Typeset with jpsj2.cls Letter

Efficiency of Energy Transduction in a Molecular Chemical Engine

Kazuo Sasaki ∗,Ryo Kanada 1and Satoshi Amari

Department of Applied Physics,Tohoku University,6-6-05Aoba-yama,Sendai 980-8579

1

Cybermedia Center,Osaka University,Toyonaka 560-0043

(Received November 10,2006)

A simple model of the two-state ratchet type is proposed for molecular chemical engines that convert chemical free energy into mechanical work and vice versa.The engine works by catalyzing a chemical reaction and turning a rotor.Analytical expressions are obtained for the dependences of rotation and reaction rates on the concentrations of reactant and product molecules,from which the performance of the engine is analyzed.In particular,the efficiency of energy transduction is discussed in some detail.

KEYWORDS:molecular chemical engine,molecular motor,ratchet,efficiency,mechanochemical coupling

The F 1motor (F 1-ATPase),which is one of biological molecular motors,1,2is a remarkable molecular machine.It works as a rotary motor when it catalyzes the hy-drolysis of adenosine triphosphate (ATP)into adenosine diphosphate (ADP)and inorganic phosphate (Pi).3,4It can also generate (synthesize)ATP from ADP and Pi when its rotor is forced to rotate in the opposite direc-tion;5,6this is analogous to a heat engine working as a heat pump.A molecular machine,like the F 1motor,that can convert chemical energy into mechanical work and vice versa will be referred to as a molecular chemi-cal engine ;it operates as a motor if it produces motion out of chemical energy,whereas it works as a generator if it generates “fuel”molecules of high chemical poten-tial from “waste”molecules of low chemical potential by consuming mechanical energy.

It has been recognized that certain fundamental fea-tures of biological molecular motors can be captured by “Brownian motor”or ratchet models in which the sys-tem undergoes Brownian motion on a potential surface that changes stochastically between two or more profiles corresponding to different chemical states;see,for ex-ample,refs.7–11for reviews.For example,simple two-state ratchet models have demonstrated how the Brow-nian motion can be rectified to produce directed mo-tion;12,13the dependence of the motor velocity on the concentration of the fuel molecule ATP have been an-alyzed with simple models 14–16and with an elaborate model;17and the efficiency of energy transduction 18and other measures of efficiency 19–21have been discussed.Al-though the effects of the concentration of fuel molecule or the input free energy on the performance of molecu-lar motors have been studied in previous investigations,little attention has been payed to the effects of waste molecules.

In this letter we propose a simple model for molecular chemical engines that explicitly takes into account the effects of both the reactant and product molecules.Our model is a variant of two-state ratchet models 7,8,11–13in general,and is closely related with the one outlined by Astumian and Bier 22in particular.The main difference between our model and other two-state models worked

energy)of the engine in state0.Then the torque exerted on the rotor by the engine in state j(j=0,1)is given by −d V j/dθ.In Fig.1,an example of the pair of potentials V0and V1(which is partly motivated by the analysis27

of experiments on the F1motor)is shown,together with “pathways”(indicated by arrows)the engine can take. For simplicity,we assume that a large barrier of poten-tial V1lies in the intervalθB<θ<θA+2π(mod2π) which cannot be surmounted by the rotor;we also as-sume that potential V0has a barrier,which may be large or small,in the intervalθA<θ<θB.If the latter bar-rier is so large that the dashed passes in Fig.1cannot be taken,a single forward revolution(increase inθby 2π)of the rotor is always accompanied by a single for-ward chemical reaction A→B;if this is the case,it is said that the“mechanochemical coupling”of the engine is tight.Otherwise,it is loose.9,28

A transition between states0and1occurs when the engine binds or releases a ligand molecule(vertical ar-rows in Fig.1).Let w j(θ)be the rate of transition from state j to the other state at angleθof the rotor.The assumption mentioned above that the binding and the dissociation occur at particular values ofθmay be ex-pressed as

w j(θ)=ωA jδ(θ−θA)+ωB jδ(θ−θB)(j=0,1),(1) whereωA j andωB j are positive constants associated with binding(j=0)and dissociation(j=1)of molecules A and B(see Fig.1),andδ(θ)is the delta func-tion.The use of the delta function is a mathematical idealization,which has been introduced by several au-thors8,14,15,19,29–31to carry out various calculations an-alytically.

The condition of detailed balance requires24thefirst equality in

ωα0

k B T =nα∂t

+

∂J j

D0

=

ρA−σρB−a D0

=

b AρA−σb BρB

Fig.2.The dependence of rotation rateν(solid line)and reaction rate r(dashed line)on load torque L for∆µ/k B T=15,20,and 25withρB=1.0.The parameters characterizing the potentials (V0and V1)and the transition rates are chosen as follows:W0= W1=10k B T,Φ0=18k B T,Φ1=40k B T,θA=π/3,θB= 4π/3,θC=π,andκA=κB=D0.The upper inset is the magnification of a region whereν∼0and r∼0for∆µ/k B T= 20.The lower inset shows the efficiencyηfor∆µ/k B T=20.

It is emphasized that the dependences ofνand r onρA andρB given by eqs.(5)and(6)are quite general in that they are independent of potential profiles(V0and V1), which affect only the values of coefficients(σ,a,b’s and c’s).We also note that bothνand r depend onµA andµB separately,while in the two-state models for molecular motors worked out previously14–16,18,22(and a single-state model32proposed recently)the chemical potentials come into play only through the difference∆µ=µA−µB (or∆µ=µATP−µADP−µPi if the reaction ATP⇄ADP+Pi is considered instead of A⇄B).

In what follows we discuss the properties of the engine extracted from eqs.(5)and(6)for a particular set of potentials V0and V1.Here,for simplicity,we consider the piecewise linear functions shown in Fig.1:the vertices of V0are located atθ=θA,θC,andθB withθC satisfying θA<θC<θB,and those of V1atθ=0,θA,andθB; the potential shapes are specified by parametersΦ0= V0(θC)−V0(θA),W0=V0(θB)−V0(θA),Φ1=V1(0)−V1(θB),and W1=V1(θA)−V1(θB).33

It can be shown that eqs.(5)and(6)are approximated by

ν≈r≈D0ρA (c0+c AρA+c ABρAρB)(7) in the absence of the load(L=0)if exp(W j/k B T)≫1 (j=0,1),exp[(W0−Φ0)/k B T]≫1,κA/D0is not too small,andρB is not too large.The dependence on the concentrationρA of fuel molecule for afixedρB in this expression agrees with the one known as the Michaelis-Menten equation,and such a dependence of the rota-tion and reaction rates on the ATP concentration was observed for the F1motor.4,27The dependence onρB predicted in eq.(7)may be observed for the F1motor as the dependence on the ADP concentration.

Examples of the dependences ofνand r on L are shown in Fig.2.Bothνand r decrease

monotonically Fig.3.The dependence of the efficiencies of motor(the upper left portion)and of generator(the lower right portion)on L and∆µis shown as a contour plot forρB=1.The other parameters are the same as the ones in Fig.2.In the hatched region the engine works as neither a motor nor a generator(it is useless).

with increasing L.The rotation rateνbecomes zero at a certain value L0of L,and the reaction rate r becomes zero at a somewhat larger value L1(see the upper inset of Fig.2).Thus,the engine works as a motor for0≤LL1,whereas it wastes both the chemical and mechanical energies for L0In the examples shown in Fig.2,the mechanochemical coupling is almost tight(ν≃r)for small load.However, the difference betweenνand r becomes apparent(the coupling becomes looser)as L is increased.This is be-cause the effective barrier heightΦ0−W0−L(θB−θC) of potential V0seen from the locationθ=θB decreases and hence the leftward dashed passes in Fig.1are taken more frequently for larger L.

The efficiencyηof energy transduction is defined as η=2πνL/r∆µfor motor andη=r∆µ/2πνL for gener-ator.18These expressions may be written as

η=χη0,(8) whereχ=ν/r(r/ν)is the“tightness”of mechanochem-ical coupling,andη0=2πL/∆µ(∆µ/2πL)is the ef-ficiency in the tight coupling limit for motor(genera-tor).Note that largerχindicates tighter mechanochem-ical coupling,and we haveχ=1in the tight coupling limit.

An example ofηas a function of L is depicted in the lower inset of Fig.2,where we observe thatηfor motor (generator)has a maximum near the“stall”load L0(L1). In the tight coupling limit,the maximum efficiency of η0=1is achieved at L=∆µ/2π∓0(the minus sign for motor and the plus sign for generator).In the case of loose coupling,the maximum value ofηtends to be larger for largerχ.Since the tightness decreases with increasing L as explained above,the maximum ofηfor motor is larger than that for generator in this example. The dependence ofηon L and∆µfor a particular choice ofρB is shown as a contour plot in Fig.3.In

Fig.4.The maximum efficienciesηm(ρB)of motor and of gener-ator for givenρB are plotted againstρB for different choices of Φ0.The other parameters are the same as the ones in Fig.2. this example,the largest efficiencies of motor and gener-ator are achieved in the condition∆µ∼2πL∼20k B T far from equilibrium(∆µ=L=0);a similar observa-tion was made for related models by Parmeggiani et al.18 Note that,in the tight coupling limit,we have the max-imum value ofη0=1on the diagonal line∆µ=2πL. We have obtained qualitatively similar patterns of con-tour lines to the one shown in Fig.3for other choices of ρB,although the locations and the hights of the peaks change asρB is varied.Letηm(ρB)be the maximum value ofηobtained by adjusting L and∆µfor a givenρB. Figure4showsηm as a function ofρB for three choices of the barrier heightΦ0of potential V0.It is noted that ηm increases withρB for both motor and generator,and saturates to a certain value—the upper limit of efficiency achieved by the engine(characterize by V0(θ),V1(θ),κA,κB and D0).

The dependence ofηm(ρB)onρB shown in Fig.4 may be understood qualitatively as follows.Remem-ber that largerηm is expected for larger tightnessχof mechanochemical coupling.As the the concentrationρB of molecule B is increased,the binding of molecule B (transition from state0to state1atθ=θB)occurs more frequently,which in effect leads to the decrease in the chance of taking the leftward dashed passes in Fig.1. Therefore,the tightnessχand henceηm will increase withρB,which is consistent with what we see in Fig.4. In the case of motor,the dissociation of molecule B oc-curs more frequently than the binding,and the former is not affected byρB,while the binding of molecule B is essential for generator.This explains the stronger depen-dence ofηm onρB for generator than for motor observed in Fig.4.

It may be worth remarking that the present model may be viewed as a motor driven by ion-flow across a membrane:34,35the transition atθA can be viewed as ion exchange with the outside of the membrane,and the one atθB as ion exchange with the inside;in this case only one chemical species is involved.

In summary,we have proposed a minimal model for molecular chemical engines that properly takes into ac-count the effects of fuel and waste molecules.The model is simple enough to work out various properties of the engine such as the efficiency of energy transduction.The detailed analysis of the model,including the derivations of various results presented here,and its extensions to situations other than the rotary motor will be reported in future publications.

We would like to thank A.Parmeggiani,J.Prost, J.-F.Joanny,K.Sekimoto, E.Muneyuki,H.Noji, H.Higuchi,F.Matsubara,and M.Sasaki for useful dis-cussions and comments.This work was supported in part by the Grants-in-Aid for Scientific Research in Priority Areas from the Japan Ministry of Education,Culture, Sports,Science and Technology.

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