Dynamic Motions of Molecular Motors in the Actin Cytoskeleton

Abstract

During intracellular transport, cellular cargos, such as organelles, vesicles, and proteins, are transported within cells. Intracellular transport plays an important role in diverse cellular functions. Molecular motors walking on the cytoskeleton facilitate active intracellular transport, which is more efficient than diffusion-based passive transport. Active transport driven by kinesin and dynein walking on microtubules has been studied well during recent decades. However, mechanisms of active transport occurring in disorganized actin networks via myosin motors remain elusive. To provide physiologically relevant insights, we probed motions of myosin motors in actin networks under various conditions using our well-established computational model that rigorously accounts for the mechanical and dynamical behaviors of the actin cytoskeleton. We demonstrated that myosin motions can be confined due to three different reasons in the absence of F-actin turnover. We verified mechanisms of motor stalling using in vitro reconstituted actomyosin networks. We also found that with F-actin turnover, motors consistently move for a long time without significant confinement. Our study sheds light on the importance of F-actin turnover for effective active transport in the actin cytoskeleton.

INTRODUCTION

Intracellular transport is a process by which cellular cargos, such as organelles, vesicles, and proteins, are moved within cells. Intracellular transport plays a crucial role in a wide variety of cellular functions (Hirokawa, Niwa, & Tanaka, 2010; Maxfield & Menon, 2006). For example, during development of neural systems, neurotransmitters and mRNAs are transported, which facilitates neuronal functions including neurogenesis and morphogenesis (Hirokawa et al., 2010). In addition, cholesterol levels in cell membranes, which affect membrane properties and signal transduction processes, are regulated via intracellular transport (Maxfield & Menon, 2006).

To understand how cells regulate intracellular transport, many researchers have characterized motions of cargos via single particle tracking (Moerner, 2007). By tracking and analyzing the motions of vesicles or microspheres inside cells, it has been discovered that they undergo dynamically distinct motions (Huet et al., 2006). The slope of the mean square displacement (MSD) calculated from their trajectories in log-log scale (α) provides information about the nature of the motions. Vesicles attached to cellular components, such as a membrane, tend to show stalled or highly subdiffusive motions (α << 1) (Caspi, Granek, & Elbaum, 2002; Johns, Levitan, Shelden, Holz, & Axelrod, 2001; Kulkarni, Castelino, Majumdar, & Fraser, 2006; Steyer, Horstmann, & Almers, 1997). When cargos are not connected physically to any cellular component, they are transported within the cytosol primarily via diffusion (Zhuravlev, Lan, Minakova, & Papoian, 2012). Due to crowded environments of cytoplasm, they exhibit subdiffusive motions (α < 1) (Dix & Verkman, 2008). Cargos can be transported by molecular motors along the cytoskeleton, which is called active transport. Cargos undergoing active transport show ballistic (α ~ 2) or superdiffusive (1 < α < 2) motions (Kulkarni et al., 2006; Reverey et al., 2015). Cargos can also show a subdiffusive or stationary behavior when multiple molecular motors are bound to a single cargo and attempt to move in different directions (Lombardo et al., 2019), which is called tug-of-war (Müller, Klumpp, & Lipowsky, 2008).

Traditional studies regarding active transport focused mainly on transport of cargos driven by two types of molecular motors, kinesin and dynein, along radially oriented microtubules. This active transport is responsible for long-distance transport between a cell nucleus and a membrane (Gross, 2004; Vale, 1987). For example, cargos are transported from the nucleus to the membrane for secretion by kinesin, and from the membrane to the nucleus for endocytosis by dynein (Ross, Ali, & Warshaw, 2008). However, cargo transport along actin structures is also of great importance for local transport and cooperates with microtubule-based transport. For example, in neuronal growth cones, transport is dependent on both microtubules in the axon and F-actin in the soma (Brown, 1999; Mallik & Gross, 2004). In addition, myosin V helps regulation of secretory processes by preventing a majority of secretory vesicles in cortical actin networks from reaching a membrane (Meunier & Gutiérrez, 2016; Trifaró, Gasman, & Gutierrez, 2008). Due to structural differences between microtubules and the actin cytoskeleton, motions of cargos are quite different depending on where they are moving. Cargos exhibit directed motions on microtubules, whereas they show more diffusive and slower behaviors on actin structures (Desnos et al., 2003; Huet et al., 2006; Rudolf et al., 2003). Studies demonstrated that MSD of myosin motors walking on random actin networks decays as measurement time increases, which corresponds to the aging process in glassy dynamics (Burov et al., 2013; Cugliandolo, Kurchan, & Ritort, 1994).

Since actin networks in cells are highly disorganized, it is not obvious how myosin motors walk along the actin networks. One study found that MSD of myosin II is quite different depending on network architecture and types of cross-linkers (Scholz, Weirich, Gardel, & Dinner, 2018); motors walking on mixed polarity bundles formed by fimbrin and α-actinin can be trapped, whereas polarity-sorted bundle structures formed by fascin lead to highly directional motions. Another study using liposomes linked to multiple myosin V molecules showed that motions of the liposomes tend to be stalled more in unbranched actin networks, compared to those in branched networks formed by Arp2/3 (Lombardo et al., 2019). These experimental studies suggest that motors that can bind to more than one F-actin can be stalled or slow down due to tug-of-war.

Computational models have also been widely used to identify governing factors for the motions of motors in actin networks. It was found that cargos are transported from a nucleus to a membrane most efficiently when actin is densely distributed near the nucleus (Ando, Korabel, Huang, & Gopinathan, 2015). Also, it was shown that the length and number of F-actins significantly affect transport efficiency (Maelfeyt & Gopinathan, 2019; Maelfeyt, Tabei, & Gopinathan, 2019; Mlynarczyk & Abel, 2019). In addition, several studies showed that specific geometry of actin networks can trap motors. For example, it was demonstrated that stalling of myosin motors walking on an actin network may originate from a cycling state in which motors keep circulating within a geometrical trap formed by more than two F-actins (Scholz et al., 2016). Another study showed that motors can be trapped near the center of clusters formed by motor-driven polarity sorting of F-actins (Freedman, Hocky, Banerjee, & Dinner, 2018). These computational studies have provided valuable insights into understanding of how myosin motors move in disorganized actin structures. However, some of the assumptions used in those studies were less biologically relevant in that the mechanics and dynamics of cytoskeletal components were neglected or oversimplified. For example, load-dependent velocity of motors have not been incorporated in many previous models (Ando et al., 2015; Maelfeyt et al., 2019; Mlynarczyk & Abel, 2019). In addition, it was assumed that a network is completely rigid, meaning that filaments constituting a network were not allowed to be displaced or deformed (Scholz et al., 2016).

We hypothesized that motions of motors can be quite different if assumptions are closer to real actin networks. We employed an agent-based computational model to investigate motions of motors in disorganized actin networks with more physiologically relevant assumptions. The model accounts for the mechanics and dynamics of cytoskeletal components, such as the force-dependent walking velocity of motors and deformability of actin networks. We quantitatively analyzed motions of myosin motors in actin networks under various conditions. We found that motor motions can be confined due to three different reasons in the absence of F-actin turnover. Two of the reasons were verified via in vitro experiments using reconstituted actomyosin networks. However, in the presence of F-actin turnover, motors consistently move for a long time without significant confinement.

METHODS

We used an agent-based computational model based on Brownian dynamics (Jung, Murrell, & Kim, 2015; Kim, 2015; Kim, Hwang, Lee, & Kamm, 2009; Mak, Zaman, Kamm, & Kim, 2016). In this model, F-actin, motor, and actin cross-linking protein (ACP) are coarse-grained by cylindrical segments (Fig. 1a). Motions of the cylindrical segments are governed by the Langevin equation. Bending and extensional forces maintain equilibrium angles and distances formed by the cylindrical segments, respectively. Repulsive force accounts for volume-exclusion effects between overlapping F-actins. Details of the model are described below and in the supplementary material, and parameter values are listed in Tables S1 and S2.

Figure 1.

Analysis of motor motions using an agent-based computational model. (A) Actin filaments (F-actins, blue) are modeled as serially-connected cylindrical segments. Adjacent segments are connected by elastic hinges. Actin cross-linking proteins (ACPs, yellow) and two-arm motors (red, bottom) are modeled as two segments connected by elastic hinges. One-arm motors (red, top) are modeled as one segment. ACPs can bind to a pair of F-actins to form a functional cross-link. One-arm motors can bind to only one F-actin unlike two-arm motors that can bind to a pair of F-actins simultaneously. Arms of motors walk toward the barbed ends of F-actins. Bending (κ_b) and extensional (κ_s) stiffness govern mechanical behaviors of these segments. (B) An example of movement of a motor walking on a thin cortex-like network. A periodic boundary condition is applied in x and y directions. A trajectory of a motor measured in a simulation is visualized using a red line. A red circle represents the initial position of the motor.

Dynamics of F-actin

Formation of F-actin is initiated by appearance of one cylindrical segment occurring at a given nucleation rate (k_n,A). The segment is quickly elongated into a filament by addition of cylindrical segments to the barbed end at a given assembly rate (k_+,A). Depolymerization of F-actin is simulated by removal of cylindrical segments from the pointed end at a given disassembly rate (k_-,A). In cases without F-actin turnover, k_-,A is zero, so F-actins do not undergo any dynamic event after all actin segments are consumed for formation of F-actins. By contrast, in cases with F-actin turnover, k_-,A is equal to k_+,A, leading to treadmilling with a balance between polymerization at the barbed end and depolymerization at the pointed end. Thus, the average length of F-actin does not change over time after the number of free actin segments reaches a steady state. An actin turnover rate, k_t,A (= k_-,A = k_+,A), indicates how fast the treadmilling takes place.

Dynamic behaviors of ACPs

ACPs bind to binding sites located on actin segments without preference of a contact angle. We modeled both permanent and transient ACPs. Permanent ACPs do not unbind after binding to F-actins, which mimic kinetic behaviors of ACPs like scruin (Gardel et al., 2004). Transient ACPs can unbind from F-actins in a force-dependent manner, following Bell’s law (Bell, 1978):

$$k_{\text{u},\text{ACP}}=\{\begin{array}{ll}{k}_{\text{u},\text{ACP}}^0\text{exp}\left(\frac{{\lambda }_{\text{u},\text{ACP}}\left|{\overrightarrow{F}}_{\text{s},\text{ACP}}\right|}{k_{\text{B}}T}\right)\hfill & \text{if }r\ge r_{0,\text{ACP}}\hfill \\ k_{\text{u},\text{ACP}}^0\hfill & \text{if }r<r_{0,\text{ACP}}\hfill \end{array}$$ (1)

where $k_{\text{u,ACP}}^0$ is a zero-force unbinding rate constant, λ_u,ACP represents sensitivity to the magnitude of applied spring force $\left(\left|{\overrightarrow{F}}_{\text{s,ACP}}\right|\right)$, and k_BT is thermal energy. The spring force affects the unbinding rate only when the length of an ACP arm (r) is longer than its equilibrium length (r_0,ACP). Note that we employed a positive value for λ_u,ACP to mimic the response of a slip bond, so k_u,ACP exponentially increases with a higher applied force. The reference values of $k_{\text{u,ACP}}^0$ and λ_u,ACP are determined based on those of filamin A (Ferrer et al., 2008).

Dynamics of motors

Two types of motors are used in this study: one-arm motors can bind to only one F-actin, whereas two-arm motors can bind to a pair of F-actins. Two-arm motors can represent a cargo transported by more than one myosin motor in context of active transport and also represent myosin thick filaments that can bind to more than one F-actin (Bond, Brandstaetter, Sellers, Kendrick-Jones, & Buss, 2011). Indeed, effects of connectivity between a motor and multiple F-actins on active transport have been considered in a recent study (Scholz et al., 2016).

After binding to F-actin, each motor arm can either unbind from F-actin or walk toward the barbed end of F-actin in force-dependent manners. The force-dependent unbinding (k_u,M) and walking (k_w,M) rates of motor arms are determined by the parallel cluster model (PCM) which accounts for cooperative behaviors of several myosin heads with consideration of mechanochemical cycles (Erdmann, Albert, & Schwarz, 2013; Erdmann & Schwarz, 2012). The implementation and benchmark of PCM into our model are explained in detail in our previous study (Kim, 2015). In general, k_u,M and k_w,M generated from PCM have tendency to decrease as forces exerted on motor arms increase (Fig. S1). Thus, a motor arm eventually stops walking if motor arms experience forces beyond stall level which is ~3.8 pN per myosin head. If motor arms reach the barbed end of F-actin, they slide off from F-actin by a next walking event. After slide-off or unbinding, motor arms can bind to different F-actin, resulting in hopping from one F-actin to the other. If all arms of a motor lose connection to F-actin, the motor diffuses in the medium until one of its arms binds to another F-actin. We explicitly account for such diffusion in a free state because diffusion may play a significant role in motor motions as suggested in a previous study (Zhuravlev et al., 2012).

One-arm motors cannot generate large forces because drag forces acting on the center of the motors are not strong enough to resist reaction forces from F-actins, as one cannot exert a large force on a rope by pulling it on a slippery surface. On the contrary, two-arm motors can develop large forces on F-actins if they are bound to relatively anti-parallel F-actins, as one can exert large forces on two ropes by pulling them with both hands in opposite directions even on a very slippery surface.

Network assembly

In each simulation, motors walk on a pre-assembled network. If motors walk during network assembly, network morphology can become highly different at the beginning of measurement of motor motions. Then, it is hard to compare cases in different conditions since motors in those cases walk on a different structure from the beginning. To avoid this issue, we assembled networks without motors, then imported the pre-assembled network into each simulation with motors. Note that preassembly of networks without motor activities has been employed in previous in vitro experiments (Linsmeier et al., 2016).

For network formation, a cross-linked actin network is assembled via dynamic events of F-actins and ACPs in a three-dimensional thin rectangular domain (5×5×0.2μm) with a periodic boundary condition only in x and y directions (Fig. 1b). In all simulations, nucleation of F-actins takes place in a random direction perpendicular to the z direction at equal probabilities, followed by fast polymerization. This initially results in formation of a cortex-like network with randomly oriented F-actins. ACPs bind to a pair of F-actins to form a functional cross-link between F-actins.

By varying cross-linking density (R_ACP) and average F-actin length (<L_f>), we are able to control network connectivity. Larger R_ACP and <L_f> lead to formation of networks with higher connectivity. If ACPs cannot unbind from F-actins, a network remains homogeneous until the end of simulations. By contrast, with transient ACPs that can unbind, it was observed that F-actins tend to gradually form bundles over time under conditions of high connectivity (Movie S1), which is consistent with previous in vitro studies (Falzone, Lenz, Kovar, & Gardel, 2012; Wachsstock, Schwartz, & Pollard, 1993) and a computational study (Nguyen, Yang, Wang, & Hirst, 2009). In general, it takes a while for such a network to reach a steady-state configuration. If the slow transition to a bundled network occurs while motors walk on the network, patterns of motor motions inevitably become time-variant because geometry of a network where motors walk keeps changing over time. In addition, it is computationally inefficient to delay activation of motor walking in each simulation until a network reaches a steady state. Thus, we ran a few simulations in the absence of motors for 1000 s to generate networks that reached a steady state in terms of network morphology. Then, we imported these steady-state networks at the beginning of the simulations in order to quantify motions of motors without significant influences of F-actin bundling.

Evaluation of motions of motors

We evaluate motions of motors in a network in three ways. First, we calculate time-averaged, ensemble-averaged mean squared displacement (TE-MSD):

$$\text{TE-MSD}(\tau )=\frac{1}{N}\sum _{i=0}^N\left(\frac{1}{T-\tau }\underset{0}{\overset{T-\tau }{\int }}{\Vert r_i(t+\tau )-r_i(t)\Vert }^{2}dt\right)$$ (2)

where N is the number of motors, T is the duration of a simulation, τ is the lag time, r_i is the position vector of the ith motor, and t is time. TE-MSD(τ) indicates how far motors are displaced for τ on average.

Second, we calculate ensemble-averaged mean squared displacement (E-MSD) that represents how far motors are displaced on average for τ from an initial position at t = 0:

$$\text{E-MSD}(\tau )=\frac{1}{N}\sum _{i=0}^N\left(\Vert r_i(\tau )-r_i(0){\Vert }^2\right)$$ (3)

If the nature of motor motions is almost time-invariant (i.e. ergodic), TE-MSD would be very similar to E-MSD (Scholz et al., 2016). By comparing E-MSD and TE-MSD with reference curves indicating ~τ¹ and ~τ², we estimate the local power-law exponent α which is indicative of the nature of motions. α < 1, α = 1, 1 < α < 2, and α = 2 indicate subdiffusive, diffusive, superdiffusive, and ballistic motions, respectively.

Third, we fix the lag time (τ) and calculate ensemble-averaged mean squared displacement during the lag time at each time point t, which is named τ-MSD:

$$\tau \text{-MSD}(t)=\frac{1}{N}\sum _{i=0}^N\left(\Vert r_i(t+\tau )-r_i(t){\Vert }^2\right)$$ (4)

τ-MSD provides information about motor motions at each time point.

TE-MSD and E-MSD have been used in previous studies with different names (Burov et al., 2013; Scholz et al., 2016; Tabei et al., 2013). τ-MSD was often averaged from the beginning to different time points (i.e. measurement time) to determine whether motions undergo an aging process induced by confinement or trapping (Burov et al., 2013; Scholz et al., 2016; Tabei et al., 2013). Unlike the previous studies, we used τ-MSD at each time point to understand when motor motions start slowing down. In the previous studies, τ used for calculating τ-MSD ranges between 5 s and 15 s (Burov et al., 2013). We found that with higher τ, τ-MSD is larger, but the shape of all curves does not change significantly (Fig. S2). Thus, we set τ to 10 s for calculating τ-MSD. We calculated TE-MSD, E-MSD, and τ-MSD using 10–100 motor trajectories under each condition (Figs. S3–S6). To have sufficient number of trajectory data, we ran multiple simulations if each simulation has no more than 10 motors. The number of simulations run for such conditions is shown in Table S3.

Evaluation of F-actin confinement

To indirectly estimate elasticity of actin networks, we calculate TE-MSD of all actin segments (Fig. 2A). In our previous study, we showed a correlation between TE-MSD of F-actins and network elasticity (Kim et al., 2009). If F-actins are well connected due to more ACPs or longer F-actins, F-actins cannot move freely, resulting in lower TE-MSD which implies higher network stiffness (Kim et al., 2009; Mason, Ganesan, Van Zanten, Wirtz, & Kuo, 1997). By contrast, if F-actins are loosely connected, F-actins can move more freely, leading to higher TE-MSD which implies lower network stiffness. We evaluate TE-MSD of actin segments only in cases with 5 motors to avoid network stiffening induced by many motors so that TE-MSD is determined largely by network connectivity regulated via R_ACP and <L_f>.

Figure 2.

Motions of one-arm and two-arm motors in networks with different connectivity. In each simulation, 5 motors were used (N_M = 5), and ACP unbinding and F-actin turnover did not take place ($k_{\text{u},\text{ACP}}^0=0{\text{\hspace{0.17em}s}}^{-1}$, k_t,A = 0 s⁻¹). (A) TE-MSD (time- and ensemble-averaged mean squared displacement) of actins in networks with low, medium, and high connectivity. Networks with higher connectivity exhibit much lower TE-MSD, implying that movements of F-actins are confined more if connectivity is higher. (B) TE-MSD of one-arm and two-arm motors. Gray dashed lines indicate the slope of MSD corresponding to diffusive motions (~τ¹) and ballistic motions (~τ²). Two-arm motors show much lower TE-MSD in networks with medium and high connectivity, whereas one-arm motors exhibit similar TE-MSD regardless of network connectivity. (C) Average force exerted on each myosin head in motor arms, which is the ensemble average of time average of forces acting on motor arms, divided by the number of heads represented by each motor arm (N_h). One arm motors bear very small force (~0.1 pN) because they cannot generate force on F-actins. By contrast, two-arm motors bound to relatively antiparallel F-actins can generate high force. (D) The fraction of stalling indicating how long motors are stalled during a simulation run up to t = 1000 s. One-arm motors do not experience force-induced stalling due to very small force acting on their arms, whereas two-arm motors are stalled significantly.

Quantification of heterogeneity and aggregation of networks

Under certain conditions, a network tends to aggregate due to motor activity, resulting in heterogeneous network morphology. We analyzed the extent of aggregation and network heterogeneity using the radial distribution function, g(r), that represents density of particles as a function of a radial distance from a particle (Freedman et al., 2018). First, a particle density ρ is calculated by assuming that all particles are uniformly distributed throughout an entire region. Then, at a distance r from a particle, the number of particles in a donut-shaped region with small thickness Δr is counted (Fig. S7A). Note that the area of the donut-shaped region is approximately 2πrΔr, and the number of particles in the donut-shaped region, P(r), is normalized by 2πrΔrρ. The normalized value at the distance r is calculated with respect to all particles, and all normalized values are averaged into g(r).

$$g(r)=\langle \frac{P(r)}{2\pi r\Delta r\rho }\rangle$$ (5)

If particles are distributed uniformly, g(r) is close to one. Conversely, g(r) > 1 represents aggregation of particles near the distance r, and g(r) < 1 indicates the sparse distribution of particles near the distance r. At every 100 s, we calculate g(r) using instantaneous positions of actin segments with Δr = 20 nm (Fig. S7B). Then, we average values of g(r) at 0.1 μm < r < 1 μm to obtain $\overline{g}(t)$. The lower limit (0.1 μm) is an approximate distance at which g(r) becomes higher than 1, and the upper limit (1 μm) is an approximate distance beyond which g(r) shows no significant change. Then, the highest value of $\overline{g}(t)$ found in a simulation $\left({\overline{g}}_{\text{max}}\right)$ is divided by the initial value calculated at the beginning of the simulation $\left({\overline{g}}_{\text{init}}\right)$ as a measure of network aggregation. Higher “${\overline{g}}_{\text{max}}/{\overline{g}}_{\text{init}}$” indicates more severe network aggregation.

Experimental methods

To verify some of the hypotheses derived from simulations, we employ an in vitro experimental model of the cell cortex. Unlike simulations where non-muscle myosin II was modeled, we used skeletal muscle myosin in experiments. Since we are interested in tracking individual motors, we could not use non-muscle myosins in experiments due to a limitation in tagging those myosins.

The in vitro model followed the method previously described (M. Murrell & Gardel, 2014). On a plasma-cleaned glass coverslip, we generate a lipid bilayer consisting of a mixture of 100:1 egg phosphatidylcholine:Oregon Green 488 DHPE lipids. F-actins are pre-polymerized at concentration of 2.64 μM G-actin in F-buffer (50 mM KCl, 1 mM MgCl₂, 10 mM Imidazole, 200 μM EGTA, and 500 μM ATP at pH 7.5) and stabilized with 1.32 μM phalloidin. We use 75:25 as a ratio of dark to fluorescently labeled G-actin. Pre-polymerized F-actins are crowded onto the lipid bilayer with 0.25% 14kDa MW methylcellulose, and then glucose oxidase (0.25% mg/mL), catalase (0.05 mg/mL), and glucose (25 mM) are added to the imaging chamber to minimize photobleaching.

α-actinin aliquots and fluorescently labeled myosin are snap frozen and freshly thawed immediately before use. We centrifuge myosins immediately before use in the presence of stabilized pre-polymerized F-actins and high ATP to remove inactive motors. α-actinin and myosin are added to the imaging chamber after F-actins are fully crowded onto the lipid bilayer.

RESULTS

In this study, we investigated effects of several factors on motions of motors, such as network connectivity, reversibility of cross-links, F-actin turnover, the number of motors, and connectivity between motors and F-actins. For network connectivity, we used three different conditions: low, medium, and high. Average filament length (<L_f>) for low, medium, and high connectivity is 1.2 μm, 2.3 μm, and 3.9 μm, respectively. Cross-linking density (R_ACP) for low, medium, and high connectivity is 0.01, 0.04, and 0.1, respectively.

Motors are stalled by locally generated forces in networks with high connectivity

First, we analyzed motions of one-arm and two-arm motors in networks with three different connectivity levels. It was assumed that ACPs form permanent cross-links without the possibility of unbinding (i.e. $k_{\text{u,ACP}}^0=0$), and F-actins do not undergo turnover (i.e. k_t,A = 0). Although this condition is less physiologically relevant since F-actins and most kinds of the ACPs turn over in cells, we used this condition as a control case to understand effects of ACP unbinding and F-actin turnover on motor motions.

F-actins in networks with higher connectivity show much lower TE-MSD of actins, implying that F-actins are confined more (Fig. 2A and Movie S2). As shown in our previous study (Kim et al., 2009), highly confined F-actins are indicative of larger network elasticity. We found that the magnitude of TE-MSD of one-arm motors is larger than that of two-arm motors in general (Fig. 2B, S8A). In addition, at τ < 100 s, the slope of TE-MSD of one-arm motors is higher than that of two-arm motors (Fig. S8B). One-arm motors cannot generate large forces because they can bind to only one F-actin (Fig. 2C). Thus, they are able to walk along F-actins for a long time without force-induced stalling (Fig. 2D and Movie S2). It is noteworthy that TE-MSD and E-MSD measured from motions of one-arm motors are similar to each other, meaning that motions are more “ergodic” (Figs. 2B, S8D). Interestingly, in the case with one-arm motors, the magnitude and slope of TE-MSD are lower in a network with low connectivity at τ > 100 s, compared to networks with medium and high connectivity (Fig. 2B). At this time range, the slope is smaller than one, indicating subdiffusive motions. With a small number of ACPs and short F-actins, thermal fluctuation of F-actins and forces exerted on F-actins by motors can make F-actins move substantially as high TE-MSD values of F-actins imply (Fig. 2A). Therefore, movement of F-actins can cause motor movement to deviate from ballistic motions even when motors are walking along F-actins without stalling, thus resulting in lower magnitude and slope of TE-MSD at τ > 100 s.

TE-MSD of two-arm motors in all cases is smaller than those of one-arm motors. As network connectivity increases, the magnitude and slope of TE-MSD in cases with two-arm motors noticeably decrease (Figs. 2B, S8A, S8B). This can be explained by force-induced stalling. Unlike one-arm motors, two-arm motors can generate large forces if they bind to two F-actins oriented in relatively anti-parallel directions. Forces are generated better by motors in networks with high connectivity (Fig. 2C) because F-actins are confined tightly (Fig. 2A); two-arm motors can easily develop forces on tightly confined F-actins by pulling them. As a result, two-arm motors in networks with medium and high connectivity spend most of time in a stalling state (Fig. 2D and Movie S2). Since TE-MSD is calculated by averaging squared displacements over an entire time range, it is hard to estimate how long motors walk on average before force-induced stalling. τ-MSD and E-MSD can provide useful information about this (Figs. S8C, D). In the network with low connectivity, the slope of E-MSD and the magnitude of τ-MSD become smaller after t ~ 50 s, implying that two-arm motors walk for ~50 s on average before stalling. In the networks with medium and high connectivity, such changes in E-MSD and τ-MSD start between ~ 1 s and ~10 s, indicating that it takes less time for two-arm motors to be stalled in these networks.

From the slope of E-MSD, we found that both one-arm and two-arm motors do not show ballistic movement even at short time scale (τ < 1 s). This is attributed to frequent unbinding of motors from F-actins which leads to diffusion of motors and hopping to other F-actin (Fig. S1B). In addition, in cases with a low-connectivity network, relatively large movement of F-actins can make the trajectory of motors less straight even when motors are very processive. To confirm this hypothesis, we analyzed motions of one-arm motors with much less frequent unbinding within a network with high connectivity (Fig. S9). We found that motors show nearly ballistic movement at an early time range (τ < 10 s).

In sum, we found that the amount of forces exerted on motors highly affects motions of motors in disordered networks. If motors can bind to only one F-actin, they tend to consistently walk for a long time without stalling because they cannot generate large forces. If motors can bind to two F-actins, motors can be stalled by locally generated forces to an extent proportional to confinement of F-actins to the network.

Motors can be stalled due to global force transmission through a network

It was shown that networks with more motors generate larger stress and show stronger contractile behaviors (Jung et al., 2015). We probed effects of the number of motors (N_M) on motions of two-arm motors in the same three networks with different connectivity. All other conditions are the same as above. TE-MSD of motors walking in a network with low connectivity shows strong dependence on N_M (Fig. 3A); with a smaller number of motors, TE-MSD is higher. Such dependence of TE-MSD on N_M is consistent with changes in average force generated by motors and the fraction of motor stalling. As N_M increases, the average force becomes larger than the stall force, and the fraction of stalling is much higher (Fig. 3B, C). This implies that motors are stalled by high forces even in a network with low connectivity if there are a larger number of motors (Movie. S3). Although motors may contribute to enhancement of cross-linking level by acting as a tentative cross-linker, 100 motors cannot make the cross-linking density high enough to reduce TE-MSD. Note that the number of ACPs in a network with low connectivity is 1,504, which is much larger than 100.

Figure 3.

Motions of two-arm motors in networks with different connectivity and different number of motors (N_M). ACP unbinding and F-actin turnover were not considered in these simulations ($k_{\text{u},\text{ACP}}^0=0{\text{\hspace{0.17em}s}}^{-1}$, k_t,A = 0 s⁻¹). (A) TE-MSD (time- and ensemble-averaged mean squared displacement) of motors. Gray dashed lines indicate the slope of MSD corresponding to diffusive motions (~τ¹) and ballistic motions (~τ²). In a network with low connectivity, higher N_M leads to lower TE-MSD. (B) Average force exerted on each myosin head in motor arms, which corresponds to the ensemble average of time average of forces acting on motor arms, divided by the number of heads represented by each motor arm (N_h). For N_M > 10, the average force increases with N_M in a network with low connectivity, resulting in higher average force at N_M =100 than that in networks with medium and high connectivity. (C) The fraction of stalling indicating how long motors are stalled during a simulation run up to t = 1000 s. Motors are mostly stalled in networks with medium and high connectivity, regardless of N_M. In a network with low connectivity, the fraction of stalling increases at N_M > 10, which is consistent with the increase in the average force shown in (B). (D) Visualization of networks at 800 s. Blue, white, and red represent low, intermediate, and high forces, respectively. Only actins and motors bearing high forces (> 25 pN) are visualized. In a network with low connectivity and high N_M, adjacent motors are connected via force-bearing F-actins, indicating force transmission between the motors.

We hypothesized that this can be attributed to mechanical interactions via force transmission between motors. Tensile forces generated by motors can be transmitted to other motors through taut F-actins and ACPs that physically connect the motors, which can result in an increase in forces acting on the other motors. This is similar to mechanical interations between myosin molecules occurring within a thick filament; a power stroke motion of one myosin molecule on F-actin can develop forces on other myosin molecules bound to different or the same F-actin via the backbone of the thick filament. To confirm this hypothesis, we visualized large forces (> 25 pN) acting on F-actins and motors (Fig. 3D). In the low-connectivity network with N_M = 100, F-actins between neighboring motors bear a large amount of forces, which clearly demonstrates mechanical interactions between the neighboring motors. Motors create a web-like structure bearing large forces, and such large forces are not observed in other parts of the network (Fig. 3D).

In networks with medium and high connectivity, dependence of TE-MSD on N_M is much weaker (Fig. 3A). In addition, forces exerted on motors and the fraction of motor stalling do not vary due to a change in N_M (Figs. 3B, C). Forces acting between motors increase with larger N_M, but the degree of the increase is less than that observed in the low-connectivity network (Fig. 3D). Indeed, fewer neighboring motors mechanically communicate (i.e. fewer motors connected by F-actin with high forces) compared to those in the low-connectivity network, so the web-like structure is not observed in networks with medium and high connectivity. This indicates that motors in networks with medium and high connecitivity tend to exert forces to their vicinity rather than other motors located distantly. τ-MSD and E-MSD show that it takes ~10 s on average for 100 motors to be stalled in the low-connectivity network (Figs. S10A, B). By contrast, in networks with higher connectivity, motors are stalled soon (~ 1 s) after they start walking (Movie S3). In networks with higher connectivity, motors can be stalled quickly by reaction forces from F-actin at cross-linking points or between anti-parallel F-actins located near their initial positions. On the other hand, in networks with low connectivity, motors keep moving until they find one of a few positions where they can generate large forces and feel forces transmitted from other motors to form the web-like structure, thus resulting in a longer time before being stalled (Movie S3).

In sum, two-motors walking in the low-connectivity network without ACP unbinding and F-actin turnover can be stalled by global force transmission between motors if the number of motors is large. However, in networks with higher connectivity, motors are stalled quickly by locally generated forces, so global force transmission plays an insignificant role in motor stalling.

Motions of motors can be confined by F-actin aggregation

Unlike the assumption used above, most of ACPs existing in cells form transient cross-links because they unbind stochastically from F-actins in a force-dependent manner (Ferrer et al., 2008; Xu, Wirtz, & Pollard, 1998). We ran simulations under the same conditions as above but incorporated the force-dependent unbinding of ACPs (Eq. 1). TE-MSD of two-arm motors in networks with medium and high connectivity is not dependent on the number of motors (N_M) as strongly as in the cases without ACP unbinding (Fig. 4A). However, the magnitude of TE-MSD is much higher, and the average force exerted on motors and the fraction of stalling are significantly lower compared to the cases without ACP unbinding (Figs. 4A–C). In addition, in networks with medium and high connectivity, τ-MSD decreases from ~5 s but reaches a plateau at later times (Fig. S11A). The slope of E-MSD is also reduced at ~5 s but increases later (Fig. S11B). These imply that motors did not remain in the stalling state for a long time (Movie S4). Since forces acting on a network are relaxed consistently by ACP unbinding, motors stalled by high local forces have an opportunity to resume walking. Although motors are stalled less, they still experience forces close to the stall force and thus walk more slowly than motors in the low-connectivity network, as seen in TE-MSD, E-MSD, and τ-MSD.

Figure 4.

Motions of two-arm motors in networks with different connectivity and different number of motors (N_M). In these cases, ACPs are allowed to unbind from F-actins with the reference unbinding rate $\left(k_{\text{u},\text{ACP}}^0=0.115{\text{\hspace{0.17em}s}}^{-1}\right)$. F-actin turnover is not considered (k_t,A = 0). (A) TE-MSD (time- and ensemble-averaged mean squared displacement) of motors. Gray dashed lines indicate the slope of MSD corresponding to diffusive motions (~τ¹) and ballistic motions (~τ²). In a network with low connectivity, higher N_M leads to lower TE-MSD after ~5 s. (B) Average force exerted on each myosin head in motor arms, which is the ensemble average of time average of forces acting on motor arms, divided by the number of heads represented by each motor arm (N_h). Motors in networks with medium and high connectivity experience much higher forces. (C) The fraction of stalling indicating how long motors are stalled during a simulation run up to t = 1000 s. Motors in networks with higher connectivity are stalled for a longer time, which is consistent with higher forces exerted on motor arms shown in (B). (D) Network heterogeneity quantified using the radial distribution function from positions of actins. Networks with low/medium connectivity high N_M exhibit aggregating behaviors. (E) TE-MSD of motors walking in reconstituted F-actin networks with low motor density. Cross-linking density is 0 (blue) or 0.1 (red). (F) A correlation between adjacent image frames showing only motors in (black and red) experiments and in (cyan and green) simulations. Cross-linking density is high in two cases represented by black and cyan and low in the other cases.

TE-MSD of motors walking in the low-connectivity network shows dependence on N_M at τ > ~5 s. With higher N_M, TE-MSD tends to be lower although forces exerted on motors are lower (Figs. 4A, B). τ-MSD and the slope of E-MSD show a noticeable drop after ~100 s, meaning that motors walked relatively fast regardless of N_M before ~100 s (Fig. S11A, B). We found that confinement of motor movements at high N_M is attributed to aggregation of a network. If many motors walk in a network with low connectivity, F-actins tend to form an aster-like structure (Figs. 4D, S11C and Movie S4). Additionally, it was shown that barbed ends are located near the center of the aster-like structure via polarity sorting (Wollrab et al., 2019). Our results also shows that at the center of the aster-like structure, motors and barbed-ends are distributed more densly at later times (Fig. S11D). Then, once motors walk into the center, they cannot escape because there are few tracks oriented radially outward. However, they continuously move around the center region without force-induced stalling (Fig. 4B, C). In the aster-like structure, many of F-actins are oriented in a relatively parallel configuration due to the polarity sorting, and they are highly curved due to buckling occurring during aster formation (M. P. Murrell & Gardel, 2012). Thus, a region near the center of the aster-like structure is not a favorable environment for two-arm motors to generate significant tensile forces. Thus, they rather keep walking along F-actins without stalling.

With 10-fold lower $k_{\text{u},\text{ACP}}^0$ (= 0.0115 s⁻¹), network aggregation still takes place in the low-connectivity network with high N_M (Fig. S12C), and similar dependences of TE-MSD and average motor forces on N_M were observed (Figs. S12A, B). However, with 100-fold lower $k_{\text{u},\text{ACP}}^0$ (= 0.00115 s⁻¹), the network does not aggregate significantly (Fig. S12F), and TE-MSD is almost independent of N_M (Fig. S12D). The average force exerted on motors increases with higher N_M (Fig. S12E), but it is not high enough to induce motor stalling for a long time. This implies that even slow ACP unbinding can prevent motor stalling induced by the global force transmission by slowly relaxing forces on a network.

In sum, two-arm motors in networks with force-dependent unbinding of ACPs are not stalled permanently even with very high network connectivity because forces are relaxed consistently by ACP unbinding. If the network has low connectivity and a sufficient number of motors, it can severely aggregate. This can induce confinement of motors at the center of aggregates for a long time although the motors keep walking around the center.

Experimental measurements support the hypotheses for motor stalling and confinement

We qualitatively verified our hypotheses regarding the stalling and confinement of motors induced by force generation and aggregation, using reconstituted two-dimensional networks with F-actin stabilized by phalloidin, α-actinin which transiently cross-links F-actins, and skeletal muscle myosin (SKMM). Skeletal muscle myosin is a highly non-processive motor which spends only ~4% of its lifetime in a bound state (Harris & Warshaw, 1993). Hundreds of them are assembled into a long thick filament structure to stay on F-actin while they walk along F-actin fast (Skubiszak & Kowalczyk, 2002). Thus, admittedly, motors used in our in vitro experiments are quite different from those used in simulations in terms of the number of F-actins that they can bind simultaneously and total maximum force generated by one motor. In simulations, we can mimic behaviors of thick filaments consisting of skeletal muscle myosins by imposing the high ATP-dependent unbinding rate and the large number of myosin heads represented by each motor arm. In addition, our model can simulate a thick filament structure with multiple motor arms (Kim, 2015). However, large forces generated by such motors prevented us from focusing mainly on motor motions without a significant change in network morphology. Thus, we decided to employ slightly non-processive, weak motors. However, due to the universal tendency of force-dependent walking and unbinding of different myosin isoforms, we can still make a qualitative comparison between some results from experiments and simulations to verify hypothesis.

First, with low motor density (0.5 nM SKMM), we conducted experiments with many ACPs (10:1 G-actin:α-actinin) and without any ACP. TE-MSD of motors is much lower in the case with high cross-linking density (Fig. 4E), which is consistent with simulation results (Fig. 4A). Note that values of TE-MSD measured in experiments are higher than those evaluated in simulations, which can be attributed to a large difference in walking velocity between SKMM (< ~5 μm/s) (Harris & Warshaw, 1993) and motors in simulations (< 140 nm/s) that intend to mimic behaviors of non-muscle myosins (Cuda, Pate, Cooke, & Sellers, 1997). Still, the decrease in TE-MSD with high cross-linking density is consistent with our observation of force-induced motor stalling occurring at high R_ACP.

We repeated experiments with higher motor density (≥150 nM SKMM). Since many motors are closely located under this condition, tracking individual motors in images is not feasible. Thus, instead of quantifying MSD, we calculated a correlation between adjacent image frames showing only myosin motors in order to evaluate the extent of motions during each time interval. A correlation close to 1 indicates that motors hardly move between frames, whereas a lower correlation value is indicative of faster myosin movements between frames. With high connectivity (10:1 G-actin:α-actinin, 220 nM SKMM) (i.e. high cross-linking density), the correlation slightly decreases at the beginning and then remains at ~1 (Fig. 4F and Movie S8). The correlation calculated using the simulation under similar conditions (N_M = 100 and high network connectivity) shows a qualitatively similar tendency. It implies that motors in both the experiments and the simulations hardly move over time. In the presence of only myosin motors (150 nM), severe network aggregation was observed (Movie S8) as in the simulation (Fig. S11C and Movie S4). In both the experiment and the simulation, the correlation significantly drops at the beginning and is recovered to 1 after ~400 s (Fig. 4F). This implies that the network aggregates mostly at t < 400 s. Overall, correlations calculated in the experiments with high and low cross-linking density are consistent with our hypotheses for local force-induced stalling and confinement due to network aggregation.

Motor motions without F-actin turnover can be confined or stalled due to three reasons

The results that we have shown so far suggest three reasons for which motor motions are stalled or confined (Fig. 5): local force-induced stalling, global force-induced stalling, and confinement by network aggregation. First, local force-induced stalling occurs in networks with high connectivity without dependence on motor density (Fig. 5, blue). If there is no unbinding at all, motors can be stalled permanently in one location. As ACPs unbind more frequently, network connectivity needs to be higher to achieve motor stalling induced by locally generated forces. Even if ACP unbinding takes place very slowly, motors repeat walking and stalling over time without permanent stalling.

Figure 5.

Phase diagram showing three mechanisms of motor stalling or confinement in a three-dimensional parametric space consisting of network connectivity, motor density (N_M), and ACP unbinding rate $\left(k_{\text{u},\text{ACP}}^0\right)$. It is assumed that F-actins do not undergo turnover.

Second, global force-induced stalling occurs in networks with low connectivity, many motors, and ACPs that unbind very slowly (Fig. 5, green). The ACP unbinding rate needs to be very low to keep the motors stalling for a long time. As network connectivity increases, more motors are required to achieve network-level force transmission between neighboring motors because it becomes harder to transmit forces over a long distance in a network with higher connectivity. Otherwise, with higher network connectivity, slightly faster ACP unbinding is necessary for bringing effective network connectivity to lower level so that global force-induced stalling can arise.

Lastly, motor motions can be confined by network aggregation if motor density and ACP unbinding rate are high, but network connectivity is low (Fig. 5, red). We do not call it stalling because motors consistently walk near the center of aggregated structures even after severe network aggregation. As connectivity increases, more motors or faster ACP unbinding events are required for confinement of motor motions via network aggregation.

Note that none of these three mechanisms for motor confinement or stalling can be found in models using motors walking in completely rigid networks in a force-independent manner. Because our model accounts for dynamics and mechanics of cytoskeletal elements rigorously, we could identify these three mechanisms.

F-actin turnover can rescue motors from confinement or stalling

The three mechanisms mentioned above were identified without F-actin turnover. Such a condition is similar to many in vitro studies where F-actin does not undergo turnover due to stabilization of F-actin via phalloidin. However, during most of the cellular processes, F-actin turns over very fast via various modes of actin dynamics (Chugh & Paluch, 2018). Recent computational studies have shown that F-actin turnover can regulate force generation/dissipation and contractile behaviors in actin networks (Chandrasekaran, Upadhyaya, & Papoian, 2019; Hiraiwa & Salbreux, 2016; Mak et al., 2016; McFadden, McCall, Gardel, & Munro, 2017; Popov, Komianos, & Papoian, 2016). Common observations in those studies are that F-actin turnover keeps relaxing forces, prevents motors from generating large forces, and suppresses severe network aggregation. To evaluate effects of F-actin turnover on the three mechanisms of motor confinement/stalling, we incorporated F-actin treadmilling by imposing identical polymerization (k_+,A) and depolymerization rates (k_-,A) of F-actin. Then, the turnover rate of F-actin, k_t,A, is equal to k_+,A and k_−,A.

First, we repeated simulations under conditions for local force-induced stalling. We used a network with the highest connectivity, N_M = 100, and without ACP unbinding. As the turnover rate of F-actin increases, TE-MSD of two-arm motors becomes larger, and the fraction of stalling and average force exerted on motors are reduced, resulting in enhanced motor motions (Figs. 6A, B and Movie S5). With k_t,A = 20 s⁻¹, TE-MSD, average motor force, and the fraction of stalling become very similar to those of two-arm motors walking in the low-connectivity network (Figs. 2B–D). If actin segments disappear due to depolymerization, all ACPs and motors that were bound to the segments are likely to lose connection to F-actin permanently because they cannot rebind to the same segments. Thus, the turnover (i.e. treadmilling) is able to facilitate force relaxation more efficiently than ACP unbinding, which can prevent local force-induced stalling from taking place. Note that these simulations imported a network without noticeable bundle formation unlike other simulations because we found that relatively fast actin turnover changes network morphology substantially if the network initially has bundles formed without actin turnover.

Figure 6.

Effects of a F-actin turnover rate (k_t,A) on motions of two-arm motors. All of the simulations used for this figure have 100 motors (N_M = 100). (A) TE-MSD of motors in a network with high connectivity in the absence of ACP unbinding $\left(k_{\text{u,ACP}}^0=0\right)$. Gray dashed lines indicate the slope of MSD corresponding to diffusive motions (~τ¹) and ballistic motions (~τ²). With higher k_t,A, TE-MSD (time- and ensemble-averaged mean squared displacement) is higher, which indicates less motor stalling. (B) Average force exerted on each myosin head in motor arms and the fraction of stalling decrease with higher k_t,A. (C) TE- MSD of motors in a network with low connectivity without ACP unbinding $\left(k_{\text{u,ACP}}^0=0\right)$. (D) Average force acting on each myosin head in motor arms and the fraction of stalling decrease with higher k_t,A. (E) TE- MSD of motors in a network with low connectivity. ACPs were allowed to unbind from F-actins with the reference unbinding rate $\left(k_{\text{u},\text{ACP}}^0=0.115{\text{\hspace{0.17em}s}}^{-1}\right)$. (F) Network heterogeneity and average force acting on ACPs decrease with higher k_t,A. For evaluating network morphology, we calculate the radial distribution function, g(r), every 10 s. Then, the average value of g(r) at 0.1 μm < r < 1 μm, $\overline{g}(t)$, is calculated for each time point. The maximum value of $\overline{g}(t)$ normalized by its initial value, ${\overline{g}}_{\text{max}}/{\overline{g}}_{\text{init}}$, is used as a measure for network heteterogeneity. All of the results shown here imply that F-actin turnover prevents motors from being stalled or confined.

Then, we investigated influences of F-actin turnover on global force-induced stalling by imposing different turnover rates on a network with low connectivity, N_M =100, and without ACP unbinding. Even with a slight increase in the turnover rate, TE-MSD significantly increases, whereas average motor force and the fraction of stalling drop significantly, resulting in enhanced motor motions (Figs. 6C, D and Movie S6). Spatial distributions of actin segments and motors with very high force (>50 pN) clearly show that force transmission between neighboring motors diminishes as the turnover rate increases (Fig. S13A). This indicates that global force-induced stalling of motors is much more sensitive to force relaxation occurring in a network than local force-induced stalling. Indeed, we have shown that global force-induced stalling does not appear even with slow ACP unbinding (Fig. S12D), indicating high sensitivity to force relaxation.

Lastly, we repeated simulations under conditions for severe network aggregation. We used a network with low connectivity, N_M = 100, and ACP unbinding $\left(k_{\text{u},\text{ACP}}^0=0.115{\text{\hspace{0.17em}s}}^{-1}\right)$. As the turnover rate increases, TE-MSD of motors increases, and network is more homogeneous without severe aggregation (Figs. 6E, F, S13B and Movie S7). To explain effects of the turnover rate, we quantified loads acting on ACPs at t < 100 s in each case. Since it is expected that only a small portion of ACPs support large loads, we focused on the largest forces (top 5%) supported by ACPs. F-actin turnover reduces the amount of loads on ACPs (Fig. 6F). As a result, the ACP unbinding rate does not exponentially increase, so network connectivity is not disrupted significantly by too frequent ACP unbinding events. In addition, clusters formed by network aggregation can be disassembled if the turnover is fast enough. Because of these two effects, network aggregation is suppressed by F-actin turnover, resulting in consistent movements of motors without noticeable confinement.

Overall, with sufficiently fast F-actin turnover, three regimes for confinement and stalling of motors in the phase diagram shown in Fig. 5 disappear, implying that myosin motors in cells can keep moving in a superdiffusive, ergodic fashion. It is expected that cells regulate motor motions differently for various physiological functions by controlling F-actin turnover rates.

DISCUSSION

A myriad of studies showed that cortical actin network and the myosin super family play a very important role in transport of secretory carriers, which are either vesicles or granules, near a plasma memebrane (Bond et al., 2011). Unconventional myosin proteins 1c, 1e, Va, VI and the non-muscle myosin II are of particular importance (Loubery & Coudrier, 2008). Myosin molecules walking on F-actins transport the carriers actively as their cargos. If there are more than one myosin molecule bound to the cargo, it will be easier for the cargo to maintain connection to F-actins and to hop from one F-actin to the other. However, there can be a tug-of-war between myosin molcules bound to different F-actins oriented in relatively opposite direcitons. By contrast, if there is only one myosin molecule bound to the cargo, the cargo may keep moving in a unidirectional manner, but it will be harder to maintain connectivity to F-actins for a long time.

Thus, understanding motions of myosin motors in disordered actin networks is very important for defining mechanisms of active transport. Motions of myosin motors have been investigated in a plethora of studies (Burov et al., 2013; Freedman et al., 2018; Scholz et al., 2016; Scholz et al., 2018). For example, confinement of myosin motions (i.e. limited mobility) in F-actin networks has been reported in several experimental studies (Lang et al., 2000; Scholz et al., 2016). Various computational studies have suggested possible mechanisms of the motor confinement (Freedman et al., 2018; Scholz et al., 2016). In this study, to show how motor motions are confined or restored, we investigated motions of myosin motors in cross-linked F-actin networks using an agent-based model. The model rigorously accounts for mechanics and dynamics of F-actin, ACP, and motor, including network deformability, force-dependent walking of motors, ACP unbinding, and F-actin turnover.

First, we showed that motors that can bind to one F-actin walk freely in a network regardless of network connectivity because they cannot generate large forces that can make the motors slow down. However, if motors can bind to more than one F-actin, motors can be stalled or slow down due to local force generation. The extent of the local force-induced motor stalling is higher if F-actins in a network are better connected to other F-actins via more ACPs or longer F-actins; motors can develop larger forces on F-actins by pulling them if movement of F-actins is restricted more, but the motors will feel the same amount of forces as reaction forces. Thus, in networks with higher connectivity between F-actins, most of the motors are stalled shortly after they start walking. This is consistent with previous experimental studies showing that motors bound to several filaments exhibit confined movements due to tug-of-war (Lombardo et al., 2019). However, a previous computational study reported that there is no significant difference in motor motions between a bundled network with high connectivity and a homogeneous network with low connectivity (Freedman et al., 2018). In this study, a few additional tracer motors were put on an actomyosin network that reached a steady-state morphology. Although it was not explained in detail, it is likely that the tracer motors interact with the network in a different way compared to pre-existing motors that drove a change in network morphology. Otherwise, the tracer motors should have been stalled in the bundled network like the pre-existing motors.

We also demonstrated that the tug-of-war may occur at network level, leading to stalling of motors in a network with low connectivity. It was shown in a previous study that forces generated by contractile elements within a network can be transmitted to other contractile elements (Ronceray, Broedersz, & Lenz, 2016). If there are a sufficient number of motors so that an average distance between neighboring motors is short enough, forces generated by motors can be transmitted to adjacent motors. Motors initially walk relatively freely, but as a result of mechanical interations between them, motors eventually find locations where they form a web-like structure that bear large tensile forces at network scale. Then, they are stalled and remain in relatively the same locations for a long time. This network-scale force transmission arises in networks with relatively low connectivity because long-range force transmission through a polymeric network is unlikely to occur in densely cross-linked networks where motors are stalled by local force generation.

In addition, we found that motor motions can be confined due to network aggregation. Motors are not stalled but keep moving around the center of aggregated structures. This is consistent with the previous computational study which demonstrated that aggregation of motors inhibits mobility of motors (Freedman et al., 2018). Based on these results, we suggested a phase diagram showing how and when motor motions can be confined in disordered networks.

Our study shows that F-actin treadmilling can prevent motors from being stalled or confined, resulting in superdiffusive motions of motors, regardless of motor density, F-actin length, and the density and unbinding rate of ACPs. F-actin turnover induces relaxation of forces acting on motors and ACPs by removing a portion of F-actins. This force relaxation can prevent force-induced stalling of motors and aggregation of F-actins more effectively than ACP unbinding. Suppression of network aggregation via F-actin turnover was demonstrated in previous computational study (Hiraiwa & Salbreux, 2016; Mak et al., 2016; Popov et al., 2016). In addition, this result is consistent with previous experimental results showing that stabilization of F-actins leads to lower myosin mobility (Lang et al., 2000). Since the cell cortex consists of F-actins with fast turnover, it is likely that the stalling and confinement of motors in the cortex are rare events under most physiological conditions. However, motors would walk slower than their unloaded walking velocity since they still bear a certain amount of forces from prestress residing in the cortex. In addition, when the cortex shows aggregating behaviors, such as formation of cytokinetic rings, it is expected that motor motions are confined.

Interestingly, we did not observe a cycling state in which motors are trapped in a network by walking along a closed loop-like structure formed by a few F-actins (Scholz et al., 2016). The previous study found the cycling state by tracing motors walking on a frozen network structure that may have such a closed loop-like structure permanenly. However, in our model, a network cannot have long-lasting loop-like structures because F-actins keep thermally fluctuating or move actively due to forces generated by motors. In cases with F-actin turnover, network morphology changes even more rapidly. Thus, although motors in our model can be trapped in a closed loop-like structure for a short time period, it would not be able to affect motor motions for a longer time.

Some of the results in this study may look similar to previous computational works. For example, a change in network morphology induced by motors has been studied actively during recent decades. In particular, aggregation (i.e. cluster formation) in actomyosin networks has been shown in several studies (M. P. Murrell & Gardel, 2012; Wollrab et al., 2019). Effects of network connectivity or the unbinding rate of ACPs on aggregation have been studied before (Freedman et al., 2018; Schuppler, Keber, Kröger, & Bausch, 2016). However, to our knowledge, these studies did not investigate and analyze motions of motors as deeply as this study. Compared to those previous works, novelty of this study is quantitative analysis of motor motions under various conditions. In addition, it was shown that movement of F-actins driven by motors is the greatest with intermediate F-actin turnover rates (Popov et al., 2016), which may look similar to our last results with F-actin turnover. Although we calculated MSDs of F-actins for some cases, our main focus is motions of motors in disordered networks. The extent of active transport can be estimated properly by motions of motors, not by F-actin motions. We also showed clearly how motions can be confined depending on conditions, which has not done in other previous studies in a quantitative manner.

In conclusion, our study demonstrated how motions of myosin motors in cortex-like networks can be regulated by network connectivity, local force generation, network-scale force transmission, and the turnover of ACPs and F-actins. The results from our study provide valuable insights into understanding mechanisms of intracellular transport driven by myosin motors in the actin cytoskeleton beyond those shown in previous studies.