Towards synthetic molecular motors

16 Pages·2016·2.46 MB·English

by  Amartya Sarkar

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This content was downloaded from IP address 54.70.40.11 on 19/11/2017 at 12:37
NewJ.Phys.18(2016)043006 doi:10.1088/1367-2630/18/4/043006
PAPER
Towardssyntheticmolecularmotors:amodelelastic-networkstudy
OPENACCESS
AmartyaSarkar1,HolgerFlechsig1,2,3andAlexanderSMikhailov1,2
RECEIVED 1 DepartmentofPhysicalChemistry,FritzHaberInstituteoftheMaxPlanckSociety,Faradayweg4-6,D-14195Berlin,Germany
25November2015
2 ResearchCenterfortheMathematicsonChromatinLiveDynamics(RcMcD)andDepartmentofMathematicalandLifeSciences,
REVISED GraduateSchoolofScience,HiroshimaUniversity,1-3-1Kagamiyama,Higashi-Hiroshima,Hiroshima739-8526,Japan
16February2016
3 Authortowhomanycorrespondenceshouldbeaddressed.
ACCEPTEDFORPUBLICATION
15March2016 E-mail:[email protected],holgerfl[email protected]@fhi-berlin.mpg.de
PUBLISHED Keywords:molecularmotor,elasticnetwork,stochasticmodel,Brownianratchet,powerstroke,unidirectionaltransport,syntheticmotor
7April2016
Supplementarymaterialforthisarticleisavailableonline
Originalcontentfromthis
workmaybeusedunder
thetermsoftheCreative Abstract
CommonsAttribution3.0
licence. Proteinmolecularmotorsplayafundamentalroleinbiologicalcellsanddevelopmentoftheir
Anyfurtherdistributionof syntheticcounterpartsisamajorchallenge.Here,weshowhowamodelmotorsystemwiththe
thisworkmustmaintain
attributiontothe operationmechanismresemblingthatofmusclemyosincanbedesignedattheconceptlevel,without
author(s)andthetitleof
addressingtheimplementationaspects.Themodelisconstructedasanelasticnetwork,similartothe
thework,journalcitation
andDOI. coarse-graineddescriptionsusedforrealproteins.Weshowbynumericalsimulationsthatthe
designedsyntheticmotorcanoperateasadeterministicorBrownianratchetandthatthereisa
continuoustransitionbetweensuchtworegimes.Themotoroperationunderexternalload,
approachingthestallcondition,isalsoanalysed.
Introduction
Single-moleculemotorsplayafundamentalroleinbiologicalcells.Iftheirsyntheticanaloguesaredeveloped
andimplemented,thismayleadtoexcitingtechnologicalapplications.Becauseoftheirimportance,protein
motorsarecurrentlysubjectofintensiveexperimentalandtheoreticalinvestigations.High-precisionsingle-
moleculeexperimentswithmyosin[1,2],kinesin[3,4],F1-ATPase[5,6],hepatitisCvirus(HCV)helicase[7–9]
andothermotorproteinshavebeenperformed;theyhaveyieldedvaluableinformationaboutthedetailsoftheir
operation.Sincechemicalstructuresofthesemoleculesareknown,theirbehaviourcanalsobestudiedbydirect
all-atommoleculardynamics(MD)simulations.ThedifficultyinvolvedinMDsimulationsisthattheyare
extremelytime-consumingand,fortypicalproteinmotors,onlythedynamicsonthescalesofuptoa
microsecondcouldhavebeenresolvedinsuchsimulations.Takingintoaccountthatasingleturnovercycleofa
molecularmotorwouldusuallytakeabout10ms,itisobviousthatfullMDsimulationsarestillfarfrom
reproducingevensinglemotoroperationcycles(thoughtheycanindeedstronglycontributetoclarificationof
thedetailsofsuchcycles).
Therearetwowaystoovercomethisdifficulty.Ontheonehand,specialhardwareforacceleratedMD
simulationsisbeingdeveloped.ByusingthespecializedANTONhardware,ithasalreadybecomepossibleto
followthedynamicsofasmallprotein,thebovinepancreatictrypsininhibitor(BPTI),overthetimespanof1ms
[10].Soon,suchsimulationsmayalreadybecomefeasibleforthelargerproteins.However,runninganall-atom
simulationovermanyturnovercyclesforanactualmolecularmotor,withallreactioneventsincluded,and
accumulatingthedataforstatisticalanalysisovermanysuchcycleswouldremainadistantchallengeforMD
simulationseveninthefuture.
Ontheotherhand,coarse-grainedproteinmodelscanbeemployedtospeedupthesimulations.While
manyvariants,suchas,e.g.,theGo-model[11,12],areavailable,theattentionhasrecentlybecomefocusedon
theclassofelastic-network(EN)models.IntheanisotropicENmodel,introducedbyI.Baharwithcoworkersin
2001[13],aproteinistreatedasanetworkofpointparticles(residues)withelasticinteractionsbetweennetwork
neighbours.Thenetworkisbuiltbyusingtheexperimentalx-raydiffractioninformationabouttheequilibrium
structureofaprotein.
©2016IOPPublishingLtdandDeutschePhysikalischeGesellschaft
NewJ.Phys.18(2016)043006 ASarkaretal
WhileENmodelsareempiricalandcouldnotsofarbederivedfromthefulldynamicalmodels,ithasbeen
checkedthat,formanyproteins,theycorrectlyreproducethermalfluctuationsinthepositionsofresidues(B-
factors)intheframeworkofthenormal-modeanalysis[14–19].Moreover,theycanbeusedtoexplore
sensitivityoftheproteinstolocalperturbations,suchasbindingofaligand[20–22].TheENmodelshavebeen
alreadyappliedtoprincipalmolecularmotors,suchasmyosinVorkinesin[23–27].DetailsofENapplications
canbefoundinthereviewvolume[28]withtheforewordbyKarplus.Inarecentpublication[10],predictionsof
anENmodelwerecomparedwiththedatafroma1msANTONsimulationfortheBPTIproteinandgood
agreementhasbeenobserved.ImprovementsoftheoriginalelasticENmodelhavebeenproposedand
investigated(see,e.g.,[29–32]).
UsingreducedENdescriptions,italreadybecomespossibletotraceentireoperationcyclesofrealprotein
machinesandmolecularmotors.Fortheenzymeadenylatekinase,completeturnovercyclesincludingthe
solventeffectscouldbereproducedandstatisticalanalysisforthesequencesofmanysuchcyclescouldbe
performed[33].ForthemolecularmotorHCVhelicase,singleturnovercyclescouldbetracedandinteractions
ofthismotorwiththedouble-strandedDNAcouldberesolved[34],confirmingtheinchwormratchet
mechanismdeducedfromtheexperiments[35].
Inadditiontostructurallyresolvedtheoreticalstudiesofproteinmotors,basedoneitherfullMDorreduced
ENsimulations,therearealsoinvestigationswheretheinternaldynamicsofamolecularmotorisnotresolved.
Instead,theentiremotorismodelledasaparticlemovinginaperiodicpotential.Thetheoreticalconceptofa
molecularmotorasaBrownianratchethasattractedmuchattention[36–41]
Differentkindsofsyntheticnanomotorshavebeenpreviouslyproposedandinvestigated.Sometimes,
micrometer-sizeparticlesarereferredtoasthe‘motors’iftheycanpropelthemselvesthroughafluidduetothe
imbalanceofsurfacetensionforces(see,e.g.,[42]).Ithasbeenconjecturedthatevensingleenzymemolecules,
whencatalyticallyactive,cangeneratepropulsionforcesandactivelymoveinthesolution[43,44].Adifferent
kindofmotorsareswimmersthatcanactivelymoveinfluids[45]andlipidbilayers[46]byactivelychanging
theirconformationorbasedonothermolecularinteractions[47].Furthermore,relativelysmallmolecules
operatedasswitchesareabletomoveactivelyandthustobehaveasmotorstoo(seereview[48]).Suchkindsof
motors,however,differintheiroperationmechanismsfromproteinmachines.
Theaimofthepresentstudyistwofold.First,wedemonstratehowamodelmolecularmotor,roughly
resemblingthepropertiesofrealproteinmachinesand,inparticular,ofthemusclemyosin,canbeconstructed.
Second,weperformextensivestatisticalanalysisofthismodelENmotor.Themeanpropagationvelocityasa
functionofthetemperatureparameter,controllingtheintensityoffluctuationsisdetermined.Themotor
operationunderloadismoreoverconsidered.Byvaryingtheparameters,weshowthatthesamemodelcanbe
operatedintheweakandstrongcouplingregimeswhichcorrespondtodeterministicandBrownianratchets;the
intermediateregimesarealsopossible.Investigationsofsuchmodelsystemcanhelptobetterunderstandthe
operationofrealproteinmotorsand,furthermore,theycanassistinthefuturephysicalimplementationof
syntheticmolecularmotorssimilarintheirfunctionalprinciplestotheiractualbiologicalcounterparts.
Results
Theelasticmachine
Theproposedmodelmotorconsistsofacyclicallyoperatingmachinethatinteractswiththefilamentby
employingaratchetmechanismwhichtransformscyclicmachinemovementsintothetranslationalmotionof
thefilament.First,weexplainhowthemachineisdesigned.Itsinteractionswiththefilamentaredescribedinthe
nextsection.
TheENmachinewhichweusehasbeenproposedbyTogashiandMikhailov[23]andhaspreviouslybeen
employedinotherapplications[49–51].Nonentheless,onlyabriefdescriptionofthemachinehassofarbeen
provided[23].Ithasbeendesigned[23]byusingevolutionaryoptimizationmethods,insuchawaythatits
conformationaldynamicsandligand-inducedinternalmotionscloselyresemblethoseofactualprotein
machines.Particularly,itsequilibriumstates,withandwithouttheligand,havelargeattractionbasins,sothat
evenafterlargeperturbationsthesystemreturnsbacktothem.Moreover,relaxationproceedsalongawell-
definedpathintheconformationalspace.
Themodelmachineismadeupoftworelativelyrigiddomainsconnectedbyaflexiblehingeregion
(figure1).Asubstrateligandcanbindinabindingpocketsituatedinthehingeregionandinduceaclosing
motionofthedomainsaboutthehinge.Furthermore,thesubstrateligandcangetconvertedtotheproductand
getreleased,thusbringingaboutthereverseopeninghingemotion.Themachinehas64identicalparticles
connectedbyasetofidenticalelasticlinks.
2
NewJ.Phys.18(2016)043006 ASarkaretal
Figure1.Elasticnetworkofthemachine.Thenetworkisformedby64particles(greenandbluebeads)connectedbyelasticlinks.
Additionally,theligand(redbead)anditslinkstothethreebindingnodesi=32,40,41(bluebeads)areshown.
Theelasticenergyofthemachineisgivenby
1 64
(R) = k å A (d - d0)2. (1)
ij ij ij
2
i=1,j>i
HereR = {R}isthesetofallcoordinatesR oftheparticlesi = 1, 2,¼,64.κistheelasticconstantwhichis
i i
thesameforalltheelasticlinksinthenetwork.Thenetworkarchitectureisspecifiedbytheadjacencymatrix
whoseelementsA areeither1or0dependingonwhetherabondexistsbetweenapairofparticlesiandj.
ij
Furthermore,d = ∣R - R∣isthedistancebetweenthepairofparticlesiandj,andd0 = ∣R0 - R0∣isthe
ij i j ij i j
equilibriumdistancebetweenthem;{R0}areequilibriumpositionsofallthenodes.Theequilibriumpositions
i
R0oftheparticlesandthematrixofconnectionsA aregivenin[23](seealsomethods).
i ij
Themachineoperatesinsideaviscousfluid.Itsdynamicsisoverdampedandisdescribedbytheequations
dR ¶
i = -g , (2)
dt ¶R
i
whereγisthemobilitycoefficient,thesameforallbeadsi = 1,¼,64.Later,thermalfluctuationswillbetaken
intoaccountbyintroducingnoisetermsintosuchrelaxationequations.Notethatinertialeffectscanalsobe
neglectedforrealproteinmachines,whenconformationalmotionsonthetimescaleslargerthanapicosecond
areconsidered.Hydrodynamiceffectsarenottreatedinthepresentstudy,buttheycanbeincludedtoo[49].
Whentheligandbinds,itformselasticlinkswiththreebeads(i = 32, 40, 41)locatedinthehingearea.The
liganditselfistreatedinourmodelasanadditionalparticle(i=65).Itisconvenienttointroduceabinary
variableswhichtakesthevalues=1,iftheligandisattached,ands=0otherwise.Theexpressionfortheelastic
energy,includingtheligandatthepositionR ,is
65
1 64 1
(R; s) = k å A (d - d0)2 + sk å (d - d0 )2. (3)
ij ij ij i,65 i,65
2 2
i=1,j>i i=32,40,41
Thelastligandinteractiontermdependsonthedistancesd = ∣R - R ∣betweentheligandandthethree
i,65 i 65
networkbeads.Moreover,d0 arethenaturallengthsofthethreeadditionallinks,
i,65
d0 = d0 = d0 = d0.Forsimplicity,weassumethattheligandlinkshavethesameelasticconstantκas
32,65 40,65 41,65 lig
theotherlinksinthenetwork.Whentheligandispresent,thenetworkdynamicsisgivenbyequations(2)where
theexpression(3)fortheelasticenergyshouldbeused.
Next,theconditionsatwhichbindingorreleaseoftheligandtakeplaceneedtobespecified.Moreover,the
initialpositionoftheligandafterbindingshouldalsobedefined.Inthepresentstudy,weusetheligandbinding
anddetachmentconditionswhichareslightlydifferentfromtheoriginalformulation[23]andaredefinedin
termsofthehingeangle;suchconditionshavealreadybeenemployedinthestudyofmembranemachines[51].
Thehingeanglefisintroducedthroughtheequation
3
NewJ.Phys.18(2016)043006 ASarkaretal
Figure2.Definitionofthehingeanglef.Here,cmdenotesthecentreofmassofthethreeligandbindingparticlesi=32,40,41.
(R - Rcm) · (R - Rcm)
cosf = 7 64 . (4)
∣R - Rcm∣∣R - Rcm∣
7 64
WhereRcmisthecentreofmassofthethreebeadsi = 32, 40and41whichformtheligandbindingpocket
1
Rcm= (R + R + R ). (5)
32 40 41
3
Becausethetwoarmsofthemachinearefairlystiff,theanglefcanbeinterpretedastheanglebetweenthetwo
arms(seefigure2).
Intheequilibriumstateoftheligand-freenetworkthehingeangleisrelativelylarge(f = f )andwereferto
0
thisstateastheopenconformationofthemachine.Weassumethatbindingofthesubstrateligandispossible
withinacertaininterval[f - D , f + D ]ofhingeanglesinthevicinityoftheopenconformation.Within
0 0 0 0
thisinterval,itcantakeplacewithsometransitionprobabilityw perunittime.Weassumethat,whentheligand
0
arrives,itbecomeslocatedinthecentreofmassofthebindingpocketformedbytheparticlesi = 32, 40, 41;
hencewehaveR0 = Rcmatthemomentofbinding.
65
Theligand-networkcomplexwillundergorelaxationtoitsownequilibriumstate.Thisrelaxationprocess
willbedescribedbyequations(2)wheretheexpression(3)fortheelasticenergywiths=1shouldbeused.Note
that,whentheligandisattachedtothenetwork,itmoveslikeanyotherparticle;weassumeforsimplicitythatits
mobilityγisthesame.
Whendesigningthemachine,thenaturallengthsoftheligandlinksarechosentobeshorterthantheir
typicalinitiallengths.Therefore,theselinkstendtoshrink.Notethat,throughtheligandlocatedinthehinge
region,thetwoarmsbecomeadditionallylinked.Whentheligandlinksgetshorter,thetwoarmsmovecloser
onetoanotherandthehingeangledecreases.Theequilibriumstateoftheligand-networkcomplexcorresponds
thereforetoaclosedconformationofthemachinewhichischaracterizedbyarelativelysmallhinge
anglef = f .
1
Figure3displaystheoperationcycleofthedesignedelasticmachine.Atthebeginningofthecycle
(figure3(a))themachineisinanopenconformation.Thesubstrateligand(redparticle)arrivesintotheligand
bindingpocketinthehingeregionandestablisheselasticlinks(figure3(b))tothreeneighbouringbeads(blue
colour).Thisinducesatransitiontothetheclosedconformation(figure3(c))whichcorrespondstothe
equilibriumstateoftheligand-machinecomplex.Nowtheligandisconvertedtotheproductparticlewhichis
instantaneouslyreleased(figure3(d)).Afterthatthefreemachinereturnstoitsequilibriumopenconformation
andthecyclecanberepeated.Thedetailedoperationofthemachinecanbeobservedinvideo1.
Thereactionisintroducedbyassumingthatinthevicinityoftheclosedconformationthenatureofthe
boundligandcansuddenlychange,sothatitgetstransformedtoaproductparticlewhichisimmediately
released.Thistransitionoccurswiththeprobabilityw perunittimewithintheinterval[f - D, f + D]of
1 1 1 1 1
thehingeangle.Thereleaseoftheproductischaracterizedbysuddendisappearanceofthethreeelasticlinks
betweentheligandandthenetwork.Afterthetransitiontheelasticenergyofthenetworkisdescribedby
equation(3)withs=0.Asweassume,theproductisimmediatelyevacuatedandthereforethereversebinding
oftheproducttothemachinecannotoccur.Wealsoneglectthepossibilitythatthesubstratedissociatesbeforeit
getsconvertedtotheproduct.
Interactionswiththefilament
Ouraimistoimitatesomeaspectsofthefunctioningofarealmolecularmotor,themusclemyosin.Ineachofits
cycles,themyosinperformsapowerstroke,inducedbybindingofATPandthehydrolysisreaction.Underthe
powerstroke,themyosinheadisattachedtotheactinfilamentandgeneratesaforcetodragit.Afterthepower
4
NewJ.Phys.18(2016)043006 ASarkaretal
Figure3.Operationcycleofthemachine.
stroketheheadgetsdetachedandtheproteinreturnstoitsinitialconformation.Becausethemyosinholdsthe
filamentduringtheforwardpartofthecycleandisdetachedfromitduringthebackwardpart,aratcheteffectis
naturallyimplemented.Aswewillshow,similaroperationcanbeachievedbyusingtheelasticmachine
describedintheprevioussection.Note,however,thatanimportantaspectofmyosinmotoroperationwould
stillbeabsentinourmodel.Themyosinactivelygraspstheactinfilament,becauseoftheconformational
changesintheactinbindingcleftthatareligand-induced.Ourmodelistooprimitivetoincorporatesucha
mechanism.Instead,adifferentinteractionbetweenthemotorandthefilamentwillbeemployed.
Beforeweproceedtothedetailedformulation,wewanttoshowhowthemotorwouldwork.Video2gives
anexampleofthemotoroperationinabsenceofthermalfluctuations.Whentheligandbinds,apowerstrokeis
executed.Asthemachinemovesforwardapproachingitsclosedconformation,theswingingarmcomesvery
nearthefilamentandestablishesanattractivebondwithit,therebydraggingthefilamentforward.While
holdingthefilament,itmovesforward,untiltheligandconversionintotheproductanditsreleaseoccur.Inthe
secondpartofthecycle,whenthemachinereturnstoitsequilibriumopenconformation,theswingingarm
movesalongapathfartherawayfromthefilamentandnoattractivebondisformed,sothemachineandthe
filamentstaydetachedfromeachother.
Figure4(a)illustratestheset-upofourmodel.Asshowninfigure4(a),thefilamentispositionedalongthex-
axisofourchosencoordinatesystem.Itcanonlymovebackorforthalongthisaxis.Theswingingfreearmofthe
elasticmachineisassumedtohaveaspecialnode(i=64)thatcaninteractwiththefilament.Theotherarmof
themachineisimmobilizedbyfixingthenodes,i = 1, 2, and10atR = {x, y, z},R = {x, y, z }
1 1 1 1 2 2 2 2
andR = {x , y , z }.
10 10 10 10
Thetrajectoryoftheendparticle(i=64)ofthemachineasitundergoescyclicconformationalchangesis
displayedinfigure4(b).Thetrajectoryoftheendparticleisanalmostplanarloop,lyingapproximatelyonthex-
zplane.Figure5(a)showsthetimedependenceofthedistanceh oftheinteractingnodefromthefilament.For
64
thechosenpositionofthemachine,theendparticleremainsclosetothefilamentduringtheforwardpartofthe
cycle(s=1)whileitisrelativelyfarawayduringthereversepart(s=0).Additionally,figure5(b)showsthe
5
NewJ.Phys.18(2016)043006 ASarkaretal
a z b z s=1
4.0
s=0
3.0
2.0
1.0
x
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
x 0.2
0.3
0.4
y y
Figure4.(a)Constructionofthemotorsystem.Threebeadsi=1,2,10areimmobilized.Thebeadi=64interactswiththe
regularlyspacedforcecentresofthefilamentwhichisfreetomovealongthex-axis.(b)Thetrajectoryofthenodei=64forone
operationcycle.
Figure5.(a)Timedependenceofthedistanceh betweentheinteractingnodeandthefilament.(b)Timedependenceoftheposition
64
x oftheinteractingnodeonthex-axisalongthefilament.Distancesaremeasuredinunitsofa 4.
64
timedependenceofthepositionx oftheendparticlealongthefilament.Duringthepowerstroke(s=1),the
64
armmovesintheforwarddirection,whereasthereversemotionoccursduringtherestoftheoperationcyclefor
themotor.
Notethatthetimeofthepowerstrokeismuchshorterthanthetimeneededforthearmtoreturntoitsinitial
position.Thisisbecause,duringthepowerstroke,additionalattractiveinteractionsbetweenthearmandthe
headofthemachinearepresentthroughtheligandlocatedinthehinge.
Realactinfilamentsareformedbyaperiodiclineararrangementofmonomersrepresentingsingleactin
proteins.Inourmodelsystem,wedonotwanttoreproducesuchcomplexstructure.Instead,itisassumedthat
thefilamentrepresentsarigidrodwhosemotionsareconstrainedtoalinechosenasthex-axisinourcoordinate
frame.Alongthefilament,periodicallyspacedforcecentresarelocated.Theirpositionsare
x (t) = X(t) + na, n = 0, 1, 2,,m, (6)
n
whereaisthespatialperiodandX(t)isthecoordinateoftheentirerigidfilamentattimet.Eachoftheforce
centresiscapableofinteractingwiththemachinetip.
Tomodeltheinteractionbetweenthefilamentandthemachinetip,Lennard–Jonespotentialsareused.The
totalinteractionpotentialintbetweentheendparticle(i = 64)ofthemachineandthefilamentisgivenbya
sumofpairinterationpotentials
m
int(R , X) = å int(r ), (7)
64 n
n=-m
where
⎧
V (r ) - V (l ) ifr < l ;
int(r ) = ⎨ LJ n LJ c n c (8)
n ⎩0 otherwise,
6
NewJ.Phys.18(2016)043006 ASarkaretal
Figure6.ProfilesoftheinteractionpotentialV inunitsofs kfordifferentseparationsh oftheinteractingnode(i=64)fromthe
int 64
filament.Separationsaremeasuredinunitsofa 4.
and
⎡⎛ ⎞12 ⎛ ⎞6⎤
V (r ) = s⎢⎜C⎟ - 2⎜C⎟ ⎥, (9)
LJ n ⎣⎢⎝r ⎠ ⎝r ⎠ ⎦⎥
n n
Herer isthedistancebetweentheendparticle(i=64)andthenthforcecentreofthefilament,
n
r2 = h2 + (X + na - x )2.Thecoefficientσspecifiestheinteractionstrength,whiletheparametersCandl
n 64 64 c
determinetherangeoftheinteraction.
NotethattheinteractionpotentialisaperiodicfunctionofthefilamentpositionXandithastheperioda.
Indeed,thetotalinteractionwiththe(infinite)filamentisnotchangedifthefilamentisshiftedbydistancea
correspondingtotheseparationbetweenindividualforcecentres.Figure6showstheinteractionpotential
(equation(7))asafunctionofthefilamentpositionXforthreedifferentdistancesh .Theperiodicityofthe
64
effectiveinteractionpotentialisdeterminedbythelatticedistancea,oftheforcecentresonthefilament.
Thecompletemodel
Thus,theequationsofmotionfortheparticles(i = 1,,64)ofthemotorandthefilamenthavetheform
dR ¶(R; s) 
i = -g + z (t); (i ¹ 1, 2, 10, 64), (10)
dt ¶R i
i
dR ¶(R; s) ¶ (R , X) 
64 = -g - g int 64 + z (t), (11)
dt ¶R ¶R 64
64 64
dR dR dR
1 = 2 = 10 = 0, (12)
dt dt dt
dX ¶ (R , X)
= -G int 64 - GF + c(t). (13)
ext
dt ¶X
Hereistheelasticenergy(3)ofthemachineand intheinteractiongivenbyequation(7).Themobilitiesof
int
themachineparticlesandofthefilamentareγandΓ,respectively.Wehaveaddedtothelastequationthe
externalforceF whichmaybeappliedtothefilament.
ext
Thelasttermsintheequations(10),(11)and(13)takeintoaccountthermalfluctuations.Inequations(10)

and(11),z (t) = {zx(t), zy(t), zz(t)}areindependentwhitevectornoiseshavingthecorrelations
i i i i
áza(t)zb(t¢)ñ = 2gk Td d d(t - t¢)wherea, b = x, y, z, (14)
i j B ij ab
k istheBoltzmannconstantandTisthetemperature.Inequation(13)c(t)isawhitenoisewiththecorrelation
B
function
ác(t)c(t¢)ñ = 2Gk Td(t - t¢). (15)
B
Thediscretevariablestakestwovalues:s=0correspondstothefreemachine,whereass=1corresponds
totheligand-boundmachine.Bindingoftheligand,i.e.,transitionfroms=0tos=1,isastochasticevent
whichtakesplacewiththeprobabilityratew withintheinterval[f - D , f + D ]ofthehingeanglegiven
0 0 0 0 0
byequation(4).Thereleaseoftheligand,i.e.,transitionfroms=1tos=0,cantakeplacewiththeprobability
ratew insidetheinterval[f - D, f + D]ofthehingeangle.Notethatgenerallytheratesw andw depend
1 1 1 1 1 0 1
ontemperatureT.Inourmodelsystem,suchpossibledependenceisnothowevertakenintoaccount.Our
investigationswillbealwaysperformedneartothesaturationregime,withbindingorreleaseofaligandalways
7
NewJ.Phys.18(2016)043006 ASarkaretal
takingplaceoncetherespectivewindowhasbeenentered.Thefocuswillbeonthefluctuationeffectscontrolled
bythetemperatureparameterinthemodel.
Whens=1,themachineincludesaligandparticle(i=65)andtheequationofmotionforthisparticlehas
theform
dR ¶(R; s) 
65 = -g + z (t), (16)
dt ¶R 65
65
where(R; s)isgivenbyequation(3).Theligandparticlehasthesamemobilityγandissubjecttothesamekind
ofnoise(equation(14))asthemachineparticles.Whentheligandarrives,i.e.,atthemomentofthetransition
froms=0tos=1,itsinitialpositionisR = RcmwhereRcmisgivenbyequation(5).Whenthereverse
65
transitionfroms=1tos=0takesplace,theligandisimmediatelyremovedandwedonottrackitsposition
anyfurther.Anewligandparticleisboundtothemachineasthesubstrateandremovedfromitastheproductin
eachnextoperationcycle.
Inournumericalsimulations,thedimensionlessformofthemodelisemployed.Toobtainit,timeis
measuredinunitsoft = 1 kgwhereasthecoordinates{R}andXaswellasthedistancesh andx are
i 64 64
measuredintheunitsofL = a 4whereaisthespacingbetweenconsecutiveforcecentresonthefilament.
Belowthesamenotationsareusedfortherescaledvariables.Inthedimensionlessform,theexplicitevolution
equationsofthemodelare
⎛ ⎞
dRi = - å64A (R - R)⎜⎜dij - di0j⎟⎟ + x (t); for i ¹ 1, 2, 10, 64, (17)
dt ij i j ⎝ d ⎠ i
j=1 ij
⎛ ⎞
dR64 =-å64A (R - R)⎜⎜d64,j - d604,j⎟⎟ + åm f (r) ¶rn + x (t) ; fori = 64, (18)
dt 64,j 64 j ⎝ d ⎠ int n ¶R 64
j=1 64,j n=-m 64
dR
i = 0; for i = 1, 2, 10, (19)
dt
⎛ ⎞
dR65 = - å (R - R)⎜⎜d65,j - d605,j⎟⎟ + x (t); fori = 65(ligand), (20)
dt 65 j ⎝ d ⎠ 65
j=32,40,41 65,j
dX m ¶r
= m å f (r) n - mf + h(t), (21)
dt int n ¶X ext
n=-m
⎡⎛ ⎞13 ⎛ ⎞7⎤
f =⎢⎜ c ⎟ - ⎜ c ⎟ ⎥H(r - l) (22)
int ⎣⎢⎝r ⎠ ⎝r ⎠ ⎦⎥ n c
n n
whered = ∣R - R∣andr = h2 + (X + 4n - x )2;weusethestepfunctionH(z)=1forz>0and
ij i j n 64 64
H(z)=0forz⩽0.Thedimensionlessinteractionstrengthcoefficientis= 12s kaCandthedimensionless
characteristicinteractiondistanceisc = 4C a.Thecoefficientm = G gistheratioofthemobilitiesofthe
machineparticlesandthefilament.Therescaledexternalforceis f = (4 ka)F
  ext ext
Therescaledequationsincludenewnoisesx (t) = (t L)z (t)andh(t) = (t L)c(t)withthecorrelation
i i
functions
áxa(t)xb(t )ñ = 2Qd d d(t - t )where a, b = x, y, z, (23)
i 1 j 2 ij ab 1 2
áh(t)h(t )ñ = 2mQd(t - t ). (24)
1 2 1 2
whereΘistherescaledtemperaturegivenbyQ = 16k T ka2.Notethatthetransitionrateconstantsforthe
B
bindingandreleaseoftheligandalsobecomerescaledandthedimensionlesstransitionrateconstantsare
n = w kgandn = w kg.
0 0 1 1
Examplesofmotoroperation
Whenconversionofthesubstrateintotheproductisexcluded,theligandbindstothemachineandstays
indefinitelylongwithinit.Therefore,themotorcanonlyexhibitthermalfluctuationscharacteristicforits
ligand-boundequilibriumstate(s=1).Invideo3,weshowasimulationofthemotorinthiscaseunder
relativelyweakthermalnoise(Q = 0.005).Theswingingarmofthemotorgetsattachedtothefilamentand
performsequilibriumthermalfluctuationstogetherwithit.Ascouldbeindeedexpectedunderequilibrium
conditions,nouni-directionalmovementofthefilamentisobserved.
Ifthemotorisstillatequilibrium(nosubstrateconversion),butthetemperatureisincreased(Q = 0.030),
thebehaviourbecomesdifferent(seevideo4).Now,thearmofthemotorcannotfirmlyholdontothefilament
and,asaresult,thefilamenteasilyslidesagainstit.Incontrasttovideo3,motionsofthearmandthefilament
seemtobeindependentinthiscase.
8
NewJ.Phys.18(2016)043006 ASarkaretal
a b c
2.5 2.5 2.5
2 s=0 2 s=0 2 s=0
1.5 1.5 1.5
h46 h46 s=1 h46
1 1 1
s=1
0.5 0.5 0.5 s=1
0 0 0
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
φ φ φ
Figure7.Trajectoriesofthebeadi=64intheplaneofvariablesh andfattemperatures(a)Q=0.005(b)Q=0.03and(c)
64
Q=0.Thedatafor10consecutivecyclesistaken.
Thebehaviourofthemotorgetschangeddramaticallywhenthereaction,i.e.,theconversionofthesubstrate
intoproduct,isincluded.Video5showsthemotoroperatingundertheseconditionsatalowlevelofthermal
fluctuations(Q = 0.005).
IfthetemperatureisincreasedtoQ = 0.030,itcanbeobserved(Video6)thatthearmofthemotorbegins
toslideagainstthefilament,similartowhatisseeninvideo4.Thismeansthatinthiscasethemotorcanonly
weaklyaffecttheintrinsicindependentBrownianmotionofthefilament.
Figures7(a)and(b)displaytrajectoriesofthebeadi=64,locatedattheendofthearm,intheplaneofthe
variablesh (distancetothefilament)andf(hingeangle)attwotemperaturesQ = 0.005andQ = 0.03,
64
respectively.Here,theredcolourindicatesthattheligandispresentinsidethemachine(s=1)andthegreen
colourcorrespondstothefreemachine(s=0).Forcomparison,wealsoshowinfigure7(c)thetrajectoriesin
absenceofthermalnoise,i.e.,Q = 0.Foreachtemperature,thedatafor10consecutivecyclesistaken.
Examiningfigures7(a)and(b),onecannoticethatthemotorarmispersistentlycyclinginacounter-clockwise
direction.Wecanalsonoticethatevenattherelativelyhightemperature(Q = 0.030)infigure7(b)themotions
ofthearmremainwell-defined,sothatthetwobrancheswiths=0ands=1donotoverlap.
Strongandweakcouplingregimes
Supposefirstthatthemotorarmisfixed,sothatthedistanceofthebeadi=64fromthefilament,h ,andthe
64
projectionofitspositiononthefilament,x ,arebothconstant.Thenthefilamentiseffectivelyperforming
64
thermalBrownianmotioninaperiodicpotentialV (X) = V (X - x , h ).Thispotentialisgivenby
int int 64 64
equation(22)anddisplayedforthreedifferentvaluesofh infigure6.Asseeninfigure6,theheightofthe
64
barrierstronglydependsonthedistance(h )betweenthearmandthefilament.Atthetypicaldistance,
64
h » 0.4,whichischaracteristicfortheligand-boundstate(s=1),thebarrierheightisaboutV » 0.02.
64 max
Whentheligandisabsent(s=0),theseparationgetsincreasedaboveh  2.0and,atsuchdistancesthe
64
interactionpotentialV vanishes.Thismeansthatintheligand-freestatethemotorcannotexhibitanyaction
int
onthefilament.
Supposenowthemotorarmmovesinsuchawaythatitsseparationfromthefilamenth remainsconstant,
64
butitspositionx withrespecttothefilamentchanges,x = x (t).Thenthefilamentwillexperiencea
64 64 64
travellingperiodicpotentialV (X, t) = V (X - x (t), h ).
int int 64 64
AtlowtemperaturesQ  V thefilamentgetslockedintoatravellingtroughandmovestogetherwithit.
max
Thismeansthatthemovingmotorarmisabletoholdthefilamentandtransportitbyapowerstrokewhenthe
ligandisbound(s=1).Intheotherpartofthecycles=0,theinteractionisabsentandthereforethearmcan
movebackwithoutthefilament.Thisoperationmodeofthemotorcanbedescribedasthestrongcouplingregime
andcanbeobservedinvideo5.Insucharegime,themotorworksalmostlikeadeterministicratchet.The
cyclingmachinegeneratesaflashingtravellingpotentialwhichispresentinonepartofthecycle,draggingthe
filament,andabsentintheotherpart.
Theaboveexplanationofthemotoroperationinthestrongcouplingregimeisonlyapproximateand
qualitative.Forinstance,onehastofurthertakeintoaccountthattheinteractionpotentialchangesgradually
withthedistanceh tothefilamentwhichvarieswithintheoperationcycle(seefigure6)andisalsoaffectedby
64
thermalfluctuations.Moreoverthemotionofthearmandthereforetheoperationofthemachinearealso
affectedbytheinteractionswiththefilament.Whenthemotorarmdragsthefilamentinthepowerstroke,the
reactionforceactsonthearmandmodifiesitsmotion.Hence,theflashingpotentialisnotexternaland
independentofthefilamentdynamics.
Figure8(a)displaysthedependenceofthefilamentcoordinateXontimeinthestrongcouplingregime.
Additionally,highlightednarrowstripesinthisfigureindicatetheintervalsoftimeduringwhichthemotorisin
9