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Conceived and designed the experiments: ZJ SPG CCY. Performed the experiments: RPE ZJ. Analyzed the data: RPE ZJ SPG CCY. Contributed reagents/materials/analysis tools: RPE ZJ CCY. Wrote the paper: RPE ZJ SPG CCY.

¶ REP and ZJ are co-first authors on this work. SPG and CCY are co-senior authors on this work.

The authors have declared that no competing interests exist.

The spatial organization of the cell depends upon intracellular trafficking of cargos hauled along microtubules and actin filaments by the molecular motor proteins kinesin, dynein, and myosin. Although much is known about how single motors function, there is significant evidence that cargos

The spatial organization of living cells depends upon a transportation system consisting of molecular motor proteins that act like porters carrying cargos along filaments that are analogous to roads. The breakdown of this transportation system has been associated with neurodegenerative diseases such as Alzheimer's and Huntington's disease. In living cells, cargos are typically carried by multiple motors. While some aspects of multiple motor function have received attention, how the cargo itself affects transport has not been considered. To address this, we developed a three-dimensional computer simulation of motors transporting a spherical cargo subject to fluctuations produced when small molecules in the intracellular environment buffet the cargo. These fluctuations can cause the cargo to pull on the motors, slowing them down and making them detach from the filament (road). This effect increases as the cargo size and viscosity of the medium increase. We also found that the presence of the cargo helped the motors to bind to a filament before it drifted away. If other filaments were present, then the cargo could bind to one of them. Our results also indicated that it is better to group the motors on the cargo rather than spread them randomly over the surface.

Cells are highly organized, and much of this organization results from motors that move cargos along microtubules. The single-molecule properties of molecular motors are relatively well understood both experimentally and theoretically. With this as a starting point, we investigated how the presence of the cargo itself alters transport. Aside from exerting viscous drag, the cargo could in principle alter single-motor based transport both by changing the motors' diffusion and ability to contact the filament (a free motor diffuses very differently from a cargo-bound one), and also by exposing the motor to the random forces resulting from thermal fluctuations of the cargo which depend on the size of the cargo and the viscosity of the environment. Whether such effects are significant are investigated here.

Recent studies show that cargos

To approach these problems requires a new theoretical framework: past studies simplified the problem using essentially one-dimensional models

The attachment of motors to a cargo of finite size, rather than an idealized point mass, has a number of ramifications. First, the function of the motor(s) might be altered by the translational and rotational diffusion of the cargo; the larger the cargo, the more effect it has on the motors' diffusion, and thus, potentially, on the motors' ability to contact/interact with a microtubule. Second, when a motor is attached to both the microtubule and the cargo, it will feel instantaneous forces due to the cargo's thermal motion. These forces will depend on the cargo's size; and the random thermal ‘tugs’ from the cargo could slow the rate of travel of a motor and, in principle, induce the motor to detach from the filament. Third, there is a relationship between the cargo size, the total number of motors present, how they are arranged, and how many can be engaged. To illustrate this, imagine one cargo that is 50 nm in diameter, and another that is 500 nm in diameter. In the first case, even if the motors are randomly distributed on the cargo, because the length of an individual motor is more than 100 nm, all of those on the lower half of the cargo, and some on the upper half, will be able to reach a nearby microtubule (

(A) Motors on a cargo with diameter of 50 nm can easily reach the microtubule. (B) Motors on a cargo with a diameter of 500 nm have difficulty reaching the microtubule from most places on the cargo. (C) Motors attached to the cargo at the same point (South Pole) can easily reach the microtubule.

We thus set out to answer the following questions:

How does a cargo affect the rate at which the motor(s) on the cargo bind to the microtubule?

For a single motor on a cargo, how does the presence of the cargo affect the motor's effective ‘on’ rate, i.e., the rate at which the motor binds to the microtubule?

How does the cargo's size and viscosity affect the probability that the cargo will bind to the microtubule before diffusing away?

Further, how does the length of the motor compared to the cargo size contribute to these properties?

What about the binding probability of a cargo with multiple motors?

How does the distance between microtubules affect the probability that a cargo with one or more motors will bind to a microtubule?

Does the cargo's Brownian motion affect the motor's function as measured by its travel distance?

For a cargo with a single motor?

For a cargo with multiple motors?

Number of engaged motors.

For randomly distributed motors, does doubling, say, the total number of motors on the cargo double the number of motors engaged in hauling the cargo along the microtubule?

What is the relationship between the cargo size, the viscosity, the number of motors present, the average number of motors actively engaged in transporting the cargo, and the cargos' mean travel distance?

For motors “clustered” on the cargo, what is the relationship between cargo size, the viscosity, the size of the cluster, the number of motors in the cluster, and the mean travel distance?

We organized the presentation of our results according to these questions.

To address these questions, we developed three-dimensional Monte Carlo simulations. Generally speaking, Monte Carlo is an approach to computer simulations in which an event A occurs with a certain probability P_{A} where 0≤P_{A}≤1. In practice, during each time step, a random number _{A}, event _{A}, event

Our simulations were carried out as follows. We started with a three dimensional spherical cargo, subject to rotational and translational diffusion according to the equations presented below and in the

We start by describing how we simulate transport of a cargo with motors attached. Our basic algorithm is as follows. Consider one or more motors attached at random points to the cargo surface. The cargo is then suspended above the microtubule, with a well-defined separation distance between the bottom of the cargo and the top of the microtubule, and the motors are each given an opportunity to attach to the microtubule. If none do (either because none can reach, or because although they can reach, they stochastically are not able to attach in the allotted time with the ‘on’ rate assumed to be ∼2/sec

The other initial condition is used if there are multiple motors and we are more interested in transport along the microtubule after the motors attach to the filament. In this case, if none of the motors attaches after being given the opportunity to do so, the cargo is rotated so that at least one motor attaches to the microtubule.

Once some subset of the motors is attached, the cargo travels along the microtubule. At each time step of the simulation, each motor on the cargo is given the opportunity to detach from the MT if it is attached, or attach if it is detached (and geometrically can reach the MT). If a motor is attached to a MT, then there is some probability that it will bind and hydrolyze ATP, and subsequently take a step. Although kinesin is a two headed motor, we model each motor by a single kinesin head that hydrolyzes ATP in such a way that Michaelis-Menten kinetics is obeyed. The probabilities of a motor detaching from the MT, releasing ATP, and taking a step are all dependent on the load on the cargo because the cargo exerts force on the motors (see

In our simulations, the spherical cargo is subjected to thermal fluctuations which we can divide into translational and rotational components. The equation of the cargo's translational motion is given by the Langevin equation:_{L} and the force of the engaged motors pulling on the cargo. We solve this equation in the

For the cargo's rotational motion, the corresponding Langevin equation is

After considering motors randomly attached anywhere on the cargo, we consider cases which have a restricted region of the cargo surface area where motors can attach. For these cases, we start each simulation with N motors randomly attached to the cargo's surface within a region specified by the cone angle as shown in

We organize our results according to the questions posed in the introduction.

Let us start with our investigation of the effect of the presence of the cargo on motor attachment to a microtubule. Kinesin is estimated to have an on-rate of approximately 2 sec^{−1}, so that one expects a free motor close to a microtubule to take roughly 0.5 seconds to attach to the microtubule. If we attached that same motor to a small (25 or 50 nm radius) cargo, and held the cargo on the microtubule so that it could rotate randomly due to thermal motion but not diffuse away, we saw that the presence of the cargo had little effect on the typical time for the motor to bind (

The motor is initially attached to either the South Pole (cone angle = 0, solid lines) of the cargo facing the microtubule or randomly attached somewhere on the cargo (cone angle = 180 degrees, dashed lines). The cargo rests on the microtubule, and rotates randomly due to thermal effects but cannot diffuse away from the microtubule. The simulations were run long enough to allow at least 95% of the cargos to bind to the microtubules. The binding time was the averaged over the cargos that bound to the microtubule in this time. 1× represents a viscosity equal to that of water and 10× means the viscosity is 10 times greater than that of water. We took the viscosity of water to be 10^{−9} pN-s/nm^{2} throughout the paper.

In

The cargos are initially resting on the microtubule, and can rotate and diffuse away due to thermal fluctuations. In the first plot (A), the simulation was started with the motor located at the South Pole facing the microtubule, and in the second plot (B), the motor was randomly placed anywhere on the cargo. Each curve represents the outcome of 6000 trials.

When considering the system of the motor plus cargo, one might wonder how important the physical length of the motor is. We investigated this in the context of the on-rate. We varied the cargo radius from 25 to 250 nm, the motor length from 25 to 1000 nm, and the viscosity from that of water to 10 times that of water. As might be expected, the higher the viscosity, the longer it took for a single motor attached to a cargo to bind to a microtubule. In addition, motors with shorter stalks on average tended to take longer to bind to the microtubule. Interestingly, we discovered that the effective on-rate improved until the motor was approximately equal to the cargo's radius; for motors much longer than the cargo's radius there was little additional improvement (

(A) Average time for a 110 nm motor to bind to a microtubule versus cargo radius. (B) Average time for a motor bound to a cargo with radius 100 nm to bind to a microtubule as a function of motor length. (C) Log-log plot of the average time to bind (in seconds) versus the ratio of motor length to the cargo radius. The solid horizontal line marks a time to bind of 5 seconds. The cargo was initially resting on the MT with the motor randomly placed on the cargo's surface. The cargo was allowed to rotate randomly due to thermal effects, but it could not diffuse away from the microtubule. The simulations were run long enough to allow at least 95% of the cargos to bind to the microtubules. The binding time was the averaged over the cargos that bound to the microtubule in this time. The solid lines are guides to the eye. In all 3 plots the blue squares correspond to the viscosity of water (1×), and the red circles correspond to 10 times the viscosity of water.

When we considered a cargo hauled along a microtubule by a single active motor without any thermal fluctuations, we found that the motor and cargo underwent an oscillating porpoise-like motion as the cargo traveled down the microtubule. This is illustrated in

The figure illustrates porpoise-like oscillations about the equilibrium point.

The cluster angle is zero. Blue lines, labeled “no thermal,” correspond to the case of no thermal rotational and no thermal translational diffusion. Red lines, labeled “thermal,” correspond to the presence of both thermal rotational and thermal translational diffusion.

Since our single motor simulations showed that the presence of a cargo can affect the on-rate of a motor, we wanted to see how the time it took for at least one motor to bind was affected by the total number of motors attached to the cargo as well as the size of the cargo. We considered the case of multiple motors randomly attached anywhere on the cargo surface. We placed motors randomly on the surface of a cargo, and measured the average time that it took to have at least one motor bind to the microtubule. An example of our results is shown in

The different lines correspond to different cargo radii ranging from 25 nm to 250 nm. The cargo is allowed to rotate due to thermal diffusion but it cannot diffuse away. The medium has the viscosity of water. Having a cone angle of 180 degrees means that the motors are randomly placed on the cargo's surface. Having a cone angle of 0 degrees means that the motors are all located at the South Pole facing the microtubule.

Above, we investigated how the motor's length affected on-rate in the single-motor case. We extended these studies to determine the time it took for a motor to bind to a microtubule, as a function of the motor length (25 to 1000 nm), the cargo radius (25 to 250 nm), and the number of motors on the cargo. For these simulations the cargo rested on the microtubule and was not allowed to diffuse away, but did rotate randomly due to thermal effects, so that if the motor(s) were initially unable to reach the cargo, they soon came within reach of the microtubule. The motors were either clustered at one random point on the cargo's surface or else they were spread randomly over the surface. We tried motors of different lengths and cargos of different sizes. Our results are shown in _{bind} that it took for the first motor to bind to the microtubule had a power law dependence on the length _{bind}∼L^{−b} where the exponent b varied between 1.3 and 1.7. This meant that the longer the motor, the lower the average binding time. When the motor was much longer than the cargo radius, the average binding time became independent of the motor length. For motors short compared to the radius of the cargo, the binding time decreased as the number of motors increased. But as the motor length increased, the time it took to bind was less sensitive to the number of motors on the cargo. We can see that from

(A) Average time for 110 nm motors to bind to a microtubule versus cargo radius at the viscosity of water. (B) Average time for motors bound to a cargo with radius 100 nm to bind to a microtubule as a function of motor length at the viscosity of water. (C) Log-log plot of the average time to bind (in seconds) versus the ratio of motor length to the cargo radius at the viscosity of water. (D) Log-log plot of the average time to bind (in seconds) versus the ratio of motor length to the cargo radius at 10 times the viscosity of water. The solid horizontal line marks a time to bind of 5 seconds. The motors are either clustered at one point on the surface of the cargo or they are randomly distributed over the surface of the cargo. The cargo was allowed to rotate randomly due to thermal effects, but it could not diffuse away from the microtubule. The log-log plots show that the time to bind goes as L^{−b} where L is the length of the motor and the exponent b varies between 1.3 and 1.7. The solid and dashed lines are guides to the eye.

So far we have only considered how long it takes a cargo that is sitting on a microtubule to actually bind to a single microtubule, but unattached cargos may not start on a microtubule. Rather they may be floating in the cytoplasm and may need to come within reach of a microtubule. In this case the time for a motor on a cargo to bind to a microtubule depends on the time to diffuse to a microtubule, and once it finds a microtubule, to bind to it before the cargo diffuses away. The time to diffuse between microtubules depends on the viscosity, the size of the cargo, and the distance between microtubules. The typical intracellular environment consists of multiple microtubules extending radially from the nucleus. In a flat 2D cell, the microtubule density decreases as 1/r where r is the distance from the microtubule organizing center (MTOC). At the periphery of

We performed simulations to see how the binding fraction (the fraction of cargos where at least one motor binds to a microtubule) depends on the distance between microtubules, the radius of the cargo, and the viscosity. The geometry was a slab (extending from z = −1 micron to z = +1 micron) with either one microtubule or an infinite number of evenly spaced microtubules running parallel to the y-axis. For the case of one microtubule, it lays along the y-axis with the plus end in the positive y-direction, and slab extended to infinity in the x and y directions. For the case of microtubules evenly spaced by a distance x_{MT}, we placed one along the y-axis, and the next one a distance x_{MT} away. To obtain an infinite number of microtubules, we used periodic boundary conditions in the x direction such that if a cargo has a position x > x_{MT} (x < x_{MT}), we mapped x to x−x_{MT} (x+x_{MT}). Initially the cargo rested on a microtubule with motor(s) attached randomly on the surface of the cargo. The cargo was allowed 60 seconds to attach; if it failed to attach in that time, the trial was deemed a failure. The cargo was able to diffuse translationally and rotationally. The z-position of cargos that entered the floor or ceiling in the z direction was reset such that the cargo just touched the floor or ceiling. We considered microtubule spacings between 400 and 1200 nm, cargo radii from 25 nm to 250 nm, and viscosities from that of water to 1000 times that of water. We ran 1000 trials for a given set of values of the parameters.

Our results for the binding fraction are shown in

(A) 3D plot of binding fraction vs. cargo radius and microtubule (MT) distance for a cargo with 1 motor randomly attached on its surface at 10 times the viscosity of water for both translational and rotational diffusion. The lower blue surface is for the case of 1 microtubule where there is obviously no dependence on microtubule distance. The upper surface is for the case of an infinite number of microtubules. (B) Binding fraction vs. microtubule distance for a cargo with radius 250 nm at 10 times the viscosity of water for both translational and rotational diffusion. The solid lines are for an infinite number of microtubules, while the single microtubule case is represented by the straight dot-dash lines which have no dependence on microtubule distance. Blue pluses, red circles, and green squares are for 1, 2, and 3 motors, respectively, placed randomly on the surface of the cargo. (C) Binding fraction vs. cargo radius for a cargo with 2 motors randomly attached to its surface at the viscosity of water for translational diffusion and 10 times the viscosity of water for rotational diffusion. MTD stands for microtubule distance, i.e., the spacing between microtubules. (D) Binding fraction vs. microtubule (MT) distance for a cargo with a radius of 250 nm and 2 motors randomly attached to its surface at 10 times the viscosity of water for rotational diffusion. The solid lines are for the case of an infinite number of microtubules and the dot-dash lines are for one microtubule. Blue, red, green, and magenta correspond to 1, 10, 100, and 1000 times the viscosity of water for translational diffusion.

The presence of the cargo could also affect motor function by amplifying thermal noise effects, leading to increased random forces acting on the motor, and thus possibly affecting travel distance. We considered a cargo being hauled along a microtubule by a single active motor. The radius of the cargo ranged from 50 nm to 500 nm, and we studied environments with the viscosity of water and 10 times the viscosity of water. In

(A) Cargo run length and (B) cargo velocity vs. cargo radius for a cargo hauled by a single motor with and without rotation at the viscosity of water and 10 times the viscosity of water.

The contribution of rotational diffusion in limiting the run length continued out to higher viscosities, though not quite so strikingly. This is seen in

(A) Run length and (B) average velocity of a cargo carried by a single motor vs. the viscosity relative to water for 50 nm and 500 nm radius cargos both with and without rotational diffusion.

As we discussed for the case of one motor on a cargo, a cargo that was tethered to a filament by one or more motors was subjected to thermal motion in the form of translational and rotational diffusion. These thermally generated forces and torques increased with the size of the cargo according to Stokes' law which says that the translational drag coefficient α_{T} = 6πηR, and the rotational drag coefficient α_{R} = 8πηR^{3}. As a result, the run lengths decreased with increasing viscosity and cargo radius for a fixed number of motors and cluster angle. We saw this for the case of one motor (

We show the case of multiple motors in ^{th} percentile run lengths L_{80} are plotted versus cargo radius for 5 motors at different values of the viscosity and cluster angle. By 20^{th} percentile run length, we mean the run length such that 80% of the motors traveled at least this far. We denote this run length by L_{80}. One can clearly see that L_{80} decreased with increasing cargo radius. This decrease was accentuated by increasing viscosity, especially in the simplest case of zero cluster angle.

The error bars underestimate the error and the curves would be made smoother by increasing the sample space. The roughness of the curves is not indicative of any physical behavior.

To further investigate the importance of rotational diffusion, we measured L_{80} for 5 motors for a sphere with R = 500 nm at the viscosity of water, and R = 250 nm at twice the viscosity of water. Note that the translational drag coefficient α_{T} was the same for both cases, but α_{R} was larger for the 500 nm sphere. We found that L_{80} was the same in both cases if we turned off rotational diffusion, but if we included rotational diffusion, then L_{80} (R = 250 nm) = 9.9 microns compared to L_{80} (R = 500 nm) = 7.7 microns. Clearly rotational diffusion of the cargo played an important role in increasing the load on the motors and decreasing the travel distance.

Since our simulations suggested that the presence of the cargo can affect both effective motor ‘on’ rates and the mean travel distance, we were also interested in how multiple motors attached to the cargo might function. We first considered motors randomly on the surface of a cargo, and during the simulation, calculated the average number of engaged motors as well as the distribution of the number of engaged motors as a function of the total number N of motors on the cargo. This was done as a function of the cargo radius R, for both the case where the cargo moved through a medium with the viscosity of water, and also for a medium with10 times the viscosity of water. The results in

Plot at 10× viscosity of water is similar.

From a biological point of view, the relationship between the number of motors engaged and the total number of motors present suggested that for most cargo sizes, doubling the total number of motors on the cargo did not double the number of engaged motors. Put differently, even though the number of actively engaged motors is linearly proportional to the total number N of motors on the cargo, it does not necessarily follow that the number of engaged motors will double if the total number of active motors on the cargo doubles because the constant of proportionality may be too small to correspond to doubling the number of active motors (

The number of engaged motors grows more slowly for larger cargos.

What this means from a regulatory point of view is that, in the absence of some higher-order organization of motors (see below), to control motion by recruitment of motors to the cargo, different numbers of motors must be recruited depending on the cargo size. In the two cases above, if the number n of engaged motors is approximately 1, to recruit enough motors to end up with n∼2 requires one to quadruple the total number of motors present in the R = 50 nm case, but increase the total number of motors by a factor of 20 in the R = 250 nm case.

Alternatively, one could hypothesize that motor recruitment could be controlled locally, so that the surface density of motors could be controlled. Interestingly, because of geometrical effects, fixed densities of motors did not equate to the same number of engaged motors on cargos of different sizes. For example, a density of 6.3×10^{−5} motors/nm^{2} corresponded to 2 total motors (and 1.5 engaged motors) on a cargo with R = 50 nm, but to 50 total motors (and 4.7 engaged motors) on a cargo with R = 250 nm. The relationship between motor density and the number of engaged motors is investigated more fully below.

Frequently, one uses Poisson statistics to estimate the number of motors that are engaged. We were able to directly test this assumption. By recording the number of engaged motors at each time step, we were able to calculate the distribution P(N_{engaged}) of engaged motors. We found that P(N_{engaged}) obeyed Poisson statistics to a good approximation that improved as the total number N of motors increased. For 30 motors or more, it was an excellent approximation. An example is shown in

30 total motors on a cargo with radius of 125 nm.

If motors were randomly distributed on a cargo, a key quantity that helped to determine the number of engaged motors, and hence the run length, was the density of motors on the cargo surface.

The lines are interpolated between points with the viscosity of water. The points at 10 times the viscosity of water have no lines. Inset is a blow-up of the region near the origin.

In our example above, 5 motors on a cargo with a diameter of 100 nm correspond to a density of 1.6×10^{−4}/nm^{2}, and 50 motors correspond to a density of 1.6×10^{−3}/nm^{2}. This is in contrast to a 500 nm diameter cargo where density of 6.4×10^{−6}/nm^{2} corresponds to 5 motors, and a density of 6.4×10^{−5}/nm^{2} corresponds to 50 motors. This is summarized in

Radius [nm] | Total number of Motors | Density [nm^{−2}] |
Number of Engaged Motors |

50 | 5 | 1.6×10^{−4} |
2.4 |

50 | 50 | 1.6×10^{−3} |
17.7 |

250 | 5 | 6.4×10^{−6} |
1.3 |

250 | 50 | 6.4×10^{−5} |
4.7 |

In addition to motor density, cargo size is also important because motors have a harder time reaching the microtubule from a large cargo. Since the motors cannot pass through the cargo, there is more excluded volume for large cargos.

The simplest non-random organization for a group of motors is to cluster them together. We investigated the effects of clustering by specifying the cluster size in terms of the cone angle subtended by the cluster (see

The run length increases exponentially with the number of motors. The error bars underestimate the error and the curves would be made smoother by increasing the sample space. The roughness of the curves is not indicative of any physical behavior.

For two or more motors, the run length decreased as the cluster angle and cargo radius increased. An example is shown in

(A) Run length vs. cluster angle. (B) Run length vs. cargo radius. The error bars underestimate the error and the curves would be made smoother by increasing the sample space. The roughness of the curves is not indicative of any physical behavior.

Another way to exhibit this data is shown in ^{th} percentile run lengths L_{80} as a function of the cargo radius and cluster angle. We can see from the figures that L_{80}, the minimum distance traveled by 80% of the motors, decreased with increasing cluster angle and cargo radius. This decrease was more rapid for large cargos and for large cluster angles.

80% of the cargos traveled at least a distance that we call the 20^{th} percentile run length. 20^{th} percentile run lengths vs. (A) cargo radius and (B) cluster angle for 5 motors. The error bars underestimate the error and the curves would be made smoother by increasing the sample space. The roughness of the curves is not indicative of any physical behavior.

In ^{th} percentile run lengths at the viscosity of water and 10 times the viscosity of water as a function of the cargo radius and the cluster angle. At the higher viscosity, one can see that L_{80} decreased faster with increasing cargo radius and cluster angle. This is to be expected since the higher viscosity medium produced a greater drag on the cargo, and hence a higher load on the motors. In our model the probability of a motor detaching from the microtubule increased exponentially with increasing load

20^{th} percentile run lengths vs. (A) cargo radius and (B) cluster angle for 5 motors on the cargo. The error bars underestimate the error and the curves would be made smoother by increasing the sample space. The roughness of the curves is not indicative of any physical behavior.

The results described above showed that run length increased if the motors were clustered rather than randomly spread over the surface of the cargo. We wondered if one cost of clustering might be to increase the time for the cargo to attach, compared to randomly placed motors. We performed simulations in which the cargo initially was resting on the microtubule and was subjected to thermal fluctuations that could cause it to diffuse away or to rotate randomly. In

Each trial was terminated when the cargo bound to the microtubule, or if the cargo did not bind after 60 seconds, or if the cargo did not bind and diffused more than 5.5 microns away from the microtubule. (5.5 microns is 50 times the length of the motor.) The dashed lines correspond to the motors being clustered, i.e., attached to the cargo at one point. The solid lines correspond to the motors being randomly distributed over the surface of the cargo. The cargo initially was resting on the microtubule, and was allowed to randomly rotate and diffuse away due to thermal fluctuations. (A) Medium had the viscosity of water. (B) Medium had ten times the viscosity of water. The curves would be made smoother by increasing the sample space. The roughness of the curves is not indicative of any physical behavior.

Our study of the effects of the cargo on transport has a number of ‘take-home’ messages. The first is that, at both the single-motor and multiple-motor levels, the presence of the cargo can significantly alter the effective ‘on’ rate/probability of successful binding of the motor(s) to the filament, because the center of mass of the cargo diffuses away from the microtubule relatively slowly, and while this is occurring, its rotational diffusion frequently brings the motor close enough to the microtubule to allow attachment. Thus, the cargo ‘helps’ the motor to attach, though the degree of assistance depends on cargo size and viscosity of the medium surrounding the cargo. Rapidly diffusing cargos might not linger long in the vicinity of a microtubule, but in a cell where there are multiple filaments available, these cargos could quickly find and bind to a filament.

Second, in order to for a motor to attach to the filament in a reasonable amount of time, the motor length needs to be longer or comparable to the radius of the cargo which may explain why motors are 60 to 110 nm in length.

Third, if motors are randomly arranged on the cargo's surface, the relationship between the number of motors present and the number of actually engaged motors depends strongly on the cargo size, so that different simple models of regulating cargo motion by recruiting motors to the cargo surface (either by a specified change in total number of motors, or by a specified change in local motor surface density) will have different effects on overall cargo motion as a function of cargo size. Thus, in order to have regulation affect a set of cargos equally, independent in variations in cargo size, it is best to have motors clustered in a small region on the cargo.

A further finding also supports the utility of motor clustering: for large cargos, if motors are randomly placed, achieving a reasonable number of engaged motors (n = 3–6) would require a large number of motors (50–100) to be present on the cargo, which appears inconsistent with biochemical characterizations of cargo-bound microtubule motors

In addition, a reasonable number of engaged motors would be required for long travel distances of several microns but not for short run lengths. Since microtubules can be tens of microns long compared to actin filaments which have a typical decay length of 1.6 microns

For the purposes of this paper, we have assumed that the points where motors are attached to the cargos are fixed on the cargo's surface. This is true in some cases, e.g., when motors bind to dynactin which in turn binds to spectrin which is a filament that coats some vesicles

Clustering does not seem to affect the rate at which the first motor of a cargo attaches to a microtubule unless the cargo is large (greater than 200 nm) and the viscosity is high. Motor proteins are sufficiently long (greater than 50 nm) and rotational diffusion sufficiently rapid that the number of motors on a cargo does not significantly affect the rate at which the cargo binds to the microtubule.

Details of Monte Carlo simulation of a cargo hauled by motor proteins along a microtubule.

(DOC)

CCY thanks the Aspen Center for Physics for its hospitality during the initial stages of this work.