HindawiPublishingCorporation AbstractandAppliedAnalysis Volume2008,ArticleID241736,13pages doi:10.1155/2008/241736 Research Article Stokes Efficiency of Molecular Motor-Cargo Systems HongyunWang1 andHongZhou2 1DepartmentofAppliedMathematicsandStatistics,UniversityofCalifornia,SantaCruz, CA95064,USA 2DepartmentofAppliedMathematics,NavalPostgraduateSchool,Monterey,CA93943,USA CorrespondenceshouldbeaddressedtoHongZhou,[email protected] Received2November2007;Accepted30April2008 RecommendedbyYongZhou Amolecularmotorutilizeschemicalfreeenergytogenerateaunidirectionalmotionthroughthe viscousfluid.Inmanyexperimentalsettingsandbiologicalsettings,amolecularmotoriselastically linkedtoacargo.Thestochasticmotionofamolecularmotor-cargosystemisgovernedbyasetof Langevin equations, each corresponding to an individual chemical occupancy state. The change of chemical occupancy state is modeled by a continuous time discrete space Markov process. Theprobabilitydensityofamotor-cargosystemisgovernedbyatwo-dimensionalFokker-Planck equation. The operation of a molecular motor is dominated by high viscous friction and large thermal fluctuations from surrounding fluid. The instantaneous velocity of a molecular motor is highlystochastic:thepastvelocityisquicklydampedbytheviscousfrictionandthenewvelocityis quicklyexcitedbybombardmentsofsurroundingfluidmolecules.Thus,thetheoryformacroscopic motorsshouldnotbeapplieddirectlytomolecularmotorswithoutcloseexamination.Inparticular, a molecular motor behaves differently working against a viscous drag than working against a conservativeforce.TheStokesefficiencywasintroducedtomeasurehowefficientlyamotoruses chemicalfreeenergytodriveagainstviscousdrag.Foramotorwithoutcargo,itwasprovedthat theStokesefficiencyisboundedby100%(cid:4)H.WangandG.Oster,(cid:5)2002(cid:6)(cid:7).Here,wepresentaproof forthegeneralmotor-cargosystem. Copyrightq2008H.WangandH.Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproductioninanymedium,providedtheoriginalworkisproperlycited. 1.Introductionandmathematicalformulation Molecular motors play a central role in many cellular functions. For example, an F motor o utilizes the transmembrane proton gradient to power the ATP synthesis in the F portion of 1 theATPsynthase;aKinesindimerhydrolyzesATPtodriveintracellularvesicletransportation; andaV-ATPasehydrolyzesATPtopumpprotonsagainstalargeprotongradienttoregulate intracellularacidity.Understandingtheoperatingprinciplesofmolecularmotorsiscrucialto comprehendingintracellularproteintransportandcellmotility. 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THIS PAGE Same as 14 unclassified unclassified unclassified Report (SAR) Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 2 AbstractandAppliedAnalysis Due to the small size of molecular motors, the inertia of the motor is negligible. As a result,themotoroperationisdominatedbyhighviscousfrictionandlargethermalfluctuations from the fluid environment (cid:4)1(cid:7). Because of these properties of molecular motors, the results for macroscopic motors, in general, do not necessarily extend to molecular motors. In both macroscopic motors and molecular motors, a chemical reaction can be used to generate a conformational change and an active force at the reaction site. The conformational change along with the active force is then mechanically delivered to drive the motor motion. This mechanism of generating directed motion is called a power stroke motor (cid:4)2, 3(cid:7). However, in a molecular motor, a unidirectional motion can be generated by a completely different mechanism.Forsimplicity,weconsiderthecasewhereamotorisrestrictedtomovinginone spatial dimension, such as moving along a polymer track. Suppose we use the free energy releasedfromthechemicalreactiontoblockthermalfluctuationstowardonedirection.Then, the motor will move toward the opposite direction. Here the unidirectional motion is not directlydrivenbyanactiveforceproducedatthereactionsite.Ratheritisdirectlydrivenby thebombardmentsofsurroundingfluidmolecules.Ofcourse,drawingthermalenergyfroman isothermalenvironmenttodrivemotormotionisnotsustainablewithoutafreeenergyinput. Thefreeenergyforblockingthebackwardfluctuationsandrectifyingtheforwardfluctuations comes from the chemical reaction. This mechanism of generating a unidirectional motion is calledaBrownianratchet(cid:4)4–7(cid:7). One of the main differences between molecular motors and macroscopic motors is manifested in the issue of efficiency. For a macroscopic motor, the efficiency is well defined no matter it is working against a conservative force or working against a friction force. For a molecular motor, we do need to distinguish these two cases. When a molecular motor is workingagainstaconservativeforce,thethermodynamicefficiencyiswelldefinedandisthe energyoutputtotheexternalagentexertingtheconservativeforcedividedbythechemicalfree energyconsumptioninthemotor.Whenamolecularmotorisworkingagainstaviscousdrag, thesituationiscompletelydifferent.Ithasnoenergyoutputatall.Onewaytodefineefficiency in this case is to proceed with the apparent energy output based on the average velocity. Theefficiencydefinedthisway,calledStokesefficiency,isdifferentfromthethermodynamic efficiency. First of all, these two efficiencies are for two different cases: the thermodynamic efficiency is for a motor working against a conservative force; and the Stokes efficiency is for a motor working against a viscous drag. But this is not the only difference. In single moleculeexperiments,amotorcanbeputtoworkagainstaconservativeforce.Inadifferent experimentalsetup,thesamemotorcanbeputtoworkagainstaviscousdrag.Foramolecular motor, both the thermodynamic efficiency and the Stokes efficiency can be measured but in twodifferentexperimentalsetups.BeforewecantreattheStokesefficiencyasavalidefficiency measurement,weneedtoshowthatitisboundedby100%.Sincetheapparentenergyoutput used in the definition of the Stokes efficiency does not have a thermodynamic meaning, the Stokes efficiency being bounded by 100% cannot be argued simply from thermodynamical pointofview.Inthispaper,weprovemathematicallythattheStokesefficiencyisboundedby 100%.TheproofisbasedontheFokker-Planckformulationofmotor-cargosystems.Belowwe willfirstdescribethismathematicalformulation. Amolecularmotor,ingeneral,hasmanyinternalandexternaldegreesoffreedom.One of these degrees of freedom is associated with the motor’s unidirectional motion, the main biological function of the motor. For example, a Kinesin dimer walks along a microtubule filamenttowardthepositiveend(cid:4)8,9(cid:7).Therearemanylevelsofmodelsformolecularmotors, H.WangandH.Zhou 3 from simple kinetic models with a few states to all atom molecular dynamics. We adopt a modeling approach of intermediate level in which the unidirectional motion is followed explicitlyandtheeffectsofotherdegreesoffreedomaremodeledinthemeanfieldpotential affecting the unidirectional motion (cid:4)2, 5, 7, 10(cid:7). To introduce this modeling approach of intermediate level, we start with the simple case of a small particle in a fluid environment, restrictedtomovinginonespatialdimension,andsubjecttoastaticpotential,φ(cid:5)x(cid:6),wherex isthecoordinatealongthespatialdimension.Thissituationisactuallyveryclosetothatofa molecularmotor.Themaindifferenceisthatamolecularmotorisdrivenbyswitchingamong a set of static potentials, each corresponding to one chemical occupancy state. The particle experiences are (cid:5)1(cid:6) the force derived from the potential, (cid:5)2(cid:6) the viscous drag force from the fluidenvironment,and(cid:5)3(cid:6)theBrownianforcealsofromthefluidenvironment.Boththedrag force and the Brownian force arise from collisions of the particle with the surrounding fluid molecules. The drag force is the mean of this stochastic bombarding force and the Brownian forceisthefluctuatingpartofthisbombardingforce.Thedragforceontheparticleisalways opposing the motion and is proportional to the velocity: drag (cid:8) −ζu, where u is the velocity and ζ is the drag coefficient of the particle. Here, we adopt the convention that stochastic processes are denoted by boldface letters. The Brownian force on the particle has zero mean andismodeledasaGaussianwhitenoise,whichistheformalderivativeoftheWeinerprocess (cid:5)theBrownianmotion(cid:6).ThemagnitudeoftheBrownianforceisrelatedtothedragcoefficient as (cid:2) dW(cid:5)t(cid:6) Brownian force(cid:8) 2k Tζ , (cid:5)1.1(cid:6) B dt wherek istheBoltzmannconstant,T istheabsolutetemperature(cid:4)11(cid:7),andW(cid:5)t(cid:6)istheWeiner B process (cid:4)12(cid:7). This is a result of the fluctuation-dissipation theorem (cid:4)13–15(cid:7). The stochastic motionoftheparticleisgovernedbytheLangevinequation (cid:2) mdu (cid:8)−ζu−φ(cid:2)(cid:5)x(cid:6)(cid:9) 2k TζdW(cid:5)t(cid:6), dx (cid:8)u, (cid:5)1.2(cid:6) B dt dt dt wheremisthemassoftheparticle.Equation(cid:5)1.2(cid:6)hastwoverydifferenttimescales:thetime scaleofthemotorforgettingaboutitscurrentvelocity(cid:5)thetimescaleofinertia(cid:6),andthetime scaleofmovingoveroneperiodofpotentialφ(cid:5)x(cid:6)(cid:5)thetimescaleofreactioncycle(cid:6).Dividing both sides of (cid:5)1.2(cid:6) by m and noticing that quantity t ≡ m/ζ has the dimension of time, we 0 rewrite(cid:5)1.2(cid:6)as du −1(cid:3) (cid:4) (cid:8) u−g(cid:5)t(cid:6) . (cid:5)1.3(cid:6) dt t 0 Hereg(cid:5)t(cid:6)isastochasticprocessgivenby φ(cid:2)(cid:5)x(cid:6) √ dW(cid:5)t(cid:6) g(cid:5)t(cid:6)(cid:8)−D (cid:9) 2D , (cid:5)1.4(cid:6) k T dt B whereD isthediffusioncoefficientoftheparticleandisrelatedtothedragcoefficientbythe Einsteinrelation:D≡k T/ζ(cid:4)1(cid:7).Thesolutionof(cid:5)1.3(cid:6)hastheform B (cid:5) (cid:6) (cid:9) (cid:5) (cid:6) −t (cid:7) (cid:8) 1 t −(cid:5)t−s(cid:6) (cid:7) (cid:8) u(cid:5)t(cid:6)(cid:8)g(cid:5)t(cid:6)(cid:9)exp u(cid:5)0(cid:6)−g(cid:5)0(cid:6) (cid:9) exp g(cid:5)s(cid:6)−g(cid:5)t(cid:6) ds. (cid:5)1.5(cid:6) t t t 0 0 0 0 4 AbstractandAppliedAnalysis Note that (cid:5)1.5(cid:6) does not provide an explicit expression for the solution of (cid:5)1.3(cid:6) since g(cid:5)t(cid:6) is unknown.Nevertheless,itisclearfrom(cid:5)1.5(cid:6)thattheeffectoftheinitialvelocityu(cid:5)0(cid:6)decays exponentiallywithtimescalet ≡ m/ζ.Forthatreason,wecallt thetimescaleofinertia.In 0 0 general,thedragcoefficientisproportionaltothelinearsizeoftheparticle(cid:4)1(cid:7),andthemass isproportionaltothevolumeoftheparticle.Forasphericalparticleofradiusr,themass,the dragcoefficient,andthetimescaleofinertiaare,respectively, 4 m 2ρ m(cid:8) πρr3, ζ(cid:8)6πηt, t (cid:8) (cid:8) r2, (cid:5)1.6(cid:6) 0 3 ζ 9η whereρisthedensityoftheparticleandηistheviscosityofthesurroundingfluid.Ingeneral, the time scale of inertia t is proportional to the square of the linear size of particle whereas 0 the coefficient varies with the shape of particle. Thus, for small particles, the time scale of inertia t is very small. For example, for a latex bead of diameter 1μm in water, we have 0 t (cid:8) 56×10−9s (cid:8) 56ps. A time period of 56ps is much smaller than the typical time scales 0 of chemical reaction cycles in molecular motors, which are of the order of milliseconds. The shorttimedynamicsoftheparticleisfulloffastdecayingoftheinstantaneousvelocityintime scaleoft andfastincreasingoftheinstantaneousvelocityexcitedbythestochasticBrownian 0 force. These short-time dynamical behaviors present significant difficulties for analysis and computations. It is desirable to capture relatively long-time behaviors of the motor in the timescalesofreactioncycles,withoutresolvingtheseshort-timedetails.Forthatpurpose,we consider the mathematical limit of (cid:5)1.3(cid:6) as time scale t goes to zero. As time scale t goes to 0 0 zero,thelimitofthesolutionof(cid:5)1.3(cid:6)satisfies dx φ(cid:2)(cid:5)x(cid:6) √ dW(cid:5)t(cid:6) (cid:8)−D (cid:9) 2D . (cid:5)1.7(cid:6) dt k T dt B The reduction from (cid:5)1.3(cid:6) to (cid:5)1.7(cid:6) in the limit of small t is called the Einstein-Smoluchowski 0 limit(cid:4)16(cid:7). Inamolecularmotor,thepotentialisnotstatic.Instead,thepotentialchangeswiththe current chemical occupancy state of the motor. In this paper, we consider the case where a motorhasonlyonecatalyticsite(cid:5)onereactioncycle(cid:6).Theextensionoftheanalysistothecase of multiple catalytic sites is straightforward but tedious. Let N be the number of chemical occupancystatesofthemotorinconsideration.Let{1,2,...,N}denotethesetofNoccupancy states,andlet{φ (cid:5)x(cid:6),φ (cid:5)x(cid:6),...,φ (cid:5)x(cid:6)}denotethecorrespondingsetofNperiodicpotentials. 1 2 N Here φ (cid:5)x(cid:6) is the periodic motor potential when the motor is in chemical occupancy state j. j Mathematically,φ (cid:5)x(cid:6)isaperiodicfunctionofperiodLwheretheperiodLiseitheronemotor j steporamultipleofmotorsteps.ForaKinesindimerwalkingonamicrotubulefilament,one motorstepisabout8nm.FortherotarymotorofF ATPase,onemotorstepisonethirdofone 1 revolution(cid:5)2π/3(cid:6).ThemechanicalmotionofthemotorisgovernedbyLangevinequation: dx (cid:8)−DφS(cid:2)(cid:5)t(cid:6)(cid:5)x(cid:6) (cid:9)√2DdW(cid:5)t(cid:6), (cid:5)1.8(cid:6) dt k T dt B where S(cid:5)t(cid:6) denotes the current chemical occupancy state, and φS(cid:5)t(cid:6)(cid:5)x(cid:6) is the periodic motor potentialcorrespondingtoS(cid:5)t(cid:6).Equation(cid:5)1.8(cid:6)caneitherbeviewedasoneLangevinequation that varies with the current occupancy state S(cid:5)t(cid:6) or be viewed as a collection of N Langevin H.WangandH.Zhou 5 equations, each corresponding to an individual chemical occupancy state. Equation (cid:5)1.8(cid:6) describes only the spatial motion of the motor. Equation (cid:5)1.8(cid:6) is not closed since it depends on the current occupancy state S(cid:5)t(cid:6). To close (cid:5)1.8(cid:6), we need to follow the time evolution of thechemicaloccupancystate.InatransitionfromoccupancystateAtooccupancystateB,the commitment time is generally much smaller than the residence time. The commitment time is the time of the actual transition process. More precisely, the commitment time is the time periodduringwhichthesystemhasleftstateAbuthasnotyetarrivedatstateB.Ifwedefine a transition state between state A and state B, the commitment time can also be viewed as theresidencetimeofthetransitionstate.Becausethecommitmenttimeoftransitionissmallin comparisonwiththeresidencetime,wecanapproximatethetransitionasinstantaneous.Thus, wemodelthestochasticevolutionofthemotor’schemicalstateasacontinuoustimediscrete spaceMarkovprocess,alsocalledajumpprocess: ⎧ (cid:7) (cid:8) (cid:3) (cid:4) ⎪⎪⎨(cid:5)δt(cid:6)ki→j x(cid:5)t(cid:6) (cid:9)o(cid:5)δt(cid:6), j /(cid:8) i, Pr S(cid:5)t(cid:9)δt(cid:6)(cid:8)j |S(cid:5)t(cid:6)(cid:8)i (cid:8) (cid:14) (cid:7) (cid:8) (cid:5)1.9(cid:6) ⎪⎪⎩1−(cid:5)δt(cid:6) ki→l x(cid:5)t(cid:6)(cid:6)(cid:9)o(cid:5)δt , j (cid:8)i, l/(cid:8)i where ki→j(cid:5)x(cid:5)t(cid:6)(cid:6) is the transition rate from state i to state j. Notice that the transition rate depends on the motor position x(cid:5)t(cid:6). In other words, in Markov process (cid:5)1.9(cid:6), the evolution of chemical state S(cid:5)t(cid:6) is affected by the motor position x(cid:5)t(cid:6). On the other hand, in Langevin equation(cid:5)1.8(cid:6),theevolutionofmotorpositionx(cid:5)t(cid:6)isaffectedbythechemicalstateS(cid:5)t(cid:6).Thus, Langevinequation(cid:5)1.8(cid:6)andMarkovprocess(cid:5)1.9(cid:6)arecoupled,andtogethertheygovernthe stochasticevolutionofboththemechanicalmotionandthechemicalreactionofthemotor.In mostofchemicalreactions,forexample,intheATPhydrolysiscycle,thesystemgoesthrough thesetofNoccupancystatessequentially.Inthiscase,theNoccupancystatesformaloop,and the only allowed transitions are either forward to the next state or backward to the previous statealongtheloop.Here,byconvention,werepresentthetransitionN → 1byN → N(cid:9)1and representthetransition1 → NbyN(cid:9)1 → N.Withthisconvention,wehavethatki→j(cid:5)x(cid:5)t(cid:6)(cid:6) /(cid:8) 0 onlywhenj (cid:8)i(cid:9)1orj (cid:8)i−1. Thederivationin(cid:4)17(cid:7)wasformotorsystemswithoutcargos.Inmanysinglemolecule experiments,themotorisnotobserved/recordeddirectly.Insteadalargelatexbeadisattached to the motor, the position of the bead is observed/controlled with the help of a laser trap or a force clamp (cid:4)9, 18–20(cid:7). Motor-cargo systems also occur in biological settings. For example, aKinesindimerwalksonamicrotubulefilament,towingavesicletowardthemembranefor exporting. To accommodatethese experimental andbiological settings, weneed tostudy the behaviorsofmotor-cargosystems.Figure1showsamotor-cargosystemcorrespondingtothe experimentalsetupsin(cid:4)9,19(cid:7).AKinesindimerwalksonamicrotubuletowardtheplusend andalatexbeadislinkedtotheKinesindimertovisualizethemotionoftheKinesindimerand toexertaforceontheKinesindimer.TheforceexertedontheKinesindimerisvariedeitherby changingthedragcoefficientofthelatexbead(cid:4)19(cid:7)orbyapplyingaconservativeforceonthe beadusingalasertrap(cid:4)9(cid:7).Intherecentlydeveloped2Dlasertrap(cid:4)20(cid:7),aforceperpendicular tothedirectionofmotormotioncanalsobeexertedontheKinesindimer. Let x(cid:5)t(cid:6) be the coordinate of the motor and y(cid:5)t(cid:6) the coordinate of the cargo along the direction of motion as illustrated in Figure1. As we discussed above, the motor is driven by switchingamongasetofN potentials,eachcorrespondingtoanindividualoccupancystate. 6 AbstractandAppliedAnalysis Link Cargo Motor − (cid:9) Polymertrack y x Cargoposition Motorposition Figure1: Amotor-cargosysteminsinglemoleculeexperiments,correspondingtotheexperimentalsetup in(cid:4)9,19(cid:7).AKinesindimerwalksonamicrotubulefilamenttowardtheplusend,andalatexbeadislinked totheKinesindimertovisualizethemotionoftheKinesindimerandtoexertaforceontheKinesindimer. Thelatexbeadiseitherloadedbytheforcefromalasertraporloadedbytheviscousdragfromthefluid media. ThestochasticmotionofthemotorisgovernedbyanoverdampedLangevinequation (cid:3) (cid:4) dx (cid:8)D −E(cid:2)(cid:5)x−y−R(cid:6)−φS(cid:2)(cid:5)t(cid:6)(cid:5)x(cid:6) (cid:9)(cid:15)2D dW(cid:5)t(cid:6), (cid:5)1.10(cid:6) M M dt k T dt B whereD isthediffusioncoefficientofthemotor,and−E(cid:2)(cid:5)x−y−R(cid:6)istheelasticforceonthe M motorexertedbytheelasticlinkbetweenthemotorandthecargo.HereE(cid:5)x−y−R(cid:6)istheelastic potentialandRtherestlengthoftheelasticlink.TherestlengthRcanbeeliminatedusinga change of variable y(cid:16) (cid:8) y(cid:9)R. So without loss of generality, we assume R (cid:8) 0. The stochastic evolutionoftheoccupancystateisgovernedbyjumpprocess(cid:5)1.9(cid:6).Inthemotor-cargosystem, theexternalloadingforcenolongeractsdirectlyonthemotor.Theexternalloadingforceacts directlyonthecargo.Thestochasticmotionofthecargoisgovernedby (cid:3) (cid:4) dy f (cid:9)E(cid:2)(cid:5)x−y−R(cid:6) (cid:15) dW(cid:5)t(cid:6) (cid:8)D (cid:9) 2D , (cid:5)1.11(cid:6) C C dt k T dt B whereD isthediffusioncoefficientofthecargo,andf istheexternalloadingforceactingon C the cargo. Note that in the motor-cargo system, the motor does not directly feel the external loadingforceonthecargo;themotorseestheelasticforcefromthecargo.Ontheotherhand, the cargo does not directly feel the internal motor force; the cargo sees the elastic force from themotor.Allcommunicationsbetweenthemotorandthecargoaredoneviatheelasticlink. Inexperiments,onlyaveragequantitiescanbemeasuredrepeatedlyandreliably.Allaverage quantitiescanbecalculatedbyfollowingtheprobabilitydensityofthemotor-cargosystem.Let ρ (cid:5)x,y,t(cid:6)betheprobabilitydensitythatthemotorhasatpositionx,thecargoisatpositiony, j andthechemicalreactionisinoccupancystatej attimet.Mathematically,ρ (cid:5)x,y,t(cid:6)isdefined j as (cid:17) (cid:18) x≤x(cid:5)t(cid:6)<x(cid:9)δx Pr ,S(cid:5)t(cid:6)(cid:8)j y ≤y(cid:5)t(cid:6)<y(cid:9)δy ρ (cid:5)x,y,t(cid:6)(cid:8) lim . (cid:5)1.12(cid:6) j δx→0 δxδy δy→0 H.WangandH.Zhou 7 Thesetofprobabilitydensities{ρ (cid:5)x,y,t(cid:6)}isgovernedbythetwo-dimensionalFokker-Planck j equationcorrespondingtoLangevinequations(cid:5)1.10(cid:6)and(cid:5)1.11(cid:6)andjumpprocess(cid:5)1.9(cid:6)(cid:4)16(cid:7). TheFokker-Planckequationisbasedontheconservationofprobability ∂∂ρtj (cid:8)−∂∂xJj(cid:5)x(cid:6)− ∂∂yJj(cid:5)y(cid:6)−(cid:7)pj(cid:9)1/2−pj−1/2(cid:8). (cid:5)1.13(cid:6) In the above, J(cid:5)x(cid:6)(cid:5)x,y,t(cid:6) is the probability flux along the x-direction (cid:5)the dimension of the j motorposition(cid:6)inoccupancystatej andithastheexpression (cid:5) (cid:6) J(cid:5)x(cid:6)(cid:5)x,y,t(cid:6)(cid:8)−D 1 ∂Φj(cid:5)x,y(cid:6)ρ (cid:9) ∂ρj . (cid:5)1.14(cid:6) j M k T ∂x j ∂x B Φ (cid:5)x,y(cid:6) is the total potential of the motor-cargo system in occupancy state j, which includes j boththeinternalmotorpotentialcausedbythechemicalreactionandtheelasticenergyinthe linkconnectingthemotorandcargo.Φ (cid:5)x,y(cid:6)doesnotcontaintheeffectoftheexternalloading j force,whichisaccountedforseparately.Φ (cid:5)x,y(cid:6)hastheexpression j Φ (cid:5)x,y(cid:6)(cid:8)φ (cid:5)x(cid:6)(cid:9)E(cid:5)x−y(cid:6). (cid:5)1.15(cid:6) j j J(cid:5)y(cid:6)(cid:5)x,y,t(cid:6) is the probability flux along the y-direction (cid:5)the dimension of the cargo position(cid:6) j inoccupancystatej andithastheexpression (cid:5) (cid:17) (cid:18) (cid:6) J(cid:5)y(cid:6)(cid:5)x,y,t(cid:6)(cid:8)−D 1 −f (cid:9) ∂Φj(cid:5)x,y(cid:6) ρ (cid:9) ∂ρj . (cid:5)1.16(cid:6) j C k T ∂y j ∂y B pj(cid:9)1/2(cid:5)x,y,t(cid:6)isthenetprobabilityfluxdensity(cid:5)netprobabilityfluxperunitareainthex−y plane(cid:6)alongthedirectionofreactionfromoccupancystatejtooccupancystatej(cid:9)1.Ithasthe expression pj(cid:9)1/2(cid:5)x,y,t(cid:6)(cid:8)kj→j(cid:9)1(cid:5)x(cid:6)ρj(cid:5)x,y,t(cid:6)−kj(cid:9)1→j(cid:5)x(cid:6)ρj(cid:9)1(cid:5)x,y,t(cid:6). (cid:5)1.17(cid:6) Note that the transition rate kj→j(cid:9)1(cid:5)x(cid:6) does not depend explicitly on the cargo position (cid:5)y(cid:6). The cargo can affect the chemical reaction indirectly by changing the motor position. In the chemical reaction governed by jump process (cid:5)1.9(cid:6), the transition rates ki→j(cid:5)x(cid:6) cannot be arbitrarily specified. These rates are constrained by detailed balance-like condition, which ensuresthatifthesystemisbroughttoanequilibrium,thentheequilibriumsolutionisgiven bytheBoltzmanndistributionandtheprobabilityfluxvanisheseverywhere(cid:4)11(cid:7).Specifically, supposestateAandstateBhavewell-definedfreeenergyG andG .Thenthetransitionrates A B betweenthesetwostatesmustsatisfy (cid:5) (cid:6) kA→B (cid:8)exp GA−GB . (cid:5)1.18(cid:6) kB→A kBT Itfollowsthatthetransitionratesusedin(cid:5)1.17(cid:6)areconstrainedby (cid:5) (cid:6) (cid:5) (cid:6) kj→j(cid:9)1(cid:5)x(cid:6) (cid:8)exp Φj(cid:5)x,y(cid:6)−Φj(cid:9)1(cid:5)x,y(cid:6) (cid:8)exp φj(cid:5)x(cid:6)−φj(cid:9)1(cid:5)x(cid:6) . (cid:5)1.19(cid:6) kj(cid:9)1→j(cid:5)x(cid:6) kBT kBT 8 AbstractandAppliedAnalysis Theboundaryconditionsfor(cid:5)1.13(cid:6)inthex-directionarepseudoperiodic: ρ (cid:5)x(cid:9)L,y(cid:9)L,t(cid:6)(cid:8)ρ (cid:5)x,y,t(cid:6), j j φ (cid:5)x(cid:9)L(cid:6)(cid:8)φ (cid:5)x(cid:6), (cid:5)1.20(cid:6) j j ki→j(cid:5)x(cid:9)L(cid:6)(cid:8)ki→j(cid:5)x(cid:6). In the y-direction, (cid:5)1.13(cid:6) extends from negative infinity to positive infinity. For fixed x and largey,theelasticenergyE(cid:5)x−y(cid:6)islargeandtheprobabilitydensityρ (cid:5)x,y,t(cid:6)isexponentially j small.Thus,atinfinity,wehave ∂ρ (cid:5)x,±∞,t(cid:6) ∂ρ (cid:5)x,±∞,t(cid:6) ρ (cid:5)x,±∞,t(cid:6)(cid:8)0, j (cid:8)0, j (cid:8)0. (cid:5)1.21(cid:6) j ∂x ∂y Theboundaryconditionsalongthereactiondirectionarealsopseudoperiodic: ρN(cid:9)j(cid:5)x,y,t(cid:6)(cid:8)ρj(cid:5)x,y,t(cid:6), φN(cid:9)j(cid:5)x(cid:6)(cid:8)φj(cid:5)x(cid:6)(cid:9)ΔG, (cid:5)1.22(cid:6) kN(cid:9)i→N(cid:9)j(cid:5)x(cid:6)(cid:8)ki→j(cid:5)x(cid:6), whereΔG<0isthefreeenergychangeinonereactioncycle. 2.Stokesefficiency,previousresult,andageneralproof In (cid:4)17(cid:7), the Stokes efficiency was proposed to measure how efficiently the motor utilizes the chemical free energy to drive through the viscous fluid. Let (cid:8)u(cid:9) denote the average velocity andletr denotethechemicalreactionrate(cid:5)averagenumberofreactioncyclesperunittime(cid:6) ofthemotorsystemTheStokesefficiencyforamotorsystemwithoutcargoisdefinedas ζ(cid:8)u(cid:9)2 η ≡ , (cid:5)2.1(cid:6) Stokes r(cid:5)−ΔG(cid:6)(cid:9)f(cid:8)u(cid:9) whereζisthedragcoefficientofthemotor.Inamotor-cargosystem,themotorparthasdrag coefficientζ (cid:8) k T/D andthecargoparthasdragcoefficientζ (cid:8) k T/D .Sinceboththe M B M C B C motorandthecargomovethroughtheviscousfluidwiththesameaveragevelocity,theStokes efficiencyforamotor-cargosystemisdefinedas (cid:7) (cid:8) ζ (cid:9)ζ (cid:8)u(cid:9)2 η ≡ M C . (cid:5)2.2(cid:6) Stokes r(cid:5)−ΔG(cid:6)(cid:9)f(cid:8)u(cid:9) The denominator of the Stokes efficiency is the total free energy consumed per unit time: r(cid:5)−ΔG(cid:6) is the chemical free energy consumed in the chemical reaction per unit time and f(cid:8)u(cid:9) is the free energy input to the motor-cargo system from the external agent exerting the conservativeforce.Iftheexternalconservativeforceisagainstthemotormotion,thenf(cid:8)u(cid:9)is negative,whichmeansthefreeenergyisactuallyoutputtotheexternalagent.Thenumerator oftheStokesefficiency,(cid:5)ζ (cid:9)ζ (cid:6)(cid:8)u(cid:9)2,hasthedimensionofenergyperunittime.Butitdoes M C H.WangandH.Zhou 9 not have a clear thermodynamic meaning. As a result, η ≤ 1 cannot be derived from Stokes simple thermodynamic arguments. In (cid:4)17(cid:7), we proved η ≤ 1 for motor systems without Stokes cargos. Below we present a proof for the general motor-cargo system described by Fokker- Planckequation(cid:5)1.13(cid:6)withdetailedbalanceconstraint(cid:5)1.19(cid:6)andboundaryconditions(cid:5)1.20(cid:6), (cid:5)1.21(cid:6),and(cid:5)1.22(cid:6).Specifically,wewanttoprove (cid:7) (cid:8) ζ (cid:9)ζ (cid:8)u(cid:9)2 ≤r(cid:5)−ΔG(cid:6)(cid:9)f(cid:8)u(cid:9). (cid:5)2.3(cid:6) M C Sinceallaveragequantitiescanbedeterminedfromthesteady-statesolution,weconsiderthe steadystateof(cid:5)1.13(cid:6)asfollows: (cid:7) (cid:8) ∂∂xJj(cid:5)x(cid:6)(cid:9) ∂∂yJj(cid:5)y(cid:6) (cid:8)− pj(cid:9)1/2−pj−1/2 . (cid:5)2.4(cid:6) Tofacilitatetheanalysisbelow,weintroduceanewfunctionH (cid:5)x,y(cid:6): j Φ (cid:5)x,y(cid:6) (cid:7) (cid:8) H (cid:5)x,y(cid:6)≡ j (cid:9)ln ρ (cid:5)x,y(cid:6) . (cid:5)2.5(cid:6) j j k T B Note that in the steady state, everything is independent of t. From the pseudoperiodic boundaryconditioninthereactiondirection,weobtainimmediatelythat ΔG HN(cid:9)j(cid:5)x,y(cid:6)(cid:8)Hj(cid:5)x,y(cid:6)(cid:9) . (cid:5)2.6(cid:6) k T B WiththehelpoffunctionH (cid:5)x,y(cid:6),werewritethefluxesJ(cid:5)x(cid:6)andJ(cid:5)y(cid:6)as j j j (cid:5) (cid:6) J(cid:5)x(cid:6) (cid:8)−D 1 ∂Φjρ (cid:9) ∂ρj (cid:8)−D ρ ∂Hj, (cid:5)2.7(cid:6) j M k T ∂x j ∂x M j ∂x B (cid:5) (cid:17) (cid:18) (cid:6) (cid:5) (cid:6) J(cid:5)y(cid:6) (cid:8)−D 1 −f (cid:9) ∂Φj ρ (cid:9) ∂ρj (cid:8)−D ρ ∂Hj − f . (cid:5)2.8(cid:6) j C k T ∂y j ∂y C j ∂y k T B B Theaveragevelocityandthechemicalreactionratehavetheexpressions: (cid:9) (cid:9) L ∞ (cid:14)N (cid:8)u(cid:9)(cid:8) J(cid:5)x(cid:6)dydx, (cid:5)2.9(cid:6) j 0 −∞j(cid:8)1 (cid:9) (cid:9) L ∞ (cid:14)N (cid:8)u(cid:9)(cid:8) J(cid:5)y(cid:6)dydx, (cid:5)2.10(cid:6) j 0 −∞j(cid:8)1 (cid:9) (cid:9) L ∞ r (cid:8) pj(cid:9)1/2dydx. (cid:5)2.11(cid:6) 0 −∞ Intheabove,wehavetwoexpressionsfortheaveragevelocity(cid:8)u(cid:9).Thefirstexpression(cid:5)2.9(cid:6) is the average velocity of the motor, and the second expression (cid:5)2.10(cid:6) is the average velocity of the cargo. Since the motor and the cargo are elastically linked, they must have the same