Myosin Va molecular motors manoeuvre liposome cargo through suspended actin filament intersections in vitro

Intracellular cargo transport relies on myosin Va molecular motor ensembles to travel along the cell's three-dimensional (3D) highway of actin filaments. At actin filament intersections, the intersecting filament is a structural barrier to and an alternate track for directed cargo transport. Here we use 3D super-resolution fluorescence imaging to determine the directional outcome (that is, continues straight, turns or terminates) for an ∼10 motor ensemble transporting a 350 nm lipid-bound cargo that encounters a suspended 3D actin filament intersection in vitro. Motor–cargo complexes that interact with the intersecting filament go straight through the intersection 62% of the time, nearly twice that for turning. To explain this, we develop an in silico model, supported by optical trapping data, suggesting that the motors' diffusive movements on the vesicle surface and the extent of their engagement with the two intersecting actin tracks biases the motor–cargo complex on average to go straight through the intersection.

A. The mechanical model. Myosin motors, when not bound to actin (red), diffuse over the surface of the rigid vesicle (blue), whose surface is fluid. The motors can bind to the rigid actin filament (bound motors in yellow, actin in green). These bound motors then step along the actin filament.
As they step, the motors experience forces, defined by a simple mechanical model. B. Simple mechanical model of a myosin Va motor. Each motor contains a linear and a torsional spring, that resist extension and bending, respectively. C. Mechanochemistry of myosin stepping and detachment. Each motor, once bound, can step forward, step backward or detach. These rates depend on the amount of force, directed along the actin filament, that is applied to the motor. D.
Mechanochemistry of myosin attachment. Myosin's attachment to a specific actin binding site depends on the mechanical energy it would take for a myosin to bind, with a higher energy cost making binding less likely. In this sample calculation, two myosin molecules are bound to actin and the relative attachment rate of a third motor at any actin binding site is color coded according to the color bar. The attachment rate is scaled so that the maximum attachment rate is 1. Negative cooperativity in myosin binding to actin. A. Calculation of attachment rate to binding sites on actin (Eq. 6) with one (top), two (middle) and three (bottom) myosin motors bound shows that the binding of myosin motors restricts the access of subsequent motors to actin binding sites.
When two or three myosin motors are bound to actin, attachment depends on the relative position of the motors. Insets show a histogram of the overall attachment rate (the sum of the attachment rate over every actin binding site), relative to the overall attachment rate of a second motor, ka.
One hundred motor configurations are shown in each histogram. B. The average attachment rate of an unbound myosin molecule strongly decreases as the number of actin bound motors increases.
The average overall attachment rate when 1-4 myosin motors were bound to actin was calculated for five simulations of motor ensembles transporting a fluid vesicle. The overall attachment rate, averaged over all observed motor configurations, is shown relative to the overall attachment rate of a second motor ka. C. Strong negative cooperativity makes it rare for more than three motors to be simultaneously bound to actin. For five simulations, we determined the proportion of time that 1-10 myosin motors were simultaneously bound to actin. Simulation parameters are given in Supplementary Table 1.

Supplementary Figure 11.
A stiff spring approximation reasonably describes the full model and greatly increases computational efficiency. A. The stiff spring approximation describes the attachment rate to each binding site along actin. For two bound myosin molecules, an exact calculation of the binding rate of a third myosin molecule to each actin binding site (top) compares well with a calculation that assumes myosin is inextensible (middle) and the difference between the two is small (bottom). B.
The stiff spring approximation captures the negative cooperativity of myosin motors binding to actin. Average overall attachment rate when 1-4 myosin motors were bound to actin was calculated for five simulations of motor ensembles transporting a fluid vesicle both using the exact model (solid line, hollow symbols) and using the stiff spring approximation (dashed line, solid symbols). The attachment rate is averaged over all observed motor configurations and shown relative to the overall attachment rate of a second motor calculated using the exact calculation (ka).
Inset shows a histogram of overall attachment rates when two myosin motors are bound to actin, demonstrating that the stiff spring approximation captures the distribution of attachment rates along with the average. C. The stiff spring approximation describes the proportion of time that different numbers of myosin motors are bound to actin. For five simulations, we determined the proportion of time that 1-10 myosin motors were simultaneously bound to actin using both the full model (solid line, hollow symbols) and the stiff spring approximation (dashed line, solid symbols).
Agreement is reasonable, and the simulations take minutes rather than days. Simulation parameters are given in Supplementary Table 1.

A model of 3D transport by multiple myosin Va motors introduction
To better understand and interpret our measurements of vesicles being transported by myosin Va motor teams, we constructed a mathematical model of the physical system. While the model makes some simplifying assumptions, our aims were to both capture the essence of the system and to make the model physically consistent. The following is a discussion of model assumptions and additional details of model results to supplement the main text.

Model and model assumptions
The model contains three components, the vesicle, the actin filament(s), and the myosin Va motors. Each myosin motor undergoes chemical reactions (i.e. stepping forward and backward along actin and detaching from actin) whose rate depends on the force applied to the motor. Thus, to define the model, we must determine the forces on the motors as they attach to, detach from and step along the actin filament(s). To determine these forces, we must create a mechanical model of each of the three components.

Component 1: Mechanical model of the vesicle
We model the vesicle as a rigid sphere, of radius R (Supplementary Figure 8A). The surface of the vesicle is fluid, so that forces tangent to its surface cause the membrane to flow. Contact between the vesicle and the actin filaments is frictionless. There is no attraction or repulsion between the vesicle and the actin filaments, other than steric constraints that keep the two from occupying the same space.

Component 2: Mechanical model of myosin
We model each myosin Va molecule as an extensible rod, of rest length l, anchored into the fluid surface of the vesicle by a deformable pivot (Supplementary Figure 8B). The motor's extensibility is linear, with spring constant k. The pivot, that connects the myosin motor to the vesicle, contains a universal joint with a torsional spring of stiffness κθ (Supplementary Table 1).
There is also a frictionless hinge that allows the motor to freely rotate about an axis normal to the surface of the vesicle. The combination of pivot and torsional spring allows us to model the bending stiffness of the myosin molecule. In particular, the spring and pivot work together to cause the motors to extend in the normal direction off of the vesicle and resist angular deflections in any direction equally.
Naturally, myosin Va is not really an extensible rod. Thus, myosin's rest length, l, and spring constant, k, might vary depending, say, on whether one or two heads are bound to actin, or on the state of each head. Vilfan estimates ~ 34 nm for the total length of each lever arm and myosin head 4 . Since the coiled-coil might add a little to this length, we estimate l = 50 nm (we examine this assumption further in section 3.4). Vilfan 4 also estimates the stiffness of a single lever arm to be around 0.25 pN/nm when force is applied perpendicular to its long axis. When both heads are bound, and force is applied vertically, stiffness should be higher than twice this value, so we estimate k = 1 pN/nm.
When attached to actin, the motors can experience forces. Since the vesicle is fluid, tangential components of these forces, FT, cause the motors to move relative to the vesicle surface at a velocity v = μ FT, where μ is a drag coefficient. The anchoring point of the motor diffuses across the surface of the vesicle with diffusion constant D = μ kBT, where kB is Boltzmann's constant and T is the absolute temperature 5 .
As we are interested in how motor teams transport fluid vesicles, we make the simplifying assumption that the vesicles are "ideally fluid." That is, μ , so that diffusion is very fast and motors experiencing tangential forces slide very quickly. This assumption contrasts with an "ideally solid" vesicle, where motors are anchored rigidly to the surface (μ 0). The measured value of the diffusion constant, D = 0.92 μm 2 /s (ref. 5) giving μ = 2.22x10 5 nm/pN . s, shows that diffusion is fast and that even modest tangential forces (~1pN) cause rapid sliding velocities (~ 200 μm/s).

Component 3: Mechanical model of actin
We model each actin filament as a rigid rod that is held rigidly in place. Myosin binding sites are arrayed every 5.5 nm along a right-handed double helix, with a periodicity of 72 nm (~ 14 monomers). Each binding site has a specific orientation, so that a myosin motor, once bound, extends rigidly from the binding site. This preferred orientation, which is orthogonal to the actin filament, rotates azimuthally with the actin periodicity (one full rotation every 72 nm, or 14 monomers).

Calculating forces: rapid mechanical equilibrium
Generally, the equations of motion for a small sphere in water are given by the Langevin where the first term on the right hand side is the viscous force on the vesicle, with γ being a drag constant, Fext is the sum of the external forces on the vesicle and Ff is a stochastic fluctuating force, due to solvent collisions. Since the fluctuating force is stochastic, this is a stochastic ordinary differential equation (ODE) and can be re-written as a partial differential equation (PDE) in terms of a probability density (ρ(x,t)), the Smoluchowski equation where V(x) is the potential giving rise to Fext, such that V = Fext. In To determine this attachment rate, we calculate E for every binding site on actin that is not already occupied with a myosin molecule. So, for a particular binding site, this requires finding the mechanical equilibrium configuration that would result from a motor binding to that binding site, calculating the energy of that configuration, and subtracting the energy of the current configuration from that value. The results of an example calculation are shown in Supplementary   Figure 8D.
Given the vesicle radius, Eq. 6 has one free parameter, k0. Rather than specifying k0 directly, it is more convenient to specify an overall attachment rate, which is the sum of katt over every binding site on actin. Generally, this attachment rate depends on the number of myosin motors on the vesicle surface that are attached to actin, and the relative position of those attached motors.
However, when a single myosin molecule is bound to actin, the vesicle is always positioned directly over it. Therefore, we define ka to be the overall attachment rate when one motor is bound (i.e. the attachment rate of a second motor). We specify this value in our simulations (which then defines k0 in Eq. 6).
To simulate a molecule undergoing these interactions, we use the Gillespie algorithm 8 .
Briefly, at each time step, we determine the time of every myosin molecule undergoing every possible chemical reaction (stepping forward, stepping backward, attaching and detaching) by picking random numbers from the appropriate distribution. These times are then sorted, and the reaction with the shortest time is implemented and time is advanced by that time step. In the original method 8 , the next shortest reaction would be implemented, time advanced, and so forth.
However, since each reaction generally changes the forces on each attached motor, and since the reaction rate constants depend on these forces, after each reaction occurs we re-calculate all reaction times. Importantly, this method gives the correct stochastic fluctuations that occur when a small number of molecular motors transport a shared cargo.

Model parameters
The model parameters are summarized in Supplementary Table 1. We can estimate or calculate all but two: κθ and ka. To estimate these parameters, we performed a sensitivity analysis (section 3.4).
We also examined the sensitivity of the model to ps and l (see section 3.4). Since the simulation results were insensitive to these parameters, we used parameter values that differ from those in Supplementary Table 1 for some simulations. For example, we performed our sensitivity analysis of κθ and ka with ps = 0.33, which gives a spiral pitch of 1.5 m. This shorter spiral pitch makes it easier to uniformly cover all possible approach angles.

Results
We performed a series of simulations of this model of myosin motor ensembles transporting a fluid vesicle in three dimensions. Here we focus on three specific results of these simulations.
First, because myosin molecules occasionally take a short step, simulated motor ensembles move vesicles along actin in a left-handed spiral. Second, simulated myosin ensembles exhibit strong negative cooperativity when binding to actin; that is, the binding of one motor strongly decreases the binding rate of the next motor. Third, simulated myosin ensembles prefer to proceed straight through 3D actin intersections, and the probability of going straight, turning and terminating agrees well with our experimental observations. We now discuss these three results, and then examine how the last result depends on parameters.

Results I: spiral trajectories
Because myosin takes an occasional short step, single myosin Va molecules follow a left-  Figure 9C). We therefore conclude that, in the model, myosin's occasional short step is responsible for the spiral trajectories observed for motor ensembles.

Results II: negative cooperativity
The model predicts strong negative cooperativity when myosin motors bind to actin. In particular, when a single myosin motor is bound to actin, the vesicle can rotate around myosin's pivot (the pivot is shown in Supplementary Figure 8B). Although this rotation is resisted by myosin's torsional spring, the second myosin molecule has access to many binding sites on actin.
After that second myosin binds to actin, the motion of the vesicle is restricted, so that it can only rotate around a single axis (the line that connects the pivots of the two attached myosin molecules), and far fewer binding sites are available. Binding of a third myosin molecule effectively freezes the configuration of the vesicle, so almost no binding sites are available ( Supplementary Figure   10A).
In order to characterize this negative cooperativity, we quantified how myosin's attachment rate depends on the number of myosin molecules bound to actin. Adding up the binding rates to each actin monomer gives an overall attachment rate. When more than one motor is bound to actin, the distribution of available binding sites depends on exactly how the motors are positioned relative to one another. Thus, there is a distribution of these overall attachment rates (Supplementary Figure 10A, insets). To characterize how overall attachment rate depends on the number of actin-bound myosin motors, we calculated overall attachment rate for a series of five simulations. We then determined when a given number of motors were bound, and found the average attachment rate. We observed that, if the overall attachment rate is ka when one motor is bound, it decreases to 0.33ka when two motors are bound and further to 0.05ka when three or more motors are bound (Supplementary Figure 10B).
One result of this negative cooperativity is that even though 10 motors are available to bind, and even though these motors can diffuse over the surface of the vesicle, the model predicts that there are typically a maximum of three motors simultaneously bound to actin. In fact, only 9% of the time are there four motors simultaneously bound to actin (Supplementary Figure 10C), while there are 1-3 motors simultaneously bound to actin the remaining 91% of the time. We never saw more than four motors simultaneously bound. Thus, at an actin intersection, we would not see a tug of war between two large groups of myosin motors, but rather between groups of motors comprised of a single motor and of two motors.

Stiff spring approximation, increasing computational efficiency
To compare the model to measurements of vesicles encountering actin intersections in 3D, we had to increase the efficiency of the simulations. In particular, calculating of the binding energy for the calculation of attachment rates is computationally expensive. It requires calculating the equilibrium position of the system for myosin binding to each available actin binding site. Finding the equilibrium position is not trivial, since even assuming mechanical equilibrium, one must simultaneously solve three non-linear equations (force balance in 3D). Further, since the solution is generally not unique, one must identify the correct mechanical equilibrium point.
To simplify the calculation, we made the approximation that myosin, as a linear spring, is very stiff. Then, extension of myosin is energetically prohibited. One can then easily find the equilibrium position of the vesicle by solving a geometry problem -i.e. if three myosin molecules are bound, where is the vesicle such that each myosin molecule is not extended? Solving similar geometry problems for two and one bound myosin, we increase the computational efficiency by roughly 10 3 , so that simulations that would take overnight can be completed in minutes. One important consequence of the approximation is that four motors cannot bind concurrently, unlike the more complex simulations, where the binding of four motors is disfavored, but possible.
To ensure that this stiff spring approximation was valid, we performed five simulations with the approximation (~15 minutes of computer time) and five simulations without the approximation (about a week). The approximation does a reasonable job capturing details of attachment and the distribution of attached motors (see Supplementary Figure 11).

Simulation results
We performed a series of experiments, observing motor teams navigating actin intersections in 3D (described in the main text and supplementary information). All actin intersections were at approximately right angles; minimum separation between the filaments varied. Since our model does not allow a single motor to step from one filament to the other, when comparing the model to these experimental data, we only considered experiments where the separation was 50 nm or greater. With these large separations, a single myosin molecule can't simultaneously bind to both filaments. There are then 75 measurements, with separations ranging from 50-250 nm.
To simulate these experiments, we started with a vesicle having a single motor bound to an actin filament. We then put the crossing filament 2.0 μm away from the initial attachment point, and added random noise from a uniform distribution with a maximum of 1 and minimum of −1 μm, so that the crossing filament was initially 1 -3 μm from the initial attachment point. We also randomized the polarity of the intersecting actin filament. Thus, the initial approach angle is dictated by the stochastic motion of the motors that transport the vesicle. Additionally, by placing the crossing filament some distance away from the initial attachment point, we ensure that the attachment/detachment of motors has reached steady-state. In the simulations, we can control the separation between the filaments, but we cannot control the approach angle of the vesicle, since it is determined by the stochastic motion of the motors.
Thus, in order to approximate the approach angles observed in the experiments, we performed each of the 75 simulations, described above, five times. From these 375 simulations, we kept the 75 whose filament separations matched and whose approach angle were closest to the experimentally observed values.
We then repeated this process six times (a total of 2,250 individual simulations

Actin polarity does not affect outcome probabilities
In our experiments, we found that the number of vesicles that turned left at an actin intersection (n=16) was almost identical to the number that turned right at an actin intersection (n=15). This result suggests that actin polarity does not affect whether a vesicle turns or goes straight through an actin intersection. In our six simulations, we also found no effect of actin polarity on the

Sensitivity analysis
The parameters used in our simulations were l = 50 nm, ps = 0.22, ka = 2.4 s -1 and κθ = 0.25 pNnm/rad. We performed a sensitivity analysis to determine how our results depend on these parameters.

Myosin rest length and spiral pitch
Our best estimate for myosin's rest length, l, is 50 nm. However, we expect that this value can vary depending on myosin's state (i.e. whether one or two heads are bound). Further, although myosin's full reach is likely around 50 nm, it might be reasonable to position the hinge in our mechanical model of myosin (Supplementary Figure 8B) at the junction of myosin's two lever arms, decreasing l to 35 nm (e.g. 4 ). We therefore performed simulations with l = 35 nm.
For each of these simulated data sets, we performed 75 simulations, with filament gaps matching a subset of our experimental measurements. For these 75 simulations, we determined the proportion of trajectories that went straight, turned and terminated, for trajectories with an approach angle where interaction was predicted. We repeated this process 10 times, and found the average and standard deviation.
At the initiation of our simulations, the crossing actin filament was randomly placed between 1 and 3 μm away from the vesicle. This variation of 2 μm ensures that, given a spiral pitch of ~ 2 μm we uniformly sample all possible approach angles. However, since the motors are stochastic, even when the average spiral pitch is 2 μm, some vesicles can follow spirals with a longer pitch over a 1-3 μm distance. Thus, to determine whether a bias in approach angle affected our results, we performed simulations with a shorter (ps = 0.33, Ps = 1.5 μm) spiral pitch.
Regardless of spiral pitch and myosin length, all simulations gave similar results (Supplementary Figure 13A). Thus, errors in our estimates of these values have a minimal effect on our conclusions.

Myosin stiffness and attachment rate
The model has two parameters, ka and κθ, that were unknown. We used ka = 2.4 s -1 and κθ = 0.25 pN nm/rad in our simulations. To determine how these variables affect our conclusions, we performed a sensitivity analysis. Because a shorter spiral pitch does not affect simulation outcomes, in these simulations we used ps = 0.33, giving a spiral pitch of Ps = 1.5 μm, in order to ensure a uniform sampling of approach angles.
We examined 17 different combinations of ka and κθ. For each parameter set, we performed ten simulations of our experiments, as described above, and, when the vesicle had an approach angle where interaction was predicted (Supplementary Figure 4, Interaction geometries of Supplementary Figure 12A), we recorded trajectories that went straight, turned or terminated. The accumulated results of these ten simulations was then compared to data, and we determined whether the simulations were different from (p<0.05, χ 2 test) or not different from (p>0.05, χ 2 test) the measurements. This data met all requirements for analysis by χ 2 test. These simulations allow us to identify a region of parameter space where simulation and experiment are not significantly different (yellow region in Supplementary Figure 13, bottom right). Note that, when ka gets too large, we expect that more than three motors will be bound. This provides an upper-bound to ka.
Generally, we find that turning probability increases with attachment rate, but rapidly saturates at around =0.5 -1 s -1 . For example, when κθ is held fixed at 0.25 pNnm/rad, increasing the attachment rate (ka = 0.4, 0.8, 1.6, 2.4, and 3.2 s -1 ) decreases the termination probability at the expense of turning, with little difference in the probability of going straight (see Supplementary Figure 13B, left). For this value of κθ, only the lowest attachment rate was significantly different from the measurements.
We also find that turning probability decreases with torsional stiffness. For example, when ka is held fixed at 2.4 s -1 , increasing the angular stiffness (κθ = 0.0625, 0.25, 1, 2, 4 pNnm/rad) leads to a decrease in turning probability, with the probability of going straight increasing and the probability of terminating remaining roughly constant (see Supplementary Figure 13B, top). For this value of ka, only the highest torsional stiffness was significantly different from the measurements.

Summary of sensitivity analysis
The sensitivity analysis demonstrates two main points. First, the qualitative result of our experiments and simulations, that motor teams prefer to go straight through 3D actin intersections, is generic. For all parameter combinations we tested, straight trajectories were always most frequent. Second, there is quantitative agreement between our simulations and experiments for a wide range of parameter values. In fact, all simulations with an attachment rate higher than ka = 1 s -1 and a stiffness lower than κθ = 2.5 pNnm/radian were not different from the experiments. Thus, agreement between model and experiment is robust, and is unlikely to be affected by model assumptions.