Surfing at wave fronts: The bidirectional movement of cargo particles driven by molecular motors

Bioscience Hypotheses (2009) 2, 428e438 available at www.sciencedirect.com

journal homepage: www.elsevier.com/locate/bihy

Surfing at wave fronts: The bidirectional movement of cargo particles driven by molecular motors ¨ler, Carla Goldman* Daniel Gomes Lichtentha Departamento de Fı´sica Geral, Instituto de Fı´sica Universidade de Sa˜o Paulo, CP 66318, 05508-900 Sa˜o Paulo, Brazil Received 23 March 2009; accepted 31 March 2009

KEYWORDS Collective effects of molecular motors; Cargo transport; Non-linear partial differential equations; Shock waves

Abstract We examine the properties of intracellular transport of particles (vesicles, organelles, virus, etc.) in the realm of models that describe the dynamics of interacting molecular motors moving along microtubules. We use a continuum description of motor distribution and argue that certain features of cargo movement have their origins on its ability to perturb the existing motor profile and to surf at the resulting shock wave fronts that separate regions of different motor concentrations. In this case, the observed bidirectionality of cargo movement is naturally associated with reversals of shock direction. Comparison of the quantitative results predicted by this model with available data suggests that the geometrical characteristics of cargo may be related to the extension and intensity of the perturbation they produce and thus, to their kinetics. Possible implications of these ideas to understand features of the movement of virus particles within the cell body are discussed in connection with their distinguished morphological characteristics. ª 2009 Elsevier Ltd. All rights reserved.

Introduction The active transport of particles (organelles, vesicles, virus) inside the cells is mediated by motor proteins such as myosin, kinesin and dynein [1]. The unidirectional motion of a single motor along protein filaments or microtubules is well characterized experimentally and it was first modeled at a microscopic level by a stochastic dynamics describing the behavior of a Brownian particle in the presence of a time-dependent

* Corresponding author. Tel.: þ55 11 3091 6766; fax: þ55 11 3814 0503. E-mail address: [email protected] (C. Goldman).

asymmetric potential field [2e5]. The idea is based on the mechanism of ‘‘ratchet and paw’’ introduced by Feynman to discuss the meaning of the second law of thermodynamics [6]. Since then, this model has been used as a prototype to explain why and in what conditions Brownian particles are able to do work against external potential gradients. More recently, it appeared in the literature attempts to describe the properties resulting from the movement of many interacting Brownian motors with particular interest on the nature of the observed bidirectional movement accomplished by cargo for which the combined action of many motors appears to be crucial [7e10]. The non-diffusive bidirectional movement is characterized by inversions of cargo direction after

1756-2392/$ - see front matter ª 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.bihy.2009.03.010

Surfing at wave fronts processive runs 1 that may be preceded by relative long resting times (intervals within which the particles remain at the same position on the microtubule). Quantitative models that describe the movement of relative extensive objects such as filaments in motility assays suggest that bidirectionality arises in this case from the collective behavior of considered grafted motors acting on a same cargo [11]. It is not clear, however, how to extend these models to describe the observed bidirectional movement of relative small particles, as vesicles or virus, that happens at microtubules where the carrying motors are free to move (not grafted). The most accepted explanation in this case comes from the referred coordination model [7,9]. According to this, the bidirectional movement would result from the coordinated action of two types of motors e plus-ended and minus-ended motors e attached simultaneously to cargo. The reversal of its direction would just reflect the fact that one or other type of motor, but not both simultaneously, happens to be active during the respective time intervals. Motor coordination, that is, the control of each motor activity, would be accomplished by an external non-motor protein complex that should be able to coordinate the movement and timing of many motors of different nature and distinct characteristics e certainly a very non-trivial job. Up to now, such an external complex still needs to be identified in real systems [12]. Another possibility within the same lines is based upon the existence of a tug-of-war competition mechanism between two types of motors that may happen, according to a recent analysis [13], in the absence of a coordinator. In any case, however, it remains the problem to explain how and in what conditions the transport can get into effect, avoiding the traffic-jam that may be expected as a consequence of the presence of many motors and cargoes dividing space on a single microtubule and, eventually, moving to opposite directions [14]. It is thus accurate to attribute to the bidirectional transport of cargo the status of a ‘‘major puzzle in the context of in vivo intracellular transport’’, as in a recent work by Kulic and co-worker [15]. At certain scales of interest, phenomena related to the collective behavior of interacting Brownian motors have also been considered using models to describe dynamic aspects of continuum motor distributions. Such approach is generally justified upon evaluation of the characteristic sizes and time scales where these molecules operate at very low Reynolds numbers [16] and it is based on the continuum versions of the ‘‘asymmetric exclusion processes’’ (ASEP) [17e20] for studying the long-time behavior of motors interacting through short-range interactions as excluded volume [16,21e23]. From this perspective, the (microscopic) asymmetric movement of individual motors is assumed a priori. The general interest in this context consists in analyzing the steady-state behavior of a defined motor density distribution. These are obtained as the solutions to the corresponding non-linear

1

Here, processive run makes reference to the movement accomplished towards a definite direction before inversion or detachment from microtubule.

429 differential equation e the viscous Burgers equation, that describes the dynamics of the system in these limits e with open boundary conditions. It is also possible to superimpose to the ASEP a non-conservative Langmuir process that allows the system to exchange motors with the exterior bulk at any position of the microtubule [22]. Apparently, however, questions related to cargo transport, have not been considered in this context. Here, we make a proposal in this direction based on considerations about interactions among cargoes and motors. It relies on the idea that a cargo is able to produce local changes onto a priori motor density distribution, as it approaches the microtubule at a defined initial time. Thereafter cargo shall take advantage of the gradients of motor density, induced by this initial perturbation, to move along microtubule by surfing at the density shock waves formed as the motor system relaxes back to the original profile. Shock waves separate regions of different motor densities and evolve in this case according to the transient solutions to the Burgers equation for the considered initial conditions (or initial perturbation). Within this view, bidirectional movement of cargo particles would follow as a direct consequence of the reversals in shocks propagation directions. Surfing of bacteria along thin nanotubes [24] and surfing of virus along the surface of filopodia [25] have been observed and are examples of systems where bidirectional movement happens in the presence of a single motor type. It is also noticeable in this respect the experimental data published recently showing the occurrence of vectorial transport of small floating objects moving at the fronts of chemical waves produced in a BelousoveZhabotinsky (BZ) type of chemical reaction [26]. This strongly supports the ideas presented here and suggests that, although the motor/cargo system is rather distinct from that considered in these experiments, it may exhibit an equivalent mechanism to that responsible for the transport within the BZ medium. In Section 4 we discuss one possibility. We use data from the movement of vesicles in Drosophila embryo [27] to perform a phenomenological analysis of our model considering the explicit expressions predicted for shock velocities. As we shall see, this allows one to associate geometrical characteristics of cargo with the extension and magnitude of the considered initial perturbation it produces on motor density profile. In view of this, we discuss on possibility to use these results to understand aspects of the virus movement within the cell body.

From the microscopic potential models to the macroscopic motor density profile In order to present our ideas in the context of the related works in the literature, it is convenient to review some aspects of the existing theoretical studies intended to describe the dynamics of interacting motor particles. In particular, we consider the usual procedure to obtain the continuum limit in the mean field approximation (the macroscopic limit) of an ASEP model representing the stochastic dynamics of motors that move by jumping from one site to another in a one-dimensional lattice. Within this

430 view, the underlying (microscopic) diffusion process is not accounted for explicitly; its effects on the movement of each motor, under the influence of the time-dependent asymmetric potential field, are assumed a priori. Yet, the parameters that define the asymmetry of the potential can be incorporated into the considered ASEP. To see this and to fix the notation used here, we briefly review the mechanisms introduced in Refs. [2e5]. Consider the profile shown in Fig. 1. This represents the state ‘‘on’’ of the asymmetric and periodic potential field. The position of each minimum is usually associated to the position of a onedimensional lattice site; the distance between neighbor sites being l Z b þ d coincides with the spacial period of the potential. The difference between parameters b and d defines the extension of the asymmetry. The time dependence is such that it alternates between the state ‘‘on’’ and the state ‘‘off’’ (flat potential). The state ‘‘off’’ allows the particle to diffuse freely starting from the position i of a local minimum of the potential that existed when it was in the state ‘‘on’’. This diffusive movement of the particle persists within a time interval t after which the potential is turned to the state ‘‘on’’ again. At this time, the particle is supposed to fall immediately into the nearest minimum of the potential that reappeared (adiabatic approximation), either to the right or to the left. But, because d > b, it is more likely that the nearest minimum achieved by the particle be that at site i þ 1. The process can be repeated and the unidirectional movement of the particle is explained in this way on the basis of a diffusive process coupled to an external chemical reaction (ATP hydrolysis), not in equilibrium with the system, that promotes the conversions between the two states of the potential. ASEP models have been introduced more recently in this context as a way to account for collective effects of many motors moving along a common lattice [21,23]. At these scales, motors are self-driven and the interactions among them are short-range represented by excluded volume, i.e., a motor cannot occupy a site already occupied by another motor. Let us now consider a stochastic description of the process at this scale. The time unit being the interval t within which a particle can jump to a neighbor site with probability pi to move from site i to site i þ 1 and probability qi to move from site i to site i  1 (Fig. 2). Neglecting correlations (mean field approximation) the average density ri(t þ t) of motors at site i, at time t þ t satisfies the following recurrence relation

D.G. Lichtentha ¨ler, C. Goldman Excluded volume interactions are introduced into the model by attributing to pi and qi an explicit dependence on the occupation of the target sites: pi ðtÞZpð1  riþ1 ðtÞÞ

ð2Þ

and qi ðtÞZqð1  ri1 ðtÞÞ

ð3Þ

Here, p and q are dimensionless constants such that p, q < 1. Now, let N be the total number of lattice sites and let L h 1 be its total length. As in Ref. [22], the discrete model can be coarse-graining with lattice constant l Z 1/N to a continuum such that for N/N the density ri(t) becomes a function r(x, t) of a continuum variable x Z i/N. In this limit, ri  1 can be expanded in powers of l: ri1 ðtÞZrðx  l; tÞZrðx; tÞ  lvx rðx; tÞ   þð1=2Þl2 v2x rðx; tÞ þ O l3 and the recurrence in (1) becomes  ðrðx; t þ tÞ  rðx; tÞÞ=tZg ððp þ qÞ=2Þlv2x rðx; tÞ þ ðp  qÞð2rðx; tÞ  1Þvx rðx; tÞ   þ O l2

Figure 1 The asymmetrical potential field introduced in Refs. [3e5]. This is defined by parameters b and d with d > b.

 ð5Þ

where we have defined gZl=t

ð6Þ

For t/0 and l/0 keeping g in (6) finite and neglecting terms of O(l ), Eq. (5) converges to the non-viscous Burgers equation vt r þ gðp  qÞð1  2rÞvx rZ0:

ð7Þ

Now, a connection between the above description and the microscopic model can be made by setting pZd=l and qZb=l

ð8Þ

so that Eq. (7) is rewritten as vt r þ Kvx ðrð1  rÞÞZ0

ð9Þ

where we have defined Khgðd  bÞ=lZðd  bÞ=t

ð10Þ

as a finite and positive constant with dimension of [length].[time]1. Eq. (9) can also be expressed as vtr þ vxj Z 0 with the particle current given by jðx; tÞZKrðx; tÞð1  rðx; tÞÞ:

ri ðt þ tÞZri ðtÞ þ riþ1 ðtÞqiþ1 ðtÞ þ ri1 ðtÞpi1 ðtÞ  ri ðtÞðpi ðtÞ þ qi ðtÞÞ ð1Þ

ð4Þ

ð11Þ

Considering open boundary conditions, the non-equilibrium steady-state solutions to this equation have already been explored in the references mentioned above to study the long-time behavior of molecular motors dynamics. This allows one to make predictions on the stationary properties of motor density profile along the microtubules as, for example, on the emergence of stable domain walls separating regions of different motor densities, which have already been confirmed by experiments [28]. Here, we consider a different problem. We ask for the short-time behavior of the system described by Eq. (9), after a perturbation that can produce local changes on the motor density distribution, with respect to the steady-state solution rN Z 1/2. Such perturbation defines a particular choice of initial conditions to be considered in this problem.

Surfing at wave fronts

431

Figure 2 The ASEP model. Each motor (diamond) is allowed to jump to its nearest neighbor site (if it is non-occupied) with probability p if the target site is at its right, or q, if target site is at its left.

Shocks as cargo drivers It is known that perturbations around the steady-state solutions of Burgers equation (9) may evolve as shock waves. The microscopic origins of shocks in this case are associated with the considered excluded volume interactions among motors at the microtubule and with the stochastic nature of individual motor dynamics described by the corresponding ASEP model [17e20]. From a macroscopic point of view, shocks arise because of the non-linear nature of Eq. (9) as motor density inhomogeneities evolve to discontinuous solutions [29,30]. Here, we want to attribute to the kinematic properties of these shock waves the origins of the observed bidirectional movement of cargo. The idea comes from the observation that shock wave fronts may end up to move in opposite direction of that of individual motor particles if the average rate of motor accumulation in already dense regions exceeds typical (average) motor velocities (Fig. 3). In order to illustrate this, we solve Eq. (9) for periodic boundary conditions considering rðx; 0ÞZ1=2 for x < 0 rðx; 0ÞZ1=2  3 for 0 < x < 2a rðx; 0ÞZ1=2 for 2a < x < 3a rðx; 0ÞZ1=2 þ 3 for 3a < x < 4a rðx; 0ÞZ1=2 for x > 4a

ð12Þ

as the initial conditions (IC) for motor profile (Fig. 3(a)). Parameter a is related to the extension of this initial perturbation and the dimensionless parameter 3 is a measure of its magnitude. The calculations are performed in the Appendix and the solutions obtained for the density profile for all t > 0 are represented in Fig. 4, where the arrows indicate the directions of the traveling shock fronts. The expressions obtained for the traveled distances and corresponding average shock velocities at each time interval, up to an eventual encounter with other fronts, are compiled in Table 1. Now, cargo is introduced into the system based on two general ideas, namely (i) an a priori set of moving motors at the microtubule provides to cargo (any cargo) an assessable density background, and (ii) the approach of cargo to this motor background at a defined initial time, modifies such motor density profile inducing in this way local inhomogeneities into motor distribution, the choice in (12) being just an example. For the properties of shocks to reproduce the kinematics of cargo observed in vivo an additional condition must be satisfied. This is related to the possibility that after perturbing the motor system, cargo resumes its movement as it gets trapped at a wave front and continue on its course by following the movement of the shock. In this case, cargo

shall exhibit a surfing type of movement, as the motor density profile globally relaxes back to the situation (steady-state) it had before perturbation.2 In the next section, we suggest a mechanism that may lead the system to satisfy these conditions based on motor number conservation. The results in Table 1 are used to examine the consequences of these hypotheses.

Discussion In general, the main difficulty to construct a theory that at the same time explains the origins and makes quantitative predictions on the dynamics of both molecular motors and cargo driven by these motors, relies precisely on the fact that the existing models have been conceived to explain the observed drift of motor movement to a preferential direction that depends on motor type. For this, it has been proposed that the observed bidirectionality of cargo movement results from a coordinated action of two types of motors, the remaining question being to explain the ways to achieve this coordination. The alternative mechanism proposed here does not rule out this possibility. It does not, however, use this as a necessary condition as required in the formulation of the coordination or tug-of-war models. Actually, there is nothing new in the above formulation about the origins and evolution of shock wave fronts or in the general description of motor dynamics considering the discrete or continuum versions of asymmetric exclusion processes. The novel idea presented here is concerned with the possibility to relate the kinematic properties of the shocks in the evolving motor density profile with the corresponding characteristics of cargo dynamics. In particular, bidirectionality of cargo movement appears as a consequence of the reversals of shocks directions while individual carrying motors perform their movement with a preferred direction towards one of the microtubule ends. In a recent publication one reports experimental data showing for the first time the occurrence of vectorial transport of macroscopic objects (small pieces of paper) as they become trapped at the wave fronts produced in chemical reactions of a BZ type [26]. In this case, trapping is attributed to existing differences between the interfacial tension gradients at the two front sides. 2 Of course, the presence of the cargo must affect the motor dynamics and thus, the relaxation process. We do not account for these effects in the present study since they should not change significantly the macroscopic structures of the shocks, at least to the extent of the phenomena (i.e., direction reversals) that we want to illustrate here. To accomplish this, it would be necessary to modify the initial microscopic ASEP model to incorporate cargo explicitly [31].

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D.G. Lichtentha ¨ler, C. Goldman

Figure 3 A macroscopic view of the motor/cargo dynamics. The evolving motor density profile is shown at different instants of time. The cargo is represented at each instant at the shock front, the arrows pointing to the direction of the movement. Fig. 3(a) shows that the considered initial condition (12) corresponds to a local perturbation on motor density with respect to its steady state r(x, t) Z 1/2.

One may argue that trapping can also be achieved in the cargo/motor model system if the average number of motors Nc that remains attached to cargo within the course of its movement, and corresponding motor density r(xc(t), t) Z rc, are conserved quantities, in average at least. Here, xc(t) represents the positions at the microtubule occupied by the motors interacting directly with cargo at each time t. Notice that the conservation requirement is just on the number (or density) of attached motors, not on the particular motor particles. Therefore, some of the motors that at certain instant of time are attached to cargo may be able to detach at a subsequent instant of time, when they are immediately replaced by other motors that bind at different sites. The cartoon in Fig. 3(a) illustrates a situation where cargo is at the edge of a dense motor region. While motors coming from the left are being accumulated in this region of the microtubule, this imposes to cargo a backward movement in order to keep r(xc(t), t) Z rc Fig. 3(b). This is equivalent to assert that while motors move according to the underlying relaxation dynamic determined by the initial value problem posed above, cargo moves to accompany the regions at the microtubule where r(xc(t), t) Z rc. But notice that such a set of points are precisely those expanded by the characteristic curves of equation (9) that emanate from the regions where r(xc(0), 0) Z rc at the initial time (see the Appendix). Therefore, the idea that cargo moves in a way to follow at all times the regions of specified motor density at the microtubule is equivalent,

from a mathematical point of view, to conceive that, at all times within the transient regime, cargo follows the characteristic lines where r Z rc. Fig. 5 shows that if this is the case then necessarily cargo ends up trapped at a shock front where the characteristic curves merge. Suppose, as an example, that rc Z 1/2 (see Fig. 3). Then, at the initial time cargo would find, or create, a region where the motor density at the microtubule equals to r(xc(0), 0) Z 1/2. Its subsequent movement will be such to follow this region as it is driven by the underlying motors. In a very short period of time, as such region moves according to the process described by Eq. (9), cargo would encounter a shock front to which it necessarily gets trapped. The process is similar to that undergone by the floating objects in the BZ medium. In the case of motor/cargo system, crossing such fronts would drive cargo to a region of different motor density. Then, in order to achieve the condition r(xc(t), t) Z 1/2, cargo must change direction by following the movement of the wave front.3

3 The waves that are considered here represent perturbations on the motor density r(x, t), obtained from the microscopic (discrete) motor distribution by the limiting procedure of Section 2.These are thus longitudinal (compression waves), not transversal, otherwise more than one motor would allow to compete for the same lattice site. Here, we do not consider this possibility.

Surfing at wave fronts

433

Figure 4 A microscopic view of the motor/cargo dynamics. Cargo interacts with a set of motors at the neighborhood of regions containing different number of motors per unit length. Local inhomogeneities of motor distribution along the track are emphasized in the figure. The particular profile shown in Fig. 4(a) e the initial condition e and in Fig. 4(b) corresponds to a microscopic view of the density distribution in Fig. 3(a) and (b), respectively. The arrows indicate the direction of the cargo movement, at the shock front as motors move forward. This evolves to the situation in Fig. 4(c), or corresponding Fig. 3(c). After a longer period of time, cargo/motor system is represented at a microscopic level once more as it evolves from the situation represented in Fig. 4(d) to that of Fig. 4(e). The continuum view of these are in Fig. 3(d) and Fig. 3(e), respectively. This time, cargo develops a movement towards forward direction in order to avoid the rarefied regions at its left.

We can argue about the origins of this effect from a microscopic perspective. For this, consider the representations in Fig. 4(a)e(e) that attempt to offer

a microscopic picture of the macroscopic situations represented in Fig. 3(a)e(e), respectively. Although the correspondence is not totally rigorous, the scheme in Fig. 4 is

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D.G. Lichtentha ¨ler, C. Goldman

Table 1 Kinematic properties of shock fronts formed within the course of time evolution of the initial density profile in (12). The average velocities were computed from the ratio between traveled distance (column 4) and the corresponding time interval (column 3) for each processive movement. Shock

Initial time

Time interval

Travel distance

Average velocity

Instantaneous velocity

1 2 3 4 5 6

0 0 a/23K 3a/43K 4a/33K 4a/3K

a/23K a/23K a/43K 7a/123K 8a/33K N

a/2 a/2 0 a/6 4a/3 N

3K e3K 0 23K/7 3K/2 e

3K 3K 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23K  3a3K=t 3K/2 pffiffiffiffiffiffiffiffiffiffiffiffi a3K=t

intended to represent ensemble (or time) averages taken over many microscopic configurations of the interacting particle system. These pictures show the movement of the cargo as it accompanies the regions of the microtubule where it is able to bound to an ‘‘optimum number’’ N Z Nc of motors. Or, equivalently, one may think that cargo moves in the presence of an effective force of attractive nature derived from a putative ‘‘potential field’’. At each time, the positions of the potential minima coincide with the positions at the microtubule where the density of motors achieves the corresponding optimum value. As the position of the particular minimum achieved by cargo moves along the microtubule due to the rearrangement of motors, cargo moves as well. From one hand, cargo avoids in this way the regions of the microtubule where it would result N > Nc. This effect may be traced from a presumable limitation on

Figure 5 Characteristics to Eq. (9) for the initial conditions given by (12). There are indicated the rarefaction regions R1 and R2, and the regions S1, S2, S3, S4 and S5 of the characteristic families x1, x2, x3, x4 and x5, respectively. The value of r is constant along each characteristic line. The encounter of characteristics at the shock fronts are represented by the dark curves. At each side of the shock fronts, r has a different value. The time t is given in units of K1.

the number of sites in the cargo that could be used for motor binding. In other words, this is equivalent to conceive that there are present excluded volume interactions among motors that are manifested at the cargo. This is shown in Fig. 4(a)e(c). The sequence represents the movement of cargo at the shock front following the region where N Z Nc avoiding in this way the dense regions on the right where N > Nc. From the other hand, one may think that the number of motors that bind to cargo at each time tends also to approach Nc from below. That is, cargo would be driven to regions of the microtubule where there is the possibility to bind to Nc motors, instead of a smaller number N < Nc. In principle, this effect can be traced from the attractive nature of the mechanisms responsible for the binding of individual motors to cargo. If these are additive, cargo would be driven to more dense regions, pointed to the direction of the resulting forces, which are known to be mediated by certain proteins as dynactin [32]. Following this, one may conclude that cargo tends also to ‘‘avoid’’ regions where it would result N < Nc. This is represented in Fig. 4(d) and (e). The sequence in this case shows the cargo at the shock front moving towards (attracted to) a region where N Z Nc avoiding this time the region where N < Nc. Notice that in the above reasoning, we do not account for the influence that cargo might have on the movement of motors. In fact, this problem has already been addressed in the literature using a mechanical model that accounts for friction effects acting on a motor/cargo complex [33]. Only one motor and one cargo have been considered in the referred work. The authors concluded that the motor in the complex diminishes its velocity when compared to the velocity of an isolated motor, i.e., moving on the microtubule but not associated with cargo. Therefore, in order to extend this quantitative analysis to apply to our model, it would be necessary to include more than one motor (actually, a density of motors) interacting with cargo as this situation approaches better the system discussed here. Because our present interests are on the origins of the bidirectional movement of cargo, we shall be restricted in the following to examine qualitatively if these effects could be influential to determine the direction of cargo movement at a shock front. For this, one may think, as a first approach, that all motors that are directly bound to cargo at each instant of time, have their velocities diminished if compared to the typical velocities of the remaining (not bounded) motors at

Surfing at wave fronts the microtubule. The effect to the motor profile would be just to smooth the local differences between the motor densities at the two sides of the shock front where cargo is placed. To see this consider, for example, the system evolving from Fig. 4(a) to (b) and then to (c). One can anticipate that if motors that are attached to cargo and to the microtubule [Fig. 4(a) and (b)], are slower than the motors bound only to the microtubule in the region of the microtubule in front of cargo, these cargo-free motors would disperse so that the situation in front of the cargo would appear more rarefied than it is represented in Fig. 4(c). By the same reasons, the motor density profile that appear rarefied at cargo’s back in Fig. 4(b) should, instead appear more dense than in Fig. 4(c). In other words, we expect that the effects on motor movement due to the interaction with cargo would be such to smoothing out (but not destroy!) the discontinuity introduced by the shock. Consequently, this introduces no qualitative changes to the results presented here regarding the mechanisms related to the bidirectionality of cargo movement. In spite of this, and in order to better estimate the magnitude of such effects, one could, in principle add to Eq. (9) a small contribution representing a local perturbation to the motor distribution due to friction with cargo. We believe, however, that eventual errors introduced in our results by neglecting these corrections, as we did in the above development, are minor if compared to the contribution already neglected, due to the (global) diffusion term that appeared in Eq. (5). The above reasoning may offer some clues to justify our hypothesis regarding the permanence of cargo at a shock wave front of the evolving motor density profile. Accordingly, the phenomena would have its microscopic origins on a balance between excluded volume interactions among motors taking place at the cargo and forces of pure attractive nature responsible for motor/cargo binding. Nevertheless, in the present work and with present available information on these systems, we are unable to prove it. This is in fact the key hypothesis made here and should be seen as a conjecture to be verified by experiments. We do not believe that this can be better discussed based only on pure theoretical arguments. Therefore, it may be helpful to analyze the consequences of such conjecture and to suggest possible directions for future investigation, using an interplay between theory and experiment.

Phenomenology As an example, consider the system of Fig. 3. The cargo shown in Fig. 3(a) and (b) is traveling driven by motors at the referred wave front 2, or second shock. Within the time interval T1 Z a/2eK, it performs a backward movement after which it stops, remaining at a rest for an interval T2 Z a/4eK. At the instant t Z 3a/4eK it resumes its movement but in the reverse direction (Fig. 3(c) and (d)). Thereafter, (Fig. 3(e)e(g)) the particle proceeds, eventually changing the magnitude of its velocity though not direction anymore (in the specific case of IC (12) direction reversal happens just once) until the motor density recovers its steady-state profile r(x, t) Z 1/2 and the particle reaches the microtubule end, as expected. Therefore, at least qualitatively, the

435 phenomenological aspects of the observed movement, as reversals of direction, processivity in both directions and the existence of resting times are accounted for by this model. Indeed, macroscopic reversals of shock directions as predicted above, are expected for a certain class of IC (12) representing local perturbations that produce both a rarefaction of motor concentration localized in the region of interaction with cargo, and a compensatory concentration increasing in a neighboring region. The choice in (12) is just a prototype. Evidently, by considering other initial conditions that exhibit these general characteristics, for example, by introducing additional parameters to represent other regions of motor profile also affected by such initial interaction with cargo, one may expect a great variety of time intervals and corresponding traveling distances by the wave fronts (or cargo) in both directions, differing thus in processivity, velocity and in the extension of the bidirectional motion. Therefore, a relevant aspect of this proposal is the possibility to attribute primarily to cargo the task for creating such initial changes in the motor density profile, and thus to induce its own surfing movement. We can test for the compatibility of the above hypothesis using existing data on cargo movement. Consider for instance the data on the movement of lipid vesicles on Drosophila embryo [27]. We can check for their compatibility with the results in Table 1 all expressed in terms of just two parameters, i.e., a and the product eK. To accomplish this, we observe that data analysis of the referred experiment identifies two typical runs in both directions: one that is short (average distance dshortw100 nm) and slow (average velocity vsloww200 nm/s) and another that is long (average distance dlongw1000 nm) and fast (average velocity vfastw400 nm/s). We use these data to estimate the values of a and eK. We proceed by identifying vslow with the average velocity of the 4th shock front v4 Z 23K/7. It results4 3Kw700 ðnm=sÞ:

ð13Þ

We can now use this to predict the magnitude of the average shock velocity for the 5th shock v5 Z 3K/2w350 nm/s which approaches very closely the values (absolute values) observed for vfastw400 nm/s. These results must still be tested with respect to the predictions for the distances traveled by the corresponding shocks. This complements the analysis by estimating possible values for the parameter a. Accordingly, the distance d5 is expected to result approximately one order of magnitude greater than d4. This is in fact in accordance with the results shown in Table 1 since from there d5/ d4 Z 8. 4 It must be remembered that the parameter 3 was defined dimensionless. It measures a deviation from the steady-state density value r Z 1/2. We can also parametrize d and b in terms of new constants a and b with units of [nm/s] such that d Z (a þ b)t and b Z (a  b)t with a > b. With this g Z 2a and K Z 2b. Using the result shown in Eq. (13), one may conclude that for 3 < 1/2 then b > 700 nm/s that may reveal a characteristic scale of velocity in the original microscopic model.

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D.G. Lichtentha ¨ler, C. Goldman

Another possibility to analyze our results is accomplished by redefining a such that

for studying the mechanisms of virus infection and also for designing drugs that must be driven into the cell body.

ah500x ðnmÞ

Conflict of interest

ð14Þ

where x is a numerical factor. From this, it results d4 Z 80x (nm) and d5 Z 650x (nm). Therefore, for x Z 1.2, then d4w100 (nm) and d5w800 (nm). These can be compared to the experimental results available for dshort and dlong given above to conclude that the characteristic times and distances traveled by shocks 4 and 5 can be related to the typical values obtained in experiments for short and long runs, respectively.

The shape and size of cargo and the transport of virus particles More interesting, however, is the fact that the approach to the experimental data is made possible with the choice in (14) for xwO(1). First, notice that 500 nm is the average diameter of the vesicles transported in the referred experiments. Thus, if motorecargo interactions are of a short-range nature, this implies that the magnitude and extension of the initial perturbation would be related to the geometrical characteristics of the cargo. In this case, differences in cargo morphology (size, specific geometry), would be related to different changes (or initial perturbations) each cargo produces into an equilibrium motor density profile and, consequently, to the diversity of movements. Such diversity is indeed observed in experiments using different motor/cargo systems [7]. The above considerations seem specially attractive to explain the behavior of virus particles moving within the cellular environment observed more recently using confocal microscopy technique [25,34,35]. According to the authors of these experiments, viruses appeared surfing along filopodia in such a way to perform a bidirectional motor-driven movement. The phenomena was observed in the presence of just a single type of motor protein, e.g., unidirectional myosin II. From these observations the authors then discuss on the possibility that, within the situations imposed by these experiments, each virus particle may be able to create the conditions to induce its own movement. These observations support the idea that viruses use (hijack) existing motor-based transport machinery to invade the cell [35]. The model presented here suggests a mechanism to explain how this can be accomplished without external and specific mechanisms of coordination. Such explanation has a focus on the morphological aspects of cargo e which, of course, is a distinguished characteristic of virus. In our view, this complements the suggestions made in Ref. [25] relative to the existence of a common mechanism that drives different viruses to their targets in cells, in spite of their morphological differences. The results presented above allows us to speculate that this common mechanism might be just related to the ability of virus to interact with the system by perturbing the existing motor profile. Specific geometries of each viral species, however, would be responsible to create specific movements, possible necessary for their efficiency. We thus believe that these ideas may be helpful

There is no conflict of interest.

Acknowledgments DGL acknowledges the fellowship by the Brazilian agency Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo ´gico (CNPq). CG acknowledges the financial support from Fundac ¸a ˜o de Amparo a ` Pesquisa do Estado de Sa ˜o Paulo (FAPESP).

Appendix Here we obtain the properties of shock waves shown in Table 1. For this, we use the method of characteristics to find the transient solutions to the Eq. (9) vrðx; tÞ=vt þ ð1  2rðx; tÞÞvrðx; tÞ=vxZ0 for the chosen initial conditions (IC) (12). In order to simplify the presentation, we set K Z 1. The final expressions obtained for the quantities of interest (Table 1) depict their explicit dependence on K. As references, we follow the textbooks by Haberman [29] and that by Evans [30]. The method allows us to find the properties of shocks which is our main concern in this work. For each pair (r(x0),x0) there corresponds a characteristic curve for (9) given by xðtÞZð1  2rðx0 ÞÞt þ x0

ð15Þ

Thus, the IC in (12) define the different characteristic families: x1 ðtÞZx0 for x2 ðtÞZx0 þ23t x3 ðtÞZx0 for x4 ðtÞZx0  23t x5 ðtÞZx0 for

x0 < 0 for 0 < x0 < 2a 2a < x0 < 3a for 3a < x0 < 4a 4a < x0

ð16Þ

and the function r(x, t) assumes a constant value (equal to r(x0)) along each of these particular curves. The set of characteristics for the present initial value problem is represented in Fig. 5 where one observes the occurrence of shock wave fronts (black lines) separating regions of different motor densities. There are six shocks (indicated in Fig. 5 as S1, S2,.). There are also shown in Fig. 5 the resulting rarefaction regions (indicated as R1, R2,.) that are localized between any two characteristic families that get apart from each other as the time evolves. The exact determination of the origin of each of these shocks (time and space) e and rarefaction regions and the course of their time evolution allows us to describe the solutions to r(x, t). This is performed below. At the initial time (t Z 0) there are two shocks:  shock 1 (xs1(t)): originates at position xs1(0) Z 2a, between the families defined by x2(t) and x3(t). The

Surfing at wave fronts

437

velocity of this shock front is determined by mass conservation conditions:

rarefaction and also cross the characteristics from family x2. Thus, t04 is determined by the relation xs3 ðt04 ÞZxr2 ðrZ1=2 þ 3; t04 Þ, resulting

dxs1 =dtZ½jðrðx3 ÞÞ  jðrðx2 ÞÞ=½rðx3 Þ  rðx2 ÞZ3 where the current j(r) is defined by (11). We then have xs1 ðtÞZ2a þ 3t

t04 Z3a=43

ð17Þ

that gives the shock position as a function of time.  shock 2 (xs2(t)) originates at position xs2(0) Z 3a, between the families defined by x3(t) and x4(t). The velocity of this shock is given by dxs2 =dtZ½jðrðx4 ÞÞ  jðrðx3 ÞÞ=½rðx4 Þ  rðx3 ÞZ  3

ð24Þ

The continuity condition in this case reads dxs4 =dtZ ½jðrr2 ðxs4 ÞÞ  jðrðx4 ÞÞ=½rr2 ðxs4 Þ  rðx4 Þ; that is dxs4 =dtZ3  ð4a  xs4 Þ=2t

ð25Þ

then xs4 is given by the solutions to the above differential equation for initial condition xs4 ðt04 Þ such that xs3 ðt04 ÞZxs4 ðt04 Þ: The result is [36] pffiffiffiffiffiffiffiffiffi xs4 ðtÞZ4a  2 3a3t þ 23t ð26Þ

from which xs2 ðtÞZ3a  3t

ð18Þ

Still at t Z 0, there originate two rarefaction regions  rarefaction 1 (xr1(t)) originates at xr1(0) Z 0, and evolves within the region between families x1(t) and x2(t). The search for entropic solutions drives one to look for the corresponding family of characteristic curves xr1(t) defined by the density rr1 at each point in this region: xr1 ðtÞZð1  2rr1 Þt for 1=2  3 < rr1 < 1=2

ð19Þ

which are also represented in Fig. 5. Conversely, one can write the solutions to rr1(xr1, t) for density profile at each position xr1 in this region as rr1 ðxr1 ; tÞZ1=2  xr1 =2t for 0 < xr1 < 23t

ð20Þ

 rarefaction 2 (xr2(t)) originates at xr2(0) Z 4a, and evolves within the region between families x4(t) and x5(t). The characteristics in this region are given by xr2 ðtÞZ4a þ ð1  2rr2 Þt for 1=2 < rr2 < 1=2 þ 3

ð21Þ

thus

The dependence on time of the corresponding shock velocity can now be determined pffiffiffiffiffiffiffiffiffiffiffiffi ð27Þ dxs4 =dtZ23  3a3=t

 shock 5 (xs5(t)) occurs between the two rarefaction regions, at the initial time t05 ; when the characteristic from the family x2 that originates at x0 Z 0 encounters xs4     xr2 rr1 Z1=2  3; t05 Zxs4 t05 resulting 4a t05 Z 33

ð22Þ

dxs5 Z½jðrr2 ðxs5 ÞÞ  jðrr1 ðxs5 ÞÞ=½rr2 ðxs5 Þ dt rr1 ðxs5 ÞZðxs5  2aÞ=t Considering that xs4 ðt05 ÞZxs5 ðt05 Þ this can be integrated to find

t03 Za=23:

ð23Þ

The velocity of the shock is given by dxs3 =dtZ½jðrðx4 ÞÞ  jðrðx2 ÞÞ=½rðx4 Þ  rðx2 ÞZ0 meaning that the shock xs3 ðt03 ÞZxs1 ðt03 ÞZ5a=2:

front

is

stationary

at

 shock 4 (xs4(t)) starts at the time t04 when the family x4 with origin at x0 Z 4a reaches the second region of

ð30Þ

thus dxs5 =dtZ3=2

 shock 3 (xs3(t)) starts at time t03 when the two families xs1(t) and xs2(t) cross. Thus, t03 is determined by the condition: xs1 ðt03 ÞZxs2 ðt03 Þ; resulting

ð29Þ

Mass conservation condition in this case (see (22) and (20)) reads

xs5 ðtÞZ2a þ ð3=2Þt

rr2 ðt; xr2 ÞZ1=2 þ ð4a  xr2 Þ=2t for 4a  23t < xr2 < 4a

ð28Þ

ð31Þ

 shock 6 (xs6(t)) starts at the time t06 when shock 5 encounters the characteristics from family x5 that starts at x0 Z 4a. Then, t06 results from the condition, xs5 ðt06 ÞZ4a. Using the expression derived above for xs5(t) one obtains t06 Z4a=3: Mass conservation imposes that dxs6 =dtZ½jðrðx5 ÞÞ  jðrr1 ðxs5 ÞÞ=½rðx5 Þ  rr1 ðxs5 ÞZxs6 =2t whose solution for the initial condition xs6 ðt06 ÞZ4a is

ð32Þ

438

D.G. Lichtentha ¨ler, C. Goldman

pffiffiffiffiffiffiffi xs6 ðtÞZ2 a3t

ð33Þ

and from which one finds the corresponding shock velocity: pffiffiffiffiffiffiffiffiffi ð34Þ dxs6 =dtZ a3=t The fact that this velocity is always positive, implies that for t > t06 Z4a=3 the solutions to Eq. (9) for the considered initial conditions (12) are ð35Þ

rðx; tÞZ1=2 for x < 0 rðx; tÞZ1=2  x=2t for 0 < x < 4a þ rðx; tÞZ1=2 for 4a þ

pffiffiffiffiffiffiffiffiffi a3=t < x

pffiffiffiffiffiffiffiffiffi a3=t

ð36Þ ð37Þ

From these, one concludes that for sufficiently long times, i.e., for t[t06 Z4a=3; the density profile assumes the uniform steady-state solution r(x, t) Z 1/2. This corresponds to the situation of motor distribution before the initial perturbation. The properties of shocks derived here are compiled in Table 1.

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