Fluctuation analysis of mechanochemical coupling depending on the type of bio-molecular motor
Masatoshi Nishikawa, Hiroaki Takagi, Atsuko H. Iwane, Toshio Yanagida
aa r X i v : . [ q - b i o . S C ] J a n Fluctuation analysis of mechanochemical coupling depending on the type ofbio-molecular motor
Masatoshi Nishikawa, Hiroaki Takagi, Atsuko H. Iwane, and Toshio Yanagida Graduate School of Frontier Biosciences, Osaka University, 1-3 Yamadaoka, Suita Osaka 565-0871, Japan Department of Physics, Nara Medical University, 840 Shijo-cho, Kashihara Nara 634-8521, Japan (Dated: October 30, 2018)Mechanochemical coupling was studied for two different types of myosin motors in cells: myosinV, which carries cargo over long distances by as a single molecule; and myosin II, which gener-ates a contracting force in cooperation with other myosin II molecules. Both mean and varianceof myosin V velocity at various [ATP] obeyed Michaelis-Menten mechanics, consistent with tightmechanochemical coupling. Myosin II, working in an ensemble, however, was explained by a loosecoupling mechanism, generating variable step sizes depending on the ATP concentration and realiz-ing a much larger step (200 nm) per ATP hydrolysis than myosin V through its cooperative natureat zero load. These different mechanics are ideal for the respective myosin’s physiological functions.
PACS numbers: 87.16.Nn, 05.10.Gg, 82.39.-k, 82.37.-j
Molecular motors play essential roles in various physi-ological functions, such as muscle contraction, moleculartransport, cell motility and cell division through displace-ments powered by the chemical energy of ATP hydrolysis.The primary goal of molecular motor biophysical stud-ies is to understand how chemical energy is convertedinto mechanical movement, i.e. mechanochemical cou-pling. Single molecule techniques are a powerful toolto investigate chemical reactions and mechanical move-ments by molecular motors because they can directly ob-serve the elemental reaction process that is undetectablein ensemble average measurements [1, 2]. This is espe-cially true for motors like myosin V, which function asa single molecule or in cooperation with a small numberof other myosin V molecules [3]. By combining singlemolecule studies with in vitro kinetics studies, myosin Vis thought to produce regular 36 nm steps coupled to oneATP hydrolysis [4]. This means that the mechanochem-ical coupling of myosin V is a one-to-one relationship,i.e. ”tight coupling”, although no direct measurementhas shown this relationship. In contrast, little is knownabout the mechanochemical coupling of ensemble func-tioning motors, such as myosin II in muscle. Becauseof its poor stepping ability, mechanical steps are eas-ily buried in thermal noise. This makes direct detec-tion of individual mechanical events difficult [5], leav-ing most to discuss only the average displacement [6].Furthermore, because in vivo myosin II work in ensem-ble, single molecule characteristics may not accurately re-flect the mechanochemical coupling of myosin II. In fact,multi-molecule systems show various characteristics un-predictable from single molecule studies [7, 8]. Therefore,the mechanochemical coupling of myosin II is still poorlyunderstood.In order to examine mechanochemical coupling of bothkinds of molecular motors, in this report, we investigatedthe characteristics of fluctuation in myosin V and myosinII motility and discuss their mechanochemical coupling D i s p l ace m e n t [ n m ] Time [s] Myosin II Myosin V ca. Single molecule systemMyosin V Actin
Step b. Multi-molecule system
ATP cycleStep
Myosin II
ATP cycle
FIG. 1: Schematic of the experiments and typical traces ofmyosin motility. a , Single molecule motility assay of myosinV. The single molecule movement of myosin V along the actinfilament was directly measured. Myosin molecules were genet-ically fused GFP (Green Fluorescent Protein) for observation(green star). Fluorescence from myosin in solution is not im-aged because of its fast diffusion whereas molecules movingalong actin filaments are imaged as moving bright spots. Theexperiments were performed at 25 ◦ C. b , In vitro actin glidingassay. The movement of an individual actin filament drivenby a myosin II ensemble was measured. Actin filaments wereattached to the fluorescent dye tetramethyl - rhodamine phal-loidin for observation (colored in orange). Myosin moleculeswere not labeled and not observed. The experiments wereperformed at 28 ◦ C. c , Typical displacement time series ob-tained from the motility assays. ATP concentration, [ATP],was 100 µ M in both assays. consistently. Since these fluctuations contain the un-derlying molecular process, we can extract pertinent in-formation of mechanochemical coupling from fluctuationanalysis, as Schnitzer et al. did to show one kinesin stepcouples to a single ATP turnover [9].In myosin V experiments, we directly observed sin-gle myosin molecule movement along actin filaments[10]. Fluorescence imaging was performed by TIRF mi-croscopy [11]. Actin filaments were adsorbed onto acover slip. Freely floating fluorescently dyed myosinsin solution bound to actin filaments commencing move-ment. The movement was tracked by fitting the fluores-cent spots with a two-dimensional Gaussian distributionfunction. On the other hand, in myosin II experiments,we performed the in vitro actin gliding assay [12] (Fig.1) and measured the sliding velocity between the actinfilament and myosin II molecules. Myosin II moleculeswere adsorbed onto the substrate. Introducing actin fila-ments with fluorescent dye and ATP resulted in a slidingmovement between the myosin molecules and the actinfilaments. The trajectories of the actin movement wereobtained by tracking the center of the brightness of theactin fluorescence. For the convenience of the analy-sis, we chose the length of the actin filament to be lessthan 600 nm. In order to obtain velocity statistics, weconverted two dimensional trajectories of both experi-ments into a displacement time series and calculated theMean Square Displacement (MSD). We fitted the MSDto MSD(∆ t ) = µ v ∆ t + σ v ∆ t + ξ , where µ v is the meanvelocity, σ v is the variance of velocity per unit time, ξ isthe measurement error, and ∆ t is the sampling time. Weperformed both experiments at various ATP concentra-tions, calculated the MSDs for each [ATP], and identifiedthe above parameter values. Here, we set ∆ t = 200 msfor myosin V and 32 ms for myosin II. For myosin V,MSDs were calculated from 100 traces, each of whichcontained more than 12 sampling points on average at[ATP] ranging from 2 to 1000 µ M; for myosin II, MSDswere calculated from 30 traces, each of which containedmore than 100 sampling points on average at all [ATP]conditions. In very low [ATP] conditions (200 nM to 1 µ M), we calculated myosin V MSDs from 40 traces, eachof which contained more than 30 sampling points on av-erage at ∆ t = 2 s.The relationship between [ATP] and the mean velocityis shown in Fig. 2 . This relationship is known as theMichaelis - Menten (MM) mechanism, which representsthe catalytic activity of the enzyme. We applied the MMmechanism to the actomyosin sliding movement :(Scheme 1) A · M + AT P ↔ A · M · AT P → A · M + P r , where M, A, and Pr denote myosin, actin, and the ATPhydrolysis products ADP and Pi, respectively. In thisscheme, an ATP binds reversibly to the actomyosin com-plex. The actomyosin catalyzes ATP hydrolysis to pro-duce a regular displacement, i.e. tight coupling. Notethat in the case of myosin II, a single actin filament inter-acting with multiple myosin II molecules follows Scheme1, whereas in the case of myosin V, single myosin Vmolecule obeys MM kinetics. Although in the formercase, the number of interacting myosin II molecules seemsto affect the frequency of the chemical reaction, it wasexperimentally confirmed that the velocity of the singleactin filament saturates and is independent of the num-ber of interacting myosin molecules except for myosindensities much low than those used here [12, 13]. In this -1 V e l o c it y [ n m / s ] ATP [ µ M]Myosin IIMyosin V
FIG. 2: The dependency of the mean velocity on the ATPconcentration in motility assays. Solid lines show the fit ofthe Michaelis-Menten equation to the experimental data. Seetext for details. scheme, the ATP dependence of the sliding velocity is de-scribed by V = V max [ AT P ] / ( K m + [ AT P ]) , where V max is the maximal velocity at saturating [ATP] and K m isthe [ATP] corresponding to the half maximal velocityof the sliding movement, called the Michaelis constant.Solid lines in Fig. 2 show the fitted MM equation with V max = 689 nm/s and K m = 24 . µ M for myosin Vand V max = 5960 nm/s and K m = 93 . µ M for myosinII, respectively. These values are consistent with earlierstudies [13, 14]. This shows that regarding mean velocity,Scheme 1 holds for both kinds of myosins.Then, is this simple MM mechanism also valid for thedescription of the velocity fluctuation? In order to ex-amine this point, we further investigated the relationshipbetween µ v and σ v (Fig. 3). In the case of myosin V,there is a linear relationship between µ v and σ v (Opencircle) except for the velocity region corresponding to the[ATP] around the Michaelis constant. To test the valid-ity of Scheme 1, we performed stochastic simulations andcompared these with the experimental results. Scheme1 requires three rate constants even though experimen-tally only two parameters, V max and K m , were obtained.Therefore we simplified Scheme 1 without losing any rel-evant statistics:(Scheme 2) A · M + AT P ↔ A · M · AT P
Since the left and right states in Scheme 1 are identicalfor the actomyosin complex, we can rewrite Scheme 1 asScheme 2. Here we assumed a regular stepping motionis accompanied by a left directed reaction. Although theleft directed arrow should include both ATP unbindingand ATP hydrolysis reaction, the former only reduces thestepping rate. This causes the reduction of the velocity ateach [ATP], but does not affect the relationship betweenthe mean and the variance of the velocity. Thus we can -6 -5 -4 -3 -2 -1 -4 -3 -2 -1 N o r m a li ze d v a r i a n ce σ v Normalized mean μ v Myosin IIMyosin VMyosin V sim.Myosin II sim. 10 nmMyosin II sim. 200 nm FIG. 3: Relationship between µ v and σ v . To show the depen-dence of the velocity variance per unit time on µ v , we normal-ized the following variables as follows : ¯ µ v = µ v / V max , ¯ σ v = σ v / V max . Lines show the results of stochastic simulationsof the MM mechanism. Parameters used in the simulationsare as follows: k − = 18.5 s − and k + = 0.750 s − µM − for myosin V, k − = 596 s − and k + = 6.40 s − µM − formyosin II with 10 nm step, and k − = 30.0 s − and k + =0.320 s − µM − for myosin II with 200 nm. ignore the ATP unbinding reaction in this analysis. Therate constant of the left directed reaction, k − , was set tobe k − = V max / step-size in order to obtain the exper-imental V max . The rate of the right directed reaction, k + , was calculated by k + = k − / K m . We calculatedthe number of stepping events for a unit time intervalin the stochastic simulation with the Gillespie algorithm[15]. The step-size of myosin V was set at 36 nm, basedon an earlier study [4]. The simulation results (red linein Fig. 3) show good agreement with the experimentaldata. This confirms that myosin V is a tight couplingmotor, consistent with earlier studies.The resulting relationship between µ v and σ v is ex-plained as follows : in very low [ATP] conditions, ATPbinding is the single rate limiting step in the reactionsequence. Thus, the number of occurring chemical re-actions in a given time interval can be described by aPoisson distribution, in which the linear relationship be-tween µ v and σ v holds. On the other hand, when [ATP] isnear the Michaelis constant, the frequency of ATP bind-ing and the hydrolysis reaction are comparable. Here, thefluctuation of each reaction cancels, and the fluctuationdecreases. In higher [ATP] conditions, however, ATP hy-drolysis becomes a single rate limiting step and σ v againbecomes proportional to µ v . (We also confirmed thechange of the number of the rate limiting steps throughthe randomness parameter analysis [9]).In the case of myosin II, the relationship between µ v and σ v is qualitatively different from that of myosin V. σ v is proportional to µ v in the lower velocity range, while σ v is proportional to µ v in the intermediate and highervelocity range. The crossover of the relationship occurs at the velocity corresponding to [ATP] = 2 µ M, whichis similar to the Michaelis constant of ATPase [13]. Wealso performed stochastic simulations for Scheme 2 forthe myosin II data. Here, the movement of a single actinfilament is assumed to be displaced 10 nm by individualmyosin II molecules, as is expected from the tight cou-pling model [16] (Fig. 3 in green). The simulation resultsof scheme 2 showed a linear relationship between σ v and µ v that could not reproduce the crossover or the largerfluctuations appearing in the experimental data. Thechange in the number of interacting myosin II moleculesi.e. the number fluctuation cannot produce the crossoverfor the following reasons: the number fluctuation has atemporal correlation in the time scale of L/V , where L is the length of the actin filament and V is the velocity.This becomes smaller as the [ATP] increases. When con-sidering the velocity fluctuation per unit time, the effectof the number fluctuation also becomes smaller as the[ATP] increases because it is averaged out by the law oflarge numbers. Furthermore, to achieve the V max valuewith a 10 nm step, we have to set the stepping rate to 600 s − , which is large enough to average out the fluctuationof reactions. Thus, it is necessary to have a displacementlarger than 10 nm per one ATP hydrolysis to reduce thenumber of stepping events per unit time and realize thelarger fluctuations (for example, a 200 nm step simulationis shown in Fig. 3 in blue). This cannot be achieved in atight coupling model because it is not physically plausiblefor the myosin molecule to generate a regular displace-ment larger than its own body ( ≈
20 nm). Therefore, toproduce a step-size more than 20 nm, multiple steppingevents are required to occur in a single ATP hydrolysis.Employing a larger step-size can reproduce the largefluctuation as shown in Fig. 3. However, it still can-not reproduce the crossover relationship, because thestochastic property of Scheme 2 only holds the µ v ∝ σ v relationship, as mentioned above. Therefore, we incorpo-rated a variable step-size depending on the [ATP], takinginto account the change in the relationship between µ v and σ v around the Michaelis constant of the ATPase,2 µ M. Here we assume that the ATPase kinetics fol-lows the Michaelis constant of 2 µ M. This assumptionleads to the relationship V mot = V AT P ase × d , where d is the step-size per one ATP hydrolysis. Followingfrom Scheme 2, V mot = V motmax [ AT P ] / ( K motm + [ AT P ])and V AT P = V AT Pmax [ AT P ] / ( K AT Pm + [
AT P ]), then we ob-tain d = d max ( K AT Pm + [
AT P ]) / ( K motm + [ AT P ]), where d max = V motmax /V AT Pmax . In this expression, d is variabledepending on the [ATP]. In fig. 4, we show stochas-tic simulations using scheme 2 with a step-size function.Rate constants were set as in the case of Fig. 3, exceptfor k + = k − / K AT Pm . We employed d max to be 100,200 and 300 nm to compare with the experimental data,making d max five to fifteen times larger than the size of amyosin molecule. The large d max is expected to be due tostochastic multiple steps by myosin molecule(s), so it isactually more plausible that d max distributes with a largemean value (100 ∼
300 nm) rather than being constant.Fig.4 shows a simulation with an exponential distribu-tion and a mean of d max . A large but constant d max (200 nm) can also explain the dependency of σ v on µ v ,but not as well as the exponential distribution assumedhere. Overall, the result shows that the model could suc-cessfully reproduce the experimental data with the mostlikely maximum step-size, d max , being 200 nm. Thus, thecharacteristic the step size dependence on the [ATP] isessential to reproduce the crossover relation, confirmingthe ”loose coupling” nature of myosin II [17]. Note thatit is not essential assumption that the step is exponen-tially distributed, because the simulation result in whichthe step size is exponentially distributed but not depen-dent on the [ATP] condition does not show the switchingbehavior (Fig. 4, in purple). Here, we should again stressthat our analysis does not need to estimate the numberof interacting myosin molecules to obtain the displace-ment of an actin filament per ATP hydrolysis cycle. Al-though earlier studies reported the step-size, they neededto estimate the number of interacting molecules and thisdiscrepancy lead to the one-order difference in calculatedstep-size [12, 13]. Estimating the number of interactingmolecules is extremely difficult and therefore unreliable.Our method can overcome this difficulty, and clarify themanner of the mechanochemical coupling. This confirmsthat our fluctuation analysis is a promising tool to inves-tigate the mechanochemical coupling in molecular mo-tors. Also, we should note that d max = 200 nm is notunrealistic because Yanagida et al. and Higuchi et al.[18, 19, 20] have already suggested the likelihood of avalue much greater than that predicted from the tightcoupling model [16].The resulting characteristics for myosin-V and myosin-II are reasonable considering their physiological func-tions. Myosin-V transports cargo over a long distanceby a single or a small number of motors. The tightcoupling mechanism guarantees a consistent step-by-stepmovement over this long distance. On the other hand,myosin-II works in an ensemble like muscle to produceadaptive motions depending on the condition. The loosecoupling mechanism is advantageous for such adaptation.For example, this mechanism can generate a large forcewith a short step at a high load but also a large step at alow load to conserve energy. Because a single myosin-IImotor cannot generate steps larger than 30 nm [5], stepsas large as 200nm should be produced not by a singlemyosin-II motor but by the cooperation of many [21]. Itis possible that the active steps by some myosin moleculescause passive steps in others through the backbone con-necting motors. To elucidate how the cooperative dy-namics arise from the interaction of multiple motors andhow this adapts to the biological environment is the focusour future work.We thank Takao Kodama, Masahiro Ueda, Tatsuo Shi- V a r i a n ce [( n m / s ) ] Mean [nm/s] Datad max = 100 nmd max = 200 nmd max = 300 nmd = 200 nm exp. -1 V e l o c it y [ n m / s ] ATP [ μ M] FIG. 4: Comparison with our proposed model to the experi-mental data of myosin II. Symbols are the same as the myosinII data in Fig. 3. Solid lines show the simulated relationshipof our model. See text for details.
Inset , The dependenceof the mean velocity on the [ATP]. Symbols are the same asmyosin II data in Fig. 2. Red solid line is the stochastic sim-ulation. The parameters are d max = 200 nm, K ATPm = 2 µ M, K motm = 93 µ M, respectively. bata, and Fumiko Takagi for valuable discussion, and Pe-ter Karagiannis for revising the manuscript. This studywas supported by ”Special Coordination Funds for Pro-moting Science and Technology: Yuragi Project” andLeading Project of the MEXT, Japan. [1] T. Funatsu, Y. Harada, M. Tokunaga, K. Saito, and T.Yanagida, Nature , 555 (1995).[2] K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M.Block, Nature , 721 (1993)[3] A. D. Mehta, R. S. Rock, M. Rief, J. A. Spudich, M. S.Mooseker, and R. E. Cheney, Nature , 590 (1999)[4] M. Rief, R. S. Rock, A. D. Mehta, M. S. Mooseker, R. E.Cheney, and J. A. Spudich, Proc. Natl. Acad. Sci. U. S.A. , 9482 (2000)[5] K. Kitamura, M. Tokunaga, A. H. Iwane, and T.Yanagida, Nature , 129 (1999)[6] J. E. Molloy, J. E. Burns, J. Kendrick-Jones, R. T.Tregear, and D. C. White, Nature , 209 (1995)[7] W. O. Fenn, J. Physiol. , 373 (1924)[8] K. Yasuda, Y. Shindo, and S. Ishiwata, Biophys. J. ,1823 (1996)[9] M. J. Schnitzer and S. M. Block, Nature , 386 (1997)[10] M. Nishikawa, S. Nishikawa, A. Inoue, A. H. Iwane, T.Yanagida, and M. Ikebe, Biochem. Biophys. Res. Com-mun. , 1159 (2006)[11] M. Tokunaga, K. Kitamura, K. Saito, A. H. Iwane, andT. Yanagida, Biochem. Biophys. Res. Commun. , 47(1997)[12] T. Q. Uyeda, S. J. Kron, and J. A. Spudich, J. Mol. Biol. , 699 (1990)[13] Y. Harada, K. Sakurada, T. Aoki, D. D. Thomas, and T.Yanagida, J. Mol. Biol. , 49 (1990)[14] J. E. Baker, E. B. Krementsova, G. G. Kennedy, A. Arm- strong, K. M. Trybus, and D. M. Warshaw, Proc. Natl.Acad. Sci. U. S. A. , 5542 (2004)[15] D. T. Gillespie, J. Phys. Chem., , 2340 (1977)[16] J. A. Spudich, Nature , 515 (1994)[17] F. Oosawa and S. Hayashi, Adv. Biophys. , 151 (1986)[18] T. Yanagida, T. Arata, and F. Oosawa, Nature , 366 (1985)[19] H. Higuchi and Y. E. Goldman, Nature , 352 (1991)[20] Although their value is about three times smaller thanthat of ours, they use the lowest value possible for d.[21] F. J¨ulicher, J. Prost, Phys. Rev. Lett.,75