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 in vivo are carried by multiple motors. While some aspects of multiple motor function have received attention, how the cargo itself —and motor organization on the cargo—affects transport has not been considered. To address this, we have developed a three-dimensional Monte Carlo simulation of motors transporting a spherical cargo, subject to thermal fluctuations that produce both rotational and translational diffusion. We found that these fluctuations could exert a load on the motor (s), significantly decreasing the mean travel distance and velocity of large cargos, especially at large viscosities. In addition, the presence of the cargo could dramatically help the motor to bind productively to the microtubule: the relatively slow translational and rotational diffusion of moderately sized cargos gave the motors ample opportunity to bind to a microtubule before the motor/cargo ensemble diffuses out of range of that microtubule. For rapidly diffusing cargos, the probability of their binding to a microtubule was high if there were nearby microtubules that they could easily reach by translational diffusion. Our simulations found that one reason why motors may be approximately 100 nm long is to improve their ‘on’ rates when attached to comparably sized cargos. Finally, our results suggested that to efficiently regulate the number of active motors, motors should be clustered together rather than spread randomly over the surface of the cargo. While our simulation uses the specific parameters for kinesin, these effects result from generic properties of the motors, cargos, and filaments, so they should apply to other motors as well.
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 in vivo are frequently moved by more than one microtubule-based motor [1], [2], [3], [4]. This raises the question of how multiple motors function together, the subject of recent theoretical and experimental work [1], [5], [6], [7]. In vitro, when more than one motor is actively hauling a cargo, the run length, i. e. , the distance that the cargo travels along the microtubule before detaching, increases with the number of active motors. However, the presence of the cargo itself may be important when there are multiple motors. In addition to possibly changing the single-molecule' s function, the cargo' s size may alter the relationship between the total number of motors present and the number of motors actively engaged in transporting the cargo (assuming random motor organization on the cargo' s surface). If motors are not randomly organized, details of this organization will also be important. How each of these factors contributes to overall transport is unknown. To approach these problems requires a new theoretical framework: past studies simplified the problem using essentially one-dimensional models [5], [6], [8], [9] that had the motors attached to the cargo at a single point, with the cargo represented by a single point (though potentially experiencing viscous drag proportional to a specific diameter). Here we have developed a bone-fide three dimensional Monte Carlo simulation that allows us to directly investigate how the presence of the cargo itself affects single-motor driven transport and motor-microtubule attachment, as well as how the relationship between cargo size and the arrangement of motors on the cargo affects ultimate cargo motion, all within the context of a cargo experiencing random Brownian translational and rotational motion. 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 (Figure 1A). In contrast for the 500 nm cargo, most motors will be unable to reach if they are randomly distributed on the cargo (Figure 1B). However, if all the motors were clumped at a single point, the size of the cargo essentially becomes irrelevant, because if one motor can reach, they all can (Figure 1C). We thus set out to answer the following questions: 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 PA where 0≤PA≤1. In practice, during each time step, a random number x is generated with uniform probability between 0 and 1. If x≤PA, event A occurs; if x>PA, event A does not occur. 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 Text S1. To this cargo, we attached kinesin motor (s) that are modeled as bungee cords, i. e. , they behave as springs with a spring constant of 0. 32 pN/nm [5], [10] when stretched beyond their relaxed length of 110 nm but produce no force when compressed. We started the simulation so that potentially one or more motors could bind to a cylindrical microtubule (25 nm diameter). The motors then moved the cargo along the microtubules, taking 8 nm steps. While technical details of the simulation are in the Text S1, the general idea is that at each time step Δt, we consider all motors present, calculate all forces acting upon them, and then ask what each of them does. 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 [11], [12], [13]), we use one of two initial conditions. If we want to find the time it takes for a cargo with a single motor to attach, then the cargo is allowed to rotate consistent with Brownian diffusion, and the procedure is repeated. Eventually, the motor binds. The time between when the simulation is started and when the motor attaches is the ‘on’ rate for the cargo; since only one motor is present, it reflects how the presence of the cargo affects the motors' on-rate. 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 Text S1. This load has contributions from the externally applied force, the other motors which are pulling the cargo, and from thermal fluctuations. The thermal fluctuations randomly rotate and translate the cargo which, in turn, can stretch the motor linkage and exert a load on the motor. (See below for further details on thermal fluctuations.) Once all the motors have been given a chance to step, the cargo is translated and rotated according to the force and torque to which it is subjected. The cargo travels along the microtubule until all the motors detach from the microtubule, and the ‘run’ ends; this then determines the run length of the cargo. The velocity is calculated by dividing the distance the cargo moves by the travel time τ, where τ is typically 1 msec but may be as long as 10 msec. Averaging over these velocities gives the average velocity. To get good statistics, we simulate a specified number of runs with the same initial conditions to get a set of runs. We also simulate a number of sets with different initial conditions to obtain good statistics. 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: (1. 1) where m is the cargo' s mass and is the cargo' s velocity. The drag force on the cargo is proportional to its velocity with the drag coefficient, where R is the cargo' s radius and is the coefficient of viscosity which is the kinematic viscosity multiplied by the specific gravity of the fluid. is the sum of the forces due to an external force of magnitude FL and the force of the engaged motors pulling on the cargo. We solve this equation in the Text S1, and quote the solution here for the position of the cargo at time step t+Δt: (1. 2) where is the standard deviation of a normal distribution and is a vector in Cartesian coordinates of the laboratory frame of reference that represents three independent random variates drawn on a normal distribution having zero mean and unit standard deviation. For the cargo' s rotational motion, the corresponding Langevin equation is (1. 3) where is the moment of inertia of a solid spherical cargo, and is the drag coefficient proportional to the angular velocity. is the torque on the cargo referenced from the center of mass due to the engaged motors. is the rapidly varying random torque due to the thermal fluctuations of the environment. We solve this equation in the Text S1 where we give the formulas for the change in orientation of the cargo at each time step. These formulas are analogous to Eq. (1. 2). As we shall see, rotational diffusion due to thermal fluctuations can play a significant role in limiting the distance that a cargo can travel. 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 Figure 2. The area available for attachment can be described by a cone with its apex at the center of the sphere. A line extends from the apex to the base of the cone. The cluster angle φ is the angle between this line and the side of the cone. The intersection of the cone with the surface of the cargo defines the allowed region of motor attachment. The cluster angle can vary between 0 and 180 degrees. A cluster angle of 90 degrees defines the lower hemisphere of the cargo. A cluster angle of 180 degrees corresponds to the entire spherical surface, and means that the motors can attach anywhere on the sphere.
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 [4], though it is consistent with biochemical characterizations of cargo-bound myosin motors [18] which are likely randomly arranged on cargos [18], [19]. Overall, our findings suggest that, in vivo, microtubule motors are likely organized into clusters when present on large cargos, but that such clustering is unnecessary for small cargos. 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 [18], we expect long travel distances along microtubules but relatively short run lengths along actin filaments. Thus we predict the microtubule motors kinesin and dynein to be clustered on cargos while we expect the actin motor myosin V to bind randomly to cargos. There is clear experimental evidence for the random arrangement of myosin on cargos in vivo, and weak experimental evidence for the clustering of kinesin and cytoplasmic dynein [19]. 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 [20], [21]. However, in other cases, the attachment points can diffuse through the fluid membrane of the vesicle and cluster at one location. An example of this is an experiment showing that motors dynamically accumulate at the tip of membrane tubes growing out of a vesicle as a consequence of the fluidity of the membrane [13], [22]. 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.
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.
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biophysic al simulations
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biophysics simulations
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How Molecular Motors Are Arranged on a Cargo Is Important for Vesicular Transport