F1 rotary motor of ATP synthase is driven by the torsionally-asymmetric drive shaft

2.3 MB·

by  O. Kulish

F rotary motor of ATP synthase is driven by the
1
torsionally-asymmetric drive shaft
O. Kulish, A. D. Wright , and E. M. Terentjev
CavendishLaboratory,UniversityofCambridge,CambridgeCB30HE,U.K.
ABSTRACT
6 F1F0 ATP synthase (ATPase) either facilitates the synthesis of ATP in the mitochondrial membranes and bacterial inner
1 membranesinaprocessdrivenbytheprotonmovingforce(pmf),orusestheenergyfromATPhydrolysistopumpprotons
0 againsttheconcentrationgradientacrossthemembrane. ATPaseiscomposedoftworotarymotors,F andF ,whichgenerate
0 1
2 the opposing torques and compete for control of their shared central γ-shaft. Here we present a self-consistent physical
n modeloftheF1motorasasimplifiedtwo-stateBrownianratchetbasedontheasymmetryoftorsionalelasticenergyofthe
a coiled-coilγ-shaft. Thisstochasticmodelunifiesthephysicaldescriptionoflinearandrotarymotorsandexplainsthestepped
J unidirectionalrotationoftheγ-shaft,inagreementwiththe‘binding-change’ideasofBoyer. Substitutingthemodelparameters,

9 allindependentlyknownfromrecentexperiments,ourmodelquantitativelyreproducestheATPaseoperation,e.g. the‘no-load’
2 angularvelocityisca. 400rad/santiclockwiseat4mMATP,incloseagreementwithexperiment. Increasingthepmftorque
exertedbyF canslow,stopandovercomethetorquegeneratedbyF ,switchingfromATPhydrolysistosynthesisataverylow
0 1
] valueof‘stalltorque’. Wediscussthemattersofthemotorefficiency,whichisverylowifcalculatedfromtheusefulmechanical
C
workitproduces-butisquitehighwhenthe‘usefuloutcome’ismeasuredinthenumberofH+pushedagainstthechemical
S
gradientintheF ATP-drivenoperation.
1
.
o
i
b
- Introduction
q
[
Adenosinetriphosphate(ATP),theuniversalfuelofthecell,issynthesizedinthemembranesofmitochondria,chloroplastsand

1 bacteriabytheATPsynthasecomplex. Alreadyinitsabbreviation,F0F1-ATPase,onecanseeitsdualfunctionality,1which
v arisesfromthetwocompetingrotarymotors(F andF )thatsitontheoppositeendsoftheshareddriveshaft.2,3 Inoneof
0 1
8 itsoperationmodes,thismulti-subunitenzymeusesfreeenergystoredinatransmembraneelectrochemicalgradient(pmf)
7
todrivethesynthesisofATPbytheF motor. Intheothermode,itreversesthereaction,hydrolyzingATPandutilizingthe
0 0
chemicalenergytodrivetheF motorandpumpprotonsagainsttheirconcentrationgradient.4 Whichofthetwocompeting
8 1
0 motors“wins”,andconsequentlyinwhichmodetheATPasecomplexwilloperate,isdeterminedbytheratiooftheATP/ADP
. availabletotheF part,andthetorqueexertedbythepmf-drivenF .
1 1 0
0 Inengineering,iftworotarymotorssharethesamedriveshaftandarenotattachedtoanysupport,theoutcomedependson
6 theirrelativemass(inertiamoments);thiswouldbeimpossibleinaheavilyoverdampedmolecularsystem. Thankfully,inits
1 naturalsetting,ATPaseisanchoredinthemembranebythea-subunitofitsstator;thisconstraintisthenpassedthroughtheb-
:
v andδ-subunitstoimmobilizetheα3β3hexamercomplexoftheF1motor. Sotheonlyrelevantmovingpartsinthecomplexare
i thecentralγ-shaft,whichismadeofacoiled-coilproteinα-helix,andthedisc-likec-subunit,whichisrigidlyattachedtoit
X
andwhichcanrotaterelativetothea-subunitanchoredinthemiddleofmembranebilayerplane,seeFig.1(a). Theotherend
r
a oftheγ-shaftisinsertedintothecentralcavityoftheα3β3complex,withthreeidenticalequilibriumpositionsseparatedby
120◦representingthecatalyticsitesontheβ-subunits. The“cam”endoftheγ-shaft: theoff-axiseccentricbulgeregionofthe
γ-subunit,5–7isabletodockintothiscavityinthreeequivalentorientations,whenitfitsinoneoftheindentations,Fig.1(b).
ThesearethedistilledstructuralfeaturesofATPasethatweshallneedtodevelopourphysicalmodeloftheBrownianratchet
F motor;thesestructuraldetailsareextensivelycharacterizedinmanyfamouspapers.1,3,8,9
1
ThemechanismoftheF motorisrelativelywellunderstood,inbothitsphysicalprinciplesandquantitativepredictions,
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followingtheoriginalworkofG.Osteretal.,10 whereitisshownhowanasymmetryofH+ ionchannelsinthea-subunit
canselectivelyprotonatetheAsp61siteonthec-rotorandthusconstrainitsrotationalBrownianmotionpreferentiallyinthe
clockwisedirectionasseenfromthemembraneside(assumingthenaturalH+concentrationgradientonthetwosidesofthe
membrane). Thepmfarisingfromthisconcentrationgradientisabletodeliverthesufficienttorqueonthec-ring,whichisthen
passedviatheγ-shaftintothecavityoftheα β complexandforcesthe3-stagesequentialstructuralisomerization(opening)
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oftheβ-subunitstofacilitatethecaptureofADPandphosphateion,theenzymaticreactionADP+Pi,andthereleaseofATP.
TheactionmechanismoftheF motorisunknown,inspiteofmanyattemptstoexplainit. Itisknowntobepowered
1
bytheATPhydrolysisintheβ-subunitsoftheα β complex(i.e. thereverseoftheADPphosphorilationsequence)and
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Figure1. ATPasestructureanditscoiledcoilconfinement. (a)Thevolume-fillingatomicmodel,withcoloursandlabels
indicatingtherelevantsubunits,adaptedfrom.3 (b)Theschemeofthe“cam”nestingofthecoiledcoilγ-shaftinoneofthe
threeequivalentslotsintheα β complex,alsoshowingthedirectionofF motorrotationΘ. Ourmodelreliesonthecamlock
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beingreleasedbyβ-subunitconformationalchangeonATPhydrolysis.
producestheopposite(anticlockwise)rotationofthec-ring,whichisthereforeforcedtorotateinspiteoftheenergybarrier
ofnon-protonatedAsp61,andthusdeliverH+ ionsintothehigh-concentrationchannel,whiletakingthemawayfromthe
low-concentrationvolume. InthiswayATPasecanpumpprotonsandbuildupthetransmembranepHgradientusingtheenergy
storedinATP.Therearemanyinterestingpapersintheliterature,attemptingtodiscernhowthisanticlockwisetorquecanbe
generatedinthejunctionbetweentheγ-shaftandα β complex,butnonearesatisfactory. Theyoftenstartfromtheoriginal
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mechanistic“bindingchange”ideasofP.Boyer11anddeveloptowardsthepower-strokemechanismofWangandOster.12,13
Thelattermodelsinthisveinareverycomplex,withupto64statesandmanycoupledstochasticvariables.12,14 Withmany
freeparameterstheydosucceedatmatchingthephenomenologyoftheF motor. Therenonethelessremainsaneedtoproduce
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aconceptmodelwhichexplainsthephysicaloriginoftheeffectivetorquegeneratedbyF ,andmanyanalogousATP-driven
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rotarymotors,andthedirectionofrotationintermsofthemicroscopicconfigurationandthephysicsofthesystem.Thereisalso
anoften-quoted“100%efficiency”oftheF -ATPasemotor,3,15,16aphenomenonthatwouldcontradictbasicthermodynamics.
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Theoriginofsuchaconfusionisthatonecanmeasuretheusefulpowerthemotorproduces(fromthevelocityanddrag,or
fromthecounter-torquedeliveredbypmf)andtherateofATPhydrolysis,whichoneassumesistheenergysourceforthemotor
–andthusestimatethemotorefficiencyfromtheirratio. WeshallshowbelowthattheenergyinfluxfromtheATPhydrolysis
onlyservestoswitchbetweenthestatesofthesystem,whilethedrivingpowerarisesfromtheBrownianmotionitself.
State of the art
OfthelargeliteratureontheATP-drivenrotarymotorssuchasF -ATPase,themostrecentandmostusefuloverviewisfound
1
inpapersbyOkazakiandHummer17andbyMukherjeeandWarshel.18 Inthefirst,thedetailedreviewofkeyexperimental
observationsaboutthemotorisassembledandtheatomic-levelresolutionoftheα β complex–γ-shaftinteractionispresented.
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Thesecondpaper18 givesacarefulcomparisonofvariousmodelsoftorquegeneration. Thepaperitselfextendstheideaof
chemical-rotationalfreeenergylandscapethatshouldgeneratethetorqueonone120◦legoftherotationcycle.
Inrelationtotheideasoftorquegeneration,wewouldliketopointoutthatinthecaseoflinearmotors,itisnowwell
establishedthatitisnotthechemical-mechanicalcouplingthatproduceswhatpeopleoftencallthe“powerstroke”,butthe
wholeprocessisdiffusive(notdynamic,drivenbytheBrownianmotion),andtheswitchingbetweenconformationalstates
maintainstheout-of-equilibriumconditionsthatallowunidirectionaltransport.19 Thislastpointneedstobeemphasizedsince
itisawidelyspreadissueintheliterature. Normally,inadynamicalsystem,ifthereisafrictiondragopposingthemotion
(rotationalfrictioninourcase,expressedviathecoefficientoffrictionandangularvelocity: −ξω),thentherehastobean
externalforceapplied(torqueormomentofforceinourcase: M)inordertosustaintheconstant-speedmotioninwhatis
calledthe‘terminalvelocity’regime: M=ξω. ThenthepowerlossagainstfrictionistheproductP=Mω,andthesource
ofthisenergyiscomingfromtheagentthatdeliversthetorque. Thesituationisdifferentonthemolecularscale,whenthe
physicsisdominatedbytheBrownianmotion,i.e. thethermalstochasticforceactingonallelementsofthesystem. Laterin
thetextwewillderivetheexpressionfortheaverageangularvelocityofrotationofthemotor,whichisstronglyaffectedbythe
rotationalfriction–butviathediffusionconstantD=k T/ξ. Intheoverdampedregimethereisnomomentumtransfer!20 It
B
isincorrecttoassumethatthefreerotationintheoverdampedviscousenvironmentrequiresexternalpowerP=ξ(cid:104)ω(cid:105)2,which
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isafrequentlypresentedestimatethatusuallyleadstoastatementaboutthehighefficiencyofF -ATPasemotorbecauseit
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givesapproximatelythesamevalueastherateofATPhydrolysisenergysupply. Itisequallyincorrecttoassumethatthemotor
generatesatorquemerelybecausethereisaconstantaveragevelocityobserved. Themotionoccursduetothethermalmotion,
whiletheATPhydrolysisisneededtotakethesystemoutofthedetailed-balanceconditions. Inordertogenerateusefulwork
therehastobeacounter-torqueagainstwhichthestochasticmotorhastowork,asitiscarefullystudiedbyProstetal.21 Later
inthetextwewillcalculatetheaveragevelocityoftheF motorworkingagainstsuchacounter-torque(arisingfromF inthe
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realATPase,butnotamerefrictionalresistance)andthusbeabletoestimatetheefficiency,whichdependsonconditions(e.g.
becomeszeroatthestalltorque)butiscertainlymuchlessthan100%inallsituations. Theefficiencyisalsozerowhenthereis
noexternaltorque(freerotationofthemotoragainstfriction)sincethereisnoworkdone.
Among the large amount of experimental work, recently there have been two very important studies of the F motor,
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examiningtheroleoftheγ-shaft,bothextremelydelicateandsophisticatedintheirtechniques: thegroupofK.Kinosita22
havesequentiallychoppedoffthelengthoftheγ-shaftandmonitoredthechangeinthemotoraction,ineachcaseattachinga
markertotheshorterandshortershaft. ThegroupofH.Noji23havemanagedtoremovetheγ-shaftaltogetherandmonitored
the unidirectional rotation of the hydrolysis site on the α β complex by imaging in a fast AFM. The latter paper claims
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disagreementwiththeformer(andmanyothers),pointingattheapparentunidirectionalmotoractionwithoutanyrotor–a
remarkable observation in itself. Yet, on careful analysis, we conclude that there is no discrepancy and the results are in
fact consistent. It was found that by making the γ-shaft sequentially shorter, the average speed of the motor has become
consistentlylower(inspiteofthefrictionresistanceoftherotatingmarkerbeingkeptthesame): changingfromwellover
100rps(revolutionspersecond)forthewildtypedownto∼0.8rpsfortheshortestmutantatahighATPconcentrationof2
mM.22Theexperimentwithγ-shaftrotorremoveddidnothaveanyfriction,andrecordedanaverageunidirectionalsequence
ofbindingonβ subunitswithaspeed∼0.4rpsat2µMATP.Thisneedstobecomparedwiththespeedof2−5rpsinthe
wild-type F motor observed at the same 2 µM ATP with an actin filament creating high friction drag15 or ∼20 rps with
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alow-draggoldbead.24 Itisclearfrombothexperimentsonγ-shaftreductionthatthereisacomplicated(andnotknown)
machineryintheα β complexthatmaintainsthetransferofthehydrolysissite,slowlybutconsistentlyincounterclockwise
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direction. Butitisequallyclearthattheγ-shaftplaysacrucialroleinachievingtherequiredspeedsandthemotorefficiency,
anditsreductiondramaticallyreducesboth. Wealsonotethatacoiled-coilfilamentisadefinitivecommonfeatureofallother
rotarymotorsdrivenbyATPhydrolysis(incontrasttopmf-drivenmotorslikeF orflagellar). Inthispaperwedevelopa
0
conceptofsucharotarymotorradicallydifferentfromwhatwasconsideredpreviously: amodelthatisbasedonthevery
genericasymmetrictorsionalelasticityofthecoiledcoilγ-shaft. Wedemonstratethatthemotormechanismisrobustand
generic,notmuchinfluencedbythedetailsofmolecularstructure(exceptthe120◦periodicityandtheparametersoftheγ-shaft,
whicharebothcrucial).
Ourideaofatwo-stateBrownianratchetmotorisnotoriginal(onlyitsrotaryaspectis): alongtimeagoitwasdeveloped
by Ju¨licher, Ajdari and Prost, and is now an accepted standard for linear motors such as myosin on actin, or kinesin on
microtubules.21,25 Forsuchamotor,atthemostbasicleveloneneedsjusttwoelements: aperiodicprofileoftheground-state
potentialenergywithleft-rightasymmetry(whichisthesourceofsymmetrybreakingforunidirectionalmotion), andthe
externalenergyinputtodisturbtheequilibriumdetailedbalanceandbringthemechanicalsystemintotheuppershort-lived
‘excited’state(whichiswhattheATPhydrolysisprovidesinallcases). ForF ATPasetherelevantpotentialenergyisthe
1
torsionalelasticenergyoftwistingofthecoiled-coilγ-shaftaboutitsaxis, Fig.1(c). Thelineartorsionalelasticityofthe
γ-subunithasbeenstudiedingreatdetailbymonitoringthethermalfluctuationsoftheangleofthec-ringfreelyrotatingwith
respecttotheα β compleximmobilizedonasubstrate. However,thefactthatitcostsmoreenergytounder-twistthecoiled
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coil,thantoover-twistit,isnotaprioriobvious. Herewemodeltheα-helicalcoiledcoilandproducethedistinctlyasymmetric
non-lineartorsionalpotentialenergyE(θ),showninFig.3below. The3-fold,120◦periodicityisinherentintheα β structure,
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andweplotthispotentialenergyagainsttheangleθ withrespecttotheneutralequilibriumpositionofitscamdockedin
agivenslot,asillustratedinFig.1(b,c). Forthemodeloftorsionalelasticityofthecoiledcoiltomatchthegeometryand
themeasuredtorsionalmodulusoftheγ-shaft,13,26,27 thereisverylittlefreedominitschoiceofparameters. Notethatthis
physicalreasonfortherotationsymmetrybreakingoftheF -ATPasemotorisdifferentfromtheasymmetrysuggestedbyOster
1
etal.,28wherethe‘culprit’wasproposedtobetheshapeoftheβ-subunitconfiningthebulge-endoftheγ-shaft. Wedonot
argueagainstsuchanasymmetryofthe‘lock’shape(infact,experimentsofNojietal.23 ontheunidirectionalbindingwith
norotorsurelyproveitsvalidity),merelypointoutthatitcannotproduceapowerstrokeinoverdampedconditionswhereno
momentumtransfercanoccur.
Thenextsectiongivesabriefdescriptionoftheasymmetrictorsionofthecoiledcoil. Wethenshowhowtherotarymotion
isinducedinsuchanasymmetricperiodicpotential,whenasecond‘excited’state(correspondingtotheundockedcamandthe
unrestrictedrotationaldiffusionofthec-ring)canbereachedbyanATPenergyinflux. Thefinalsectiondemonstrateshowsuch
arotarymotorwouldoperateagainstacounter-torquearisingfromthepmf-drivenF motor,andfindthestallconditions. Atthe
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endwecomparetheoreticalpredictionswithexperimentalmeasurementdataonATPsynthaseF motoraction,24,29,30 andfind
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aremarkablelevelofagreementgiventhatwehavepracticallynofittingparameters(meaningthatmostofthemodelparameters
areinfactdeterminedfromindependentexperiments). Thekeyobservationsthatwewishtomatchare: (1)Atlow[ATP]there
isanalmostlineardependenceofaveragerotationvelocitywithATPconcentration,practicallyindependentonthedegreeof
frictionalresistance,withaslope∼6·106rps/M.24,31(2)Athigh[ATP]therotationratesaturatesatavaluedeterminedbythe
frictionresistance,inmanydifferentexperimentsthisrotationalfrictioncoefficienthasspannedtherange10−4-102pN.nm.s.24
NodoubtmotivatedbyMichaelis-Menten,peoplefrequentlyuseafittingfunction(cid:104)ω(cid:105)=(1/v +2πξ/M)−1 whereM
noload
isthe“assumedtorque”generatedbythemotor. Atahigh[ATP]=2mMandlowfriction,thevalueofv wasfoundto
noload
be∼130rps.24 (3)Thevalueofthe“assumedtorque”isfrequentlyquotedas∼40pN.nm,16,31althoughtheearlierworkof
Jungeetal. produced∼56pN.nm.3 Wealreadyexplainedtheerrorininterpretingaproductξ(cid:104)ω(cid:105)ofastochasticmotorasan
actualtorquegenerated,nevertheless,theactualexperimentalvaluesofξ and(cid:104)ω(cid:105)needtobematched. Allofthishappenson
ATPhydrolysisandthebindingratesarequotedintherange2-6·107M−1s−1byanumberofdifferentgroups.24,30–32 These
valuesareourinitial‘target’inthispaper.
Asymmetric torsional energy of the coiled coil
Thecentralγ-shaftisaleft-handedcoiledcoiloftwoα-helicalproteins. Herewecalculatehowtheelasticenergyincreasesasa
functionoftheangleoftwistofsuchacoiledcoilfilament. Notethatthereareother,sometimesmoresophisticated(andalways
morecomplex)elasticmodelsofhelicalfilaments,33,34butwechoosetostaywithverysimple,qualitativelycleardescription.
Wecanregardtheα-helixasanelasticfilamentwithknowncharacteristics. Itsouterdiameteris1.2nm;itspersistence
lengthhasbeenextensivelystudied,providingareasonablyaccuratevalueforthebendingmodulusB≈570pN·nm2(equivalent
toapersistencelengthof∼100nmatroomtemperature35,36). Surprisinglyforacylindricalelasticfilament,thetorsional
elasticityofanα-helixisgreater:C≈(3/2)Bwiththecorrespondingtorsionalpersistencelengthof∼150nm. The‘surprise’
isbecauseinahomogeneouscylindermadeofanisotropicelasticmaterial,thetorsionalmodulusisrelatedtothebending
modulusbyanequationC=B/(1+σ),whereσ isthePoissonratioofthematerial.37 Soforanincompressiblematerialone
expectsC=(2/3)BandforatypicalPoissonratioofcrystallinesolids(σ=0.3)wewouldhaveC=0.77B,whiletheobserved
factorof3/2impliesanegativePoissonratioσ =−1/3. Thisisaveryunusualcaseinelasticityoftenreferredtoasan“auxetic
material”.38,39 Ontheotherhand,itmaynotbesosurprisingifweconsiderthemainelasticelementsintheα-helixarethe
hydrogenbondsintheoutershellofthecylinder,whicharealignedwiththehelixaxis: suchaconfigurationhasallthemakings
ofalocallyauxeticmaterial. ThesefactsandadiscussionarewellpresentedinapaperbySunetal.36
Whenapairofα-helicesiswoundintoatwo-strandcoiledcoiltertiarystructure,theaxisofeachoftheα-helicesmakes
a helical curve of its own, with a radius R=0.46 nm and a pitch p in the range between 11 nm40 and 14 nm.41 For our
calculationsweshalltake p=11nm. Thereisanextensiveknowledgeofthecoiledcoilgeometry,goingbacktotheworkof
PaulingandCrickinthe1950s;42,43 thisgeometryanditsparametersarewellsummarizedinrecentpapersbySunetal.44,45
Wemighttakeaviewthattheα-helixisanelasticfilamentthatisstraightinequilibrium,butisforcedtomakealeft-handed
helicalcurveinspacewhenitistwistinginthecoiledcoilconfiguration. Incalculatingtheelasticenergyweusetheapproach
ofYamakawa,46byexpressingthegeometryofahelicalcurveviathetwocharacteristicparameters: curvatureandtorsion(κ
andτ),whicharedirectlyrelatedtotheradiusRandpitch pofthecoiledcoil,Fig.2:
R p/2π
κ = , τ =− . (1)
R2+(p/2π)2 R2+(p/2π)2
Inequilibriumeachα-helicalfilamenthasκ =0andτ =0,butwhentheyaretwistedintothecoiledcoil,thetwospacecurves
r andr representthecenterlinesofeachfilament:
1 2
   
Rsinωs −Rsinωs
r1= Rcosωs , r2= −Rcosωs , (2)
qs qs
wheresisthearclengthalongthecurvesofα-helixcenterline,ω =2π/pistherateofhelicalwindingandq=pω/2π isthe
√
rateofadvanceofthecurvealongz-axis. Notethatsincetheα-helixisessentiallyinextensible,36theconstraintq= 1−R2ω2
(cid:112)
orω =1/ R2+(p/2π)2isinplace. Thewindingangleattopofthecoiledcoiliswhathasbeenacquiredwhens=L,thatis,
θ =ωL. TheFrenet-Serretequationsallowcalculatingthecurvatureandtorsionforagivenhelixconfiguration:
(cid:113)
κ =R(θ/L)2, τ =−(θ/L) 1−R2(θ/L)2, (3)
(cid:112)
whichshowsthatforsuchoverconstrainedspacecurvesthecurvatureandtorsionarenotindependent: τ =− κ/R−κ2.
Theelasticenergyofeachsuchfilament(asitisforcedtowindintoacoiledcoil)asafunctionofdeviationfromitsnatural
equilibriumismeasuredbyκ andτ: E =L(cid:2)1Bκ2+1Cτ2(cid:3).
α 2 2
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Figure2. Geometryofthetwo-strandedcoiledcoil. (a)Thedimensionsoftheγ-shaft(from5). (b)Thetwoelastic
α-helicalfilamentswoundaroundeachotherinatertiaryhelixwiththepitch p. (c)Anillustrationofbondingbetweenthetwo
nearly-parallelα-helicesviathe‘heptadrepeat’residues,whichinducestheequilibriumtwist.
Thereasonthecenterlineofanα-helixformsahelicalcurveinthetertiarystructureofcoiledcoilisbecauseamechanical
frustrationisimposedonitbythehydrophobicbondingofresidueswiththeparallelsecondα-helix.42,44,47 Atypicalα-helix
hasahelicalpitchof0.6nmand3.6residuesperturn,makingthestepalongitsaxisofh=0.6/3.6≈0.167nmperresidue.
Hencethelengthoftheα-helicalfilamentwithNresidueshasafixedvalueofL=Nh,andwealreadystatedthatitisessentially
inextensible.36 However,whenasecondα-helixisalignedparalleltoit,thehydrophobicpairingofapolarresiduesofthe
so-calledheptadrepeat(7:2pairs)43,48 formsacontactlineor“seam”betweentwoα-helices, whichhas3.5residuesper
turn,bondingthetworesidues(a,d)ofeachheptad(thestudyofcoiledcoilelasticity33callsthislinethe“interfacecurve”).
Sincethelengthalongthiscontactlineofpairedresiduesonthesideofthecylindricalα-helicalfilamentisshorterthanthe
naturallengthofthecenterlineofthiselasticfilament,therestofthefilamentcoilsintoahelix–aphenomenonfamiliarin
telephonecords,planttendrilsandcurlyhair. Thekeytotheresultinggeometryisthattheheightofthecoiledcoil(definedas
(cid:112)
H inFig.2: thelengthalongtheseamline)isshorterthanthecontourlengthL= H2+(Rθ)2,andtheirobservedratiois
H/L=3.5/3.6=0.97. SincetheheightH oftheγ-shaftisknownfromexperiment5(H≈6.5nm),theserelationsdetermine
theequilibriumpitch(p=11nm)ortheequilibriumtwist(θ =2πH/p=3.71rad,or210◦)ofthecoiledcoilpairinthe
0
γ-shaft. Theelasticallyactivelengthofα-helicesisL≈6.72nm,whichmakesapproximatelym=2(6.72/0.167)/7≈15
hydrophobicbondsalongtheseamlineoftheγ-shaft,assumingthattworesidues(a,d)ofeachheptadarepaired.
However,wecannotassumethehydrophobically-bondedpairsofmatching(a,d)heptadmembersexperienceaharmonic
potentialwithrespecttostretching/compressingtheseamline. Thislineismadeofasequenceofpairsofhydrogenbondsin
eachα-helix,linkedbythematching(a,d)heptadmembersandnearlyparalleltothelineitself: itismuchhardertostretch
thesequenceofbondedpairsthantocompressit. Instretching,orelongatingthedistancebetweenresiduesalongtheaxisof
theα-helix,theexistinghydrogenbondsthatarealignedalongtheaxisoftheα-helixneedtobestretched,whichisahard
proposition. Incontrast,theshorteningofthislinecanbemuchmoreeasilyaccommodatedbytiltingoftheα-helicalbonds
awayfrombeingparalleltothehelicalcenterline. OnecanseeananalogywiththeclassicalEulerproblemofstrutelasticity:
itishardtostretchanelasticrod,butoncompressionitcaneasilybuckleandthusrespondwithamuchlowerforce. The
expressionforsuchastretching/compressingenergy,asfunctionofthelengthofinterfacecurve,x,givenitsequilibriumlength
H ,andtheoveralllengthofthefilamentL,shouldtaketheformanalogoustothegeneralenergyofastifffilament:49
0
(cid:20) 1−(x/L)2 1−(H /L)2(cid:21)
E =mε + 0 . (4)
seam 1−(H /L)2) 1−(x/L)2)
0
ThisstretchingenergyE isplottedinFig.3(a),illustratingatypicalresponseofastifffilament,whichishardtostretchbut
seam
easytobuckle. TheexperimentallyobservedcoiledcoilheightH=6.5 nm(orthecorrespondingtwistingangleθ =3.71 rad)
0
(cid:112)
areobtainedbyconvertingfromtheseamlinelengthxtothecoiledcoiltwistingangleθ (viax= L2−(Rθ)2)andadding
thestretchingenergyE (θ)tothetwistingenergyoftwoα-helices2E (θ),whichisplottedasatotalinFig.3(b). The
seam α
minimumofthisenergydefinestheequilibriumshapeofthecoiledcoil: apairofnaturallystraightelasticfilaments(α-helices)
frustrated(contorted)bytheaddedelasticenergyoftheseamline.
It may seem that there are many unknown (free) parameters in this simple elastic model, but in fact the published
experimentalresultsconstrainthemveryaccurately. Jungeetal. havemeasuredthelineartorsionalmodulusoftheγ-shaft
byobservingfreethermalfluctuationsofitsend.6,26,29 Awiderangeofvaluesisreported,dependingonwhichsectionof
thefilamentoneistrackinginelaborateanddelicateexperiments,buttheorderofmagnituderemainswelldefined. Abigger
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Figure3. Seamlinecontortstwoα-helicesintoacoiledcoil. (a)Plotofthestretchingenergyofseamline,Eq. (4)forthe
parmetersoutlinedinthetext. (b)Plotsofthetotalelasticenergyofthetwofilamentsboundalongtheseamline,forincreasing
valuesofthebondingenergyε. Theminimumofthisenergygivestheequilibriumtwistoftheleft-handedcoiledcoil
θ ≈3.71[rad]forthecurvewithmε =250k T (withm=15thisgivesthecharacteristicparameterε =16k T).
0 B B
Figure4. Asymmetryoftorsionalelasticityofcoiledcoil. TheenergyE (θ)isplottedforthechoiceofparametersleading
1
totheobservedcoiledcoilgeometry(H,θ andtorsionalmodulusaroundtheequilibrium). Theanglestounwindandto
0
over-windthecoiledcoilaredifferent: thevaluesa≈1.197andb≈0.898arechosenforthetotalwindowofvariationtobe
120◦. Theenergybarreirthenis∆E ≈136k T.
0 B
6/15
problemisthatthetorsionalmodulushastobepresentedintheunitsof[energy/angle2]whileallthereportedvaluesforthe
‘modulus’aregiveninthedimensionsofenergyalone[pN.nm]. TheMDsimulationstudyofγ-shaftelasticity27alsohasatypo,
presentingthemodulusin[pN.nm/rad].Inspiteofallthat,thecoreexperimentalresultofJungeetal.isclearandunambiguous:
theyhavemappedtheprobabilitydistributionofangleθ ofthefreelyrotationalklyfluctuatingγ-shaftandobtainedaGaussian
shape,whichallowstodeterminethemodulusdirectly,givingavalue∼0.4pN.nm/deg2=1300pN.nm/rad2(infact,arange
ofvalueswithinthisorderofmagnitude,dependingonwhichsegmentofγ-shaftisexamined). Acombinationofthismodulus
(i.e. thecurvatureofthetorsionalenergyneartheminimumθ ,andthepositionofthisminimumatθ =3.71rad,areenough
0 0
todeterminetheparametersmε ≈250k T (or147kcal/mol,i.e. ε =9.8kcal/mol=69pN.nmperbondedpairoftheheptad),
B
andH ≈6.33nm. Soinequilibrium,theseamlineisslightlystretched(to6.5nm)attheexpenseofcoilingthetwoα-helix
0
filaments. Figure4zoomsintotheregionneartheminimumofthistorsionalenergy,whichwelabelasE (θ)asthefirstlevel
1
ofthetwo-statemodelforthemotor,andwecanclearlyseetheasymmetry: theenergyrisessteeperforunwindingthecoiled
coil(i.e. attemptingtostretchtheseamline). Thetorsionalfluctuationsofthecoiledcoilaboutthisminimumaremarkedly
asymmetric,whichisthekeyingredientofourmodelofrotarymotor. Itisremarkableinthehindsightthattheexperimental
resultsofJungeetal. areinfactshowingtheslightasymmetryoftheirprobabilitydistributions!
Theanalysisinthissectionprovidesthevaluesofasymmetryabouttheminimumoftorsionalelasticenergy. Aswehave
discussedearlierandillustratedinFig. 1(b),thereisa120◦-periodicityofthecamendoftheγ-shaftconfinementinsidethe
α β cavity. Takingthis‘window’of120◦=2π/3foreachcycle,theasymmetryofthetorsionalpotentialmakestheangular
3 3
intervals: a≈1.197andb≈0.898[rad],asillustratedinFig. 4. Ofcourse,thesuma+b=2π/3,butthecrucialparameteris
thedifference: a−b≈0.3rad,or17◦. Theheightofenergybarrierattheboundariesofeachangularcycleis∆E ≈136k T
0 B
withinthismodel. Thisisahighenergybarrier,soonecanbeassuredthatnospontaneous‘slippage’oftheoftheγ-shaftcan
occurwhileitscamisconfinedinsidetheα β cavity.
3 3
Two-state stochastic Brownian ratchet motor
Inthissectionwere-writetheoriginalmodelofJu¨licher,AjdariandProst21,25forthecaseofrotarymotion,inthesimplest
possibleform. Inthermodynamicequilibrium,withoutexternaltorque,theasymmetryofpotentialenergyE (θ)isirrelevant:
1
thedetailedbalanceensuresthatthediffusionfluxinbothdirectionsmustbeequal. Soinspiteoftherotationalsymmetry
brokenbytheslantedpotentialenergyineachperiod,thethermalmotionalonecannotproduceunidirectionalrotation. Inthis
120◦-periodiclow-energystateE (θ)thebulgeintheγ-shaft(thecamtooth)islockedinoneofthethreepositionsaroundthe
1
circle–andthec-ringonlyfluctuatesconfinedneartheminimumofthistorsionalelasticenergy. Thesearethefluctuations
measuredbyJungeetal.6,26,29 andsimulatedbyCzubandGrubmu¨ller,27whichwediscussedintheprevioussection.
WhenanATPmoleculeishydrolyzedintheα β complex(withthebindingratek ),theconfinementoftheγ-shaftcam
3 3 on
changes: hereweassumeitisreleasedandbecomesfreetorotateinsidetheα β cavity. Thismeansthatthesystementers
3 3
the“excitedstate”E withnorotationalconfinement–thisassumptionisanobviouslimitationmadetosimplifythemodel
2
andmakeiteasilytractable,andweshalldiscusslaterinthetexthowrelevantitis(itturnsoutthatitisnot). Freerotational
diffusionofthec-ringandγ-shaftoccursthen,withtheGaussiandistributionofprobability p(∆θ)∼exp[−(θ−θ )2/4Dt]. In
0
thewildtypeATPaseembeddedinitspropermembrane,therotationaldiffusionconstantofthec-ringrotorwasmeasuredby
severalauthorsandreported,e.g.,byElstonetal.:10 D≈2·104rad2/s. InmanyfamousexperimentsobservingtheF rotation
1
invitro,peoplewereattachingeitheranactinfilamentoragoldbeadinplaceofthec-ring–andobservingtherotationofsuch
amarkeragainstviscousfrictioninsolution.15,24,31 ThediffusionconstantD=k T/ξ hasbeenvariedovernearly6ordersof
B
magnitude,dependingonthegeometryofthemarker.24 Forinstance,atroomtemperaturethe40-nmgoldbeadofYasudaet
al.24 hasξ =2·10−4pN.nm.s,givingthediffusionconstantD≈2.1·104rad2/s: andalmostexactmatchwiththemembrane
frictionofthec-ring. Usinga1-µmlongactinfilamentasamarkergivesξ ≈2pN.nm.s,15 producingD≈2.1rad2/s. The
authorsinthepasthaveincorrectlyusedtheideaof“no-load”velocityatverylowfriction,butweseethatalltheincreasing
frictiondoesistoslowdownthefreerotationaldiffusionintheupperunconstrainedstateR . Weshallexaminetheeffectof
2
changingfrictiononthemotorvelocitylaterinthetext.
Afteracharacteristiclife-timeτ ofthisisomerizedstatewheretheγ-shaftcouldfreelydiffuse,theα β complexreturns
2 3 3
toitsnaturalstateandthecamreturnsbacktoalockedposition;thismaybethesamepositionaswhenitwasreleased–but
duetothefreerotationaldiffusionintheE state,itwillmorefrequentlydropintotheneighboringpositiontotheleft(the
2
closesttotheoriginal,asb<a). Wecanestimatetheaverageangularvelocityoftheresultingrotationastheratioofangular
step(a+b=120◦)andthetotaltimeofthiscycle(∆t =1/k [ATP]+τ ),giving(cid:104)ω (cid:105)=(a+b)(p −p )/∆t,wherethe
on 2 0 + −
probabilitiesofdroppingintotheright-andleft-sidepocketsofE aredeterminedbythecorrespondingerrorfunctionsfromthe
1
limitedintegralsof p(∆θ). Inthelimitwhenthetorsionalasymmetry(a−b)issmall,thisvelocitytakesamoresimpleform:
a+b (cid:18) (cid:20) b (cid:21) (cid:20) a (cid:21)(cid:19) k [ATP] a2−b2
(cid:104)ω (cid:105)≈ Erf √ −Erf √ ≈ on √ e−a2/4Dτ2, (5)
0
1/kon[ATP]+τ2 2 Dτ2 2 Dτ2 1+τ2kon[ATP] 4πDτ2
7/15
Figure5. Aschemeoftwo-stateBrownianratchetmotor. Inthelow-energystatethebulgeintheγ-shaftislockedinone
ofthethreepositions120◦aroundthecircle–andthec-ringonlyfluctuatesconfinedneartheminimumofthistorsionalelastic
energy. Inthermalequilibrium,therecanbenounidirectionalmotioninspiteofthechiralanisotropy. Anenergyinputfrom
ATPhydrolysiscausesα β isomerizationandcandisengagetheconfinement,allowingthec-ringtofluctuatefreelywhilein
3 3
thisunconstrainedexcitedstate. TherateofE →E transitionisk [ATP],whichisanactivationfunctionofATPchemical
1 2 on
potential∆µ,whiletherateofthereverseE →E transitioniscontrolledbythelife-timeoftheisomerizedstateofβ-subunit
2 1
(τ ). Onreturntoitscam-engagedstate,thecoiledcoilhasahigherprobability(p )toendupat−120◦fromtheoriginalstate,
2 −
thaninthe+120◦state(p ),whichresultsintheaverageanticlockwiserotation.
+
wherethesubscriptin(cid:104)ω (cid:105)indicatestheabsenceofanexternaltorque: atrue“no-load”conditionofthemotor. Thisrotation
0
velocityisnominallymeasuredin[rad/s]; toconvertitinto[rps]onehastodivideitby2π. HeretherateofATPbinding
k [ATP]=κ exp[∆µ /k T]canbeexpressedviathechemicalpotential∆µ =k Tln[ATP]. Takingthespecificrateof
on 0 ATP B B
ATPbindingk =4·107M−1s−1allowsconvertingtheATPconcentrationof2mMtotheactualrateofbindinggivenbythe
on
productk [ATP]≈8·104s−1,orthetimeintervalbetweentheATPbindingeventsof∼13µs.
on
Figure5showsthepictorialschemeofthisprocess,whiletheplotsinFig. 6(a,b)showhowtheaverageanticlockwise
“no-load”velocity(cid:104)ω (cid:105)dependsonthekeyparameters. Firstofall,thereisthelife-timeintheisomerizedfree-rotationstate
0
τ . Thistimeisnotgenerallyknown,andtheonlyclosemeasurementreportedbyFuruikeetal.30 isthatofadwelltimeof
2
severalmilliseconds(alsosee18). Weassumethattheactuallifetimeoftheisomerizedβ-subunitismuchshorter. Theplot6(a)
ismadeforthelow-frictionrotationcorrespondingtoa40nmgoldbeadmarker,orthewildtypemotor,D=2·104rad2/s,
andseveralrealisticvaluesof[ATP]giventhek =4·107M−1s−1. Firstofall,weseethatourconceptmodel,inspiteofall
on
itssimplifications,isquantitativelyproducingthevaluesofaveragerotationvelocityintheobservedrange(spinningatover
a100rpsisclearlyrealistic). Ontheotherhand,thedependenceonthepoorlyknownlifetimeτ isverysharpandifone
2
‘misses’theoptimalvalue,therotationvelocitydropsrapidly. Thisdependenceiseasilyunderstood: atveryshortlife-timethe
camcannotdiffusetoofarfromitsoriginalpositionintheE periodicpotential,andit’smostlikelytore-engageinthesame
1
positiongivingnonetrotation–atverylongτ thecamhastimetodiffuseveryfaranditbecomesrelativelyequal-probability
2
tore-engagetotheleftandtotheright. Itisclearthatthisisacrucialparameterforevolutiontoadjustforanygivenmotorand
itsspecificuseinthecell.
Plot6(b)showsthedependenceon[ATP]concentration,whichentersinourmodelastherateofE →E transition. As
1 2
expected, andingoodagreementwith‘canonical’experiments,24,31 atlowATPconcentrationtheaveragevelocityvaries
linearlywithconcentration–whileathighconcentrationitsaturatesatamaximalvalueω whichiscontrolledbytherateof
max
thereverseE →E transition,1/τ . Thechangebetweenthetworegimesoccursatk [ATP]≈1/τ . Itiscommoninthe
2 1 2 on 2
literaturetofitthistypeofcurvetoaMichelis-Mentenequation,whichisindeedwhatEq. (5)gives,with
a a
linear: (cid:104)ω0(cid:105)≈ √πDτ2(a−b)e−a2/4Dτ2·kon[ATP], ωmax≈ √πDτ3/2(a−b)e−a2/4Dτ2, (6)
2
(stillinthenominalunitsof[rad/s])maintainingthesimplifiedexpandedformforthesmalltorsionalasymmetry(a−b)(cid:28)1.
Notethattheratio2a2/Disapproximatelythetimeforanon-engagedγ-shafttodiffuseasingle120◦period,theratioofwhich
toτ iswhateffectivelycontrolsthemagnitudeoftheseexpressions.
2
Letusnowexaminehowthismotoroperatesindifferentfrictionregimes. Asexplainedabove,forthestochasticmotor
likethisthe(rotational)frictiondoesnotprovidea‘load’,butsimplyactsthroughthechangingrateofdiffusion,D=k T/ξ.
B
Wishing to compare with the famous experiments of Yasuda et al.15,24 which studied a wide range of markers providing
differentrotationalfrictionconstant,weplotthedependenceofaveragevelocity(cid:104)ω (cid:105)onthisfrictioncoefficient(assuming
0
8/15
Figure6. Theaveragespeedoffree-spinningmotor. (a)Therotationvelocityasafunctionoflife-timeτ atseveralvalues
2
of[ATP](givenintheplot)andthelow-frictionregimecorrespondingtothewild-typeF ATPaseandthein-vitroexperiments
1
witha40nmgoldbeadmarker. Aprominentmaximuminthevelocityindicatestheoptimalvalueforthemotoroperation
τ ≈20µsforthisresistanceregime. (b)TherotationvelocityasafunctionofATPconcentration,expressedhereviathe
2
transitionratek [ATP].Thesamevalueofthediffusionconstant(rotationalfriction)isused,andseveralvaluesofthe
on
isomerizationlife-timeτ (inseconds)arelabelledontheplot. Theonsetoftheplateauandthemagnitudeofω aredirectly
2 max
determinedbyτ .
2
Figure7. Theaveragespeedchangeswithfrictionresistance. Bothplotsshowthesamedataofhow(cid:104)ω (cid:105)varieswiththe
0
rotationalfrictioncoefficientξ,atT =23◦Cand[ATP]of2mM.Severalvaluesofτ arelabelledontheplots. Theplot(a)has
2
thelinearscaleoftheξ-axisandhighlightstheexponentialdependenceof(cid:104)ω (cid:105)(ξ)athighfriction,whilethelog-logplot(b)
0
allowsamuchwiderrangeoffrictionvaluestobeexamined.
roomtemperatureof23◦C),ratherthanthediffusionconstantD, seeFig. 7. Thelog-linearversionofthisplot(a)shows
theexponentialdecayof‘no-load’(cid:104)ω (cid:105)withincreasingfriction,whichisinfactobviousfromexaminingEqs. (5)-(6). The
0
log-logplot(b)enhancesthesmallchangein(cid:104)ω (cid:105)atverylowfriction(perhapsatunreasonablylowvalueusedmerelyfor
0
completenessofexposure). ThismayseemnotinperfectagreementwiththeresultsofYasudaetal.24 thatappeartoplateauat
ξ →0. However,wouldpointthatourmodelwasultimatelysimplifiedforclarityofconcept,anditcertainlyunderestimates
thefrictioninthe‘free-diffusion’excitedstate(whenwetookthattheγ-shafthasnoconstraintwhatsoever). Equally,thereare
difficultiestoactuallyachievevanishingfrictioninexperiment,sothissmalldecreasemightnothavebeennoticed(aswellas
thereliabilityofaccuratelydeterminethefrictionconstantlimited).
Yasudaetal.15,24 fitthecurveof(cid:104)ω (cid:105)vs. frictionconstantξ byaninterpolatedformula: (cid:104)ω (cid:105)=(1/V +2πξ/M)−1,
0 0 noload
wheretheir‘assumedtorque’was(cid:39)40pN.nm. Wealreadyexplainedthatastochasticmolecularmotordoesnotexertatorque
intheunloadedstate–itjustmakesmorerandomstepsinonedirectionthanintheother. Inthehigh-frictionlimitwhen
ξ (cid:29)k Tτ ,oursimplifiedEq. (5)predictsanexponentialdropintheaveragerotationvelocity(cid:104)ω (cid:105),whilethedataofYasuda
B 2 0
etal.15,24 isprobablygoingbroadlyoverthecrossoverregion.
9/15
Figure8. Twocompetingmotors: theroleofcounter-torque. Themodifiedsketchofthetwostatesofthemotor,skewed
bytheexternaltorqueMfromtheF motor. Now,inadditiontotheATP-activatedratchetmotordrivingtheshaft
0
anticlockwise,therearetwofactorsthatpromotetheclockwiserotation: thebiaseddiffusioninthetiltedperiodicpotential
E (θ,M)andthedriftofthediffusionenvelopewiththeconstantrateMD/k T.
1 B
F motor operating against external torque
1
Inrealcircumstances,theF motordrivenbythepmf,originallydescribedbyElstonetal.10 andreviewedinseveraltopical
0
reviewsonthesubject,exertsatorqueonthec-ring,whichispassedthroughtheγ-shafttotheconfiningregioninsidethe
α β complex. Thistorque,ortheturningmomentM,isinthedirectionoppositetothenaturalanticlockwiserotationofthe
3 3
F motor,andsothetwohavetocompete. Figure8showsthepictorialeffectofsuchanexternaltorqueontheF operation,
1 1
slantingbothenergystatesE andE andaddingtwoadditionalcontributionstotheaverageangularvelocityoftheshaft:
1 2
(cid:104)ω(M)(cid:105). Oneoftheseisadriftintheclockwisedirectioninthefree-diffusionexcitedstateE ,theotherisdiffusiontothe
2
rightintheground-stateperiodicpotentialE (θ)becausetheenergybarriersfortheforwardandbackwardmotionarenow
1
different. Thefirsteffect(ofthefreediffusionenvelopedrift)isaccountedforbytheshiftintheGaussianexponentofthe
probability p(∆θ)=exp[−(θ−θ )2/4Dt+Mθ/2k T]andisnaturallyextendingtheBrownianratchetapproachthathasled
0 B
toEq. (5). Thesecondeffectofthediffusioninatiltedperiodicpotentialisaclassicalproblemcarefullystudiedinthetextbook
byNelson,50improvingontheoriginalFeynman’streatmentoftheratchetandpawlproblem.51 ThesimpleFeynmanformula
hastheexponentiallydivergingvelocityatlargetorque,whiletheNelsonapproachcorrectlyaccountsfortheviscousdragand
resultswithalinear‘force-velocity’relationshipintheoverdampedmolecularsystem. TheformulafortheS-ratchetderived
in50hastobemodifiedbecausetherearetwoslopesofthepotentialfromitsminimum(towardsaandb),sothefinalexpression
becomesalittlebitmoreinvolved,althoughtheunderlyingprincipleofdiffusivemotionacrosstheeffectiveperiodiclandscape
remainsthesame. Itisalsoinstructivetocomparethetwoversionsofthispartofthisclockwiseangularvelocityatsmall
torqueM,i.e. inthelinearresponseregime:
(cid:104)ω+Feynman(cid:105)≈ kBDTe−∆E0/kBT·M; (cid:104)ω+Nelson(cid:105)≈ kBDT (∆E(e0∆/Ek0B/TkB)T2e−∆E10)/2kBT ·M. (7)
However,forourpracticalpurposesinthispaper,itturnsoutthatwiththeenergybarrier∆E (cid:39)136k T andtheexternaltorque
0 B
notexceedingM(cid:39)50pN.nm(whichisequalto12k T)thiscontributiontothemotoractioniscompletelynegligible.
B
Thedriftofthefree-diffusingdistributioncanbedirectlyincorporatedintotheexpressionfortheaverageangularvelocity
duetotheProstmechanism,25shiftingthemid-pointofthespreadingfree-diffusiondistribution p(θ,t)intheexcitedstateE ,
2
givingtheexpression,whichatM=0reducestoEq. (5):
(cid:18) (cid:20) (cid:21) (cid:20) (cid:21)(cid:19)
a+b b+MDτ /k T a−MDτ /k T
(cid:104)ω(M)(cid:105)≈ · Erf √ 2 B −Erf √ 2 B . (8)
1/kon[ATP]+τ2 4πDτ2 4πDτ2
TheplotsinFig. 9illustratethepredictionsofEq. (8)whentheF motorissubjectedtoacounter-torqueM(clockwise,
1
arisingfromtheF motor). Bothplotsarecomputedforthe‘near-optimal’valueofexcitedstatelife-timeτ =50µs,seeFig.
0 2
6(a)andthefrictionconstantξ =2·10−4pN.nm.s,correspondingtotherotationaldiffusionD=2·104rad2/s(asinthewild
type,orlow-friction40-nmgoldbeadofYasudaetal.24). Theplot9(a)showsa‘bigpicture’forawiderangeoftorques,but
notpermittingtoseewhathappensaroundM=0. Weseethattheexternaltorqueessentiallydrivestheshaft–intheopposite
clockwisedirectiionshowingasthenegative(cid:104)ω(M)(cid:105)intheplot,whenthetorqueisagainstthenaturalanticlockwisedirection
oftheF spin,orinthepositiveanticlockwisedirectionwhentheexternaltorqueworksinthesamedirectionasF . Athigh
1 1
enoughtorquethevelocitysaturatesataconstantplateauvalue,whichisdeterminedbytherateofATPbinding(wecontinue
usingthe‘standard’valueforthespecificbindingratek =4·107M−1s−1),andalsotheexcitedstatelife-timeτ .
on 2
10/15