DTIC ADA488475: Stokes Efficiency of Molecular Motor-Cargo Systems

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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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Stokes Efficiency of Molecular Motor-Cargo Systems
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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