Kinematics of the lever arm swing in myosin VI

Significance

Myosin VI (MVI), a molecular motor whose malfunction is linked to deafness, moves on the actin filament fueled by ATP. The chemomechanical transduction culminates in a power stroke, in which the motor domain undergoes a conformational transition exaggerated by the lever arm. We performed simulations of the MVI power stroke, showing that the lever arm undergoes a nearly free rotational diffusion that is only weakly biased by the rest of the motor. Our model yields a molecular picture of the MVI power stroke that is in quantitative agreement with experiments showing evidence for the pliancy of the lever arm. Our findings provide insights into the broad step-size distribution of MVI.

Abstract

Myosin VI (MVI) is the only known member of the myosin superfamily that, upon dimerization, walks processively toward the pointed end of the actin filament. The leading head of the dimer directs the trailing head forward with a power stroke, a conformational change of the motor domain exaggerated by the lever arm. Using a unique coarse-grained model for the power stroke of a single MVI, we provide the molecular basis for its motility. We show that the power stroke occurs in two major steps. First, the motor domain attains the poststroke conformation without directing the lever arm forward; and second, the lever arm reaches the poststroke orientation by undergoing a rotational diffusion. From the analysis of the trajectories, we discover that the potential that directs the rotating lever arm toward the poststroke conformation is almost flat, implying that the lever arm rotation is mostly uncoupled from the motor domain. Because a backward load comparable to the largest interhead tension in a MVI dimer prevents the rotation of the lever arm, our model suggests that the leading-head lever arm of a MVI dimer is uncoupled, in accord with the inference drawn from polarized total internal reflection fluorescence (polTIRF) experiments. Without any adjustable parameter, our simulations lead to quantitative agreement with polTIRF experiments, which validates the structural insights. Finally, in addition to making testable predictions, we also discuss the implications of our model in explaining the broad step-size distribution of the MVI stepping pattern.

Like their counterparts dyneins and kinesins, myosins are molecular motors that transform chemical energy harvested in the hydrolysis of ATP into mechanical work. They do so by undergoing a reaction cycle (Fig. 1) involving ATP hydrolysis coupled with binding to and unbinding from the filamentous actin (F-actin) (1, 2). Myosins loaded with a hydrolyzed ATP bind to F-actin, release the products of ATP hydrolysis, and undergo a structural change known as a power stroke. The structural change of the N-terminal part of the motor domain, where the actin and nucleotide binding sites reside, is communicated to the converter domain that moves from the prepower stroke (PrePS) state to the postpower stroke (or rigor, R) state conformation. The movement of the converter is exaggerated by the large swing of the lever arm, an oblong domain bound to light chains or calmodulins (CaMs). When a new ATP molecule binds the nucleotide-free myosin (in the R state), the motor detaches from actin and it is ready to begin a new cycle.

Fig. 1.

The reaction cycle of MVI. Color code: slate blue, motor domain; red, ins1; dark blue, converter; orange, ins2 and ins2-bound CaM; green, IQ domain and IQ-bound CaM; black, actin. The Protein Data Bank (PDB) structures used to make this figure are 4ANJ (41) and 2BKI (33) for the motor domain, the MVI lever arm was extracted from 3GN4 (23), and the F-actin was adopted from 1MVW (48). Both PrePS and R models were aligned against a myosin bound to actin in 1MVW.

Much of the work on nonmuscle myosins has focused on myosin V (MV). However, since the discovery that myosin VI (MVI) has an unusual structure, there has been an increasing interest in the motility of MVI. In addition to its biophysical importance, MVI has been implicated in a wide variety of cellular functions in different organisms (3, 4). For instance, MVI is involved in endocytosis, spermatogenesis, cell migration, the organization of the cytoskeleton and the Golgi apparatus, and protein localization. It also plays a role in the maintenance of stereocilia of the inner ear cell. Mutations of the myo6 gene induce deafness in mice and in humans (3). The effect of a deafness-inducing mutation on the MVI cycle has been elucidated recently, thus suggesting a mechanism by which the functioning of the mutant is hampered (5). MVI is overexpressed in cancerous ovarian cells, and the inhibition of its expression reduces the tumor propensity to disseminate (6). MVI is also overexpressed in cancerous prostate cells (7).

Some myosins perform their physiological function as monomers, and others form dimers that walk processively, i.e., they take multiple consecutive steps on the polar track F-actin, without detaching. MVI, monomeric in solution (8), can form dimers (9–13) capable of processive movement (9, 10).

Processive motors display high duty ratio (the motor spends most of its cycle tightly bound to F-actin), gating (communication between two heads), and a power stroke (2). Although gating in a high duty ratio motor might not be necessary, it is likely to increase motor efficiency and run length (14). A recent coarse-grained model of MVI provided structural evidence that ADP release is gated (15), in agreement with experiments (16) and kinetic models (14), although it was also proposed that blocking ATP binding constitutes the gating mechanism in MVI (17).

In addition to displaying gating and a high duty ratio (17, 18), MVI shows a number of striking features that distinguish it from other processive myosins, such as the well-studied MV: (i) For starters, whereas MV and all other members of the myosin superfamily move toward the barbed end (or plus end) of F-actin, MVI steps toward the pointed end (or minus end) (2, 19). (ii) The lever arm architecture differs significantly between MV and MVI (20). In MV there are six CaMs or light-chain–bound IQ domains that constitute the lever arm, with length that is commensurate with the 36-nm spacing in the F-actin repeat. MVI has only one IQ domain, but in a step covers the same distance as MV, suggesting that other elements contribute to the lever arm. Although their origin remains unclear (20–26), recent studies investigating MVI function suggest the importance of the proximal tail as a lever arm extension in vivo (27). (iii) Furthermore, the step-size distribution of the wild-type MV is much narrower than in MVI (9, 10, 20). (iv) And finally, both MV and MVI move processively by a hand-over-hand mechanism (28–30), but in MVI there is also evidence of inchworm-like steps (31).

Most of the unconventional mechanical properties of MVI are attributed to insert 1 and insert 2 (ins1 and ins2) (Fig. 2), two unique fragments within the myosin superfamily (19). Single-molecule experiments (32) and crystal structures of the R (33) and PrePS (34) states indicated ins2 as the key structural element responsible for the reversal of MVI directionality during stepping. Indeed, ins2 wraps around the converter domain, effectively turning the direction of the swing backward, and its removal changes the directionality of MVI motion. Kinetics experiments suggest that ins1 plays a key role in determining MVI high duty ratio and gating (17, 35).

Fig. 2.

Sequence and structural models of MVI. (A) Sequence of the modeled MVI with bound CaMs. (B and C) PrePS (B) and R (C) models used in the simulations. The color code is the same as in Fig. 1. The total number of residues in the protein is 834. Each CaM has 145 residues. The total number of residues for our PrePS and R models is 1,124. F-actin was not part of the model, but it is shown here for reference. The red spheres show the residues N785 and K834, the beginning and the end of the lever arm in our model. The gray and black spheres show P66 and A73 of the IQ-bound CaM. The size of the four spheres is exaggerated to enhance visibility. In ref. 38 a bifunctional rhodamine used to reveal the orientation of the lever arm is attached to these two residues. The three axes, $\widehat{x}$, $\widehat{y}$, and $\widehat{z}$ are shown as a red, a green, and a blue arrow, respectively. The position of the observer is shown pictorially as an eye. The observer stands on F-actin behind MVI, parallel to the $\widehat{x}$ axis and oriented in the negative direction, looking toward the pointed end of the filament.

Some of the essential features of the MVI stepping mechanism are not fully understood, including the unusually large step-size distribution and the alternation between hand-over-hand and inchworm steps. Although models of the dimer incorporating the flexibility of the MVI lever arm are capable of reproducing known features of MVI motility (36), experiments using a chimeric MVI with the MV lever arm showed that the step-size distribution was similar to the wild type (32, 37), suggesting that the elusive structure of the MVI lever arm might not be the key ingredient needed to explain the peculiar stepping mechanism of MVI. Following experimental studies, structural modeling, and simulations, it is tempting to conclude that the uncoupling of the MVI leading-head lever arm from the movement of the converter, or the pliancy of the lever arm, and the conformational transition in the converter may contribute to the large step-size distribution (30, 38–41). Furthermore, according to a recently proposed model (31, 37), short steps occur if the free, trailing head of a myosin dimer binds F-actin while the lever arm of the actin-bound, leading head remains in the prestroke orientation. Conversely, a rotation of the bound-head lever arm to the R-state orientation results in large steps.

Although experimental evidence supports the model of the uncoupling or pliancy of the lever arm (30, 38–41), direct evidence requires a structural model capable of describing the dynamics of the lever arm swing. We use a combination of coarse-grained (CG) simulations and theory to probe the dynamics of the swing of the lever arm in MVI. In recent years, numerous simulations on a variety of systems have shown the reliability of CG models in extracting the salient features of the dynamics of macromolecules (42–47). We prepared a model for the PrePS and R conformation that includes the lever arm up to ins2 and the IQ domain (Fig. 2 B and C). We modeled the power stroke by inducing the transition from the PrePS to the R conformation, ignoring any intermediate configuration and F-actin (see Materials and Methods for a fuller discussion). We generated 96 trajectories of the PrePS$\to$R transition to study the power stroke—for a discussion about the convergence of the results see SI Appendix, section 4 and SI Appendix, Fig. S14. To understand the effect of backward load on the rotation of the lever arm, we generated 96 trajectories of the PrePS$\to$R transition with the lever arm subject to a resistive force of 6 pN. Our results suggest that the PrePS$\to$R transition occurs in two major steps: In the first step, the motor domain reaches the R state, while the lever arm is uncoupled. In the second step, the lever arm diffuses toward the R-state configuration with a small guidance from the motor domain. Our model leads us to propose that the inchworm-like steps might occur when the free head binds F-actin before the uncoupled lever arm rotates to the R-state conformation, which identifies a direct connection between short (or inchworm-like) steps and the uncoupling of the lever arm. The results of our simulations not only compare favorably with experiments, but also yield precise, testable predictions.

Results

The Lever Arm Rotation Occurs in Two Major Steps.

We monitor the dynamics of the PrePS$\to$R transition using the structural overlap function ($\chi$) with respect to the R state given by (49)

$$\chi (t)=\frac{2}{N_{\mathrm{B}}^2-5N_{\mathrm{B}}+6}\sum _{i=1}^{N_{\mathrm{B}}-3}\sum _{j=i+3}^{N_{\mathrm{B}}}\mathrm{\Theta }(|r_{ij}(r)-r_{ij}(\mathrm{R})|-a),$$ [1]

where $N_{\mathrm{B}}$ is the number of beads in the CG model (excluding those that were not resolved in the crystal structure, see SI Appendix, section 1), $r_{ij}(t)$ is the distance between the beads $i$ and $j$ at time $t$, $r_{ij}(\mathrm{R})$ is the distance in the R state, $\mathrm{\Theta }$ is the Heaviside function, and $a=2\AA$ is the tolerance. The summation in Eq. 1 is over pairs of beads that are at least three residues apart in the sequence (the total number of such pairs is the inverse of the prefactor in Eq. 1). In the PrePS state, $\chi \approx 0.62$, and in the R state, $\chi \approx 0.34$. The time trace of $\chi$ shows that the PrePS$\to$R transition occurs in two steps (black line in Fig. 3A). Within a few microseconds, $\chi$ decreases to $\approx 0.55$ (Fig. 3A, Inset) and it fluctuates around this value for a long time until it undergoes another rapid transition leading to the R state. The structural overlap function fluctuates around $0.34$ until the end of the simulation. Similar patterns are found in all of the trajectories.

Fig. 3.

Dynamics of the PrePS$\to$R transition. (A) The dependence of $\chi$ on time is in black. The red arrow indicates the end of the first transition, and the green and blue arrows correspond to the start and end of the second transition, respectively. A also shows the time traces of the distance between C63 and R708 (red squares), T754 and D24 of the ins2-bound CaM (green circles), and V140 and D58 of the ins2-bound CaM (blue triangles). A, Inset shows the initial transition in $\chi$. (B, Left) Histogram of ${\tau }_{\chi 1}$. (B, Center) Histogram of ${\tau }_{\chi 2}-{\tau }_{\chi 1}$. (B, Right) Histogram of ${\tau }_{\chi 3}-{\tau }_{\chi 2}$. Here, ${\tau }_{\chi 1}$ is the end of the first transition, and ${\tau }_{\chi 2}$ and ${\tau }_{\chi 3}$ are, respectively, the start and the end of the second transition monitored by $\chi$.

Let us refer to the time of the completion of the first step as ${\tau }_{\chi 1}$, the time to the beginning of the second step as ${\tau }_{\chi 2}$, and the time to complete the second step as ${\tau }_{\chi 3}$. In practice, ${\tau }_{\chi 1}$ (${\tau }_{\chi 3}$) is measured as the first time $\chi$ reached 0.55 (0.34) (red and blue arrows in Fig. 3A). The time at which the second step begins, ${\tau }_{\chi 2}$, is measured as the last time the trajectory crosses 0.55 before reaching 0.34 (green arrow in Fig. 3A). The histograms of ${\tau }_{\chi 1}$, ${\tau }_{\chi 2}-{\tau }_{\chi 1}$, and ${\tau }_{\chi 3}-{\tau }_{\chi 2}$ are shown in Fig. 3B. Note the difference in the scale of the abscissa. Clearly, the first transition occurs rapidly (${\tau }_{\chi 1}$ average is $⟨{\tau }_{\chi 1}⟩=2.6\mathrm{\mu }$s, with SD ${\sigma }_{{\tau }_{\chi 1}}=1.2\mathrm{\mu }$s), followed by a long waiting time ($⟨{\tau }_{\chi 2}-{\tau }_{\chi 1}⟩=178\mathrm{\mu }$s, ${\sigma }_{{\tau }_{\chi 2}-{\tau }_{\chi 1}}=122\mathrm{\mu }$s) before another rapid transition ($⟨{\tau }_{\chi 3}-{\tau }_{\chi 2}⟩=4.9\mathrm{\mu }$s, ${\sigma }_{{\tau }_{\chi 3}-{\tau }_{\chi 2}}=2.4\mathrm{\mu }$s). The origin of the large distribution of ${\tau }_{\chi 2}-{\tau }_{\chi 1}$ is discussed in the next section.

Structural Transitions During the Two Steps.

To characterize the structural origin of the two steps observed in the time trace of $\chi (t)$ during the PrePS$\to$R transition, we monitor the following three distances between beads that form contacts in the R state, but not in the PrePS state: (i) dRED is the distance between C63 and R708 (red spheres in Fig. 4), (ii) dGREEN is the distance between T754 and D24 of the ins2-bound CaM (green spheres in Fig. 4), and (iii) dBLUE is the distance between V140 and D58 of the ins2-bound CaM (blue spheres in Fig. 4). When the converter moves from the PrePS to the R-state conformation, the dRED contact is formed. The formation of the dGREEN contact indicates the initial closure of the lever arm onto the motor domain, which is completed when the dBLUE contact is formed. The time traces of these three distances for a particular trajectory are shown in Fig. 3A, from which it appears that the formation of the dRED contact coincides with the first step in the decrease of $\chi (t)$ and the second step of $\chi (t)$ occurs around the same time when dGREEN and dBLUE are formed. To determine whether this observation holds for all trajectories, we extract from each trajectory the time for stable formation of the three contacts. By stable we mean that once the contact is formed, it remains intact with small fluctuations (in practice, we monitor stable contacts by checking the first time at which the contact is formed such that for the rest of the simulation the contact stays closer to the R-state ideal distance than to the largest distance explored during the PrePS$\to$R transition). We refer to the time for stable formation of dRED, dGREEN, and dBLUE as $t_1$, $t_2$, and $t_3$, respectively, and we investigate the correlations between $t_1$ and ${\tau }_{\chi 1}$, $t_2$ and ${\tau }_{\chi 2}$, and $t_3$ and ${\tau }_{\chi 3}$. Fig. 5A shows that there is good correlation between $t_1$ and ${\tau }_{\chi 1}$, with the exception of a few solid circles, which represent all those trajectories in which the lever arm begins to close onto the motor domain before the rotation of the converter domain occurs, that is, before dRED is formed. In these cases, the initial drop in $\chi (t)$ reflects the formation of the dGREEN contact. In three of five cases (blue solid circles in Fig. 5A), the initial dGREEN contact is transitory, which means that it rapidly breaks, the converter rotates, and it finally reforms; thus ultimately the stable formation of the dRED contact precedes the dGREEN contact. In the two cases shown as black solid circles in Fig. 5A the dGREEN contact is stably formed before dRED, and it does not break. The Pearson correlation coefficient and the linear fit, performed in the absence of the solid circles, illustrates that in most cases ($91/96$ trajectories) dRED forms around ${\tau }_{\chi 1}$. This suggests that at $t\approx {\tau }_{\chi 1}$ the converter is on the plus-end side of the MVI motor domain (Fig. 2 B and C). A close examination of the crystal structures before and after the PrePS$\to$R transition (33, 34) reveals that the converter undergoes a rotation and also is subject to a conformational change from the so-called P-fold to the R-fold. The distance dRED can monitor only the translation of the converter, but we show in SI Appendix, sections 2 and 3 and SI Appendix, Figs. S6–S8 that around ${\tau }_{\chi 1}$ the converter and the motor domain undergo the needed rotation and the structural transition to reach the R-state conformation. We deduce that, in most of the cases, the first step in the lever arm swing corresponds to the movement of the converter from the PrePS to the R configuration.

Fig. 4.

Monitoring the key structural changes in the PrePS$\to$R transition. The locations of the ${\mathrm{C}}_{\alpha }$ atoms of C63 and R708 are shown as red spheres, and those of T754 and D24 of the ins2-bound CaM are shown as green spheres. The blue spheres show the ${\mathrm{C}}_{\alpha }$ positions of V140 and D58 of the ins2-bound CaM, respectively. The dashed lines connect the beads involved in the dRED, dGREEN, and dBLUE contacts. To enhance visibility, the location of the spheres is also highlighted by a label R, G, or B. A shows the PrePS state, and B shows the R state.

Fig. 5.

Correlation between transition in $\chi$ and the structural changes in MVI. (A) Plot of ${\tau }_{\chi 1}$ against $t_1$. Blue (black) solid circles indicate results from simulations in which temporarily (permanently) the dGREEN contact forms before the dRED contact. The slope of the linear fit is $\approx 0.78$, whereas the intercept is $\approx 0.86\mathrm{\mu }$s. (B) Plot of ${\tau }_{\chi 2}$ vs. $t_2$. The slope of the linear fit is $\approx 1.00$, and the intercept is $\approx 2.6\mathrm{\mu }$s. (C) ${\tau }_{\chi 3}$ plotted against $t_3$. The slope of the linear fit is $\approx 1.00$, and the intercept is $\approx 0\mathrm{\mu }$s. The linear fits and the correlation coefficients are computed without the blue solid circles in A and with all of the circles in B and C.

The Pearson correlations of $t_2$ with ${\tau }_{\chi 2}$ and of $t_3$ with ${\tau }_{\chi 3}$ are shown in Fig. 5 B and C. The two correlation coefficients are very close to unity, and the linear fit yields small intercepts (Fig. 5 legend). Hence, the dGREEN contact forms around ${\tau }_{\chi 2}$, followed by the formation of the dBLUE contact at $t\approx {\tau }_{\chi 3}$.

The analysis conducted so far monitoring $\chi$ and the formation of the dRED, dGREEN, and dBLUE contacts suggests that we can divide our trajectories into two classes: (i) In most cases ($91/96$), the first step of $\chi$ corresponds to a movement of the converter domain from the PrePS to the R position (formation of the dRED contact). Rarely ($3/96$), we observed that the movement of the converter is preceded by the transitory formation of a R-state contact between the lever arm and the motor domain (dGREEN contact). Once the converter has moved, after a long waiting time, ins2 closes onto the motor domain to form the R state (stable formation of dGREEN and dBLUE contacts). Hence, the first step is a power-stroke transition of the motor domain (including the converter), and during the second step the lever arm attains the R-state conformation. This suggests that, in all these trajectories ($94/96$), the movement of the lever arm is “uncoupled” from the PrePS$\to$R transition of the converter. We refer to these trajectories as uncoupled. Examples of uncoupled trajectories are shown in Movies S1 and S2 and SI Appendix, Figs. S11 and S18. (ii) In a minority of trajectories ($2/96$) an R-state–like stable contact between ins2 and the motor domain is formed before the switch of the position of the converter. These trajectories are classified as “coupled,” because in this mechanism the interaction between the lever arm and the motor domain leads to a rapid swing of the lever arm directed by the movement of the converter. Movies S3 and S4 and SI Appendix, Figs. S12 and S19 show examples of coupled trajectories.

The number of coupled trajectories can be artificially increased by modifying the potential that induces the PrePS$\to$R transition (SI Appendix, section 1 and SI Appendix, Figs. S2–S4). However, the conformation of the lever arm generated in the coupled trajectories did not reproduce the experimental data in ref. 38 (SI Appendix, section 1 and SI Appendix, Fig. S5). Hence, in the next two sections we focus our analysis exclusively on the uncoupled trajectories.

The Rotation of the Ins2-IQ Domain in the Uncoupled Trajectories.

The reference system is shown in Fig. 2 B and C. The $\widehat{z}$ axis is nearly parallel to the actin filament, with the minus end of F-actin directed toward the positive side of the axis. MVI is approximately parallel to the $\widehat{x}$ axis and oriented toward the negative direction. The view of the observer used as reference is described in Fig. 2 B and C. To follow the movement of the lever arm, we monitor the position of N785 (in ins2, shown as a red bead in Fig. 2 B and C) and K834 (last residues of the IQ domain, shown as a red bead in Fig. 2 B and C). Sample trajectories (SI Appendix, Figs. S9, S11, and S18 and Movies S1 and S2) show that N785 moves during $t<{\tau }_{\chi 1}$ from the PrePS state nearly to the R-state configuration. When $t$ exceeds ${\tau }_{\chi 1}$, it rapidly finds the R-state equilibrium position, fluctuating around this value for the rest of the simulation. On the other hand, K834 undergoes a large rotary movement around the converter from the value in the PrePS state to the one in the R state (SI Appendix, Figs. S9, S11, and S18 and Movies S1 and S2). We conclude that, after the initial step in the PrePS$\to$R transition, N785 serves as the hinge around which the distal part of ins2 and the IQ domain rotate until they reach the R state. To monitor the rotation of the lever arm in the most intuitive way, we describe the vector connecting N785 to K834 in spherical coordinates. Because the distance between N785 and K834 is roughly constant during the PrePS$\to$R transition (SI Appendix, Fig. S10, Top), we describe the movement to the R state of the uncoupled lever arm as a rigid rod rotation with changes in the altitudinal angle, $\theta$, and the azimuthal angle, $\varphi$ (Fig. 6A). Because the relevant motion of the lever arm is in the direction of F-actin, we choose $\theta$ as the order parameter and describe the dynamics of the lever arm on a unit circle.

Fig. 6.

Extraction of the potential driving the rotation. (A) Definition of the angles $\theta$ and $\varphi$. The Cartesian axes are shown in red, green, and blue. The direction of the lever arm is shown as a thick red line, with the position of N785 and K834 shown as spheres. Note from Fig. 2 B and C that the z axis is parallel to F-actin, so the $\theta$ angle indicates the angle between the lever arm and the actin filament. The source and sink of the Fokker–Planck (FP) model are approximately shown in purple and orange, respectively. (B) The probability ${\rho }_r(\theta )$. The result from CG simulation is shown as blue solid circles. The fit of the data, obtained using SI Appendix, Eq. S11, is in red. The gray thick lines show the values of ${\theta }_i=2.25$ and ${\theta }_f=0.25$. (C) The free energy obtained to fit ${\rho }_r(\theta )$. The gray thick lines are the same as in B.

Energetics of the Lever Arm Swing.

The rotating lever arm is subject to a free energy $F(\theta )$ that governs its motion. We extract this free energy from the trajectories to decipher the extent of the coupling between the lever arm and the motor domain. Assuming that at the start of the rotation the lever arm is at an angle $\theta ={\theta }_i$ and at the end the altitudinal angle is $\theta ={\theta }_f$, and ignoring the role of the azimuthal angle, a large ($F({\theta }_i)-F({\theta }_f)>>k_{B}T$), downhill free energy suggests that the swing is guided. In contrast, if the free energy is flat ($F({\theta }_i)-F({\theta }_f)\approx k_{B}T$), then the lever arm is uncoupled in the $\theta$ direction. To extract the free energy $F(\theta )$ we cannot simply compute the logarithm of the equilibrium probability distribution $\rho (\theta )$, because once the lever arm reaches the R state we do not observe transitions to the PrePS state. Hence, the system is out of equilibrium, and we need to adopt a different strategy to extract $F(\theta )$. We adopt a procedure put forward in ref. 50 (details in SI Appendix, section 4). We generate a stationary probability distribution on a unit circle ${\rho }_r(\theta )$ from the simulated PrePS$\to$R transitions. The trajectories are injected at an angle ${\theta }_i=2.25$ rad (close to the PrePS state, ${\theta }_{\mathrm{PrePS}}\approx 2.55$ rad) and are removed when they cross ${\theta }_f=0.25$ rad (close to the R state, ${\theta }_{\mathrm{R}}\approx 0.04$ rad). We assume that the stationary probability ${\rho }_r(\theta )$ is the solution of a Fokker–Planck (FP) equation subject to appropriate boundary conditions to account for the injection and removal of the trajectories (50). Because we know ${\rho }_r(\theta )$ from simulations, we obtain the free energy $F(\theta )$ by solving the FP equation. To solve the FP equation we assume that the diffusion coefficient $D$ is a constant. In Fig. 6B we show the probability distribution ${\rho }_r^{\mathrm{CG}}(\theta )$ obtained from the analysis of the CG simulations (blue solid circles). As described in detail in SI Appendix, section 4, we fit the free energy $F(\theta )$ to obtain a probability distribution ${\rho }_r(\theta )$ (Fig. 6B, red line) that closely recovers the results from simulations.

The profile of $F(\theta )$ shows that from $\theta =2.25$ rad ($\approx {129}^{\mathrm{o}}$) to $\theta \approx \pi /4$ ($\approx {45}^{\mathrm{o}}$) a large part of the swing occurs on an almost flat free-energy profile (Fig. 6C). For $\theta <\pi /4$ there is a shallow minimum of $\approx$1 kcal/mol, that drives the last part of the swing. We conclude that the swing occurs without a strong interaction with the motor domain, implying that the lever arm movement is mostly uncoupled, which is qualitatively revealed in the trajectories (SI Appendix, Figs. S11 and S18 and Movies S1 and S2). This means that for a large part of the swing, the lever arm rotates stochastically while maintaining a hinge around N785. The capture toward the R state occurs only when the lever arm is sufficiently close to the motor domain. The stochasticity of the dynamics explains the large width in the distribution of ${\tau }_{\chi 2}-{\tau }_{\chi 1}$.

To extract the diffusion constant we compare the simulated mean first passage time (MFPT) ${\tau }_r^{\mathrm{CG}}(\theta )$ and the one resulting from a diffusing pseudoparticle in the potential $F(\theta )$. The theoretical MFPT is obtained by solving the one-dimensional FP equation using a constant diffusion coefficient (51), ${\tau }_{\mathrm{FP}}(\theta )=D^{-1}{\int }_{\theta f}^{\theta }d\theta \prime e^{\beta F(\theta \prime )}{\int }_{\theta \prime }^{\pi }d{\theta }^{″}{e}^{-\beta F({\theta }^{″})}.$ Fitting $D$ in ${\tau }_{\mathrm{FP}}(\theta )$ vs. ${\tau }_r^{CG}(\theta )$ we obtain $D\approx 0.011/\mu \mathrm{s}$ (SI Appendix, section 4). The good agreement between ${\tau }_{\mathrm{FP}}(\theta )$ and ${\tau }_r^{CG}(\theta )$ reported in SI Appendix, Fig. S13 suggests that it is reasonable to assume that D is constant.

Although we derived a one-dimensional model that captures the features of the CG simulations, we stress that the rotation occurs on a sphere, which would be better captured by a 2D model. Indeed the azimuthal component might play an important role in the rotation of the lever arm.

Comparison with Experiments.

We use our simulations to compare with experiments probing the stepping mechanism of MVI dimers (38). To follow the orientation of the lever arm during the stepping, the CaM bound to the IQ domain of MVI was labeled with a bifunctional rhodamine (BR) probe (38). The points of attachment are P66 and A73 and are shown in Fig. 2 B and C in black and gray, respectively. Sun et al. (38) inferred the rotational dynamics of the lever arm from the orientational changes in BR, using polarized total internal reflection fluorescence (polTIRF). This yielded the $\theta$ angles (along the direction of actin, referred as $\beta$ in ref. 38) and the $\varphi$ angle (the azimuthal angle is named $\alpha$ in ref. 38). [Note that $\theta$ and $\varphi$ are angles of the BR probe with respect to the reference system of actin (Fig. 2 B and C). In ref. 38, the Greek letters $\theta$ and $\varphi$ are used to identify the angle of the lever arm in the reference system of actin, and $\alpha$ and $\beta$ are used for the probe. We compare here with $\alpha$ and $\beta$ of ref. 38]. The heads of the dimers in the leading and the trailing positions were identified, and the corresponding distributions of the $\theta$ and $\varphi$ angles were extracted. According to kinetics experiments (52), both the trailing head (TH) and the leading head (LH) are in the ADP-bound state. In the TH, the lever arm swing has already occurred, and it is directed toward the pointed end of F-actin. In contrast, the LH, held under tension by the actin-bound TH, is in the uncoupled (U) state.

To compare our simulations with experiments, we reproduced the measurements by monitoring the orientation of the unit vector connecting A73 with P66 (Fig. 2 B and C). The $\theta$ and $\varphi$ angles of the BR probe in the R-state crystal structure differ between the computational model (${\theta }_{\mathrm{R}}^{\mathrm{CG}}$ and ${\varphi }_{\mathrm{R}}^{\mathrm{CG}}$) and the values used in ref. 38 (${\theta }_{\mathrm{R}}^{\mathrm{EXP}}$ and ${\varphi }_{\mathrm{R}}^{\mathrm{EXP}}$). Two factors likely contribute to this discrepancy: (i) The alignment with F-actin and the choice of the reference system are not identical in experiments and simulations (our axes are rotated compared with those in ref. 38, so $\varphi \approx \alpha +\pi /2$), and (ii) we extracted the angles from the relative position of the $\alpha$-carbons of P66 and A73 from CaM, whereas experiments, in all likelihood, detect the orientation of the $\beta$-carbons (53).

To describe the TH, we consider the R-state conformations sampled in the simulations only at $t>{\tau }_{\chi 3}$, to ensure that the R state is reached. To sample the U state of the leading head, we ran 96 simulations of the swing of the lever arm by applying to the tip of the lever arm (K834) a backward load of 6 pN (details in SI Appendix, section 1), which mimics the effect of the interhead tension. The mechanical force ($f_L$) adopted to describe the U state resists the rotation of the lever arm by applying a momentum of the force of magnitude $\approx r_{LA}{f}_L$, where we consider a lever arm $r_{LA}\approx$7.5 nm (SI Appendix, Fig. S10, Top). If we consider the entire lever arm of MVI ($r_{LA}\approx$18 nm), the force that generates the same resistive momentum is $f_L\approx 2.5$ pN, which is close to the upper bound of the interhead tension estimated experimentally [$f_L\approx 2.2$ pN (16)]. This suggests that the magnitude of our backward load is not unrealistically high. We show in SI Appendix, sections 2 and 3 that the backward load of 6 pN does not affect the first step of the PrePS$\to$R transition (SI Appendix, Figs. S6–S8 and S20, SI Appendix, section 7, and Movies S5 and S6), but it is sufficient to prevent the occurrence of the forward rotation of the lever arm in 95 of 96 cases (SI Appendix, section 1 and SI Appendix, Fig. S3). Thus, we generated the ensemble of U configurations as the collection of structures from the simulations conducted with a 6-pN backward load, sampling only the configurations for $t>{\tau }_{\chi 1}$ for which the rotation of the lever arm was not completed.

Comparison of experimental data and simulations is provided in Fig. 7. We compared the following quantities reported in ref. 38: (i) the probability distribution for the LH and TH of the $\theta$ angle (Fig. 7A), (ii) the probability distribution for the change in the $\varphi$ angle after one step ($\mathrm{\Delta }\varphi$, Fig. 7B), and (iii and iv) the probability distribution for the change after two steps of both the $\theta$ angle (iii) (${}^2\mathrm{\Delta }\theta$, Fig. 7C) and the $\varphi$ angle (iv) (${}^2\mathrm{\Delta }\varphi$, Fig. 7D). Because in experiments there is no control on the landing azimuthal angle of MVI on actin, the experimental distribution of the $\varphi$ angle for the LH and the TH is almost flat across ${180}^{\mathrm{o}}$. In simulations, we always start from a MVI parallel to the x axis; thus the comparison of calculated and measured distributions could be misleading (SI Appendix, section 5 and SI Appendix, Fig. S15).

Fig. 7.

Comparing simulations and experimental results. In A–D the histograms show the data extracted from figures 3 and 4 of ref. 38, whereas the lines show the results of our simulations and analyses. The red and blue lines are obtained directly from simulations and the black lines are from our model of a straight step of MVI dimer (SI Appendix, section 5 and SI Appendix, Eqs. S17 and S18), based upon the fit of the simulated distributions in A and in SI Appendix, Fig. S15. (A) Distribution of $\theta$ ($\beta$ in ref. 38) of the trailing (red) and leading (blue) heads. The data from simulations were fitted using normal distributions. The values of the means and dispersions are shown. The average of the distribution is denoted with a bar, and the SD is labeled as $\sigma$. (B) Distribution of the change of $\varphi$ after one step (SI Appendix, Eq. S17). (C) Distribution of the change of $\theta$ after two steps (SI Appendix, Eq. S18). (D) Distribution of the change of $\varphi$ after two steps (SI Appendix, Eq. S18).

For comparison i, we extracted directly the probability distributions from the CG simulations of the U state and the R state to describe the LH and TH, respectively. We did not simulate consecutive steps. Thus, for comparisons ii–iv described above we created a simple model of the changes in the angles after one or two steps using two assumptions (details in SI Appendix, section 5): (i) The dimer always steps by a hand-over-hand mechanism, and (ii) the values of the angles after a step do not depend on the values before the step.

We found a remarkable agreement between the $\theta$ angles obtained in simulation and experiment for both the LH and TH (Fig. 7A). Our CG model clearly reproduces the orientation and the fluctuations of the BR probe in the LH and TH, which validates the model. It should be stressed that we did not adjust any parameter in the CG model to obtain agreement with experiments.

Although the calculated changes in the $\varphi$ angle after one step agree with of the experimental data (Fig. 7B), there is a difference in the interpretation of the results based on simulations and experiments. According to experiments, in both the transitions LH$\to$TH and TH$\to$LH, $\mathrm{\Delta }\varphi$ can be positive or negative with roughly equal probability (table 1 in ref. 38). From simulations we find that $\mathrm{\Delta }\varphi$ is mostly ($\approx$97$\%$ of the time) positive (negative) in the R$\to$U (U$\to$R) transitions. Thus, in the simulations there is a contribution to each peak of $P(\mathrm{\Delta }\varphi )$ from either R$\to$U or U$\to$R transitions, whereas in experiments each transition contributes to the two peaks almost equally. Furthermore, in our stepping model we assume that the direction of the interhead tension is always aligned with F-actin, which implies that only a 36-nm step occurs. But the measured broad step-size distribution in MVI suggests that this is not always the case.

Comparison of the probability distribution after two steps yields accurate results for the $\theta$ angle (Fig. 7C), whereas our model for the distribution of ${}^2\mathrm{\Delta }\varphi$ results in a somewhat more peaked distribution (Fig. 7D). This suggests that, in our model of stepping, MVI is not nearly as “wiggly” as inferred from experiments (38). A more quantitative comparison with experiments requires a CG model of MVI dimer in complex with F-actin.

Discussion

Position of the Hinge of the Rotatory Movement.

We suggest that the location of the hinge around which the lever arm rotates is close to residue N785 (located in ins2), at the beginning of a region identified as pliant in the PrePS state (41). Within a few microseconds from the start of the PrePS$\to$R transition N785 binds the converter and then fluctuates around an average position (SI Appendix, Figs. S9 and S18–S20). Experiments have shown that chimeric constructs in which ins2 was truncated before (after) N785 and replaced with an artificial lever arm are plus-ended (minus-ended) motors (14, 54). Thus, from the structural viewpoint the location of the hinge is physically reasonable.

The Uncoupling of the Lever Arm During the Power Stroke Is Intrinsic to the Motor Head.

The two-step PrePS$\to$R transition with uncoupling of the lever arm at zero backward load suggests that it is not the presence of an actin-bound TH that induces the uncoupling of the lever arm. Our simulations illustrate that there is a possibility of a coupled swing to occur. However, this is unlikely at zero force and was observed only once when the 6-pN backward load was applied to the tip of the lever arm. Hence, the mechanism of uncoupling is inherent in the power stroke of MVI, and it is likely not due to gating or rearward tension. The flexibility of the LH lever arm was ascribed to the pliancy of the region between the lever arm and the converter (and perhaps the lever arm extension) (38) or to the uncoupling of the lever arm from the motor domain (30, 39, 41). In this paper we use “uncoupling” because we observe that in the majority of the trajectories the rotation of the lever arm is not synchronous to the power stroke. On the other hand, we observe that part of ins2 (approximately up to N785) is tightly attached to the converter during the swing (SI Appendix, Figs. S8, S18, and S20 and Movies S1, S2, S5 and S6), suggesting that the initial part of ins2 is not uncoupled from the motor domain. Thus, our simulations agree with the picture of a pliant lever arm around ins2.

The Two-Step Mechanism.

It has been suggested that the power stroke of MVI occurs with a two-step mechanism, in which first the converter rotates, and then, once the backward load from the actin-bound trailing head is relieved, the lever arm rotates toward the R-state configuration (41). In this picture, it was argued that at the end of the first step the converter domain is in a PrePS state-like configuration, but it has moved toward the R-state position on the motor domain. We find that the converter domain not only translates and rotates to the R-state position on the motor domain (SI Appendix, section 3 and SI Appendix, Fig. S7), but also assumes the conformation observed in the R-state crystal structure (the so-called R-fold of the converter; SI Appendix, Figs. S7 and S8). The flexibility around the hinge allows for these changes to occur even in the presence of a large backward load (SI Appendix, Figs. S7 and S8 and SI Appendix, section 3). It is worth noting that we are not considering the effects of an azimuthal component of the backward load, which might add more complexity to the picture, as we suggested in comparing simulations and experimental results. Note that evidence from single-molecule experiments of MVI (8) and cryo-EM images of MV (55) suggests that the lever arm swing consists of two substeps: a large transition to the ADP-bound conformation triggered by Pi release and a smaller step occurring upon ADP release. Our two-step mechanism ignores the ADP-bound intermediate, and thus these two substeps suggested by experiments are incorporated into our second transition occurring between ${\tau }_{\chi 2}$ and ${\tau }_{\chi 3}$. We cannot exclude that the converter domain occupies the P-fold conformation in the poststroke state (41). We believe that a poststroke R-fold converter provides a more succinct picture of the MVI power stroke. Additional simulations and experiments are needed to resolve this subtle issue.

The Lever Arm of MVI Rotates Stochastically, Preferentially on the Side of F-Actin.

Both the coupled and uncoupled trajectories suggest that the structure of the converter domain and the PrePS$\to$R transition favor a rotation of the MVI lever arm on the side of the actin filament, that is, on the right side as seen by the observer in Fig. 2 B and C (Movies S1–S4 and SI Appendix, Figs. S11, S12, S18, and S19). As already pointed out in ref. 38, this might confer to MVI the experimentally observed right-handed twirling around the actin filament (20, 38). In the uncoupled simulations the rotation is stochastic: The azimuthal component of the lever arm undergoes large changes, and the flat free-energy landscape explored by the $\theta$ component suggests that the rotational diffusion is almost free for a large part of the lever arm swing.

The Connection Between the U State and the Occurrence of Short (Inchworm-Like) Steps.

A model was proposed to explain the presence of short inchworm-like steps in MVI motility (31). The authors suggest that a short step occurs if the free MVI head binds actin while the lever arm of the leading, actin-bound head is still in PrePS orientation. If the actin-bound MVI lever arm rotates to the R-state orientation, the dimer takes a long step. This scheme provides a structural picture that encompasses both the long and short steps of MVI, thus explaining the broad step-size distribution. Based on the two-step mechanism for the MVI power stroke, we propose that the short (inchworm-like) steps occur if the TH rebinds F-actin before the rotation of the uncoupled lever arm, that is, before the completion of the second step of the PrePS$\to$R transition. If the second step of the transition is completed, then the dimer takes a long step. Because ATP hydrolysis and the recovery stroke occur before the rebinding of the free head to F-actin, for our proposition to be valid the rotation of the uncoupled lever arm must occur on a timescale comparable to the rate of ATP hydrolysis, which was estimated to be $\approx 10-17{\mathrm{s}}^{-1}$ (56). The timescale of transition from the U state to the R state is $⟨{\tau }_{\chi 3}-{\tau }_{\chi 1}⟩\approx 183\mu$s, which is obtained for a lever arm of about 7.5 nm. The large steps of MVI dimers demand a combination of traditional lever arm and lever arm extension of roughly 36 nm for the dimer (ignoring the hinge between the two lever arms and the free myosin head). If we assume that the timescale of rotation scales with the length of the rotating rod as $L^3/\ln (L/2a)$ (57), where $a$ is the thickness of the rod (considering CaM it is ∼3 nm), the resulting timescale becomes about $2.5$ ms, which is only $\approx 20$ times faster than the highest bound for the measured rate of ATP hydrolysis. This suggests that it is plausible that the TH might bind before the lever arm of the leading head reaches the R state (that is, after ${\tau }_{\chi 1}$, but before ${\tau }_{\chi 3}$).

Uncoupling of the Lever Arm and Phosphate Release.

It is suggested that after the Pi is released and the power stroke is completed, the pathway for Pi escape closes, thus avoiding Pi rebinding and making the transition irreversible (2). Interestingly, a recent cryo-EM structure of MV bound to F-actin indicates that the nucleotide binding site in the poststroke, ADP-bound conformation cannot bind Pi even in the absence of backward load (55). Under high forces the power stroke of MV appears to be reversed (58). Because structural evidence suggests that the converter is mechanically coupled to the nucleotide binding site via the relay helix, the load-induced reversal of the power stroke would facilitate the recovery of a conformation of the motor domain capable of binding Pi and detaching more easily from F-actin, thus explaining the experimentally observed increase in the rate of detachment as a function of phosphate concentration (58).

The two-step model for the MVI power stroke suggests that at ${\tau }_{\chi 1}$ the lever arm uncouples and the converter moves backward to the poststroke position. Thus, the backward load in MVI opposes the rotation of the converter back to the PrePS orientation instead of assisting it as in the case of MV (SI Appendix, Fig. S16). Hence, assuming that in MVI also only the prestroke conformation can bind Pi, we predict that, in contrast to MV, the detachment rate of MVI under load should not depend on the concentration of phosphate in solution.

Predictions.

Our simulations lead to the following predictions: (i) As surmised elsewhere (41), the swing of the MVI lever arm occurs in two steps: first the converter reaches the R-state conformation while the lever arm points toward the barbed-end of F-actin and is uncoupled from the motor; subsequently the uncoupled lever arm rotates until it reaches the R-state conformation. The first step is only mildly dependent on backward load (SI Appendix, Figs. S6–S8). The uncoupling of the lever arm from the motor domain after the first step suggest that, in contrast to MV (58), the addition of Pi in solution should not lead to a faster detachment from F-actin of a backward-pulled MVI. (ii) The hinge around which the lever arm rotates is close to N785. (iii) The energetics of the lever arm during the rotation suggest that it is impeded to fully stretch backward. Furthermore, it is essentially uncoupled from $\theta \approx 2.25$ rad to $\theta \approx \pi /4$ and only in the last part is captured by the motor domain. (iv) The swing of the MVI lever arm occurs primarily on the side of actin. (v) The typical time of rotation from the U state to the R state for a 7.5-nm lever arm is about 0.2 ms. This is consistent with a scheme in which short steps are taken whenever the trailing, free head binds F-actin before the rotation of the uncoupled lever arm is completed, that is, before the second step of the PrePS$\to$R transition.

Conclusions

We studied the power stroke of MVI modeled as the transition from the PrePS state to the R state, focusing on the dynamics associated with the converter, ins2, and the IQ domain.

Our simulations allow us to draw two major conclusions: (i) The power stroke of MVI occurs in two steps. In the first step, the motor domain undergoes the PrePS$\to$R transition, characterized by the movement of the converter, and then the lever arm rotates to the R-state conformation. (ii) During the rotation, the lever arm is largely uncoupled from the motor domain. The rotation of the lever arm is stochastic and occurs largely on the side of F-actin.

Our findings are based on a CG model of the PrePS$\to$R transition in which (i) the side chains of the residues were not modeled, (ii) the Pi-release and ADP-bound structures of MVI were ignored, (iii) the role of F-actin (in the simulations carried out under backward load) is approximated with some tethering potentials restraining the external degrees of freedom of MVI, and (iv) the presence of a TH is modeled as a backward load of 6 pN directed parallel to F-actin. Although we believe that some of the restrictions could be relaxed in future studies, we do not expect our conclusions to be affected by a more comprehensive description of the MVI power stroke.

Materials and Methods

Preparation of the PrePS and R States.

We prepared the PrePS and R states by combining multiple PDB structures. The sequence and the two structures in Fig. 2 show that the model of MVI is made of the motor domain, the lever arm up to residue K834 of the IQ domain, and two bound CaMs.

The PrePS and R structures up to residue H786 (located in the middle of ins2; Fig. 2A) were taken from the PDB structures 4ANJ (41) and 2BKI (33), respectively. Because the distal parts of PDB structures 4ANJ and 2BKI were either partially substituted by GFP (4ANJ) or only partially solved (2BKI), we completed our model using the ins2, the IQ domain, and related CaMs of the PDB structure 3GN4 (23). To generate a continuous and seamless model, the residues 776–816 (a large part of ins2 and the beginning of the IQ domain; Fig. 2) of 4ANJ and 2BKI were superposed with the same residues in 3GN4, resulting in a rmsd of $0.38\AA$ and $0.47\AA$, respectively. The resulting PrePS (R) structure in Fig. 2B (Fig. 2C) was used in our simulations.

Experiments (8, 59) suggest that the lever arm of MVI undergoes the following sequence of transitions: (i) From the ADP- and phosphate-bound PrePS state to a conformation in which the lever arm has not moved, but the motor domain is now primed for phosphate release (the PiR state) (59). (ii) Following the release of phosphate, the MVI head reaches an ADP-bound state. During this transition, most of the swing occurs, but the lever arm conformation does not correspond to the one in the R state (8) (see ref. 19 for a cryo-EM image). (iii) Following ADP release, the motor finally reaches the R state.

We set PrePS as the initial state and R as the final state, ignoring the intermediates that are either poorly structurally characterized (ADP bound) or close to our initial (PiR) or final (ADP-bound) states.

We did not model F-actin explicitly. Although binding to the track affects the way MVI proceeds through the cycle (5), we believe that it does not significantly affect our findings, because they depend on the conformational transition of the converter and only to a lesser degree on what triggers the movement of the converter. For the same reason, we presume that the absence of the intermediate states from our model will not impact significantly our conclusions.

CG Model.

We created a CG representation of MVI, in which each amino acid is represented as a single bead centered at the ${\mathrm{C}}_{\alpha }$ position. We used the self-organized polymer (SOP) version of the CG model (60, 61), which is described in SI Appendix, section 1. Of relevance here is that the nature of the interaction between two beads depends on their distance in the native state. If their distance in the native state is below $R_{\mathrm{C}}=8\AA$, and if they are at least three residues away from each other in the sequence, the two beads interact through an attractive potential of the Lennard–Jones type, with minimum at their native distance and well depth of ${ϵ}_{\mathrm{NAT}}=2\mathrm{kcal}/\mathrm{mol}$. The connectivity of the chain is ensured by a finitely extensible nonlinear elastic potential between consecutive beads in the sequence. The interaction between all other pairs of beads is purely repulsive, with strength of the repulsion at the diameter of the beads $\sigma =3.8\AA$ equal to ${ϵ}_{\mathrm{NNAT}}=1\mathrm{kcal}/\mathrm{mol}$.

PrePS$\mathbf{\to }$R Transition.

To simulate the PrePS$\to$R transition, we adopted the method in ref. 61 with minor variations, which were consistent with the results of fully atomistic molecular dynamics simulations of MVI carried out using the special-purpose Anton computer (SI Appendix, Fig. S1). This method (see SI Appendix, section 1 for details) uses an energy function that simultaneously incorporates the native interactions in PrePS and in R states, and it mimics the effect of phosphate release by making the interactions associated with the PrePS state weaker, thus facilitating the PrePS$\to$R transition.

We generated 96 trajectories for long enough times to ensure that MVI has completed the PrePS$\to$R transition. In these simulations we did not restrain the motor domain. Hence, the overall rotation of MVI was removed using the Kabsch algorithm (62, 63) to align the motor domain before the converter (residues 1–703, up to the end of the SH1 helix) to the motor domain of the R structure, which had been previously aligned to an actin-bound myosin II from PDB 1MVW. The rotation of the lever arm was monitored after carrying out this alignment. The converter domain and the lever arm, which are subject to the largest change, were not considered in the alignment.

We also generated 96 trajectories with backward loads of 6 pN. In this case, to avoid the rigid body movement of MVI due to the external force, we mimic the effect of F-actin by restraining the position of some residues of the motor domain (see SI Appendix, section 1 for details). We did not perform any alignment before monitoring the rotation of the lever arm. We found that performing such an alignment does not affect the quality of our results (SI Appendix, Fig. S17), suggesting that the details of the actomyosin interface are unlikely to affect the uncoupled lever arm swing. Finally, we performed 32 simulations of the PrePS state without inducing the transition. These simulations are discussed in the SI Appendix.