Model
15.0
10.0
5.0
0.0 Ste
ady-
Sta
te [T
ubul
in] (µM
)
[Total Tubulin] (µM) 0 5 15 10 25 20
B 4.0
Q1 Q1
Q2
MT
Leng
th (µ
m)
0.5
1.5
2.0
0.0
C
Time (min) 0
D
MT
Leng
th (µ
m)
2.0
3.0
5.0
0.0
Time (min) 0 2 4 10 8
1 2 2.5 < Q1 ≈ Q2 < 3 5 7 µM
Free Tubulin
Polymerized Tubulin
Q2
15 30 0 1.0
4.0
6
30 60 0 0
70
35
1.0
8
16
0
2 4 10 8 6
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-
Sta
te [T
ubul
in] (µM
)
2.0
3.0
0.0
C
[Total Tubulin] (µM) 0 2 4 6
D
Ste
ady-
Sta
te [T
ubul
in] (µM
)
3.0
9.0
12.0
0.0
[Total Tubulin] (µM) 8 10 12 16 14
Q2
Q1
Q2
1.0
5
40.0
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
0
Rat
e of
Cha
nge
in
Ave
rage
MT
Leng
th (µ
m/s
ec)
Rat
e of
Cha
nge
in
Ave
rage
MT
Leng
th (µ
m/s
ec)
A Simplified Model Detailed Model
C
Figure 5
20.0
10.0
5.0
0.0
M
T Le
ngth
(µm
) 15.0
B
D
Time (min) 0 5 15 10
40.0
20.0
0.0
M
T Le
ngth
(µm
)
10.0
30.0
Time (min) 0 10 30 20
µM 7.00 5.00 4.00 3.25 3.00 2.75 2.50 2.00 1.50 1.00
E F
[Free Tubulin] (µM) 0 5 10 25 20
0.08
[Free Tubulin] (µM) 0 2 4 6
0.10
0.04
0.00
0.02
0.00
Rat
e of
Cha
nge
in
[Pol
ymer
ized
Tub
ulin
] (µM
/sec
)
0.06
Q5a (o) & Q5b (+)
15
0.010
0.000
0.020
0.005 Rat
e of
Cha
nge
in
[Pol
ymer
ized
Tub
ulin
] (µM
/sec
)
0.025
Q5a (o) & Q5b (+)
0.06
0.02
0.04
0.01
Q5a (o) & Q5c (x)
[Free Tubulin] (µM) 0 2 4 6
[Free Tubulin] (µM) 5 10 25 20 15
Q5a (o) & Q5c (x)
0.09
0.12
0.15
0.06
0.03
0.00
Rat
e of
Cha
nge
in
Ave
rage
MT
Leng
th (µ
m/s
ec)
Rat
e of
Cha
nge
in
Ave
rage
MT
Leng
th (µ
m/s
ec)
50.0
Non-Competing Simulations
Q1≈Q2 Q1≈Q2 Q
1 ≈
Q2
Q1 ≈
Q2
JDI equation
Net rate of change
0.03
0.05
0.08
0.10
0.04
0.00
0.06
0.02
Net rate of change Drift coefficient
Net rate of change Drift coefficient
JDI equation
Net rate of change 0.030
0.015
0.09
0.12
0.15
0.06
0.03
0.00
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
0.10
0.05
-0.05
[Pol
ymer
ized
Tub
ulin
] Fl
ux (µ
M/s
ec)
-0.15
-0.10
-0.20
-0.25
0.18
0.00
-0.09
-0.18
-0.27
-0.45
-0.36
Rat
e of
Cha
nge
in
Ave
rage
MT
Leng
th (µ
m/s
ec)
0.00
0.09
CC
Pop
Gro
w
(fro
m Q
1 ≈
Q2)
C
[Free Tubulin] (µM) (Dilution or Constant)
2 4 6
Q4 ( ) & Q5a (o)
0.10
0.05
-0.05
[Pol
ymer
ized
Tub
ulin
] Fl
ux (µ
M/s
ec)
-0.15
-0.10
-0.20
-0.25
Rat
e of
Cha
nge
in
Ave
rage
MT
Leng
th (µ
m/s
ec)
0.00
0.18
0.00
-0.09
-0.18
-0.27
-0.45
-0.36
0.09 Non-Competing
Dilution
Detailed Model
Dilution [Free Tubulin] (µM)
5 10 20 15
Q4 0.04
0.00
-0.04
[Pol
ymer
ized
Tub
ulin
] Fl
ux (µ
M/s
ec)
-0.12
-0.08
0.0
-0.1
Rat
e of
Cha
nge
in
Ave
rage
MT
Leng
th (µ
m/s
ec) 0.1
-0.3
-0.5
0.02
-0.02
-0.06 -0.2
-0.10 -0.4
-0.6 -0.14
[Free Tubulin] (µM) (Dilution or Constant)
5 10 20 15
0.04
0.00
-0.04 [P
olym
eriz
ed T
ubul
in]
Flux
(µM
/sec
)
-0.12
-0.08
0.02
-0.02
-0.06
-0.10
-0.14
0.0
-0.1
Rat
e of
Cha
nge
in
Ave
rage
MT
Leng
th (µ
m/s
ec)
0.1
-0.3
-0.5
-0.2
-0.6
-0.4
Q4 ( ) & Q5a (o)
Non-Competing
Dilution
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
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
µM
Dilution (and Non-Competing) Simulations
CC
Pop
Gro
w
(fro
m Q
1 ≈
Q2)
CC
Pop
Gro
w
(fro
m Q
1 ≈
Q2)
CC
Pop
Gro
w
(fro
m Q
1 ≈
Q2)
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
0.04
0.06
0.02
0.00
[Free Tubulin] (µM) 2 4 6
[Free Tubulin] (µM) 5 15 20 0 0 10
Q3 Q3
Gro
wth
Vel
ocity
dur
ing
Gro
wth
Pha
ses
(µm
/sec
)
Gro
wth
Vel
ocity
dur
ing
Gro
wth
Pha
ses
(µm
/sec
)
Dilution [Free Tubulin] (µM) 2 4 6 0
Q6
Gro
wth
Vel
ocity
dur
ing
Gro
wth
Pha
ses
(µm
/sec
)
C
A
0.00
D
Dilution [Free Tubulin] (µM) 5 15 20 0 10
Q6
Gro
wth
Vel
ocity
dur
ing
Gro
wth
Pha
ses
(µm
/sec
)
0.12
0.10
0.04
0.06
0.02
Simplified Model Detailed Model
Dilution Simulations Dilution Simulations
DI analysis Time-step analysis Extrapolation
DI analysis Time-step analysis Extrapolation
0.10
0.08
0.04
0.06
0.02
CC
Pop
Gro
w
(fro
m Q
1 ≈
Q2)
DI analysis Time-step analysis Extrapolation
DI analysis Time-step analysis Extrapolation
CC
Pop
Gro
w
(fro
m Q
1 ≈
Q2)
CC
Pop
Gro
w
(fro
m Q
1 ≈
Q2)
0.08
0.00
0.12
0.10
0.04
0.06
0.02
CC
Pop
Gro
w
(fro
m Q
1 ≈
Q2)
0.08
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
Simplified Model 0.12
0.10
0.08
0.04
0.06
0.02
0.00
[Free Tubulin] (µM) 2 4 6 0
Rat
e of
Cha
nge
in
MT
Leng
th (µ
m/s
ec)
[Free Tubulin] (µM) 2 4 6 0
Q7
Rat
e of
Cha
nge
in
MT
Leng
th (µ
m/s
ec)
A
C
Rate of change in avg. length (J)
Vg from time-step analysis -0.02
-0.04
Rate of change in avg. length (J)
Extrapolation from J at high Tu
0.12
0.10
0.08
0.04
0.06
0.02
0.00
-0.02
-0.04
CC
Pop
Gro
w
(Q1 ≈
Q2)
CC
IndG
row
(Q3)
CC
Pop
Gro
w
(Q1 ≈
Q2)
CC
IndG
row
(Q3)
Detailed Model B
[Free Tubulin] (µM) 5 15 20 0 10
Rat
e of
Cha
nge
in
MT
Leng
th (µ
m/s
ec)
D
[Free Tubulin] (µM) 5 15 20 0 10
Rat
e of
Cha
nge
in
MT
Leng
th (µ
m/s
ec)
Rate of change in avg. length (J)
Vg from time-step analysis
Q7
0.12
0.08
0.04
0.00
-0.04
Rate of change in avg. length (J)
Extrapolation from J at high Tu
CC
IndG
row
(Q3)
CC
Pop
Gro
w
(Q1 ≈
Q2)
0.12
0.08
0.04
0.00
-0.04
CC
IndG
row
(Q3)
CC
Pop
Gro
w
(Q1 ≈
Q2)
0.010
0.000
0.020
0.005
Rat
e of
Cha
nge
in
[Pol
ymer
ized
Tub
ulin
] (µM
/sec
)
0.025
-0.005
0.015
0.010
0.000
0.020
0.005
Rat
e of
Cha
nge
in
[Pol
ymer
ized
Tub
ulin
] (µM
/sec
)
0.025
-0.005
0.015
Rat
e of
Cha
nge
in
[Pol
ymer
ized
Tub
ulin
] (µM
/sec
)
0.02
0.00
0.06
0.04
0.01
0.03
0.05
-0.01
-0.02
Rat
e of
Cha
nge
in
[Pol
ymer
ized
Tub
ulin
] (µM
/sec
)
0.02
0.00
0.06
0.04
0.01
0.03
0.05
-0.01
-0.02
50
Figure9(legendonnextpage)
Figure 9
A B
D C
E F
G H
Q3
Q3 Q3
Q3 Q3
Q5a
Q5a
Q5a
Q5a
Q5a
Q3, Q5a
kTonT [Free Tu] – kToffT
Rate of Change in Average Length
Growth Velocity (Vg) Vg Extrapolation
kTonT [Free Tu] – kToffT
Rate of Change in Average Length
Growth Velocity (Vg) Vg Extrapolation
kTonT [Free Tu] – kToffT
Rate of Change in Average Length
Growth Velocity (Vg) Vg Extrapolation
kTonT [Free Tu] – kToffT
Rate of Change in Average Length
Growth Velocity (Vg) Vg Extrapolation
kTonT [Free Tu] – kToffT
Rate of Change in Average Length
Growth Velocity (Vg) Vg Extrapolation
kTonT [Free Tu] – kToffT
Rate of Change in Average Length
Growth Velocity (Vg) Vg Extrapolation
Rat
e of
Cha
nge
in
MT
Leng
th (µ
m/s
ec)
0.005 0.005
Rat
e of
Cha
nge
in
MT
Leng
th (µ
m/s
ec)
0.010
0.015
0.000
0.020
0.010
0.015
0.000
0.020
0.020
0.040
0.060
0.000
0.080
Rat
e of
Cha
nge
in
MT
Leng
th (µ
m/s
ec)
Rat
e of
Cha
nge
in
MT
Leng
th (µ
m/s
ec)
Rat
e of
Cha
nge
in
MT
Leng
th (µ
m/s
ec)
Rat
e of
Cha
nge
in
MT
Leng
th (µ
m/s
ec)
0.04
0.08
0.12
0.00
0.4
0.8
1.2
0.0
0.4
0.6
0.2
0.0
2 4 6 0 [Free Tubulin] (µM)
kH = 10 sec-1 kH = 5 sec-1
kH = 0.1 sec-1 kH = 0 sec-1
kH = 0.5 sec-1 kH = 1 sec-1 [Free Tubulin] (µM) [Free Tubulin] (µM)
[Free Tubulin] (µM)
[Free Tubulin] (µM) [Free Tubulin] (µM)
Varying kH Varying kH
kH (sec-1) kH (sec-1)
Crit
ical
C
once
ntra
tion
(µM
)
Crit
ical
C
once
ntra
tion
(µM
)
20 40 60 0 10 20 30 0
1 2 3 0 4
1 5 0 10 1 0.1 10 0.5 5
0.5
1
5
0.1
10
5
10
15
0
20
CCKD_GTP = kToffT/kTonT
0.25 0.5 0.75 0 1 0.25 0.5 0.75 0 1 Growth Velocity (Vg)
Growth Velocity (Vg)
Growth Velocity (Vg)
kTonT [Free Tu] – kToffT
Growth Velocity (Vg)
Growth Velocity (Vg)
Growth Velocity (Vg)
kH colors: 0 0.1 0.5 1 5 10 sec-1
Simplified Model, Non-Competing Simulations
kH colors: 0.1 0.5 1 5 10 sec-1
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
[Free Tubulin] (µM) 2 4 6 0
1.0
0.8
0.4
0.6
0.2
0.0
1.0
0.8
Pro
porti
on o
f See
ds O
ccup
ied
by M
T of
at l
east
200
nm
0.4
0.6
0.2
0.0
[Free Tubulin] (µM) 5 15 20 0 10
Pro
porti
on o
f See
ds O
ccup
ied
by M
T of
at l
east
200
nm
C
[Free Tubulin] (µM) 1 3 5 0
1.0
0.8
0.4
0.6
0.2
0.0 Pro
porti
on o
f See
ds O
ccup
ied
by M
T ≥
Leng
th T
hres
hold
Length Threshold:
8.0 9.0 10.0 10.2 10.4 10.6 10.8 11.0 11.2 11.4 11.6 11.8 12.0 12.2 12.4 12.6 12.8 13.0 15.0 20.0 µM
D
[Free Tubulin] (µM) 2 6 16 0
1.0
0.8
0.4
0.6
0.2
0.0 Pro
porti
on o
f See
ds O
ccup
ied
by M
T ≥
Leng
th T
hres
hold
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
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
4 2
Simplified Model Detailed Model
Simplified Model
Detailed Model
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
µM
Non-Competing Simulations
CC
Pop
Gro
w (
from
Q1 ≈
Q2)
CC
IndG
row
(fr
om Q
3)
CC
Pop
Gro
w (
from
Q1 ≈
Q2)
CC
IndG
row
(fr
om Q
3)
CC
Pop
Gro
w (
from
Q1 ≈
Q2)
CC
IndG
row
(fr
om Q
3)
CC
IndG
row
(fr
om Q
3)
CC
Pop
Gro
w (
from
Q1 ≈
Q2)
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
Pop
Gro
w
CC
Pop
Gro
w
CC
IndG
row
1
CC
I ndG
row
Figure 11
Poc
c (pr
opor
tion
of
see
ds o
ccup
ied)
Poc
c (pr
opor
tion
of
see
ds o
ccup
ied)
Poc
c (pr
opor
tion
of
see
ds o
ccup
ied)
[Free Tubulin]
C A B
Schema(cs(notsimula(ondata),Non-Compe(ng
1
0 [Free Tubulin] [Free Tubulin]
CC
IndG
row
CC
Pop
Gro
w
0
1
0
CCIndGrow
[Free Subunit] CCPopGrow = CCIndGrow = CCKD_GTP
Figure 12
A B
Q1
Q2
Indi
vidu
al F
ilam
ent L
engt
h
Time
CCPopGrow < [Free Subunit]
[Total Subunit]
Free Subunit (few seeds) Free Subunit (many seeds) Polymerized Subunit (few seeds) Polymerized Subunit (many seeds)
Ste
ady-
Sta
te
[Fre
e S
ubun
it] o
r [P
olym
eriz
ed S
ubun
it]
CCPopGrow
Competing
CCPopGrow<<< [Free Subunit]
CCIndGrow< [Free Subunit] < CCPopGrow
Indi
vidu
al F
ilam
ent L
engt
h
Time
Rat
e of
Cha
nge
in
Indi
vidu
al L
engt
h or
Ave
rage
Len
gth
Competing Non-Competing
Non-Competing, Varying kH
Low kH
Medium kH High kH
E
D C
Rate of Change in Average Filament Length Growth Velocity during Growth Phases of Individuals
[Free Subunit]
(ii) Rate of Chg. in Avg. Length (Constant [Free Subunit])
Rat
e of
Cha
nge
in
Indi
vidu
al L
engt
h or
Ave
rage
Len
gth
[Free Subunit] range where populations of filaments grow persistently, perhaps with DI
CCIndGrow (Q3, Q6, Q7)
CCPopGrow (Q4, Q5abc)
Non-Competing
[Free Subunit] range where steady-state
DI occurs
(iii) Rate of Change in Average Length (Dilution)
(i) Growth Velocity during Growth Phases of Individuals
CCPopGrow <<< [Total Subunit]
CCPopGrow < [Total Subunit]
CCIndGrow
kH = 0
[Total Subunit] range where steady-state DI occurs
CCIndGrow < [Total Subunit] < CCPopGrow
CCIndGrow CCIndGrow CCPopGrow CCPopGrow CCPopGrow
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.
Recommended