Jamming of molecular motors as a tool for transport cargos along microtubules
JJamming of molecular motors as a tool for transportcargos along microtubules.
Lucas W. Rossi and Carla GoldmanDepartamento de F(cid:237)sica Geral - Instituto de F(cid:237)sicaUniversidade de Sªo Paulo CP 6631805315-970 Sªo Paulo, Brazil.January 2012
Abstract
The hopping model for cargo transport by molecular motors introduced in Refs. [12, 13] isextended in order to incorporate the movement of cargo-motor complexes (C-MC). Hoppingprocesses in this context expresses the possibility for cargo to be exchanged between neighbor-ing motors at a microtubule where the transport takes place. Jamming of motors is essentialfor cargos to execute long-range movement in this way. Results from computer simulationsof the extended model indicate that cargo may indeed execute bidirectional movement in thepresence of motors of a single polarity, con(cid:133)rming previous analytical results. Moreover, theseresults suggest the existence of a balance between cargo hopping and the movement of thecomplex that may control the e¢ ciency of cargo transfer and cargo delivery. An analysis of theenergy involved in this transport process shows that the model presented here o⁄ers a consid-erable advantage over other models in the literature for which cargo dynamics is subordinatedto the movement of the C-MC.keywords - intracellular transport by molecular motors; bidirectional movement of cargo;tra¢ c jam on microtubules; ASEP models. Introduction
Cargo particles move bidirectionally as they are transported by molecular motors along micro-tubules. A current explanation of this phenomenon, expressed by the so called coordination model,relies on the idea that motors of di⁄erent polarities are coordinated by external agents to workon the same particle at di⁄erent times. In a related explanation, the tug-of-war model, the twokinds of motors would act simultaneously by pulling the cargo to one or to the other side of themicrotubule [1, 2, 3, 4].A general di¢ culty encountered in any of these views concerns the presence of other particles onthe microtubules, including other motors, that may impose restrictions on cargo·s motion. In fact,as noticed in Ref. [5], there are diverse "physical barriers" at the cytoskeleton where intracellulartransport takes place. The cytoskeleton itself consists of a highly structured composition of crossed(cid:133)laments on which there are present associated proteins and other motors (and other cargos) thatmay limit both motor and cargo·s motion [6]. Because of this, the origins of the bidirectionalmovement of cargo, including organelles, vesicles, virus and other particles, on microtubules is stilla matter of intense debate [7]. Other models have been proposed in the literature as improvementson the coordination or tug-of war models and are formulated by attributing a dynamic role to themicrotubules due to their elastic properties and intrinsic polarity [8, 9, 10, 11]. Nonetheless, amore complete consideration of questions related to the tra¢ c of motors in these contexts is stillrequired.The occurrence of motor jamming on crowded microtubules would impose di¢ culties to thecoordination-like models even if there were present motors of just a single polarity. In fact, de-scriptions of the transport phenomenon in such contexts are based on the premise that cargos canmove only if attached to motors arranged to form a cargo-motor complex (C-MC). Each C-MCis supposed to follow the dynamics of the constituting motors. We shall refer to these as C-MCmodels. As noted in Ref.[11], considering even that motors may eventually be detached from andthen reattached to microtubules in order to temporarily create more space for the C-MC·s, it is notclear how this would help the system to achieve the expected e¢ ciency in the transport process.Motor attachment and detachment occur at random positions on the microtubules, not necessarilyat the places or times that would be required for cargo passage.Another related problem concerns the nature (type, strength, etc.) of the bond (linkage) betweencargos and motors. In carrying cargos along relatively long distances, it would be necessary for2-MC complexes to maintain a stable attachment between cargo and motor as they move alongtracks. On the other hand, it would seem that a strong attachment between the two particles inthis context would restrain cargo release from motors at the required places and at the right times.Thus, the reality of tra¢ c jamming and the mechanisms through which local coordination mightbe achieved still challenge the current views of the transport processes based on C-MC models.Motivated by this, we have been exploring the problem from a di⁄erent perspective as analternative to the idea that C-MC assembly is a necessary condition for transporting cargo inthis context [12, 13]. According to this alternative view, cargo transport would result from asequence of elementary hopping processes taking place on a microtubule represented by a one-dimensional lattice. Introduced in the pioneering work by Kolomeisky and Widom [14], one-dimensional hopping models have ever since been used to describe the dynamics of molecularmotors along microtubules. Many adaptations of the original idea have contributed to unraveldetails of the phenomenon and specially, the collective character of the related processes. Althoughrepresenting simpli(cid:133)ed descriptions of the reality, it is believed that these 1D hopping modelscapture essential and relevant features of motor dynamics. It should not be expected, however, toobtain from them detailed quantitative predictions about the system. Using a reasonable numberof parameters, stochastic models of this sort are expected to o⁄er restricted although importantproposals regarding the physical mechanisms of interest. In a sense, our model extends the originalidea to account also for cargo hopping in addition to the underlying motor hopping. It was conceivedoriginally on the following basis:(i) Motors and cargos would not assemble to form stable C-MC complexes. A weak and intrin-sically (cid:135)exible attachment (or "(cid:135)oppy linkage", as coined in Ref. [15]) that might eventually beestablished between motors and cargos would be short-lived. The relevant degrees of freedom ofsuch transiently assembled structures would be excited by thermal (cid:135)uctuations (noise).(ii) Because of these thermal (cid:135)uctuations, cargo may be exchanged (or "hop") between motorsoccupying neighboring sites on the lattice.It is worth mentioning here that both elements, namely thermal (cid:135)uctuations and cargo exchangehave already been observed in experiments. Fluctuations in the relative positions of cargo andassociated motors have been detected as they introduce di¢ culties in characterizing experimentallythe movement of motors by following the movement of the cargo [16]. Cargo exchange (or cargoswitch, or cargo hop, as we call here) between motors of di⁄erent polarities, moving on di⁄erentstructures like actin (cid:133)laments or microtubules, has been observed in vivo [5]. Actually, cargo3witching was found to be a useful mechanism to move cargo across the diverse structures ofcytoskeleton . Therefore, the scheme in Fig.1 expresses the idea of combining (i) and (ii) in orderto examine their e⁄ect on cargo transport, restricted to a one-dimensional space, taking place oncrowded microtubules and involving motors of a single polarity.We have shown in our previous studies that long-range movement of cargo may be achieved inthis way if (and only if) motors become jammed. Cargo would then be able to move through longdistances as it undergoes a sequence of these elementary (short-range) steps, hopping from motorto motor, either forwards or backwards. Thus, in this view, and contrarily to common expectations,motor jamming along microtubules would not impede cargo (cid:135)ow. On the contrary, jamming wouldbe desirable, as a condition for the whole process to attain a relatively high degree of e¢ ciency.Originally, the stochastic lattice model proposed in [12] and extended in [13] to explain theobserved bidirectional cargo movement was conceived on the basis of ASEP models (asymmetricsimple exclusion processes) already formulated in these contexts to describe the dynamics of a col-lection of single polarity motors [17, 18]. Because our interests focus on the study of mechanismsresponsible for transport carried out by motors, it was necessary to include cargos and their inter-actions with the motors on the same track. To that end, we have made a few assumptions in orderto de(cid:133)ne the nature of such multiparticle interacting system in conformity with (i) and (ii) above:(I) existence of steric interactions among particles;(II) restrictions to cargo movement if not by hopping process;(III) restrictions to motor movement if attached to cargos;Both (II) and (III) ensure that the C-MC·s are immobile in this model. Indeed, one expects thatthe presence of cargos on the microtubules a⁄ects motor motility. In turn, changes in motor motilityshould a⁄ect the transport of cargos. The hopping model accounts explicitly for this interplay ando⁄ers a way to examine the conditions for long-range cargo transfer in di⁄erent contexts. Observethat the analysis of the properties of such a model must necessarily be of a global nature sincethe relevant phenomenon investigated is motor jamming which is intrinsically non-local. Fromsuch analysis, we have concluded that the bidirectional movement of cargo can indeed be achievedthrough hopping under jamming conditions even in the presence of motors of a single polarity, butonly if more than one cargo participate in the dynamics [13]. As jamming takes place, a givencargo may become able to hop over large clusters assembled either behind it or at the back end ofa cargo in front, covering in this way relative long distances in both directions. We then suggested4hat the conditions for these events may be controlled by adjusting the density of motors (numberper unit volume) attached to the microtubule. Accordingly, no external agents would be necessaryto determine the direction of cargo movement.Here, we extend the hopping model by adding an extra process to the original dynamics. Weconfer to motors the ability to move to a neighbor lattice site even if attached to cargo. This meansthat we incorporate into the model the idea of the movement of the C-MC complexes. From aformal point of view, this recovers the ergodicity of the model, a question raised in Ref.[13]. Inpractical applications, this would allow one to investigate the e⁄ects on transport properties of thiscombination of two processes - the one dependent on C-MC complexes and the other based only onhopping. Yet we maintain the choice regarding the presence of single polarity motors in the system.Computer simulations of this extended model show that bidirectionality of cargo may result fromthis combination of processes even if there were present only a single cargo on the track. Estimatesof related energy costs indicate that hopping may introduce signi(cid:133)cant advantage over mechanismsthat rely exclusively on the movement of C-MC complexes.The paper is outlined as follows. The original hopping model is brie(cid:135)y reviewed in Section 2.In Section 3 we present results for cargo displacement and average velocity obtained by computersimulation of the extended version. Energy estimates and (cid:133)nal remarks are in Section 4.
The original stochastic lattice model represented in
Figure 1(a-c) has been mapped into anASEP (asymmetric simple exclusion process) [19], [20], [21] for describing the following elementaryprocesses that take place on a one-dimensional lattice (microtubule): ( a ) 10 ! with rate k; probability kdt ( b ) 12 ! with rate w; probability wdt ( c ) 21 ! with rate p; probability pdt (1)Label is assigned to a site of this lattice that is occupied by a motor carrying no cargo; label is assigned to a site occupied by a motor weakly attached to a cargo; and a label is assignedto an empty site. Notice that the above dynamics preserves the number of motors and cargoson the lattice. In principle, fast processes describing motors attachment and detachment from5he microtubule could be added to this as for example in (cid:28) with appropriate rates. Suchprocesses, however, should not modify the general characteristics of results presented below reachedin the limit of very large number (or average number) of motors on the microtubule at stationaryconditions. Because of this, we decided to keep the model as simple as possible in order to captureits essential features and understand the e⁄ects of C-MC movement on the already consideredhopping process . Process (a) represents an elementary step of a biased motor as it moves forward, towardsthe microtubule minus end according to the convention adopted here (Fig.1(a)). To de(cid:133)ne motorstepping is, of course, essential in building the model dynamics taking place on the microtubulesince it is the primary source of jamming. This in turn may create the conditions necessary to thelong-range transport of cargo at high motor densities. It is exactly this possibility that we wish toinvestigate here. Processes (b) and (c) represent the exchange of cargo between neighboring motors,to the left and to the right, respectively (Fig. 1(b,c)) . Notice that each of these elementary stepsoccurs with a certain probability and under certain conditions. For process (a) to occur with theindicated probability it is required that the site to the right of the motor stays empty during thetime interval dt . The other two processes depend on the presence of a motor to the left (b) or theright (c) of the motor attached to the cargo within dt: The stationary properties of this model arederived in [12] and [13], in the limit for which the number n of motors and the number N of siteson the microtubule are both very large in such a way that the ratio n =N ! (cid:26) , i.e. converges to a(cid:133)nite density (cid:26) of motors. The analysis performed there focuses on the behavior of cargo averagevelocity v m : For a broad range of values for the parameters, v m presents two distinct behaviorsas (cid:26) varies, characterizing the occurrence of a phase transition in this system. Moreover, in casesfor which there are present more than one cargo on the lattice, as considered in [13], v m changessign after relative long runs. Thus, in contrast with a local coordination or local dispute conceivedin the context of tug-of-war models, the phenomenon predicted in [13] emerges from the globalproperties of the system, related to the tra¢ c and associated clustering of motors, which, in turn,can be controlled by tuning the amount of motors bound to the microtubule.6 Combining hopping with the movement of the complex.
The idea here is to relax the condition used both in [12] and [13] under which the movement of cargowould take place exclusively through hopping. Accordingly, we shall add to the above dynamicsthe following process ( d ) 20 ! with rate (cid:11); probability (cid:11)dt (2)in order to let cargo to move also by means of a C-MC complex. Consistent with the fact thatwe have assumed the presence of motors of a single polarity, the complex shall be biased so as tomove in a single direction, the same as that of the motors in (a). In general, however, the numericalvalues of (cid:11) and k need not to be the same. In fact, in a recent study using Monte Carlo simulation,it was concluded that an attached cargo can indeed modify signi(cid:133)cantly the rates at which motorsbind to the microtubule, especially at high viscosities [22].We also observe that as in our original model, the attachment between cargo and motors shouldbe weak in order to allow cargo to be exchanged between neighboring motors. Notice that the ideaof combining the movement of the complex with cargo hopping does not diminish the relevance ofthe tra¢ c jam in this context. As we shall argue below, hopping and jamming conditions continueto play a crucial role in explaining the long-range movement of cargo, especially at high motordensities. We consider the extended hopping dynamics taking place at a one-dimensional lattice of N sitesalong which n motors and n cargos, with n n < N; are initially distributed at random. Thetime evolution of the system is then carried out with the aid of computer simulation through asequence of global runs considering periodic boundary conditions. Within a global run, the N sites,one at time, are tested for updating. The procedure is made sequential and the sites are selectedat random. If a selected site, say j , has not already been updated during the run, then an attemptshall be made to interchange its occupancy with site j + 1 according to the rules set in (1) and(2). If, however j + 1 has already been updated during this run, then j would remain unchanged.Subsequently, a new site is selected at random and the process is repeated until all sites are tested,which ends the run. A new run starts with its initial condition set by the (cid:133)nal con(cid:133)gurationattained in the previous run. The time unit (cid:1) t is de(cid:133)ned as one global run. The total number T
7f global runs sets the time interval for evaluating the average values for the quantities of interestat stationary conditions. T is a parameter of the algorithm. We seek stationary conditions byrepeating the entire procedure with an increasing number of runs until the average pro(cid:133)les becomeinvariant.When the simulation starts, one of the cargos in the system - the one whose properties willbe evaluated - is selected at random. At (cid:133)xed values of the parameters, the movement of thisselected cargo is marked at the end of each global run as (cid:0) ; +1 or to indicate that it executed,respectively, a step to the right, to the left or not changed its position with respect to the previousrun. The algebraic sum of all of these steps performed along the set of runs accounts for the totaldisplacement d ( T ) of the selected particle within each de(cid:133)ned time interval T . Cargo averagevelocity v m is then estimated as the ratio d ( T ) =T . Fig.2 shows the results obtained in this way for the variation of v m as a function of (cid:26); at (cid:133)xedvalues of parameters k; (cid:11); w; p; and n ; as indicated. The choice of parameters in each of theseexamples was not guided by pre-existing experimental data. Our main interest here is simply tounderstand the behavior of the model, specially regarding the relative contribution of each of thetwo modes considered to promote cargo movement. This allows us to identify the origins of someof the observed properties as, for example, the fact that v m may change sign as (cid:26) varies. Thiscon(cid:133)rms our predictions made elsewhere, based on analytical calculations of v m using the modelin [13] . This particular result suggests that motor density at the microtubule may indeed playan important role as a control parameter to set cargo·s direction and thus the ability to changethe course of its movement along the considered microtubule. Fig.3 o⁄ers a more complete viewof the behavior of v m with respect to a broader region of model parameters, at (cid:133)xed number ofcargos : Parameters p and w are both related to oscillations of the attached cargo with respect to themotor·s main symmetry axis. Therefore, if a bound cargo is able to induce a change onto motorwith respect to its symmetry axis, it is conceivable that such change might well be representedthrough a choice of numerical values for these parameters such that p = w ( Fig.3a) . On the otherhand, the reasoning behind a choice that sets k = (cid:11) has already been mentioned above. It is basedon studies of the e⁄ects of viscosity on the motor motility in the presence of an attached cargo[22]. We must emphasize, however, that in spite of these possibilities, we notice in Fig. (3b) that it is not necessary to have p = w neither k = (cid:11) in order to observe changes in the sign of We have introduced in [13] a procedure to compensate for the lack of ergodicity, as the movement of the C-MCcomplexes is not considered explicitly. Such a procedure, however, does not introduce drifts to cargo movement.Thus, the characteristics of the long-range displacements predicted there are due exclusively to motor clustering. m at varying values of (cid:26): Although it becomes clear in these (cid:133)gures that in case k = (cid:11) (Fig.3b) the region of densities within which the signal of v m remains unchanged becomes larger than thecorresponding region in case k = (cid:11) , still there are uncountable possibilities for cargo to reverse itsdirection of movement, either by changing (cid:26) or by changing p (or w ). In other words, a choice ofparameters such that p = w and/or k = (cid:11) is not a necessary condition for our model to describethe bidirectional movement because it depends mainly on clustering, a phenomenon displayed byASEP models even in the presence of a single type of particles (for example, in the absence ofcargos).Data in Fig.2 can be better appreciated with the aid of the accompanying cargo displacementpro(cid:133)les d ( t ) for t (cid:24) T measured in units of global runs (cid:1) t . These are shown in Fig.4 for thesame set of model parameters used to evaluate v m in Fig.2(a), as indicated, for di⁄erent choices ofmotor densities. The observed long-range displacements in each direction result from an interplaybetween two processes. One of those is motor clustering that enables cargo to execute long-rangemovements by hopping to both directions, and it is predominant at high motor densities. Theother process is related to pure C-MC movements. It allows cargo to move steadily in the forwarddirection if there were no impediments on the microtubule; thus, it is predominant at su¢ cient lowmotor densities. Nevertheless, the results achieved here suggest that both processes play importantroles at all motor densities. In fact, at high motor densities C-MC dynamics provides cargo with amechanism to overcome the empty spaces between clusters and reach the next cluster so that it canresume its hopping-based movement. On the other hand, at low densities hopping allows cargo toovercome the problem of having a low number of motors or clusters of motors already assembled,in order to resume its C-MC based movement.The examples of
Fig.4 illustrate these possibilities.
Fig.4(i) displays the trajectory of thecargo under consideration at relatively low motor densities. Within this region it develops a straightmovement, i.e. toward the forward direction (microtubule minus end) at a near constant averagevelocity. As just mentioned, this is likely to be due mainly to the C-MC-based movement. In fact,as shown in
Fig 5(i) , the corresponding average size of the assembled clusters at such low motordensities is very small compared to the typical sizes of clusters assembled at higher densities
Fig5(ii-iv) . Therefore, hopping is not expected to contribute to the observed long-range movementwithin this region. As the density of motors increases, cargo decreases its velocity. Clusteringthen begins to contribute as a mode of cargo transport leading it to display forward as well asbackward movements as it is able to hop over the small clusters
Fig 5(ii) in both directions, as9xplained. Thus, at the point at which the average velocity v m becomes e⁄ectively zero, cargomovement is characterized by large (cid:135)uctuations (Fig 4(ii)) because then hopping would competewith the C-MC-based transport. This situation lasts until motor density becomes su¢ ciently largesuch that large clusters take over (Figs. 5(iii)) enabling cargo to overcome long distances, thistime through hopping. This explains the movement of the cargo toward the plus direction as theclusters are assembled behind it (Fig 4(iii)) . At very high densities, once again cargo switchesthe direction of the drift (Fig 4(iv)) which, in the considered situation, is likely to be due tohopping over large clusters that are assembled at the back end of another cargo present in thesystem. In this case C-MC dynamics just allows cargo to overcome the gap and reach the clustersin front. A similar analysis can be performed for the case shown in Fig.2(b) with a large number ofcargos. The typical sizes of the assembled clusters in this case are much smaller than those shownin
Fig.5 (data not shown). Therefore, although hopping mode still operates at su¢ ciently highmotor densities, specially along clusters at the back of a neighboring cargo, the movement shouldbe imposed by that of the C-MC.In view of this, we may suggest that changes of cargo·s drift direction in long-range displace-ments can be regulated by small variations in the density of motors attached to the microtubuleunder stationary conditions. This might explain the observed bidirectional movement in real sys-tems.
The hopping model for long-range cargo transfer by molecular motors is reviewed and extended inorder to incorporate the dynamics of C-MC complexes. The results for the average cargo velocityobtained by numerical simulation indicate that the bidirectional movement displayed by cargo canbe explained by this extended version of the model, even if there were in the system just a singlecargo driven by single polarity motors.Actually, this can be the case in real systems. Very recently, Roostalu et al. observed bidi-rectional motion of cargos in experiments performed in vitro with single type minus-end directedkinesin-5 Cin8 motor proteins [23]. Although the mechanisms that would trigger the phenomenonare not detailed in their paper, the suggestion made there is that it might be due to a reversal ofCin8 intrinsic polarity in situations in which many motors work together as a team.10e claim here that these new experimental (cid:133)ndings can be accounted for by the hopping modelwith no changes to the properties of motors required. More precisely, if transport by single polaritymotors takes place in the presence of noise that promotes cargo exchange, as explained above,then it would be possible to observe long-range movement of cargos in both directions. We havealready predicted bidirectional movement through this mechanism in model systems possessingtwo or more cargos [13] . Here, we obtain similar results considering, in addition to hopping,unidirectional movement of just a single cargo through a C-MC interacting with the set of othermotors present.As noticed above, these two elements, namely cargo switching and noise have already beenreported in the literature. Here we suggest a way to use them in order to built a model thatdescribes the dynamics displayed by many interacting particles occupying the sites of a 1D lattice.We then disclose the conditions under which motors assemble into relatively large clusters. Theseclusters, in turn, allow cargos to endure a sequence of such elementary hopping steps resulting inlarge displacements in either direction. It is known that ASEP models with only one type ofparticle undergo a dynamic phase transition at which clustering appears to be controllable by theparticle density in the lattice [21]. We have shown that this also happens when cargos are added tothe system. This allows us to conclude that 1) long-range cargo transport can be explained by themechanism of hopping along such clusters, and also that 2) the relative amount (but not necessarilythe polarity) of motors bound to the microtubule, i.e. the de(cid:133)ned motor density parameter (cid:26) , cancontrol the direction of such large displacements. These conclusions come from the study of thebehavior of average cargo velocity with respect to (cid:26) as depicted in
Fig.2.
In addition, the resultssuggest the existence of limiting values for motor densities to control transport operation. Cargodirection and therefore the e⁄ectiveness of cargo delivering would be self-regulated by small changesof motor density, especially if the system is close to the jamming transition.Regarding this point, it is not clear to us how and even if the study performed in Ref.[23] atvarying motor density in gliding assays can be compared to the results achieved here. Those studiesfocus on the properties of cargo-motor interactions; thus, in principle, the results could be used toinvestigate the magnitude of cargo (cid:135)uctuations around a motor·s position as described here. Onthe other hand, the fact that the relatively large cargos considered in these experiments are notallowed to move through the C-MC introduces di¢ culties for a direct comparison between the dataobtained there and the theory discussed here. Notice that due to their (cid:133)nite extension, the cargosconsidered in the experiments can indeed traverse the gaps between clusters of motors with no11eed for the C-MC mechanism. Yet, it is noticeable the similarity between the qualitative behaviorshown in the experimental results and the predictions made here, within the considered motordensity range. In any case, as argued by Roostalu et al., the quantity of motors at the microtubuleseems to be an important tool for controlling cargo movement and direction. This is completelyconsistent with our previous claims [12, 13, 24] and it is emphasized by the results presented here.In the data referred to above, one observes motor accumulation near the cargo being observedas it moves towards the plus-end side of the microtubule, in the opposite direction of individualCyn-8 Kinesin motors. To explain these data the authors have suggested that i) the e⁄ect re(cid:135)ectssome collective properties since motors work as a team to move the cargo, and ii) such collectiveproperty would then induce motors to change their intrinsic polarity. They concluded that theCin-8 motors themselves may behave as bidirectional motors - individually, they would follow theirminus-end intrinsic polarity, whereas, if working as a team, they would move and transport cargoaccording to the C-MC mechanism toward the plus-end direction. There is no attempt in theirwork to elucidate the mechanisms responsible for the alleged change in polarity.We argue that there is another way to think about this data based on the ideas discussed here.In the context of the hopping model one does not require changes in individual motor polarity toexplain the observed movement of the team of motors, although the relevant e⁄ect would indeed beattributed to collective properties developed by the system due to the global nature of the jammingprocess. Jamming depends on the dynamics and interactions of all motors and cargos present inthe system, not just on the properties of the motors participating in the local team. Accordingly,we do understand why and how motors can accumulate next to cargo, as observed, just becausethe presence of cargo, although not being a necessary condition, enhances the conditions for motorjamming in its neighborhood.The fact that a motor cluster can indeed move toward the opposite direction from that of theconstituting motors may be better appreciated with the aid of
Fig.6 . It illustrates a situation inwhich a cluster is being formed. Motors moving toward the minus end would encounter the cargoand get jammed behind it. In turn, the presence of this cluster would induce cargo to step overit, toward the plus direction. Therefore, motors that were previously accumulated behind a cargowould pass to a position in front of it (because cargo moves back) and this would tend to dispersethe cluster, as these motors, now free to move, would continue their movement toward the minusend of the microtubule. On the other hand, motors continuously reaching the cluster at its backend would tend to increase the cluster. Thus, at the same time that the cluster loses motors in12ront of it, it gains motors behind so as to appear that it is moving toward the plus-end direction,opposite to the intrinsic motor polarity. A balance between the tendency of losing motors andacquiring motors would eventually equilibrate a cluster·s size. In conclusion, the cluster (not themotors!) would appear to move to the plus-end direction due to a dynamic process of losing andgaining motors, but not because of changes in individual motor polarity. The cargo, on the otherhand, if able to hop over the cluster, it may move in either direction, but the drift would be towardthe plus-end, accompanying overall cluster movement.Of course the dynamics exempli(cid:133)ed above can be understood as a microscopic descriptionof a shock wave [19] in this context, similar to the continuum version studied in Ref. [24]. Wemay then say that the hopping dynamics discussed here indeed expresses the relevance of thecollective behavior of motors and cargos to the transport process and o⁄ers a novel description tothe phenomenon. The results are simple, although nontrivial, in many respects, and they includea description of bidirectional e⁄ects.Nonetheless in more realistic cases within the cell environment, one should not exclude thepossibility of the presence of motors of both polarities on the same microtubule. Notice, however,that in the context discussed here, we understand that the presence of both kind of motors wouldsimply enhance the conditions for motor jamming [25]. Thus, in principle, the presence of di⁄erentmotors would in fact create more possibilities for cargo to move in both directions but not necessarilyas an e⁄ect of local coordination or competition between motors of di⁄erent polarities, but instead,as a consequence of the combined e⁄ects that these two kinds of motors would have on tra¢ cjamming and thus on motor clustering. Once again, this emphasizes the idea that the kind of long-range movement discussed here expresses collective e⁄ects involving all particles on the microtubulesince jamming is not a local phenomenon.Finally, we should notice that the mechanism discussed here does not require special stabilityof cargo-motor binding. On the contrary, the exchange of cargos would be facilitated both by anunsteady attachment between cargos and motors and, of course, by the (cid:135)exibility of the motor tail.
It is interesting to estimate the energy cost E h associated with cargo transport in the context ofthe extended hopping model to compare with an equivalent quantity E c (cid:0) mc for pure C-MC models.This can be done, for example, by estimating the energy required in each case to drive a cargo13etween two lattice sites that are far apart. For simplicity, we restrict the analysis to the casein which the movement of cargos takes place only in a de(cid:133)nite direction, to the minus-end, say.This condition is accomplished in the context of the extended hopping model by setting ! = 0 in (1). The corresponding condition in the context of C-MC models exists in the case in whichsingle polarity motors are present. Our aim is to obtain lower and upper bounds to the quantity E h =E c (cid:0) mc , the ratio between the corresponding energies in the two models.Let (cid:14) and (cid:14) be the energies required in the processes ! and ! , respectively, andlet (cid:14) be the energy cost for exchanging a cargo between neighboring motors, process ! .Considering that the energy necessary for a motor protein to move one step forward is of the orderof the energy released by the hydrolysis of one ATP molecule [26], we estimate (cid:14) (cid:24) (cid:1) (cid:0) J: For (cid:14) we use the energy associated with thermal (cid:135)uctuations needed for hopping. Accordingly, (cid:14) (cid:24) k B T (cid:24) ; (cid:1) (cid:0) J at body temperatures. It is more di¢ cult to estimate (cid:14) . Yet, becauseboth (cid:14) and (cid:14) are related to the step of a motor we may consider that (cid:14) & (cid:14) .The energy required in pure C-MC models is that for carrying a cargo along a distance com-prising the whole set of N sites, starting and ending at site since the system presents periodicboundary conditions. Let n be the total number of cargos to be transported and n the totalnumber of motors distributed along the N sites. For simplicity, we suppose that cargos are allowedto attach to a single motor each, and also that there is present only one cargo in the system. Con-sequently, n = 1 and n = n (cid:0) is the number of motors bound to the microtubule at each timethat carries no cargo. We now consider the possible con(cid:133)gurations that may be reached by thesystem in a delivering process, starting from a con(cid:133)guration in which the n motors are distributedin sequence between sites labeled N (cid:0) n + 1 and N: The C-MC starts at position N + 1 = 1 followingperiodic boundary conditions. In this situation the energy cost to move the cargo by means of theC-MC exclusively would assume a minimum value. This is because the motors arranged in this waywould need to move forward just along a minimum number n + 1 of sites each, in order to provideenough space to the C-MC for reaching site again, after completing the cycle. Any other initialarrangement would require that motors move along a larger number of sites than n + 1 . Then, theminimum energy E (0) c (cid:0) mc required for the complex to complete its way across the N sites can beestimated as E (0) c (cid:0) mc ’ (cid:14) n ( n + 1) + (cid:14) N (3)where the index (0) in E (0) c (cid:0) mc refers to the initial con(cid:133)guration under consideration. The (cid:133)rst termat the RHS accounts for the energy to move the n unbounded motors across ( n + 1) lattice sites;14he term in (cid:14) accounts for the energy to move the complex with its cargo.Starting from the same initial con(cid:133)guration, we are now able to determine upper and lowerbounds for the corresponding energy E (0) h . In the context of the extended hopping model, cargo isalso allowed to move by hopping, and thus spending less energy per step ( (cid:24) (cid:14) ) if compared to themovement through the C-MC ( (cid:24) (cid:14) ). A minimum amount E (0) h (min) = (cid:14) n + (cid:14) ( N (cid:0) n ) of energyis required to complete the circle in cases for which cargo hops (instead of moving with the aid ofC-MC) along the maximum number n of unbounded motors. This is accounted for by the term in (cid:14) . The term in (cid:14) accounts for the energy to move cargo along the unoccupied sites as it attachesto a motor to form a C-MC complex. If, however, all of the n unbounded motors move in orderto provide space to the C-MC, the energy involved in completing the circle would be exactly thesame as E (0) c (cid:0) mc given by (3). This means that (cid:14) n + (cid:14) ( N (cid:0) n ) E (0) h E (0) c (cid:0) mc or, in terms of (cid:14) =(cid:14) (cid:17) " << ; f N ( "; (cid:26) ) (cid:20) E (0) h E (0) c (cid:0) mc (cid:20) (4)where we have de(cid:133)ned f N ( "; (cid:26) ) (cid:17) " (cid:0) (cid:26) N (cid:26) (5)A remark is in order here in respect to the multiplicity of cases for which hopping may becombined with the C-MC movement. Within the pure C-MC context, any attempt by the complexto complete the path would necessarily involve energies equal to E (0) c (cid:0) mc : When hopping is addedto the dynamics, it creates a large number of possibilities for trajectories that can be followed bycargo, most of them accomplished by spending energies that are signi(cid:133)cantly less than E (0) c (cid:0) mc : Highenergies would be required only in the very rare occasions in which the path is accomplished withno hopping or just a few events of hopping.
Fig.7 shows the behavior of the gap g N ( "; (cid:26) ) = 1 (cid:0) f N ( "; (cid:26) ) (6)between the upper and lower bounds expressed by Eq. (4) as (cid:26) varies, and at di⁄erent values of N .At the scales being considered, the function g N ( "; (cid:26) ) is practically insensitive to variations of " inthe range : (cid:20) " (cid:20) : (results not shown). Notice, however, that g N ( "; (cid:26) ) increases with (cid:26) . Thismeans that by increasing motor density, the number of possibilities for cargo to follow a path thatrequires less energy than E (0) c (cid:0) mc increases. There is an accompanying increase in the "entropy",namely in the number of di⁄erent paths involving the same number of hops and the same numberof C-MC steps that may be followed in di⁄erent combinations. Thus, if the multiplicity of paths is15ccounted for, one might conclude that in the context of the extended hopping model, the eventsthat do not involve hopping do not occur in practice. The curves in Fig.7 suggest that, regardingenergy costs, the two schemes - extended hopping and pure C-MC- become comparable only atarti(cid:133)cially low motor densities .
Acknowledgements
This work is supported by Funda(cid:231)ªo de Amparo (cid:224) Pesquisa do Estado de Sªo Paulo (FAPESP).
References [1] Gross S. P., Hither and yon: a review of bidirectional microtubule-based transport, Phys. Biol.,1, R1-R11 (2004).[2] Welte M. A., Bidirectional Transport Along Microtubules, Curr. Biol. 14, R525-R537 (2004).[3] Zeldovich K.B. , J.F., Joanny, J. Prost, Motor proteins transporting cargos, Eur. Phys. J. E17, 155-163 (2005).[4] Klumpp S., M. J. M(cid:252)ller, R. Lipowsky, Cooperative transport by small teams of molecularmotors, Biophys. Rev. and Lett, 1, 353-361 (2006).[5] Ross, J. L., M, Yusef-Ali, D. M. Warshaw, Cargo Transport: molecular motors navigate acomplex cytoskeleton, Curr. Opin. Cell Biology 20, 41 - 43 (2008).[6] Seitz A., T. Surrey, Processive movement of single kinesins on crowded microtubules visualizedusing quantum dots, EMBO J. 25, 267 - 277 (2006).[7] Mallick R., S. P. Gross, Molecular Motors: Strategies to Get Along, Curr. Biol., 23, R971-R982(2004). 168] Kulic I. M., A. E. X. Brown, H. Kim, C. Kural, B. Blehm, P. R. Selvin, P. C. Nelson, V. I.Gelfand, The role of microtubule movement in bidirectional organelle transport, Proc. Natl.Acad. Sci. USA 105, 10011-10016 (2008).[9] Ally S., A. G. Larson, K. Barlan, S. E. Rice, V. I. Gelfand, Opposite-polarity motors activateone another to trigger cargo transport in live cells, J. Cell Biol. 187, 1071 - 1082 (2009).[10] Gur B., O.Farago, Biased Transport of Elastic Cytoskeletal Filaments with Alternating Polar-ities by Molecular Motors, Phys. Rev. Lett 104, 238101 (2010).[11] Ebbinghaus M., C. Appert-Rolland, L. Santen, Bidirectional transport on a dynamic lattice,Phys. Rev. E 82, 040901(R) (2010).[12] Goldman C., E. T. Sena, The dynamics of cargo driven by molecular motors in the context ofan asymmetric simple exclusion process, Physica A 388, 3455-3464 (2009).[13] Goldman C., A Hopping Mechanism for Cargo Transport by Molecular Motors on CrowdedMicrotubules, J. Stat. Phys. 140, 1167-1181(2010).[14] Kolomeisky, A.B, B. Widom, A simpli(cid:133)ed ratchet model of molecular motors. J. Stat. Phys.,93 (1998), 633(cid:150)645.[15] Petrov, D. Y., R. Mallik, G. T. Shubeita, M.Vershinin, S. P. Gross, C.C. Yu, Studying Molecu-lar Motor-Based Cargo Transport: What Is Real and What Is Noise?, Biophys, J. 92, 2953-2963(2007) . [16] Kunwar, A., M.Vershinin, J. Xu, S. P. Gross, Stepping, strain and the unexpected force-velocitycurve for multiple-motor-based transport, Curr. Biol. 18, 1173 - 1183 (2008).[17] Aghababaie Y., G. I. Menon, M. Plischke, Universal properties of interacting Brownian motors,Phys. Rev. E 59, 2578(cid:150)2586 (1999).[18] A.Parmeggiani, T. Franosch, and E. Frey, Phase Coexistence in driven one-dimensional trans-port. Phys. Rev. Lett.
90, 086601-1 (cid:150)086601-4 (2003).[19] Ferrari P.A. , C. Kipnis, E. Saada, Microscopic Structure of Travelling Waves in the Asym-metric Simple Exclusion Process, Ann. Prob. 19, 226 - 244 (1991).[20] Derrida B., S.A. Janowsky, J.L. Lebowitz, E.R. Speer, Exact solution of the totally asymmetricsimple exclusion process: shock pro(cid:133)les, J. Stat. Phys. 73, 813 - 842 (1993).1721] Blythe R. A., M.R. Evans, Nonequilibrium steady states of matrix-product form: a solver(cid:146)sguide, J. Phys. A: Math Theor. 40, R333-R441 (2007).[22] Erickson, R. P., Z. Jia, S. P. Gross, C. C. Yu, How molecular motors are arranged on a cargois important for vesicular transport, PLOS Comp. Biol.,7, e1002032 (2011).[23] Roostalu, J., C. Hentrich, P. Bieling, I. A. Telley, E. Schiebel, T. Surrey, Directional switchingof the Kinesin Cin8 through motor coupling, Science 332, 94 - 99 (2011).[24] Lichtenth(cid:228)ler, D.G., C. Goldman, Sur(cid:133)ng at the wave fronts: the bidirectional movement ofcargo particles driven by molecular motors, Bioscience Hypotheses 2, 428-438 (2009).[25] Lee, H. W., V. Popkov, D. Kim, Two-way tra¢ c (cid:135)ow: exactly solvable model of tra¢ c jam,J. Phys. A: Math Gen. 30, 8497 - 8513 (1997).[26] Howard J.,
Mechanics of Motor Proteins and the Cytoskeleton , Sinauer Associates Inc., 2001.[27] K. Kawaguchi, S. Uemura , S. Ishiwata, Equilibrium and transition between single- and double-headed binding of kinesin as revealed by single-molecule mechanics, Biophys. J. 84, 1103 - 1113(2004). 18 igure Caption
Figure 1 -
Dynamics of motors and cargos. (a) Step of a motor. The time spent by the motorwith the two heads attached to the microtubule is much larger than the time it spends with justone of the heads attached to it [27]. This is part of the "hand-over-hand" mechanism proposedto explain the kinetics of two-headed motor proteins [26]. In view of this, we shall consider thatoccupation of a site by a motor occurs whenever it is occupied by the two heads. Cargo hoppingoccurs through a mechanism of exchange between neighboring motors. Due to the (cid:135)exibility of themotor tails, the attached cargo may be caught either (b) by the motor at its right or (c) by themotor at its left. (d) Elementary dynamics of a C-MC complex.
Figure 2 -
Average cargo velocity v m as a function of motor density (cid:26) . The parameters usedare N = 100 ; T = 10 . The rates and number of cargos are such that w = 0 : ; p = 0 : ; k = 0 : ;(cid:11) = k= ; (a) n = 2 and (b) n = 15 . Insert in Fig.(2a) for T = 10 at a region of low motordensities shows that the behavior of v m in this example remains essentially the same as T increases fold suggesting that stationary conditions have been achieved in the course of the simulations. Figure 3 -
An enlarge view of the behavior of v m as a function of (cid:26) and the rate p chosensuch that p = 1 (cid:0) w; for (a) k = 0 : ; (cid:11) = k= and (b) k = (cid:11) = 0 : : Notice that although notnecessary, such a relation between p and w o⁄ers a better view of the di⁄erent possibilities forcargo·s behavior for p = w: Figure 4 -
Cargo displacement d ( t ) as function of time t , at speci(cid:133)c values of (cid:26) chosen fromthe velocity pro(cid:133)le in Fig.(2a) for N = 100 ; T = 10 and (i) (cid:26) = 0 : (ii) (cid:26) = 0 : (iii) (cid:26) = 0 : (iv) (cid:26) = 0 : : The inserts in (i) and (ii) illustrate the magnitude of the (cid:135)uctuations of d ( t ) withinthe respective regions of motor densities. Figure 5 -
Average cluster size distribution at the corresponding points (i), (ii), (iii) and (iv)of Fig.(2a), for N = 100 and T = 10 . 19 igure 6 - Cluster dynamics and a microscopic view of a shock wave. Cluster and cargo presenta drift toward the plus end whereas motors move in the opposite direction.
Figure 7 -
The behavior of the gap g N ( "; (cid:26) ) as a function of (cid:26); for " = 0 : and N as indicated.20 Figure 1(a) opping
Figure 1(b) opping
Figure 1(c) igure 1(d)
20 02 ig. 2 motor density v m ( s i t e s / t ) (a) (i) (ii) (iii) (iv) motor density v m ( s i t e s / t ) (b) ig. 3 (motor density) p = 1 - w v m ( s i t e s / t ) -0.3-0.2-0.100.10.20.3 (motor density) p = 1 - w v m ( s i t e s / t ) -0.4-0.3-0.2-0.100.10.20.3 (a) (b) ig. 4 Rho=0.19: Rho=0,50: Rho=0,90: t) d ( s i t e s ) (i) -6000-5000-4000-3000-2000-10000 t ( t ) d ( s i t e s ) (i) t) d ( s i t e s ) (ii) t) d ( s i t e s ) (iii) t) d ( s i t e s ) (iv) t) d ( s i t e s ) (i) t ( t) d ( s i t e s ) (ii) t) d ( s i t e s ) (ii) ig. 5 f r equen cy ( i ) f r equen cy ( ii ) f r equen cy (iii) f r equen cy (iv) f r equen cy
10 15 20 25 30 3500.0020.0040.0060.0080.010.012