Simple mechanochemistry describes the dynamics ofkinesin moleculesMichael E. Fisher*†and Anatoly B. Kolomeisky‡*Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742; and‡Department of Chemistry, MS60, Rice University,6100 Main Street, Houston, TX 77005-1892Edited by Thomas D. Pollard, The Salk Institute for Biological Studies, La Jolla, CA, and approved May 9, 2001 (received for review February 16, 2001)Recently, Block and coworkers [Visscher, K., Schnitzer, M. J., &Block, S. M. (1999) Nature (London) 400, 184–189 and Schnitzer, M. J.,Visscher, K. & Block, S. M. (2000) Nat. Cell Biol. 2, 718–723] havereported extensive observations of individual kinesin moleculesmoving along microtubules in vitro under controlled loads, F ⴝ 1to8 pN, with [ATP] ⴝ 1M to 2 mM. Their measurements of velocity,V, randomness, r, stalling force, and mean run length, L, reveal aneed for improved theoretical understanding. We show, present-ing explicit formulae that provide a quantitative basis for compar-ing distinct molecular motors, that their data are satisfactorilydescribed by simple, discrete-state, sequential stochastic models.The simplest (N ⴝ 2)-state model with fixed load-distributionfactors and kinetic rate constants concordant with stopped-flowexperiments, accounts for the global (V, F, L, [ATP]) interdepen-dence and, further, matches relative acceleration observed underassisting loads. The randomness, r(F,[ATP]), is accounted for by awaiting-time distribution,1ⴙ(t), [for the transition(s) followingATP binding] with a width parameter⬅具具t典典2兾具具(⌬t)2典典⯝2.5, indica-tive of a dispersive stroke of mechanicity ⯝0.6 or of a few (ⲏⴚ1) further, kinetically coupled states: indeed, N ⴝ 4 (but not N ⴝ 3)models do well. The analysis reveals: (i) a substep of d0ⴝ 1.8–2.1nm on ATP binding (consistent with structurally based sugges-tions); (ii) comparable load dependence for ATP binding andunbinding; (iii) a strong load dependence for reverse hydrolysisand subsequent reverse rates; and (iv) a large (ⲏ50-fold) increasein detachment rate, with a marked load dependence, followingATP binding.Kinesins are motor proteins that play an important role incellular transport (1). They use the energy of hydrolysis ofATP molecules for moving vesicles and organelles along micro-tubules (MTs). Understanding the mechanism of motor proteinmotion is a serious challenge of modern biology.Experimental investigation of motor proteins includes thedetermination of biochemical cycles (1), the measurement ofrate constants by standard chemical kinetic methods (2), and theelucidation of molecular structure by x-ray crystallography, etc.(1, 3, 4). Also important are measurements of mechanicalproperties by laser-based optical trap spectrometry or by the useof microneedles (5–10).Theoretical modeling of the motion of motor proteins hasinvolved mainly two approaches. The first is based on thermalratchet models in which a motor is viewed as a Brownian particlemoving in two (or more) periodic but spatially asymmetricstochastically switched potentials (11). A different approach usesa multistate chemical kinetic description and postulates that themotor protein molecule steps through a sequence of discretechemical states, possibly with branches, etc., linked by rateconstants (12–16).Recently, precise and extensive observations of the mechan-ical behavior of individual kinesin molecules moving in vitrounder controlled external loads have been reported by Visscher,Schnitzer, and Block (9). In their unique experiments the tail ortether of a (squid axon) kinesin molecule was bound chemicallyto a silica bead while the head moved along an immobilized MT.An optical force clamp, using a feedback-driven optical trap,monitored the displacement, x(t), of a single kinesin moleculewhile keeping the load on the motor close to a fixed value, F. Theprincipal findings of Block and colleagues were: (i) the stallingforce, FS, which brings the mean velocity V to zero, depends onthe concentration of ATP; (ii) under increasing external loadsthe maximum velocity of the motor protein decreases while theeffective Michaelis–Menten constant increases; (iii) the force-velocity plots exhibit different shapes depending on [ATP]; and(iv) the randomness parameter, r, which is a dimensionlessmeasure of the dispersion of the motion along the track (5, 13,14), as a function of external load at saturating [ATP] is almostconstant at low and intermediate loads but increases rapidly nearthe stalling force. Block and coworkers concluded that theirexperimental data necessitated revisions to the theoretical un-derstanding of kinesin motor function. Subsequently, they pub-lished (10) processivity data over similar force and [ATP] ranges,specifically, mean run-lengths, L (along the MT, before individ-ual kinesin motors irreversibly detach). They also proposedvarious theoretical兾mathematical descriptions of varying de-grees of elaboration. However, their analysis did not address theprevious observations of randomness or describe stall forces.Our aim here is to show that these striking observations (9, 10)can be described well qualitatively and with reasonable quanti-tative precision by using simple sequential stochastic models,which have been extended recently and analyzed critically (12–16).We fit all of the experimental data of Block and colleagues andshow that our analysis is consistent with other experiments (7) inwhich kinesin molecules move on MTs under negative (F ⬍ 0) orassisting external loads (for which the analysis of ref. 10 fails).Summary of Theoretical ApproachFollowing refs. 12–16, we suppose that a motor protein moleculesteps a distance d (equal to 8.2 nm for kinesins on MTs) betweenconsecutive binding sites located at positions x ⫽ ld (l ⫽ 0, ⫾1,⫾2,...) on a linear track (the MT) by passing through asequence of N intermediate biochemical states, j ⫽ 0, 1,...,N ⫺ 1. The motor in state jl(at site l) can jump forward to state(j ⫹ 1)lat a rate ujand can move backward to state (j ⫺ 1)lata rate wjas described by the stochastic reaction scheme共0兲lL|;u0w1共1兲lL|;u1关1⫹共t兲兴w2共2兲l...L|;uN⫺2wN⫺1共N ⫺ 1兲lL|;uN⫺1w0共0兲l⫹1,兩 4 d03 兩 4 d13 兩 4 ... 3 兩 4 dN⫺13 兩[1]where the significance of the waiting-time distribution function,1⫹(t), which extends the scheme, is explained below. The
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