Bidirectional cooperative motion of myosin-II motors on actin tracks with randomly alternating polarities
Barak Gilboa, David Gillo, Oded Farago, Anne Bernheim-Groswasser
aa r X i v : . [ c ond - m a t . s o f t ] M a r Bidirectional cooperative motion of myosin-II motors on actin tracks with randomlyalternating polarities
Barak Gilboa ∗ ,
1, 2
David Gillo ∗ , Oded Farago, and Anne Bernheim-Groswasser Department of Physics, Ben Gurion University, Be’er Sheva 84105, Israel Department of Chemical Engineering, Ben Gurion University, Be’er Sheva 84105, Israel Department of Biomedical Engineering, Ben Gurion University, Be’er Sheva 84105, Israel
The cooperative action of many molecular motors is essential for dynamic processes such as cellmotility and mitosis. This action can be studied by using motility assays in which the motion ofcytoskeletal filaments over a surface coated with motor proteins is tracked. In previous studies ofactin-myosin II systems, fast directional motion was observed, reflecting the tendency of myosinII motors to propagate unidirectionally along actin filaments. Here, we present a motility assaywith actin bundles consisting of short filamentous segments with randomly alternating polarities.These actin tracks exhibit bidirectional motion with macroscopically large time intervals (of theorder of several seconds) between direction reversals. Analysis of this bidirectional motion revealsthat the characteristic reversal time, τ rev , does not depend on the size of the moving bundle or onthe number of motors, N . This observation contradicts previous theoretical calculations based ona two-state ratchet model (Badoual et al., Proc. Natl. Acad. Sci. USA , 2002, , 6696), predictingan exponential increase of τ rev with N . We present a modified version of this model that takes intoaccount the elastic energy due to the stretching of the actin track by the myosin II motors. Thenew model yields a very good quantitative agreement with the experimental results. PACS numbers:
1. INTRODUCTION
Cells utilize biological motors for active transport ofcargo along their respective filaments to specific desti-nations [1]. Various types of motor proteins have dif-ferent preferred directions of motion. Most kinesins andmyosins, for instance, move towards the plus end of mi-crotubules (MTs) and actin filaments, respectively [2].Others, such as Ncd and myosin VI, move towards theminus end [3, 4]. While some processes, such as the trans-port of cargoes is achieved mainly by the action of indi-vidual motors, other processes, such as cell motility andmitosis, require the cooperative work of many motors.Muscle contraction, for instance, involves the simulta-neous action of hundreds of myosin II motors pullingon attached actin filaments and causing them to slideagainst each other [5]. Similarly, groups of myosin IImotors are responsible for the contraction of the con-tractile ring during cytokinesis [6]. In certain biologicalsystems, cooperative behavior of molecular motors pro-duces oscillatory motion. In some insects, for instance,autonomous oscillations are generated within the flightmuscle [7]. Spontaneous oscillations have also been ob-served in single myofibrils in vitro [8]. Finally, dyneinmotors could be responsible for the oscillatory motion ofaxonemal cilia and flagella [9, 10].The directionality of individual motors stems from in-teractions between different parts of the motor and frominteractions between the motor and the track filament[11, 12]. The direction of motion of a large collection of ∗ Authors with equal contribution motors may also be influenced by their cooperative modeof action. Specifically, in several recent experiments theability of motors to cooperatively induce bidirectionalmotion has been demonstrated. These in vitro experi-ments were performed by using motility assays in whicha filament glides over a dense bed of motors. In one suchexperiment, unidirectional motion of actin filaments dueto the action of myosin II motors was transformed intobidirectional motion by the application of an externalstalling electric field [13]. Under such conditions the ex-ternal forces acting on the actin filament nearly balancethe forces generated by the motors. Electric field was alsoused to bias the direction of motion in kinesin-MT sys-tems [14]. In another experiment, bidirectional motionof MTs was observed when subjected to the action of anensemble of NK11 motors [15]. These motors are a mu-tant form of the kinesin related Ncd, which individuallyexhibit random motion with no preferred directionality[15]. More recently, the motion of MTs on a bed of amixed population of plus-end (kinesin-5 KLP61F) andminus-end (Ncd) driven motors was shown to exhibitdynamics whose directionality depends on the ratio ofthe two motor species, including bidirectional movementover a narrow range of relative concentrations around the“balance point” [16]. Similarly, bidirectional transport ofmicrospheres coated with kinesin (plus-end directed) anddynein (minus-end directed) on MTs was also reported[17].Several aspects of cooperativity in molecular motorsystems have been addressed using different theoreticalmodels [16, 18, 19, 20, 21, 22, 23, 24, 25, 26]. One fea-ture which has not been treated in these studies is thedependence of the motion on the number of acting mo-tors. A notable exception is the work of Badoual et al. [20], where a two-state ratchet model has been used toexamine the bidirectional motion observed in the NK11-MT motility assay described in ref. [15]. The model ofBadoual et al. [20] demonstrated the ability of a largegroup of motors working cooperatively to induce bidi-rectional motion, even when individually the motors donot show preferential directionality. (The model also pre-dicts that directional motors can also induce bidirectionalmovement, if the filaments are close to stalling condi-tions in the presence of an external load.) Accordingto this model, the characteristic time in which the fila-ment undergoes direction reversal (“reversal time”), τ rev ,increases exponentially with the number of motors, N .Thus, the reversal time diverges in the “thermodynamiclimit” N → ∞ , and the motion persists in the directionchosen at random at the initial time.In this work, we present an in vitro motility assay inwhich myosin II motors drive the motion of globally a-polar actin bundles. These a-polar bundles are generatedfrom severed (polar) actin filaments whose fragments arerandomly recombined. When subjected to the action ofa bed of myosin II motors, these a-polar bundles exhibitbidirectional motion with characteristic reversal timesthat are in the range of τ rev ∼ −
10 sec. The reversaltimes of the dynamics show no apparent correlation withthe size of the gliding bundles, or equivalently, with thenumber of motors N interacting with the track (which,because of the homogeneous spreading of the motors onthe bed, is expected to be proportional to the size of themoving bundle). This observation is clearly in disagree-ment with the strong exponential dependence of τ rev on N , predicted by Badoual et al. [20].Here, we propose a modified version of this modelthat explains the experimentally observed independenceof τ rev on N . We argue that the origin of this behaviorcan be attributed to the tension developed in the actintrack due to the action of the attached myosin II motors.An increase in the number of attached motors leads to anincrease in the mechanical load which, in turn, leads to anincrease in the detachment rate of the motors, as alreadysuggested in models of muscle contraction [23, 24, 25, 26].Unlike most previous studies where the myosin conforma-tional energy was calculated, in this work we consider theelastic energy stored in the actin track and demonstratethat the detachment rate increases exponentially with N .This unexpectedly strong effect (which is another, indi-rect, manifestation of cooperativity between the motors)suppresses the exponential growth of τ rev with N .
2. MATERIALS AND METHODSA. Protein purification
Actin was purified from rabbit skeletal muscle acetonepowder [27]. Purification of myosin II skeletal muscle isdone according to standard protocols [28]. Actin labeledon Cys374 with Oregon Green (OG) purchased from In- vitrogen.
B. NEM myosin II
N-ethylmaleimide (Sigma, Co.) inactivated myosin IIwas prepared according to standard protocol of Khun andPollard [29].
C. Optical Microscopy
Actin assembly was monitored for 30 minutes by fluo-rescence with an Olympus IX-71 microscope. The labeledactin fraction was 1 /
10 and the temperature at which theexperiments were conducted was 23 ◦ C. Time-laps imageswere acquired using a DV-887 EMCCD camera (AndorCo., England).
D. Motility assay
Protocol for this assay was adopted from Kuhn et al. [29]. The assay includes two essential steps: (a) immo-bilization of actin filaments on a bed of NEM myosin IIinactivated motors, and (b) addition of active myosin IImotors at a defined concentration. For that purpose,7.5-8.5 µ l of 0.2 µ M NEM myosin II is introduced intoa flow chamber (26mm × µ M 10%O.G. labeled). Finally, the cell was supplemented with8 µ l of 0.6 µ M myosin II motors (in 2X myosin solutioncontaining: 3.3mM MgCl . 2mM EGTA, 20mM HEPESpH=7.6, 1% MethylCellulose, 3.34mM Mg-ATP, 400mMDTT, 17.6mM Dabco), supplemented with 0.133M KCl,5 µ M Vitamin D, and an ATP regenerating system con-taining 0.1mg/ml Creatine Kinase (CK) and 1mM Crea-tine Phosphate (CP). At the KCl concentrations used inthis assay, the myosin II motors are assembled in smallmotor aggregates ( ∼
16 myosin II units/aggregate) alsoknown as mini-filaments [30]. Fluorescent images weretaken every 2 seconds for 30 minutes.
E. Data Analysis
The position of fluorescent bundles was determinedas the intensity center of mass using METAMORPH(Molecular Devices) software. The position was analyzedusing a custom MATLAB (The MathWorks, Inc.) pro-gram. The data was corrected for stage drift. We firstmeasured the fluctuations of the positions of the bun-dles in the absence of ATP (i.e., when the motors arenot active). Under such conditions, the positions of thebundles measured every 1 sec exhibit a Gaussian distri-bution with zero mean and standard deviation ∆ ∼ F. Estimation of number of interacting motors
In order to evaluate the number of acting motors, thedimensions of the bundles and the motors surface concen-tration, C m , must be determined. We estimate C m byassuming that all the motors that were introduced intothe flow chamber adhere to the top and bottom glasssurfaces of the flow cell (total surface, 104 mm ). Thisgives C m ∼ µm ) − , which corresponds to denselypacked motor beds (typical distance of a few nanome-ters between motor heads). At such high densities, inho-mogeneities associated with the assembly of motors intomini-filaments can be ignored.The length of a bundle, L , was measured using META-MORPH software. The width of a bundle was estimatedby dividing its fluorescence intensity by the intensity ofsingle actin filaments, which gives an estimate for thenumber of filaments, N f , composing the bundle. Assum-ing that the shape of the bundle is cylindrical, its radiuscan be estimated as R = p N f · r , where r ∼ .
75 nm isthe actin filament radius. The motors can only interactwith the part of the bundle that faces the myosin bedcovered surface. Assuming that this part corresponds toroughly a quarter of the surface of the bundle, we findthat the area that comes into contact with the motors A ∼ ( π/ RL . The number of interacting motors is,thus, given by N = C m A . Using this approximation andthe measured surface concentration and bundle dimen-sions, we estimate (see Fig. 6) that N ∼ − G. Computer simulations
A detailed discussion on the computational model isfound in the section 4, below. The model is based onthe model presented in ref. [20], where N rigidly coupledequidistant motors interact with a one-dimensional peri-odic potential representing the actin track. The spacingbetween the motors q is larger than and incommensu-rate with the periodicity of the potential, l . The trackconsists of M ≃ ( q/l ) N periodic units, which are repli-cated periodically. In each unit of the track, a force ofmagnitude f ran and random directionality is introducedwhich defines the local polarity of the track. Globally a-polar tracks were generated by setting the total randomforce to zero (i.e., choosing an equal number of periodicunits in which the random forces point to the right andleft). The motion of the motors on these tracks was cal-culated by numerically integrating the equations of mo-tions [ dx = ( F tot /λ ) dt ] (see Eq. 1 and following text)with time step dt = 0 .
05 msec. The position of one ofthe motors along the track was recorded every 0.25 sec,and changes in the direction of the motion of the motors
FIG. 1: (A) Schematic diagram of the system before additionof active motors. The surface of a microscope slide was sat-urated by BSA (blue balls) and NEM myosin II (long, two-headed, brown objects). Actin filaments/bundles (thin yel-low line) are attached to NEM myosin heads above the sur-face. (B,C) Images of the system before the addition of activemyosin II minifilaments. (B) shows the thick actin bundlesformed at a high concentration of MgCl (1.67 mM), while in(C) the thin bundles/filaments formed at a low MgCl con-centration (0.5 mM) are shown. Bar size is 5 µ m. were identified by analyzing the position of this motor.The distribution of reversal times, t , follows an expo-nential distribution: p ( t ) = (1 /τ rev ) exp( − t/τ rev ), fromwhich τ rev was extracted. The error bars in Fig. 7Crepresent one standard deviation of the distribution ofreversal times measured for different realizations of glob-ally a-polar tracks of similar size. For each value of N ,the number of simulated realizations is 40.
3. RESULTS
A diagrammatic representation of the system is dis-played in Fig. 1. The protocol, based on that of Kuhn et al. [29], is described in detail above. In brief, thesurface of a microscope slide was saturated with NEM-inactivated myosin II motors (drawn as long, two headed,brown objects at the bottom of Fig. 1A) and passivatedby BSA (blue balls in Fig. 1A). Subsequently, actin fila-ments/bundles (thin yellow line, Fig. 1A) were grown andheld firmly on the underside of the NEM-myosin II bed.Fig. 1B shows a characteristic fluorescent microscope im-age of the system which, at this stage, consisted of alarge number of long actin bundles. The bundles wereformed due to the presence of free Mg ions (concen-tration 1.67 mM), which induced attractive electrostaticinteractions between the actin filaments [31]. Unlike bun-dles formed by certain actin-binding proteins, filamentsformed by condensation in the presence of multivalentcations are randomly arranged within the bundles with-out any specific polarity [32, 33]. At lower concentrationsof Mg (0.5?mM), both thinner bundles and single fila-ments were observed (Fig. 1C).After the initial step, myosin II minifilaments (multi-headed brown objects, Fig. 2A) were added to the cell FIG. 2: (A) Schematic diagram of the system after addi-tion of active myosin II motors. After the initial step (seeFig. 1), myosin II minifilaments (multi-headed brown objects)were added to the cell sample. The motors that landed onthe BSA surface created a homogeneous bed of immobile,yet active, motors. Other motors landed on the actin fila-ments/bundles (long yellow line) present on the surface. Themyosin II minifilaments started to move along the actin fil-aments/bundles. During their motion, the motors exertedforces on the actin filaments, which caused severing of smallactin fragments (short yellow lines). The ruptured actin frag-ments could move rapidly on the bed of active myosin IIminifilaments and fuse with other bundles. One fusion eventis demonstrated in the sequence of snapshots (B-E). Here, weshow (B) two bundles moving oppositely to each other, gettingcloser (C) and then fusing (D-E) to create one larger object.Time is given in minutes, bar size is 5 µ m. (F) The bundlescontinue to grow in size through multiple fusion processes, un-til eventually a large, highly a-polar bundle is formed (thickyellow tube - the inset illustrates the internal structure ofsuch a bundle, consisting of individual actin filaments withrandomly orientated polarities). sample. The motors that landed on the BSA surfacecreated a homogeneous bed of immobile, yet active, mo-tors. Other motors landed on the actin filaments/bundlespresent on the surface. These motors started to movealong the actin tracks, thereby exerting forces on theactin filaments, which led to the severing of small actinfragments [34, 35] (Fig. 2A). The ruptured actin frag-ments were then free to move rapidly on the bed of ac-tive myosin II motors. When gliding bundle fragments FIG. 3: Sequence of snapshots showing actin bundles movingdirectionally on a bed of active myosin II motors, at a lowmotor concentration ([myosin II motor]= 0.3 µ M). As a visualguide, we have marked one such actin bundle in orange. Thisbundle moved directionally across the image plane from leftto right at times (A) 0:00, (B) 1:40 min, and (C) 7:20 min(typically at a velocity of 10 µ m/min). Bar size is 5 µ m. came into close proximity to other bundles, they couldfuse, creating new, a-polar, bundles (Fig. 2A. See alsoSupplementary Information - movie 1, showing a smallactin piece severed from a filament, moving rapidly andfusing with a distant existing bundle). These newly cre-ated bundles could further fuse with each other to formeven larger objects (see Fig. 2F and the sequence of snap-shots in Figs. 2B-E depicting one such event of fusionof bundles). The rate of fusion events decreased withtime and, after several minutes, the system relaxed intoits final configuration, shown schematically in Fig. 2F).Notice that the severing and rearrangement of the origi-nally formed actin filaments/bundles (Fig. 1B) led to theformation of much shorter bundles (Figs. 2B-E). More-over, the random nature of the multiple fusion processesinvolved in the generation of these shorter bundles en-sured that the final actin tracks were highly a-polar. In-deed, the motion of most of the bundles shown in Supple-mentary Information - movie 2 was bidirectional (“backand forth” motion), and only those bundles undergoingrare fusion events exhibited unidirectional motion. It isimportant to emphasize that bidirectional motion wasobserved only above a certain concentration of addedmyosin II motors (0.6 µ M) and only in the presenceof ATP. At lower concentrations of motors (0.3 µ M),the motion of actin bundles was directional (see Fig. 3and Supplementary Information - movie 3. Note that inmovie 3, the motion takes place both along pre-existingactin tracks, as well as on the BSA bed, both which arecovered by active myosin II mini-filaments. The motionbetween these two areas is continuous, demonstratingthat the whole surface is covered uniformly with motors.)We, therefore, conclude that the bidirectional movementoriginates from the action of the active myosin II motorswhich (i) severed actin pieces, (ii) transported the severedfragments, which fused into actin tracks with randomlyalternating polarities, and (iii) moved these a-polar actintracks bidirectionally (see Supplementary Information -movie 2).Fluorescence microscopy was used to follow the bidi-rectional motion of the actin tracks. Fig. 4A shows theposition of center of mass of one bundle (three snapshots
FIG. 4: Position of a bundle over a time interval of 800 sec.The time interval between the consecutive data points is 2 sec.(B-D) Pseudo-color images of the actin bundle. The yellowarrows indicate the instantaneous direction of motion of thebundle. Bar size is 5 µ m. are shown in Figs. 4B-D) during a period of more than 10minutes of the experiment. The dynamics of this bun-dle are representative of the motion of the other actinbundles. Specifically, the one-dimensional motion of thebundle does not persist in the initial direction, but ratherexhibits frequent direction changes. Measurements of theposition of the center of mass of the bundle were takenat time intervals of ∆ t = 2 sec, and the mean velocity ineach such period of motion was evaluated by v = ∆ x/ ∆ t ,where ∆ x is the displacement of the center of mass (seesection 2). Fig. 5A shows the velocity histogram of thebundle shown in Fig. 4. The velocity histogram is bi-modal indicating bidirectional motion. The speed of thebundle varies between | v | = 1 − µ m/min, which is 2orders of magnitude lower than the velocities measuredin gliding assays of polar actin filaments on myosin IImotors [36]. The fact that the typical speed of the bidi-rectional motion is considerably smaller than those ofdirectionally-moving polar actin filaments can be par-tially attributed to the action of individual motors work-ing against each other in opposite directions. The bidi-rectional movement consists of segments of directionalmotion which typically last between 2 to 10 time inter-vals of ∆ t = 2. The statistics of direction changes issummarized in Fig. 5B which shows a histogram of thenumber of events of directional movement of duration t .The characteristic reversal time, τ rev , can be extractedfrom the histogram by a fit to an exponential distribu-tion: p ( t ) = (1 /τ rev ) exp( − t/τ rev ). This form (which, asexemplified in Fig. 5B, fits the data well) is expected ifthe probability per unit time to ”turn” in the oppositedirection is independent of the time since the beginning FIG. 5: Velocity histogram of the bundle whose motion isshown in Fig. 4 (based on 900 sampled data points ? Figure4 shows only 400 of those points), exhibiting a clear bimodaldistribution. (B) Distribution of the reversal time for thesame bundle. The distribution is fitted by a single exponentialdecay function with a characteristic reversal time: τ rev ∼ of motion in a given direction.Although Figs. 4 and 5 summarize the results corre-sponding to the movement of a single actin bundle, theseresults are representative of several tens of bundles whosemotion we followed in several repeated experiments. Inthese experiments we observed that essentially all a-polarbundles exhibited bidirectional motion. For the sake ofour quantitative analysis, we picked a smaller group of19 bundles (about 25% of all bundels) for which boththe reversal time and the number of acting motors N (which is proportional to the surface area, see section 2)could be determined with sufficiently high precision. Theresults corresponding to the motion of this sample of rep-resentative bundles are plotted in Fig. 6. The choice ofwhich bundles to include in Fig. 6 is based on the follow-ing practical reasons: The fluoresces intensity of smallbundles is too low (compared with the background) and,thus, the accurate position and dimensions are hard todetermine. Very large bundles are practically immobileand their motion is smaller than the experimental spatialresolution (see Material and Methods - section 2). Thus,the data in Figure 6 includes only bundles of “interme-diate” size. One can see in Fig. 6 that while N variesover half an order of magnitude, the corresponding τ rev FIG. 6: Characteristic reversal time, τ rev , of 19 different bun-dles as a function of the number of working motors N . Thereversal time for each bundle is obtained by an exponentialfit. are similar to each other (3 < τ rev <
10 sec) and show noapparent correlation with N .
4. DISCUSSION
The bidirectional motion of motors was previously seenin systems consisting of motors that lack specific direc-tionality [15, 37], mixtures of motors working in oppositedirections [16], or under the action of external forces closeto stalling conditions (forces acting on the filament thatnearly balance the forces generated by the motors) [13].One commonly used model for the dynamics of molecu-lar motors in the biophysics literature is the Brownianratchet mechanism [38]. Within this modeling approach,the motion of individual motor proteins is studied byconsidering the motion of a particle in a periodic, locallyasymmetric, potential. It follows from the second law ofthermodynamics that if the system is coupled to a ther-mal bath, the particle subjected to the periodic potentialwill not exhibit large scale directed motion. Directed mo-tion is possible only if the system is (i) locally asymmet-ric, and (ii) driven out of equilibrium by an additionaldeterministic or stochastic perturbation. This perturba-tion is used in the model to represent the consumptionof ATP chemical energy by the motors. Ratchet models are not molecular in nature but rather present a way toidentify the minimal physical requirements for the mo-tion of motor proteins. However, by choosing properlythe parameters of system, they may be employed to de-rive quantitative predictions for specific motor-filamentssystems. Ratchet models have been extended for describ-ing and analyzing the collective motion of groups of mo-tors. The motion of several motors is influenced by themotor-motor interactions [39] and mechanical coupling[20]. The model proposed by Badoual at al. [20] (andwhich, below, we present a slightly modified version of)demonstrates that mechanical coupling between the mo-tors is sufficient for the generation of highly cooperativebidirectional motion, even if the motors attach to/detachfrom the track in an uncorrelated fashion.A key prediction of the model in Ref. [20] is the ex-ponential increase of the mean reversal time of the bidi-rectional motion, τ rev , with N , the number of motors.This prediction is in contradiction with our experimentalresults (Fig. 6). Here, we show that this disagreementcan be resolved by considering the stretching energy in-volved in the interactions between the actin track andthe “walking” motors. Accounting for this effect elimi-nates the exponential dependence of τ rev on N . More-over, when values representing myosin II-actin systemswere assigned to the parameters of the model, we found τ rev ∼ −
12 sec, which in a very good quantitative agree-ment with the experimental data.Our model is illustrated schematically in Fig. 7A: Weconsider the 1D motion of a group of N point particles(representing the motors) connected to a rigid rod withequal spacing q . The actin track is represented by a peri-odic saw-tooth potential, U ( x ), with period l and height H . We choose q = (5 π/ l ∼ . l , which satisfies therequirements of the model [20] for q to be larger thanand incommensurate with the periodicity of the poten-tial. The locally preferred directionality of the myosin IImotors along the actin track is introduced via an addi-tional force of size f ran exerted on the individual motors.In each unit of the periodic potential, this force randomlypoints to the right or to the left (the total sum of theseforces vanishes), which mimics the random, overall a-polar, nature of the actin bundles in our experiments.The instantaneous force between the track and the mo-tors is given by the sum of all the forces acting on theindividual motors: F tot = N X i =1 f motor i = N X i =1 (cid:20) − ∂U ( x + ( i − q ) ∂x + f ran ( x + ( i − q ) (cid:21) · C i ( t ) , (1)where x i = x + ( i − q is the coordinate of the i -th motor. The two terms in the square brackets represent FIG. 7: (A) N point particles (representing the motors) are connected to a rigid rod with equal spacing q . The motors interactwith the actin track via a periodic, symmetric, saw-tooth potential with period l and height H . In each periodic unit, there isa random force of size f ran , pointing either to the right or to the left (red arrows). The motors are subject to these forces onlyif connected to the track. The detachment rate ω is localized in the shaded area of length 2 a < l , while the attachment rate ω is located outside of this region. The off rate ω is permitted only outside the gray shaded area. (B) A set of n + 1 pointparticles connected via N identical springs with spring constant k . Each particle is subjected to a random force f i that takesthree possible values: − f, , + f (blue arrows). The mean force acting on the particles is given by ¯ f = f CM / ( n + 1), where f CM = P N +1 i =1 f i is the total force acting on the center of mass and causing the movement of the system. The force F i stretchingthe i -th spring (red arrow) is equal to the sum of the excess forces, f i − ¯ f , acting on the particles located to the right of the i -thspring, F i = ( P ij =1 f j ) − i ¯ f . Because a similar expression can be written taking into account the forces acting on the particleslocated to the left of the spring, one can readily show that P Ni =1 F i = 0. (C) The mean reversal time, τ rev , as a function of thenumber of motors N , computed for different realizations with α = 0 . α = 0 (open circles). In the latter case, the effect of actin stretching is neglected, and τ rev grows exponentially with N (dash-dotted line), in agreement with ref. [20]. The half-filled circles denote the experimental results, also presented in Fig. 6.(D) The fraction of attached motors, N c /N , as a function of the total number of motors, for α = 0 . N c /N for α = 0 are indistinguishable. the forces due to the symmetric saw-tooth potential andthe additional random local forces acting in each periodicunit. The latter are denoted by red arrows in Fig. 7A.The function C i ( t ) takes two possible values, 0 or 1, de-pending on whether the motor i is detached or attachedto the track, respectively, at time t . The group veloc-ity of the motors (relative to the track) is determinedby the equation of motion for overdamped dynamics: v ( t ) = F tot ( t ) /λ . The friction coefficient, λ , dependsmainly on motors attached to the track at a certain mo-ment and is therefore proportional to the number of con-nected motors, N c ≤ N at time t : λ = λ N c .To complete the dynamic equations of the model, weneed to specify the transition rates between states (0 -detached; 1 - attached). The motors change their statesindependently of each other. We define an interval of size2 a < l centered around the potential minima (the grayshaded area in Fig. 7A). If located in one of these regions, an attached motor may become detached (1 →
0) with aprobability per unit time ω . Conversely, a detached mo-tor may attach to the track (0 →
1) with transition rate ω only if located outside this region of size 2 a . However,we also allow another independent route for the detach-ments of motors, which may take place outside the grayshaded area in Fig.7A (i.e., around the potential max-ima) and is characterized by an off rate ω . The rates ω , ω , ω (see blue arrows in Fig. 7A), represent theprobabilities per unit time of a motor to (i) detach aftercompleting a unit step, (ii) attach to the track, or (iii)detach from the track without completing the step.Generally speaking, the rates of transitions betweenstates depend on many biochemical parameters, most no-tably the types of motors and tracks, and the concentra-tion of chemical fuel (e.g., ATP). They may also be af-fected by the forces induced between the motors and thefilament, which result in increase in the configurationalenergy of the attached myosin motors [23, 24, 25, 26]and in the elastic energy stored in the S2 domains of themini-filaments, as well as an increase in the stretchingenergy of the actin filament. The latter contribution canbe introduced into the model via a modified detachmentrate given by: ω = ω exp( − ∆ E/k B T ), where ∆ E isthe change in the elastic energy of the actin track dueto the detachment of one motor head. The dependenceof ∆ E on the number of connected motors N c (out of atotal number of motors, N ) can be estimated in the fol-lowing manner: Consider a series of N + 1 point particlesconnected by N identical springs (representing a seriesof sections of actin filaments) having a spring constant k (see Fig. 7B). Let us assume that random forces act onthe particles and denote the force applied on the particlewith index i (1 ≤ i ≤ N + 1) by f i . Assume that eachof these forces can take three possible values: − f (repre-senting attached motors locally pulling the track to theleft), + f (attached motors pulling the track to the right),and 0 (detached motors not applying force). Defining the”excess force” with respect to the mean force acting onthe particles: f ∗ i = f i − ¯ f [where ¯ f = P N +1 i =1 f i / ( N + 1)],one can show that the force stretching (or compressing)the i -th spring in the chain is given by the sum of excessforces acting on all the particles located on one side ofthe spring F i = i X j =1 f ∗ j = − N +1 X j = i +1 f ∗ j . (2)From Eq. 2 it can be easily verified that P Ni =1 F i = 0.We thus conclude that the excess forces acting on theparticles, f ∗ i , represent a series of random quantities withzero mean. Therefore, the size of F i can be estimatedby mapping the chain of springs into the problem of a1D random polymer ring [40], where the elastic energystored in the i -th spring, ǫ i = F i / k , plays the role of thesquared end-to-end distance between the i + 1 monomerand the origin. From this mapping we readily concludethat the energy of most of the springs (except for thoselocated close to the ends of the chain) scales linearly withthe number of attached motors: ǫ ∼ N c ( f / k ). Thetotal elastic energy of the chain scales as E ∼ N ǫ ∼ N N c ( f / k ) , (3)and when a motor detaches from the track ( N c → N c − E/k B T = − αN (4)where α is a dimensionless prefactor.We simulated the dynamics of an N -motor system,choosing parameters corresponding to the myosin II-actinsystem. The period of the potential l = 5 nm corre-sponds to the distance between binding sites along theactin track [41, 42, 43], and the amplitude of the sym-metric potential is set to H = 6 k B T . Thus, the forcegenerated by each motor head on the track is 2 H/l = 10 pN (first term in square brackets in Eq. 1). The magni-tude of the random force that defines the local polarityof the track (second term) is given by f ran = 4 . ω is chosen to be 2 a = 3 . ω − = 0 . ω − = 33 ms [46, 47, 48, 49]. Withthis choice of parameters, we obtain a system with a lowfraction of attached motors N c/N ∼ . λ = 1 . · − kg/s, which yields the experimentallymeasured velocity v ∼ .
03 nm/ms ∼ µ m/min. Therate ω expresses the probability of a single motor headto detach from the track without advancing to the nextunit. The probability p of such an event is 1-2 orders ofmagnitude smaller than the complementary probability(1 − p ) to execute the step. We take p = 1 /
30 [49], whichyields ?( ω ) − ∼ pv/l = 7500 ms. Finally, the exponent α appearing in Eq. 4 is evaluated by: α ∼ ( f / kk B T ) = ( f l/ Y Ak B T ) , (5)where Y ∼ Pa is Young’s modulus for actin and A ∼
35 nm is the cross sectional area of an actin filament[49]. For the model parameters: f ∼
10 pN, l = 5 nm,we find α ∼ . τ rev as a function of N for 800 ≤ N ≤ N are replot-ted in Fig. 7C and denoted in half-filled circles). Foreach N , the computational results represent the aver-age τ rev computed for 40 different realizations of ran-dom, overall a-polar, tracks. The error bars representthe standard deviation of τ rev between realizations, wherefor each realization τ rev is estimated by fitting the his-togram of turning times to an exponential decay func-tion (as in the experimental part - see Fig. 5B). Two setsof computational data are shown in Fig. 7C: one cor-responding to α = 0 . α = 0 (open circles), i.e., without considering the ef-fect of actin stretching, but when all the other systemparameters mentioned above are kept unchanged. Thelatter case is qualitatively similar to the model presentedin ref. [20], exhibiting a very strong exponential depen-dence of τ rev on N (indicated by the straight dashed-dotted line in Fig. 7C). In contrast, the data correspond-ing to α = 0 . τ rev upon changing N . The mean reversal times computed for1200 ≤ N ≤ ≤ τ rev ≤ α = 0. The validity of our ratchetmodel is quite remarkable in view of its extreme sim-plicity; but one must be aware of the following pointsof disagreement between the experimental and compu-tational results (which illustrate the limitations of themodel): (1) The computed reversal times show weak,non-monotonic, dependence on N which is not observedexperimentally. (2) The largest computed τ rev ( τ rev = 12sec for N = 2000) is slightly larger than the experimen-tally measured reversal times. (3) The computational re-sults for N <
N > τ rev < N > ω , i.e., to our assumption that (for a given N ) the detachment of each motor head leads to the sameenergy gain (see Eq. 4). In reality, the energy changeupon detachment of a motor depends, in some complexmanner, on a number of factors such as the positionsand chemical states of the motors. Motors which releasehigher energy will detach at higher rates, and the de-tachment of these “energetic” motors will lead to the re-lease of much of the elastic energy stored in the actintrack. We, therefore, conclude that within the meanfield approach, the number of disconnecting motors andthe frequency of detachment events are probably over-estimated. This systematic error of the mean field cal-culation increases with N , and the result of this is thedecrease of τ rev in this regime, which is not observed ex-perimentally. For even larger values of N ( N > ω > ω and the effective at-tachment rate outside the gray shaded area in Fig. 7A, ω on ≡ ω − ω , becomes negative, i.e., motors detachfrom the track faster than they attach to it. In con-trast, for < < N < ω ≪ ω ≪ ω , andthe effective attachment rate ω on barely changes uponchanging the model parameter from α = 0 (ref. [20]) to α = 0 . ω on (which, nevertheless, involves a dramatic increasein ω ) leads to the following non-trivial outcome: On theone hand, the fraction of attached motors remains un-changed. The data shown in Fig. 7D corresponds to both values of α for which the results for N c /N are indistin-guishable. On the other hand, the reversal times dropby as much as three orders of magnitude (for N = 3000)when α is modified from 0 to 0.0018. This spectaculardecrease in τ rev is, therefore, not the result in the changein the number of attached motors (since, for each valueof N , the same fraction of motors is attached for bothvalues of α ), but rather can be related to the less regularmanner by which the motors detach from the track. Themore frequent stochastic detachments of motors from theactin track increases the probability per unit time of mo- tion reversal.
5. CONCLUSION
We have investigated the dynamics of myosin II mo-tors on actin tracks composed of small filamentous seg-ments with randomly alternating polarities. The absenceof global polarity leads to a bidirectional relative motionbetween the motors and the tracks. The characteristic re-versal time of this motion is of the order of a few secondsand exhibits no particular dependence on the number ofacting motors. Bidirectional motion with macroscopicreversal times has been previously observed for NK11motors on microtubules and has been attributed to thecooperativity of the motors. According to previously pro-posed models, the signature of such a bidirectional coop-erative motion is the strong exponential dependence of τ rev on N . The contradiction of this prediction with ourexperimental results can be reconciled by incorporatingan additional feature into the model, namely, the effectof actin stretching by the walking motors. To reduce theassociated elastic energy, the off rate of motors increases,and many of them detach from the track before complet-ing a unit step. This effect reduces τ rev considerably andeliminates its exponential growth with N .Single molecule experiments have led to a dramaticincrease in our understanding of the structure and dy-namics of individual molecular motors. However, manybiological processes such as muscle contraction, cytokine-sis, and the motion of axonemal cilia and flagella, involvecooperative action of many motors, which may be af-fected by the structure of the underlying track. This con-cept is clearly demonstrated in this work dealing with thebidirectional motion of myosin II motors on actin trackswith randomly alternating local polarities, but without anet preferred directionality at the mesoscopic level. Thisunique type of motion is induced by the forces of in-dividual motors whose collective effect is manifested inmacroscopically large reversal times. At the same time,the cooperativity of these forces also increases the elasticenergy of the track, and thereby limits the growth of τ rev .
6. ACKNOWLEDGMENT
We thank Nir Gov, Yoav Tsori and Oleg Krichevskyfor useful discussions. A.B.G wishes to thank the Josephand May Winston Foundation Career Development Chairin Chemical Engineering, the Israel Cancer Association(grant No. 20070020B) and the Israel Science Foundation(grant No. 551/04). [1] R. Vale,
Cell , 2003, , 467-480.[2] H. Higuchi and S. A. Endow, em Curr. Opin. Cell Biol,2002, , 50-57. [3] H. B. McDonald, R. J. Stewart and L. S. Goldstein, Cell ,1990, , 1159-1165.[4] M. Schliwa, Nature , 1999, , 431-432. [5] M. A. Geeves and K. C. Holmes, Annu. Rev. Biochem. ,1999 , 687-728.[6] B. Feierbach and F. Chang, Curr. Opin. Microbiol. , 2001 , 713-719.[7] K. E. Machine and J. W. S. Pringle, Proc. R Soc. Lond.B Biol. Sci. , 1959, , 204-225.[8] K. Yasuda, Y. Shindo and S. Ishiwata,
Biophys. J. , 1996, , 1823-1829.[9] S. Camalet and F. J¨ulicher New J. Phys. , 2000, , 1-24.[10] C. J. Brokaw, Proc. Natl. Acad. Sci. USA , 1975, , 3102-3106.[11] S. A. Endow, Nat. Cell Biol. , 1999, , 163-167.[12] .J. M´en´etry, A. Bahloul, A. L. Wells, C. M. Yengo, C.A. Morris, S. A. Sweeny and A. Houdusse, Nature , 2005, , 779-785.[13] D. Riveline, A. Ott, F. J?icher, D. A. Winklemann, O.Cardoso, J. J. Lacap?e, S. Magn´usd´ottir, J. L. Viovy, L.Gorre-Talini and J. Prost,
Eur. Biophys. J. , 1998, ,403-408.[14] M. G. L. van den Heuvel, M. P. de Graaff and C. Dekker, Science , 2006, , 910-914.[15] S. A. Endow and H. Higuchi,
Nature , 2000, , 913-916.[16] L. Tao, A. Mogliner, G. Civelekoglu-Scholey, R. Woll-man, J. Evans, H. Stahlberg and J. M. Scholey,
Curr.Bio. , 2006, , 2293-2302.[17] R. Yokokawa, M. C. Terhan, T. Kon and H. Fujita, Biotechnol. Bioeng. , 2008, , 1-8.[18] F. J¨ulicher and J. Prost,
Phys. Rev. Lett. , 2005, , 2618-2621.[19] F. J¨ulicher and J. Prost, Phys. Rev. Lett. , 1997, , 4510-4513.[20] M. Badoual, F. J¨ulicher and J. Prost, Proc. Natl. Acad.Sci. USA , 2002, , 6696-6701.[21] Y. Shu and H. Shi, Phys. Rev. E. , 2004, , 021912.[22] C. J. Brokaw, C. J, Cell Motil. Cytoskeleton , 2005, ,35-47.[23] D. A. Smith and M. A. Geeves, Biophys. J. , 1995, ,524-537.[24] T. A. J Duke, Proc. Natl. Acad. Sci. USA , 1999, ,2770-2775.[25] G. Lan and S. X. Sun, Biophys. J. , 2005, , 4107-4117.[26] R. E. L De Ville and E. Venden-Eijnden, Bull. Math.Bio. , 2008, , 484-516.[27] J. A. Spudich and S. Watt, J. Biol. Chem. , 1971, ,4866-4871. [28] S. S Margossian and S. Lowey,
Methods Enzymol. , 1982, , 55-71.[29] J. R. Kuhn and T. D. Pollard, Biophys. J. , 2005, ,1387-1402.[30] B. Kaminer and A. L. Bell, J. Mol. Biol. , 1966, , 391-401.[31] E. Grazi, P. Cuneo and A. Cataldi, Biochem J. , 1992, , 727-732.[32] P. A. Janmey,
Proc. Natl. Acad. Sci. , 2001, , 14745-14747.[33] G. C. L. Wong, A. Lin, J. X. Tang, P. A. Janmey andC.R. Safinya, Phys. Rev. Lett. , 2003, , 018103.[34] N. A. Medeiros, D. T. Burnette and P. Forscher, Nat.Cell Biol. , 2006, , 215-226.[35] L. Haviv, D. Gillo, F. Backouche and A. Bernheim-Groswasser, J. Mol. Biol. , 2008, , 325-330.[36] S. J. Kron and J.A. Spudich,
Proc. Natl. Acad. Sci. USA ,1986, , 6272-6276.[37] J. L. Ross, K. Wallace, H. Schuman, Y. E. Goldman andE. L. F. Holzbaur, Nat.Cell.Biol. , 2006, , 562-570.[38] P. Reimann, Phys. Rep. , 2002, , 57-265, and refer-ences therein.[39] F. Slanina,
Europhys. Lett. , 2008, , 50009.[40] P. G. de Gennes, Scaling Concepts in Polymer Physics ,1997 (Cornell Uni. Press, Ithaca).[41] J. T. Finer, R. M. Simmons and J. A. Spudich,
Nature ,1994, , 113-119.[42] J. E. Molloy, J. E. Burns, J. Kendrick-Jones, R. T.Tregear and D.C. White,
Nature , 1995, , 209-212.[43] A. D. Mehta, J. T. Finer, and J. A. Spudich,
Proc. Natl.Acad. Sci. USA , 1997, , 7927-7931.[44] C. M. Veigel, M. L. Bartoo, D. C. White, J. C. Sparrowand J. E. Molloy, Biophys. J. , 1998, , 1424-1438.[45] M. J. Tyska, D. E. Dupuis, W. H. Guilford, J. B. Patlak,G. S. Waller, K. M. Trybus, D. M. Warshaw, and S.Lowey, Proc. Natl. Acad. Sci. USA , 1999, , 4402-4407.[46] M. A. Geeves, Biochem. J. , 1-14[47] J. A. Millar and M. A. Geeves, Nature , 1983, , 232-237.[48] J. W. Cardon and P. D. Boyer,