Proc. Natl. Acad. Sci. USAVol. 96, pp. 6597–6602, June 1999Chemistry, BiophysicsThe force exerted by a molecular motorMichael E. Fisher* and Anatoly B. KolomeiskyInstitute for Physical Science and Technology, University of Maryland, College Park, MD 20742Contributed by Michael E. Fisher, March 18, 1999ABSTRACT The stochastic driving force exerted by a sin-gle molecular motor (e.g., a kinesin, or myosin) moving on aperiodic molecular track (microtubule, actin filament, etc.) isdiscussed from a general viewpoint open to experimental test.An elementary “barometric” relation for the driving force isintroduced that (i) applies to a range of kinetic and stochas-tic models, (ii) is consistent with more elaborate expressionsentailing explicit representations of externally applied loads,and (iii) sufficiently close to thermal equilibrium, satisfies anEinstein-type relation in terms of the velocity and diffusioncoefficient of the (load-free) motor. Even in the simplest two-state models, the velocity-vs.-load plots exhibit a variety ofcontrasting shapes (including nonmonotonic behavior). Pre-viously suggested bounds on the driving force are shown tobe inapplicable in general by analyzing discrete jump modelswith waiting time distributions.Molecular motors are protein molecules such as myosin, ki-nesin, dynein, and RNA polymerase, that move along lineartracks (actin filaments, microtubules, DNA) and perform tasksvital to the life of the organism—muscle contraction, cell divi-sion, intracellular transport, and genomic transcription (1–5).Understanding how they operate represents a significant chal-lenge. The hydrolysis of adenosine triphosphate (ATP), withthe release of adenosine diphosphate (ADP) and inorganicphosphate (Pi), is known to be the power source for manymotor proteins. An activated motor may well be in a dynami-cal or, better, a stochastic steady state but it cannot be in fullthermal equilibrium.Striking in vitro experiments observing individual motor pro-teins moving under controlled external loads (6–11) have stim-ulated enhanced theoretical work aimed at understanding themechanisms by which a biological motor functions. From abroad theoretical perspective, a molecular motor is a micro-scopic object that moves predominantly in one direction alonga “polarized” one-dimensional periodic structure, namely, themolecular track (1–11). In recent years, in addition to tradi-tional chemical kinetic descriptions (see, e.g., ref. 12 and ref-erences therein) and various more detailed schemes (11, 13,14), so-called “thermal ratchet” models have been proposedto account for the mechanics: see the review (15).A common feature of most approaches is that a mo-tor protein molecule is associated with a labeled site l(= 0; 51; 52, :::) on the track and is pictured as being inone of N essentially discrete states j, which may be free of(say, j = 0) or bound to ATP and its various hydrolysis prod-ucts (j = 1; 2;:::;N− 1). Thus, for a kinesin molecule, K, ona microtubule, M, the (N = 4) states identified might be M·K,M·K·ATP, M·K·ADP·Pi, and M·K·ADP (8, 12). Transitionrates between these states can be introduced viau1u2uN−1uN0l*)1l*)···*)N − 1l*)0l+1;w1w2wN−1wN[1]where the subscripts indicate that the states j are associ-ated with successive sites, l and l + 1, on the track spacedPNAS is available online at www.pnas.org.at distances 1x = xl+1− xl= d: this defines the step sized. Of course, states jl;jl+1;:::;jl+ndiffer physically onlyin their spatial displacements d; 2d;:::;nd, along the track.By the same token, the rates ujand wjare independent ofl (or x = ld); however, in the subsequent developments itproves useful to allow for spatially dependent rates ujl andwjl.To properly represent physicochemical reality (that is, mi-croscopic reversibility) none of the forward rates, uj, or back-ward rates, wj, may strictly vanish even though some, suchas the last reverse rate, wN, might be extremely small (11,12). On the other hand, if, as one observes in the presenceof free ATP, the motor moves under no external load to theright (increasing x), the transition rates cannot (all) satisfythe usual conditions of detailed balance that would charac-terize thermal equilibrium if Eq. 1 were regarded as a setof chemical reactions (near equilibrium) between effectivespecies jl(15). [Notice that one may envisage a second-orderrate process, e.g., M·K + ATP*)M·K·ATP, to concludeu1= k1[ATP]; this can then lead to Michaelis–Menten typerate-vs.-concentration relations (6). However, one might alsocontemplate a small “spontaneous” or first-order backgroundrate, u1;0, 0, that exists even in the absence of ATP.]Now, within statistical physics, the kinetic scheme in Eq. 1represents a one-dimensional hopping process of a particle ona periodic but, in general, asymmetric lattice. After initial tran-sients, the particle will move (16) with steady (mean) velocityV and diffuse (with respect to the mean position,x = Vt,at time t) with a diffusion constant D. Complicated, but ex-act, equations for V and D in terms of ujand wjhave beenobtained for all N (16), as exhibited in the Appendix. A di-mensionless, overall rate factor that, rather naturally, appears(see Eq. A1), is given by the product0 =N−1Yj=0ujwjA e·: [2]This will play an important role in our discussion. Note, in-deed, that viewing Eq. 1 as a standard set of chemical reac-tions and requiring detailed balance would impose 0 A 1 (or· = 0), whereas 0 , 1 (or · , 0) is needed for a positivevelocity V . [One might comment, however, (17) that as re-gards the full chemistry, the complex of motor protein plustrack may be regarded simply as catalyzing the hydrolysis ofATP: the reaction rates for this overall process may then beexpected to satisfy detailed balance.]The simplest or “minimal” physical models have N = 2,and one can then calculate analytically not only the steadystate behavior but also the full transient responses, specifi-cally, the probabilities, Pjly t, of being in state jl(“at” site l)at time t having started, say, at site l = 0 in state j = 0 at timet = 0 (see ref. 17). In ref. 17 only the special (limiting) caseswith wNA w2= 0 were treated; but as seen below, this limitcan be misleading, and so for completeness (and for possi-ble comparisons with experiment and simulation), we presentthe general N = 2 results (see Appendix). In particular, the*To whom reprint requests
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