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**|**J Gen Physiol**|**v.139(2); 2012 February**|**PMC3269788

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J Gen Physiol. 2012 February; 139(2): 101–120.

PMCID: PMC3269788

The Pennsylvania Muscle Institute and Department of Physiology, University of Pennsylvania School of Medicine, Philadelphia, PA 19104

Correspondence to Yale E. Goldman: Email: ude.nnepu.dem.liam@ynamdlog

Received 2011 September 6; Accepted 2011 December 21.

Copyright © 2012 Lewis et al.

This article is distributed under the terms of an Attribution–Noncommercial–Share Alike–No Mirror Sites license for the first six months after the publication date (see http://www.rupress.org/terms). After six months it is available under a Creative Commons License (Attribution–Noncommercial–Share Alike 3.0 Unported license, as described at http://creativecommons.org/licenses/by-nc-sa/3.0/).

See "Tilting and twirling as myosin V steps along actin filaments as detected by fluorescence polarization" on page 97.

This article has been cited by other articles in PMC.

Myosin V (myoV) is a two-headed myosin capable of taking many successive steps along actin per diffusional encounter, enabling it to transport vesicular and ribonucleoprotein cargos in the dense and complex environment within cells. To better understand how myoV navigates along actin, we used polarized total internal reflection fluorescence microscopy to examine angular changes of bifunctional rhodamine probes on the lever arms of single myoV molecules in vitro. With a newly developed analysis technique, the rotational motions of the lever arm and the local orientation of each probe relative to the lever arm were estimated from the probe’s measured orientation. This type of analysis could be applied to similar studies on other motor proteins, as well as other proteins with domains that undergo significant rotational motions. The experiments were performed on recombinant constructs of myoV that had either the native-length (six IQ motifs and calmodulins [CaMs]) or truncated (four IQ motifs and CaMs) lever arms. Native-length myoV-6IQ mainly took straight steps along actin, with occasional small azimuthal tilts around the actin filament. Truncated myoV-4IQ showed an increased frequency of azimuthal steps, but the magnitudes of these steps were nearly identical to those of myoV-6IQ. The results show that the azimuthal deflections of myoV on actin are more common for the truncated lever arm, but the range of these deflections is relatively independent of its lever-arm length.

Myosin V (myoV) is an unconventional myosin that exists as a dimer in cells and is able to step processively along filaments of actin for relatively large distances (~1 µm) through a cytoplasm that is densely packed (Mehta et al., 1999; Reck-Peterson et al., 2000; Sakamoto et al., 2000; Walker et al., 2000; Veigel et al., 2002; Sweeney and Houdusse, 2010). How myoV is able to navigate through such a complex environment has been the subject of several in vitro studies that focused on dissecting the basic stepping mechanism of individual myoV motors on isolated actin filaments (Rief et al., 2000; Ali et al., 2002, 2007; Forkey et al., 2003; Toprak et al., 2006; Kodera et al., 2010). myoV, as with other members of the myosin family, binds in a fixed orientation to filamentous actin (F-actin) through its N-terminal motor domain. At the C-terminal end of the motor domain is the light chain domain (LCD; also known as the lever arm), which is defined by six calmodulin (CaM)-binding IQ motifs, each bound to a CaM or CaM-like subunit. This domain amplifies small motions in the motor head associated with each ATP turnover resulting in forward motion along the actin filament track, a concept known as the swinging lever-arm hypothesis (Cooke, 1986). The LCD has been found to have additional degrees of freedom, giving the myosin conformational flexibility (Dobbie et al., 1998; Corrie et al., 1999; Veigel et al., 2002).

Because F-actin is a two-stranded right-handed helix, there is a potential for myoV to take off-axis steps that would require a high level of flexibility. Electron microscope images of myoV frozen during processive motility along F-actin indicate that its two heads bind to actin predominantly spanning 13 actin subunits, with smaller subpopulations spanning 11 and 15 subunits (Walker et al., 2000). The distance spanned by 13 actin subunits corresponds to the 36-nm half-helical repeat of F-actin, consistent with the 36-nm step size of myoV (Mehta et al., 1999). Two actin monomers 13 subunits apart have approximately the same azimuthal orientation (Amos and Amos, 1991), leading myoV to walk relatively straight by stepping typically 13 subunits (~36 nm) per step.

This view is supported by many single-molecule experiments in which myoV walks along actin bound to a rigid glass surface, which inhibits large azimuthal changes around the actin (Forkey et al., 2003; Toprak et al., 2006; Kodera et al., 2010). When myoV was observed to walk along filaments suspended away from the glass surface, though, the motors either walked straight or with a gradual left-handed pitch, suggesting that the leading head of myoV lands on the 13th and, sometimes, on the 11th subunit from its trailing head (Ali et al., 2002). Twirling assays, which monitor the rotation of actin gliding along multiple motors fixed to a surface, also found that myoV walks straight along actin or it follows a gradual left-handed helical path (Beausang et al., 2008a).

Polarized total internal reflection microscopy (polTIRF) and defocused orientation and position imaging (DOPI) measurements have also shown myoV walking relatively straight (Forkey et al., 2003), with some sudden sideways tilts (Syed et al., 2006). polTIRF and DOPI are methods that directly measure the three-dimensional orientation of fluorescent rhodamine probes bifunctionally attached to a CaM (BR-CaM) bound to one of the myoV LCDs. In polTIRF, several polarized light intensities are detected and used to estimate the polar and azimuthal angles *β _{P}* and

Schematics of relevant frames of reference and polarized intensities. (A) Definitions of the orientation of the vector representing the rhodamine probe absorption and emission dipoles, *β*_{P}, *α*_{P}, in the Actin Frame (1; *x*_{A}, *y*_{A}, *z*_{A}) and *θ* **...**

polTIRF and DOPI measurements report angles of the fluorescent probe. Because the rhodamine probe is not generally aligned with the myoV LCD, measurements of the probe orientation do not directly correspond to the orientation of the lever arm. In this paper, however, we developed a way to determine the local angle of the probe relative to the lever arm and to use that information to calculate the actual lever-arm orientation.

The step sizes of recombinant myoV constructs with different lever-arm lengths have been shown to depend directly on the length of their LCDs (Purcell et al., 2002; Sakamoto et al., 2005). In this work, the orientation of the BR-CaM–labeled lever arm of a truncated (4IQ) and full-length (6IQ) construct was compared using polTIRF microscopy. The truncated construct, indeed, walked much less straight along actin, wobbling from side to side as it progressed. We also determined the influence of the lever-arm length on the polar rotation (“twirling”) of an actin filament translocated by multiple myosins in a gliding filament assay. The truncated construct led to more frequent actin twirling, surprisingly demonstrating either left- or right-handedness.

G-actin was obtained from rabbit skeletal muscle (Pardee and Spudich, 1982). Biotinylated Alexa Fluor 647–labeled F-actin was prepared from G-actin, Alexa Fluor 647 actin (Invitrogen), and biotin-actin (Cytoskeleton) in a 5:1:1 ratio with 1 µM of total actin subunit concentration and stabilized with 1.1 µM phalloidin (Invitrogen). 0.3% rhodamine-labeled F-actin was prepared from unlabeled actin and 6′-IATR rhodamine-actin (Corrie and Craik, 1994) and stabilized with 1.1 µM phalloidin. Recombinant chicken myoV, with its full-length lever arm (myoV-6IQ; amino acids 1–1099) or a truncated lever arm (myoV-4IQ; amino acids 1–863) and with a FLAG affinity tag at its C terminus, was coexpressed with CaM in SF9 cells (Fig. S1) (Purcell et al., 2002). Chicken CaM, with residues Pro66 and Ala73 mutated to cysteine, was expressed in *Escherichia coli* (Putkey et al., 1985), purified, and labeled with bifunctional rhodamine (BR; provided by J.E.T. Corrie, National Institute for Medical Research, Mill Hill London, UK; Corrie et al., 1998). myoV was labeled by exchanging endogenous CaM with exogenous BR-CaM at low stoichiometry (Forkey et al., 2003).

M5 buffer (pH 7.6) contains 25 mM KCl, 20 mM HEPES, 2 mM MgCl_{2}, and 1 mM EGTA in deionized water. M5^{+} buffer is M5 plus 10 mM dithiothreitol (DTT) and 100 µg/ml wild-type CaM (expressed in bacteria; Putkey et al., 1985). The motility buffer for single-molecule myoV motility assays is M5^{+} buffer plus 4 µM ATP and 50 mM DTT. The motility buffer for actin twirling assays is M5^{+} buffer plus 100 µM ATP, 10 mM phosphocreatine (Sigma-Aldrich), 0.3 mg/ml creatine phosphokinase (prepared daily from powder; Sigma-Aldrich), and 50 mM DTT.

In the original polTIRF setup developed in our laboratory, four time-multiplexed laser polarizations, each predominately aligned along the laboratory *x*, *y*, or *z* axis, illuminated the sample with two intersecting beam paths. The fluorescence emission was directed through a polarizing beam-splitting cube and onto two avalanche photodiodes (giving polarized detection along the *x* and *y* directions over an area corresponding to a 2-µm diameter disc at the sample) resulting in eight measured polarized fluorescence intensities (PFIs) that were used to estimate the probe orientation during every 40-ms excitation cycle. Symmetric orientations of the probe related by reflections across the Cartesian planes resulted in the same intensities, leading to an eightfold ambiguity in probe orientation (i.e., angles were limited to an octant of a sphere). In this work, additional laser polarizations aligned at ±45° between the original polarizations were included (Beausang et al., 2008a), either in one beam path for a total of 12 PFIs or in both beam paths for a total of 16 PFIs, measured in 80-ms time intervals (Fig. 1 B). The angular range determined in these two cases improved to a quarter-sphere and a hemisphere, respectively. Dipole symmetry of the probe prevents further increase in unambiguous angular detection range for polTIRF and polarized fluorescence methods in general.

As described previously (Forkey et al., 2000, 2003, 2005; Beausang et al., 2008a), polarization data from single myosin molecules was recorded and used as input to an analytical model of the probe’s fluorescence emission to determine its orientation (*θ _{laboratory}*,

To compare molecules moving on actin filaments aligned in different directions in the sample plane, we transformed the orientation of the probe at each time point from the laboratory frame to a frame of reference based on the actin filament (*x _{A}*,

A precleaned fused silica slide (Corning HPFS grade; Quartz Scientific, Inc.) was further treated in an ion plasma cleaner for 5 min and then spin coated with 2 mg/ml polymethyl methacrylate (PMMA; Sigma-Aldrich) in methylene chloride. The PMMA-coated slide was assembled into a 10–20-µl flow chamber with a glass coverslip and double-sided adhesive tape. Actin was adhered to the slide and aligned approximately with *x _{laboratory}* by successive additions and 1 min of incubations of 1 mg/ml biotinylated BSA (Sigma-Aldrich), 0.5 mg/ml streptavidin (Sigma-Aldrich), and 100 nM biotinylated Alexa Flour 647–labeled F-actin, each followed by washes with M5

Only molecules with clearly changing polarization recordings from a single probe, as indicated by step photobleaching to the fluorescence intensity baseline, were chosen for analysis, corresponding to 298 and 378 recordings for myoV-6IQ and myoV-4IQ, respectively. After determining the maximum likelihood orientation for each cycle of these recordings (described below), only molecules with at least two clear orientational steps were retained for further analysis (*n* = 73 molecules [25%] for myoV-6IQ; *n* = 49 [13%] for myoV-4IQ). The percentage of myoVs stepping twice before the photobleach is 1.9-fold higher for 6IQ than for 4IQ, which is likely related to the 1.9-fold higher run length of myoV-6IQ reported earlier (Sakamoto et al., 2005).

A flow chamber was assembled as above, except that the fused silica slide was coated with 2.5 mg/ml poly-l-lysine (Sigma-Aldrich) instead of PMMA, and then 0.2 mg/ml anti-FLAG antibody (Sigma-Aldrich) was added and incubated for 1 min. 20 µl of ~0.2 mg/ml of unlabeled myoV-6IQ or myoV-4IQ (i.e., without BR-CaM) was flowed in and incubated for 8 min. Exposed poly-l-lysine and fused silica were blocked with 2 × 20–µl washes of 0.5 mg/ml BSA. Two 20-µl aliquots of pre-sheared, unlabeled F-actin (5 µM) were incubated for 1 min each to block any inactive myosin heads, and then unbound actin was flushed out with 20 µl of 2 mM ATP in M5^{+} buffer. After two further washes of M5^{+} to remove residual ATP, 5 nM actin filaments, sparsely (0.3%) labeled with rhodamine, in M5^{+} buffer were incubated for 1 min, and then motility buffer was added to initiate filament gliding. Moving fluorescent spots were selected for polTIRF recording, and then the probe orientation was determined as described above.

myoV is composed of two heavy chains bound together through a coiled-coil region located past the C-terminal end of their lever arms (Reck-Peterson et al., 2000). Our simplified model of myoV treats each half of the molecule as an actin-binding motor domain connected via a compliant joint to a rigid, inextensible lever arm, which is, in turn, connected to the other half of the molecule via a free swivel at the lever-tail junction (Fig. 1 A). The orientation of the lever arm relative to the actin filament Ω_{L} = (*β _{L}*,

A prediction of the hand-over-hand model is that the *β _{P}* and

Fig. 2 is a diagram of a labeled molecule of myoV stepping straight along actin, showing that even when the myosin walks straight, the azimuthal orientation of the probe relative to the actin (*α _{P}*) changes after a step because it is not, in general, located within the plane of rotation. However,

To directly transform the orientation of the probe in the actin frame (*β _{P}*,

$$\begin{array}{l}{v}_{A,x}=\left(\mathrm{sin}{\alpha}_{L}\cdot \mathrm{cos}{\varphi}_{P}+\mathrm{cos}{\alpha}_{L}\cdot \mathrm{cos}{\beta}_{L}\cdot \mathrm{sin}{\varphi}_{P}\right)\cdot \mathrm{sin}{\theta}_{P}+\mathrm{cos}{\alpha}_{L}\cdot \mathrm{sin}{\beta}_{L}\cdot \mathrm{cos}{\theta}_{P}\\ {v}_{A,y}=\left(-\mathrm{cos}{\alpha}_{L}\cdot \mathrm{cos}{\varphi}_{P}+\mathrm{sin}{\alpha}_{L}\cdot \mathrm{cos}{\beta}_{L}\cdot \mathrm{sin}{\varphi}_{P}\right)\cdot \mathrm{sin}{\theta}_{P}+\mathrm{sin}{\alpha}_{L}\cdot \mathrm{sin}{\beta}_{L}\cdot \mathrm{cos}{\theta}_{P}\\ {v}_{A,z}=-\mathrm{sin}{\beta}_{L}\cdot \mathrm{sin}{\theta}_{P}\cdot \mathrm{sin}{\varphi}_{P}+\mathrm{cos}{\beta}_{L}\cdot \mathrm{cos}{\theta}_{P}\end{array}$$

(1)

and

$$\begin{array}{c}{\beta}_{P}\left({\theta}_{P},{\varphi}_{P},{\beta}_{L}\right)=\mathrm{acos}\left({v}_{A,z}\right)\\ {\alpha}_{P}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{L},{\beta}_{L}\right)=\mathrm{atan}2\left({v}_{A,x},{v}_{A,y}\right)\end{array}.$$

(2)

The atan2(*x*, *y*) function is similar to atan(*y*/*x*), except that atan2(*x*, *y*) is single valued over a larger range $(-\pi \le {\alpha}_{P}\le \pi ).$

Inverting Eqs. 1 and 2 gives the lever-arm vector (** w_{A}**) and orientation (

$$\begin{array}{l}{w}_{A,x}=\left(\mathrm{cos}\left({\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P}\right)\right)\cdot \mathrm{sin}{\varphi}_{P}\cdot \mathrm{sin}{\theta}_{P}+\mathrm{sin}\left({\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P}\right)\right)\cdot \mathrm{cos}{\theta}_{P}\right)\\ \cdot \mathrm{cos}{\alpha}_{P}-\mathrm{sin}{\theta}_{P}\cdot \mathrm{cos}{\varphi}_{P}\cdot \mathrm{sin}{\alpha}_{P}\\ {w}_{A,y}=\left(\mathrm{cos}\left({\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P}\right)\right)\cdot \mathrm{sin}{\varphi}_{P}\cdot \mathrm{sin}{\theta}_{P}+\mathrm{sin}\left({\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P}\right)\right)\cdot \mathrm{cos}{\theta}_{P}\right)\\ \cdot \mathrm{sin}{\alpha}_{P}+\mathrm{sin}{\theta}_{P}\cdot \mathrm{cos}{\varphi}_{P}\cdot \mathrm{cos}{\alpha}_{P}\\ {w}_{A,z}=\left(\mathrm{cos}{\theta}_{P}\cdot \mathrm{cos}{\beta}_{P}\pm \sqrt{\mathrm{sin}{\theta}_{P}^{2}\cdot \mathrm{sin}{\varphi}_{P}^{2}\cdot \left(\mathrm{sin}{\beta}_{P}^{2}-\mathrm{sin}{\theta}_{P}^{2}\cdot \mathrm{cos}{\varphi}_{P}^{2}\right)}\right)\\ /\mathrm{cos}{\theta}_{P}^{2}+\mathrm{sin}{\theta}_{P}^{2}\cdot \mathrm{sin}{\varphi}_{P}^{2}\end{array}$$

(3)

and

$$\begin{array}{c}{\beta}_{\mathit{L}}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P}\right)=\text{acos}\left({w}_{A,z}\right)\\ {\alpha}_{L}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{P},{\beta}_{P}\right)=\text{atan}2\left({w}_{A,x},{w}_{A,y}\right)\end{array}.$$

(4)

Eqs. 3 and 4 imply that calculating *β _{L}* and

Differences in probe and lever-arm azimuth after one step are given by Δ*α _{P,n}* =

$${}^{2}\Delta {\alpha}_{P,n}=\Delta {\alpha}_{P,n}+\Delta {\alpha}_{P,n+1}=\Delta {\alpha}_{L,n}+\Delta {\alpha}_{L,n+1}={}^{2}\Delta {\alpha}_{L,n},$$

(5)

and thus

$$\begin{array}{c}\Delta {\alpha}_{P,n}-\Delta {\alpha}_{L,n}=\Delta {\alpha}_{L,n+1}-\Delta {\alpha}_{P,n+1}\\ ={\left(-1\right)}^{m-1}\left(\Delta {\alpha}_{P,n+(m-1)}-\Delta {\alpha}_{L,n+(m-1)}\right)\equiv R,\end{array}$$

(6)

where *m* is an integer describing the interval of steps being considered, and *R* is a constant for a given molecule that depends on the local probe angle (*θ _{P}*,

Our strategy for calculating (*θ _{P}*,

$${\alpha}_{L}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{P,n+1},{\beta}_{P,n+1}\right)-{\alpha}_{L}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{P,n},{\beta}_{P,n}\right)=\Delta {\alpha}_{L,n}$$

(7)

$${\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P,n+1}\right)+{\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P,n}\right)={}^{\mathit{Tot}}\beta _{L,n}.$$

(8)

The value for *R* for each molecule is obtained by calculating the average of (−1)* ^{n}*·Δ

$$\stackrel{-}{\Delta {\alpha}_{P}}=\frac{{\displaystyle \sum _{n=1}^{N-1}{(-1)}^{n}}\cdot {W}_{n}\cdot \Delta {\alpha}_{P,n}}{{\displaystyle \sum _{n=1}^{N-1}{W}_{n}}}\cong R,$$

(9)

with the following definition for the weighting function, *W* (Section 4 of the Appendix):

$${W}_{n}=\frac{1}{{\left(\left|\Delta {\alpha}_{P,n}-\stackrel{-}{\Delta {\alpha}_{P}}\right|+\epsilon \right)}^{r}},$$

(10)

where a small value, *ε*, is chosen to avoid division by 0, and *r* is an even positive integer. The results of the procedure are not sensitive to the values of *ε* and *r*. Setting them to *ε* = 0.0001 rad and *r* = 4 results in a robust stable value of *R* for each molecule.

Given *N* − 1 values of Δ*α _{P}* measured from a molecule taking

When using 16 channels of polarized intensity, the solution of the probe’s orientation Ω_{P} (*β _{P}*,

These four equivalent probe pairs can be used to calculate four corresponding pairs of the lever-arm orientation for the *N* angular states of Ω_{L,n} (*β _{L}*

For each of the four configurations described above, there is an additional equivalent solution for the lever-arm orientation that corresponds to their mirror images (Fig. S4). Given any probe solution Ω_{P,i}, two equivalent lever solutions (Ω_{L,n} = Ω′_{L,n}) can be calculated that satisfy Eqs. 7 and 8, where Ω_{L,n} is described above and Ω′_{L,n} (180° − *β _{L,n}*,

Fig. S1 provides a basic schematic of the myoV-6IQ and myoV-4IQ recombinant constructs. Fig. S2 describes the different rotation operations performed by the rotation matrix used in deriving Eqs. 1 and 2 (Section 1 of Appendix). Figs. S3 and S4 demonstrate the degeneracy of the lever angle solution (Materials and methods). Figs. S5, S8, and S9 give various probe and lever angle distributions for myoV-6IQ measured using 16 PFIs (Materials and methods). Tables S1–S3 give the best-fit values from the Gaussian fits to these distributions. Figs. S6 and S7 give the distributions for Δ*α _{P}* and

polTIRF measurements were made with BR probes (Forkey et al., 2003, 2005) bound to CaM subunits on two recombinant constructs of myoV derived from chicken: myoV-6IQ with its native number of CaMs (six) bound, and myoV-4IQ, truncated to contain four CaMs (Purcell et al., 2002). Both constructs have a coiled-coil region at the C terminus for dimerization but no cargo-binding domain (Fig. S1). Probe angular measurements for most of the data were made using 12 PFI channels (Figs. 1 B and and33–8, and Tables 2–4); further data for myoV-6IQ was taken using 16 PFI channels (Figs. S5–S9 and Tables S1–S4). Full sets of these PFIs were recorded at 80-ms intervals.

Representative traces from the processive runs of rhodamine-labeled myoV-6IQ (left) and myoV-4IQ (right) motors along actin filaments. (A and B) Measured intensities: The total measured intensity is shown (A), along with its 12 component intensities (B). **...**

Analysis of the distributions of probe *α* and *β* angles for myoV-6IQ and myoV-4IQ in the actin frame of reference

Δ*α*_{L} distributions for myoV-6IQ and myoV-4IQ. Histograms of the Δ*α*_{L} angles calculated from the mean angles of *α*_{L} that were visited during individual runs of myoV-6IQ (A) and myoV-4IQ (B). The histograms are normalized **...**

Analysis of the *θ*_{P} versus *ϕ*_{P} distributions calculated from the processive runs of myoV-6IQ and myoV-4IQ

Video sequences of BR on single myoV molecules moving processively on actin attached to the microscope slide gave average translocation velocities, *V _{ave}*, of 103 ± 3 nm/s (mean ± SEM;

The traces of the axial and azimuthal angles of the probe, *β _{P}* and

*α _{P}* usually visits two values during individual runs of myoV-6IQ and multiple values for myoV-4IQ (Fig. 3 E). Distributions of

The change of the azimuthal angle of the probe after one step of myoV, *α _{P,n+1}* −

The azimuthal angle of the probe, *α _{P}*, is related to that of the lever arm of myoV,

The orientation of the BR probe relative to a frame of reference fixed within the lever arm is described by the polar angles (*θ _{P}*,

Scatter plots of *θ _{P}* versus

*β _{L}* and

There are two possible unique solutions for the lever angles that describe different lever-arm configurations, A and B (Materials and methods and Fig. S3). The distributions of *β _{L}* calculated for these two configurations for myoV-6IQ (measured with 16 channels) each have two distinct peaks that are not substantially different (Fig. S5 A and Table S2). Therefore, although there are four equivalent probe orientations from a given run of a labeled myoV, each of these results in roughly similar values for

Most of our data were measured using 12 channels of polarized intensity, which limits the probe to a quarter-sphere of space and leads to further degeneracies in the probe and lever-arm orientations, as mentioned earlier. We present a single solution for the 12-channel data for each molecule by constraining the probe and lever orientations to be close to those found for the 16-channel myoV-6IQ data.

Distributions of Δ*α _{L}* (=

Unlabeled myoV-6IQ or myoV-4IQ was attached to the microscope slide using a C-terminal Flag tag (Fig. S1) and anti-Flag antibody. Actin, very sparsely labeled with (monofunctional) rhodamine (Materials and methods), glided on the myosin (Fig. S10) with velocity of 112 ± 5.8 and 133 ± 7.0 nm/s for myoV-6IQ and myoV-4IQ, respectively, at 0.1 mM MgATP. The angle of the fluorescent probe in actin was determined during gliding by recording 16 PFIs (Materials and methods) (Beausang et al., 2008a). As with the other myosins measured in the twirling assay, for both myoV-6IQ and myoV-4IQ, the polTIRF intensities gradually rose and fell, and the individual polarizations exhibited various relative phase shifts (Fig. 9 B). Total intensity was constant until the probe photobleached, indicating that the changes of the individual polTIRF traces are caused by probe rotation as the actin twirled around its axis during gliding (Beausang et al., 2008a). *β _{P}* often changed very little over the course of the experiment, as expected from a probe fixed to the actin (Fig. 9 D). On the other hand,

Representative traces from rhodamine-labeled actin filaments twirling over myoV-6IQ (left) or myoV-4IQ (right). Measured intensities (A and B): The total measured intensity is shown (A), along with its 16 component intensities (B). The rhodamine probe **...**

myoV-6IQ and myoV-4IQ show two clear differences in their twirling. A significant portion of filaments gliding over myoV-6IQ–coated surfaces did not twirl (|pitch^{−1}| < 0.4 µm^{−1} = 50%), whereas most filaments on myoV-4IQ slides twirled (|pitch^{−1}| < 0.4 µm^{−1} = 6%). From those filaments that did twirl, a left-handed pitch of 1.4 ± 0.13 µm (mean ± SEM) was calculated, in agreement with earlier measurements on native myoV (Beausang et al., 2008a). When the pitches of both the twirling and non-twirling filaments from myoV-6IQ are considered, an average left-handed pitch of 2.7 ± 0.64 µm is calculated. myoV-4IQ, surprisingly, twirls either to the left (56%) or to the right (38%), with pitches of 1.2 ± 0.14 µm and 1.0 ± 0.19 µm, respectively. myoV-4IQ is the first myosin tested that can twirl robustly with either handedness, suggesting a coupling between the step sizes and azimuthal changes among motors mechanically connected through actin and the glass slide.

In this study, we were able to calculate the changes in orientation of the lever arm of myoV as it steps along actin using a novel analysis method. Previous single-molecule studies using polTIRF characterized the angular changes of fluorescent probes on the lever arm, denoted *α _{P}* and

myoV-4IQ, with a truncated lever arm, has a Δ*α _{L}* distribution with a central peak near zero and two shoulders to the left and right, all with means similar to those for the native myoV-6IQ (Fig. 8 and Table 3). However, the two constructs differ significantly in the relative frequencies of these three subpopulations; myoV-4IQ has an approximately equal probability for stepping left, right, or straight for any given single step, whereas myoV-6IQ walks straight more often. These results are consistent with those of the myoV-4IQ gliding filament assays, where most of the observed actin filaments twirled (94%) with a left-handed (56%) or right-handed pitch (38%), a loss of azimuthal bias in the twirling of actin that has not been observed before for any other myosin family member (Sun et al., 2007; Beausang et al., 2008a). In the kinesin family of microtubule motors, a somewhat analogous observation has been made for Ncd, where a point mutation in its neck linker region led to a loss of directional bias in microtubule gliding assays (Endow and Higuchi, 2000). Individual microtubules were driven by this mutant in either the plus- or minus-end direction. Such behavior in the ensemble assay suggests that cooperation between the motors that are linked through the cytoskeletal filament promotes directional concordance of individual motors that have limited directional bias, thereby maintaining an initial directionality. In the case of myoV-4IQ observed here, the axial direction along the actin filament (toward the barbed end) is not variable, but the frequency for an individual motor to step sideways is high, as discussed below, and leftward and rightward steps have similar probabilities. Linking the motors through the filament apparently coordinates this helical direction, leading to either left- or right-handed twirling.

Subunits of F-actin form a right-handed helix with a helical half-pitch repeat of ~13 subunits that span ~36 nm, the same distance as the average step size of myoV (Mehta et al., 1999). Because both motor domains of myoV bind stereo-specifically to subunits of actin, the azimuthal difference between the two heads of myoV is equal to the difference between the two bound actin subunits. Electron microscopy (EM) studies have shown that a single step of myoV can span 11, 13, or 15 actin subunits (Walker et al., 2000; Oke et al., 2010), corresponding to discrete changes in the myoV azimuth of Δ*α _{L}* = −28°, 0°, and +28°, respectively, values reasonably close to the three Δ

The principal previously reported effect of shortening the lever arm is that it proportionally shortens the step size; in the case of myoV-4IQ, the step size averaged 24 nm (Purcell et al., 2002; Sakamoto et al., 2005). Based on its mean step size alone, myoV-4IQ should span, on average, approximately eight to nine actin subunits, from which we would expect a Δ*α _{L}* distribution with peaks centered at Δ

It is interesting that, despite the broad step-size distribution of myoV-4IQ, it visits actin subunits within a narrow azimuthal range, Δ*α _{L}* ≈ ±40°. This connection between myoV step-size and predicted azimuthal change was addressed in simulations performed by Vilfan (2005a), who examined likely actin-binding sites of myoV with lever arms containing two, four, six, or eight IQ motifs. The model used in this analysis relied on calculating the bending energies of the four myoV constructs. Because of high bending energies in the myoV lever arm associated with large azimuthal changes, all of the constructs were limited to spanning 2, 11, 13, or 15 actin subunits, for which the azimuthal changes fall in the range of Δ

The mean step size (Purcell et al., 2002; Sakamoto et al., 2005) and actin subunit span (Oke et al., 2010) of myoV are clearly dependent on the length of its lever arm. It is therefore surprising that the means of the Δ*α _{L}* distributions, related to the span over actin subunits, are so similar between myoV-6IQ and myoV-4IQ (Fig. 8 and Table 3). Our results on myoV-6IQ and myoV-4IQ, and the Vilfan model, strongly suggest that the azimuthal range of myoV on the actin filament is largely independent of the length of its lever arm. This constraint on its azimuth is likely a result of the relatively high stiffness of the myoV lever arm, probably leading to approximately equal azimuthal stiffness as the actin, which would energetically exclude large bending angles (Vilfan, 2005a; Sun and Goldman, 2011) and may even straighten the actin helix somewhat between the two bound heads. Although the lever arm constrains the azimuthal motions of myoV-4IQ to a range similar to that of myoV-6IQ, the myoV-4IQ construct takes more sideways steps than myoV-6IQ, presumably because its truncated lever arm makes straight steps that necessitate stretching 36 nm, less energetically favorable. We speculate that the consequent extra strain increases the likelihood of stepping sideways. Nevertheless, the range of azimuthal angles is limited by the discrete angles of the actin helix, relatively independent of the neck length. A different isoform, myosin VI, has a much larger range of step sizes and azimuthal positions because its lever arm is very flexible (Sun et al., 2007).

The distribution of *θ _{P}* versus

We can more closely identify the IQ sites that likely correspond to our data by using a recently constructed molecular mechanical model of a myoV dimer with the two motor domains bound to actin 13 subunits apart (Parker et al., 2009). From this model, we calculated the parameters *θ _{P}*,

Two important assumptions about myoV stepping are required for determining (*θ _{P}*,

To determine how a curved or kinked lever arm could perturb our analysis, we again turned to the molecular model of myoV (see above and Parker et al., 2009), which has a curved leading lever arm. The sum *β _{L}*

The second assumption made in the analysis is that there is no azimuthal twist (γ) of the lever arm around its own axis relative to the motor domain. γ and *ϕ _{P}* sum together in all coordinate calculations, so as long as γ is constant, its value is equivalent to setting an offset for

The analysis described here can be applied to other biological molecules, particularly molecular motors, rotary energy convertors, and nucleic acid–processing enzymes. Many enzymes and macromolecules undergo rotational motions as part of their functional mechanisms. For the detailed analysis described here to work, some of the features of molecular motor stepping should apply, including discrete orientations that are repeated in successive enzymatic cycles, a fixed or detectable rotational frame of reference, and labeling methodology that fixes a probe dipole relative to a domain that undergoes functionally relevant rotations. Other myosins, kinesins, or dyneins that walk along filaments with a hand-over-hand mechanism could use the analysis presented in this paper with little modification. The assumptions used in the analysis (straight lever arm and constant local twist angle) would need to be valid. In other myosins and in kinesins, the structure of the waiting state between steps and the dynamic changes during steps are not defined well (Block, 2007; Sweeney and Houdusse, 2010). With dynein, the basic motion of the step or stroke is not understood well, such as whether its AAA^{+} ring rotates during a step (Numata et al., 2008; Gennerich and Vale, 2009). These applications of detailed polTIRF analysis are relatively straightforward. When the local orientation of the probe (*θ _{P}*,

AAA^{+} ring proteins, besides dynein, constitute a large family of macromolecular machines requiring functionally important rotational motions. In the mitochondrial F1 ATP synthase, the coupling between the hydrolysis or reformation of ATP and tilting of the subunits is described by several different models (Nakanishi-Matsui et al., 2010; Okuno et al., 2011) that might be distinguished using polTIRF. Similarly, the coupling between ATPase activity of viral portal motors and pumping of DNA into the viral capsid is not settled (Moffitt et al., 2009).

The ribosome and ribosomal elongation factors undergo rotational motions, but the relationship between these motions to proofreading of aminoacyl-tRNA selection and to translocation of the messenger RNA are unknown (Schmeing and Ramakrishnan, 2009). DNA- and RNA-processing enzymes are natural applications of orientational analysis resulting from the helical nature of duplex nucleic acids. Topoisomerases and helicases exhibit large rotational motions, whose coupling to topological adjustments of the DNA and to unwinding of secondary structure is uncertain (Bustamante et al., 2011; Klostermeier, 2011). In all of these cases, details of the structural dynamics in real time may be determined and correlated with the functional outputs by analyzing polTIRF recordings as we have done for myoV.

We found that the removal of two IQ motifs from the LCD of myoV led to myoV-4IQ taking fewer straight steps than the native myoV-6IQ construct, as we expected. However, considering the 24-nm step size of myoV-4IQ, the observation that the myoV-4IQ and myoV-6IQ took steps to the left and right with similar magnitude but different frequencies, as measured by Δ*α _{L}*, was unexpected. These findings suggest that the step-wise changes in the azimuth of myoV are limited by the stiffness of its lever arm to Δ

The intensities measured in our experiments are used to calculate the orientation of the probe attached to myoV relative to the laboratory frame of reference (*θ _{laboratory}*,

In deriving an expression relating the orientations of the probe to the lever arm in the actin frame, we note that this expression must also account for the orientation of the probe relative to the frame of the lever arm. We construct the rotation matrix ^{A}**R*** ^{P}*, which is derived from four separate rotation operations that rotate the probe frame (P) to the actin frame (A) (Fig. S2). The combined rotation matrix

$${}^{0}\mathbf{R}^{1}=Rot\left({z}_{0},-\left(\frac{\pi}{2}-{\alpha}_{L}\right)\right)=\left[\begin{array}{ccc}\mathrm{sin}{\alpha}_{L}& \mathrm{cos}{\alpha}_{L}& 0\\ -\mathrm{cos}{\alpha}_{L}& \mathrm{sin}{\alpha}_{L}& 0\\ 0& 0& 1\end{array}\right].$$

(A.1)

This rotation results in a coordinate frame defined by *x _{1}*,

$${}^{1}\mathbf{R}^{2}=Rot\left({x}_{1},-{\beta}_{L}\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}{\beta}_{L}& \mathrm{sin}{\beta}_{L}\\ 0& -\mathrm{sin}{\beta}_{L}& \mathrm{cos}{\beta}_{L}\end{array}\right].$$

(A.2)

This rotation results in a coordinate frame defined by *x _{2}*,

$${}^{2}\mathbf{R}^{3}=Rot\left({z}_{2},-\left(\frac{\pi}{2}-{\varphi}_{P}\right)\right)=\left[\begin{array}{ccc}\mathrm{sin}{\varphi}_{P}& \mathrm{cos}{\varphi}_{P}& 0\\ -\mathrm{cos}{\varphi}_{P}& \mathrm{sin}{\varphi}_{P}& 0\\ 0& 0& 1\end{array}\right].$$

(A.3)

This rotation results in a coordinate frame defined by *x _{3}*,

$${}^{3}\mathbf{R}^{4}=Rot\left({x}_{3},-{\theta}_{P}\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}{\theta}_{P}& \mathrm{sin}{\theta}_{P}\\ 0& -\mathrm{sin}{\theta}_{P}& \mathrm{cos}{\theta}_{P}\end{array}\right].$$

(A.4)

The total rotation operation is described by:

$${}^{A}\mathbf{R}^{P}={}^{0}\mathbf{R}^{4}={}^{0}\mathbf{R}^{1}\cdot {}^{1}\mathbf{R}^{2}\cdot {}^{2}\mathbf{R}^{3}\cdot {}^{3}\mathbf{R}^{4}.$$

(A.5)

The result of this last rotational operation is that the *z _{3}* axis is aligned with the vector of the probe ${\stackrel{\rightharpoonup}{\mathbf{v}}}_{P}$ (Fig. S2 E), where the

$${\stackrel{\rightharpoonup}{\mathbf{v}}}_{P}=\left(\begin{array}{c}{v}_{P,x}\\ {v}_{P,y}\\ {v}_{P,z}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right).$$

(A.6)

${\stackrel{\rightharpoonup}{\mathbf{v}}}_{P}$is a vector defined in the probe frame (P) and can be redefined in the actin frame (A) as the vector${\stackrel{\rightharpoonup}{\mathbf{v}}}_{A}$:

$$\left(\begin{array}{c}{v}_{A,x}\\ {v}_{A,y}\\ {v}_{A,z}\end{array}\right)={\stackrel{\rightharpoonup}{\mathbf{v}}}_{A}={}^{A}\mathbf{R}^{P}\cdot {\stackrel{\rightharpoonup}{\mathbf{v}}}_{P}.$$

(A.7)

This is converted to *α _{P}* and

$${\alpha}_{P}=\text{atan}2\left({v}_{A,x},{v}_{A,y}\right)$$

(A.8)

$${\beta}_{P}=\text{acos}\left({v}_{A,z}\right)$$

(A.9)

Solving the above relationships, we find the two expressions for *α _{P}* and

$${\beta}_{P}\left({\theta}_{P},{\varphi}_{P},{\beta}_{L}\right)=\text{acos}\left(-\mathrm{sin}{\beta}_{L}\cdot \mathrm{sin}{\theta}_{P}\cdot \mathrm{sin}{\varphi}_{P}+\mathrm{cos}{\beta}_{L}\cdot \mathrm{cos}{\theta}_{P}\right),$$

(A.10)

where:

$$0\le {\beta}_{P}\le \pi $$

(A.11)

$${\alpha}_{P}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{L},{\beta}_{L}\right)=\text{atan}2({v}_{A,x},{v}_{A,y}),$$

(A.12)

where expressions for *v _{A,x}* and

$$-\pi \le {\alpha}_{P}\le \pi $$

(A.13)

and the atan2(*x*, *y*) function is similar to the standard arctan(*y*/*x*) function, except that atan2(*x*, *y*) is single valued over a larger range (−π ≤ *α _{P}* ≤ π vs. −π/2 ≤

We wish to find an expression for the orientation of the lever arm in terms of the orientation of the probe, both in relation to the actin frame of reference (*x _{A}*,

$$-\mathrm{sin}{\beta}_{L}\cdot \mathrm{sin}{\theta}_{P}\cdot \mathrm{sin}{\varphi}_{P}=\mathrm{cos}{\beta}_{L}\cdot \mathrm{cos}{\theta}_{P}-\mathrm{cos}{\beta}_{P}$$

(A.14)

$${\left(-\mathrm{sin}{\beta}_{L}\cdot \mathrm{sin}{\theta}_{P}\cdot \mathrm{sin}{\varphi}_{P}\right)}^{2}={\left(\mathrm{cos}{\beta}_{L}\cdot \mathrm{cos}{\theta}_{P}-\mathrm{cos}{\beta}_{P}\right)}^{2}$$

(A.15)

$$\begin{array}{l}\left(1-\mathrm{cos}{\beta}_{L}^{2}\right)\cdot \mathrm{sin}{\theta}_{P}^{2}\cdot \mathrm{sin}{\varphi}_{P}^{2}\\ =\mathrm{cos}{\beta}_{L}^{2}\cdot \mathrm{cos}{\theta}_{P}^{2}+\mathrm{cos}{\beta}_{P}^{2}-2\cdot \left(\mathrm{cos}{\beta}_{L}\cdot \mathrm{cos}{\theta}_{P}\cdot \mathrm{cos}{\beta}_{P}\right)\end{array}$$

(A.16)

$$\begin{array}{l}\mathrm{cos}{\beta}_{L}^{2}\cdot \left(\mathrm{cos}{\theta}_{P}^{2}+\mathrm{sin}{\theta}_{P}^{2}\cdot \mathrm{sin}{\varphi}_{P}^{2}\right)\\ -\mathrm{cos}{\beta}_{L}\cdot \left(2\cdot \mathrm{cos}{\theta}_{P}\cdot \mathrm{cos}{\beta}_{P}\right)+\left(\mathrm{cos}{\beta}_{P}^{2}-\mathrm{sin}{\theta}_{P}^{2}\cdot \mathrm{sin}{\varphi}_{P}^{2}\right)=0\end{array}$$

(A.17)

We then find an expression for *β _{L}*:

$$\begin{array}{l}{\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P}\right)\\ =\mathrm{acos}\left(\frac{\mathrm{cos}{\theta}_{P}\cdot \mathrm{cos}{\beta}_{P}\pm \sqrt{\mathrm{sin}{\theta}_{P}^{2}\cdot \mathrm{sin}{\varphi}_{P}^{2}\cdot \left(\mathrm{sin}{\beta}_{P}^{2}-\mathrm{sin}{\theta}_{P}^{2}\cdot \mathrm{cos}{\varphi}_{P}^{2}\right)}}{\mathrm{cos}{\theta}_{P}^{2}+\mathrm{sin}{\theta}_{P}^{2}\cdot \mathrm{sin}{\varphi}_{P}^{2}}\right),\end{array}$$

(A.18)

where:

$$0\le {\beta}_{L}\le \pi .$$

(A.19)

To solve for *α _{L}*, we rearrange Eq. A.12:

$$\begin{array}{l}\mathrm{tan}{\alpha}_{P}\\ =\frac{\left(-\mathrm{cos}{\varphi}_{P}+\mathrm{tan}{\alpha}_{L}\cdot \mathrm{cos}{\beta}_{L}\cdot \mathrm{sin}{\varphi}_{P}\right)\cdot \mathrm{sin}{\theta}_{P}+\mathrm{tan}{\alpha}_{L}\cdot \mathrm{sin}{\beta}_{L}\cdot \mathrm{cos}{\theta}_{P}}{\left(\mathrm{tan}{\alpha}_{L}\cdot \mathrm{cos}{\varphi}_{P}+\mathrm{cos}{\beta}_{L}\cdot \mathrm{sin}{\varphi}_{P}\right)\cdot \mathrm{sin}{\theta}_{P}+\mathrm{sin}{\beta}_{L}\cdot \mathrm{cos}{\theta}_{P}}.\end{array}$$

(A.20)

$$\begin{array}{l}\mathrm{tan}{\alpha}_{L}\cdot \left(\mathrm{sin}{\theta}_{P}\cdot \mathrm{tan}{\alpha}_{P}\cdot \mathrm{cos}{\varphi}_{P}-\mathrm{sin}{\theta}_{P}\cdot \mathrm{cos}{\beta}_{L}\cdot \mathrm{sin}{\varphi}_{P}-\mathrm{sin}{\beta}_{L}\cdot \mathrm{cos}{\theta}_{P}\right)\\ =\mathrm{tan}{\alpha}_{P}\cdot \left(\mathrm{sin}{\theta}_{P}\cdot \mathrm{cos}{\beta}_{L}\cdot \mathrm{sin}{\varphi}_{P}+\mathrm{sin}{\beta}_{L}\cdot \mathrm{cos}{\theta}_{P}\right)-\mathrm{sin}{\theta}_{P}\cdot \mathrm{tan}{\alpha}_{P}\cdot \mathrm{cos}{\varphi}_{P}\end{array}$$

(A.21)

Rearranging this yields an expression for

$${\alpha}_{L}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{P},{\beta}_{P}\right)=\text{atan2(}{w}_{A,x},{w}_{A,y}),$$

(A.22)

where expressions for *w _{A,x}* and

$$-\pi \le {\alpha}_{L}\le \pi .$$

(A.23)

We consider the difference in orientation of the probe between two angular states:

$${}^{\mathrm{m}}\Delta {\beta}_{n}={\beta}_{n+m}-{\beta}_{n}$$

(A.24)

$${}^{\mathrm{m}}\Delta {\alpha}_{n}={\alpha}_{n+m}-{\alpha}_{n},$$

(A.25)

where:

$$n=1,2\dots N-m$$

(A.26)

$$m=1,2\dots N-n.$$

(A.27)

Note that *m* denotes the number of the intervals in the difference * ^{m}*Δ and is even when two leading or two trailing states of myoV are considered. Given the conditions of our model for myoV, where

$${\beta}_{P,n}={\beta}_{P,n+m}.$$

(A.28)

Under the conditions given above, and when *m* is even, the trigonometric identity

$$\mathrm{tan}\left(A-B\right)=\left(\frac{\mathrm{tan}A-\mathrm{tan}B}{1+\mathrm{tan}A\cdot \mathrm{tan}B}\right)$$

(A.29)

is applied to a modified Eq. A.25:

$$\mathrm{tan}\left({}^{m}\Delta {\alpha}_{P,n}\right)=\mathrm{tan}\left({\alpha}_{P,n+m}-{\alpha}_{P,n}\right),$$

(A.30)

and Eq. A.12, simplifying to:

$$\begin{array}{l}\mathrm{tan}\left({\alpha}_{P,n+m}-{\alpha}_{P,n}\right)\\ =\frac{\mathrm{tan}{\alpha}_{L,n+m}\cdot {\mathrm{sec}}^{2}{\alpha}_{L,n}-\mathrm{tan}{\alpha}_{L,n}\cdot {\mathrm{sec}}^{2}{\alpha}_{L,n+m}}{1-{\mathrm{tan}}^{2}{\alpha}_{L,n}\cdot {\mathrm{tan}}^{2}{\alpha}_{L,n+m}}.\end{array}$$

(A.31)

Using the trigonometric identity

$${\mathrm{sec}}^{2}A=1+{\mathrm{tan}}^{2}A,$$

(A.32)

Eq. A.31 simplifies to:

$$\mathrm{tan}\left({\alpha}_{P,n+m}-{\alpha}_{P,n}\right)=\frac{\mathrm{tan}{\alpha}_{L,n+m}-\mathrm{tan}{\alpha}_{L,n}}{1+\mathrm{tan}{\alpha}_{L,n}\cdot \mathrm{tan}{\alpha}_{L,n+m}}.$$

(A.33)

Using the identity given in Eq. A.29, we find that:

$$\mathrm{tan}\left({\alpha}_{P,n+m}-{\alpha}_{P,n}\right)=\mathrm{tan}\left({\alpha}_{L,n+m}-{\alpha}_{L,n}\right),$$

(A.34)

which is equivalent to:

$${\alpha}_{P,n+m}-{\alpha}_{P,n}={\alpha}_{L,n+m}-{\alpha}_{L,n}$$

(A.35)

when the tan function is single valued; i.e.:

$$-\frac{\pi}{2}\le \left({\alpha}_{n+m}-{\alpha}_{n}\right)\le \frac{\pi}{2}.$$

(A.36)

We complete the derivation by noting that using Eq. A.25, Eq. A.35 can be identically expressed as:

$${}^{\mathrm{m}}\Delta {\alpha}_{P}={}^{\mathrm{m}}\Delta {\alpha}_{L}$$

(A.37)

when *m* is even.

We consider Eq. A.37, under the conditions given in Section 3 of this Appendix, when *m* = 2:

$${}^{2}\Delta {\alpha}_{P,n}={}^{2}\Delta {\alpha}_{L,n}.$$

(A.38)

When *m* = 1, Eq. A.25 can be expressed as:

$${}^{1}\Delta {\alpha}_{n}={\alpha}_{n+1}-{\alpha}_{n}=\Delta {\alpha}_{n},$$

(A.39)

and when *m* = 2, Eq. A.25 is:

$${}^{2}\Delta {\alpha}_{n}={\alpha}_{n+2}-{\alpha}_{n}=({\alpha}_{n+1}-{\alpha}_{n})+({\alpha}_{n+2}-{\alpha}_{n+1})=\Delta {\alpha}_{n}+\Delta {\alpha}_{n+1}.$$

(A.40)

In general, the difference between two angular states separated by *m* intervals can be expressed as:

$${}^{m}\Delta _{n}={\Delta}_{n}+{\Delta}_{n+1}+\dots +{\Delta}_{n+\left(m-1\right)}={\displaystyle \sum}_{i=1}^{m-1}{\Delta}_{n+i}$$

(A.41)

over all values of *m* and *n*. Using Eq. A.41, Eq. A.38 is then expressed as:

$$\Delta {\alpha}_{P,n}+\Delta {\alpha}_{P,n+1}=\Delta {\alpha}_{L,n}+\Delta {\alpha}_{L,n+1},$$

(A.42)

which is rearranged to define a factor, *R*, which expresses the effect of the probe orientation out of the plane of lever-arm rotation on Δ*α _{P}* and is constant over all

$$\begin{array}{l}\Delta {\alpha}_{P,n}-\Delta {\alpha}_{L,n}=\Delta {\alpha}_{L,n+1}-\Delta {\alpha}_{P,n+1}\\ ={\left(-1\right)}^{m-1}\cdot \left(\Delta {\alpha}_{P,n+(m-1)}-\Delta {\alpha}_{L,n+(m-1)}\right)\equiv R.\end{array}$$

(A.43)

This allows us to express Δ*α _{P}* as the sum of Δ

$$\Delta {\alpha}_{P,n}={\left(-1\right)}^{n}\cdot R+\Delta {\alpha}_{L,n}.$$

(A.44)

Therefore, calculating Δ*α _{L}* from Δ

$$\stackrel{-}{\Delta {\alpha}_{P}}=\frac{1}{N-1}\cdot ({\displaystyle \sum _{n=1}^{N-1}{(-1)}^{n}}\cdot \Delta {\alpha}_{P,n}).$$

(A.45)

Using Eq. A.44, this is then expressed as:

$$\begin{array}{l}\stackrel{-}{\Delta {\alpha}_{P}}=\frac{1}{N-1}\cdot {\displaystyle \sum _{n=1}^{N-1}({(-1)}^{n}}\cdot \Delta {\alpha}_{P,n}+R)\\ =R+\frac{1}{N-1}\cdot {\displaystyle \sum _{n=1}^{N-1}({(-1)}^{n}}\cdot \Delta {\alpha}_{P,n}).\end{array}$$

(A.46)

From this relation we find that:

$$\stackrel{-}{\Delta {\alpha}_{P}}=R$$

(A.47)

is true for all values of *n* and *m* when:

$$\Delta {\alpha}_{L,n}=\Delta {\alpha}_{L,n+m},$$

(A.48)

and the maximum value of *n* = *N* − 1 is even, where *N* is the total number of angular states.

We then extend this to the general case in which Eq. A.48 does not hold. Because of this, the mean of these Δ*α _{P}*s does not necessarily equate to

$${W}_{n}=\frac{1}{{\left(\left|\Delta {\alpha}_{P,n}-\stackrel{-}{\Delta {\alpha}_{P}}\right|+\epsilon \right)}^{r}},$$

(A.49)

where Δ*α _{P}*

$$\stackrel{-}{\Delta {\alpha}_{P}}=\frac{{\displaystyle \sum _{n=1}^{N-1}{(-1)}^{n}}\cdot {W}_{n}\cdot \Delta {\alpha}_{P,n}}{{\displaystyle \sum _{n=1}^{N-1}{W}_{n}}}.$$

(A.50)

Because the weighting function excludes values of Δ*α _{P}*

$${\stackrel{-}{\Delta {\alpha}_{P}}}_{weighted}\cong R.$$

(A.51)

Eq. A.50 therefore allows us to calculate a reasonable value for *R*, even in the more general case where Eq. A.48 does not hold for all *n*, as long as there are a significant number of Δ*α _{L}*

We again consider the probe at two angular states, *n* and *n*
*+*
*m* (Eqs. A.24–A.27), in the case where the condition for *β _{P}* given in Eq. A.28 is satisfied. Solutions for

$${\alpha}_{L,n}={\alpha}_{L}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{P,n},{\beta}_{P,n}\right)$$

(A.52)

$${\beta}_{L,n}={\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P,n}\right)$$

(A.53)

We first consider the general case where:

$${\alpha}_{L,n+m}-{\alpha}_{L,n}=\Delta {\alpha}_{L,n}$$

(A.54)

$${\alpha}_{L}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{P,n},{\beta}_{P,n}\right)-{\alpha}_{L}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{P,n+m},{\beta}_{P,n+m}\right)=\Delta {\alpha}_{L,n}$$

(A.55)

$${\beta}_{L,n}+{\beta}_{L,n+m}={}^{\mathit{Tol}}\beta _{L,n}$$

(A.56)

$${\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P,n}\right)+{\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P,n+m}\right)={}^{\mathit{Tol}}\beta _{L,n},$$

(A.57)

where *m* is nonzero and odd.

We then look at the specific case where *m* = 1, * ^{Tot}β_{L}* = π, and Δ

$${\alpha}_{L}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{P,n},{\beta}_{P,n}\right)-{\alpha}_{L}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{P,n+1},{\beta}_{P,n+1}\right)=0$$

(A.58)

$${\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P,n}\right)+{\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P,n+1}\right)=\pi $$

(A.59)

Eqs. A.58 and A.59 represent a system of two equations that share the six parameters *θ _{P}*,

For a given set of *β _{P}* and

$$\left({\alpha}_{L}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{P,n},{\beta}_{P,n}\right)-{\alpha}_{L}\left({\theta}_{P},{\varphi}_{P},{\alpha}_{P,n+m},{\beta}_{P,n+m}\right)\right)-\Delta {\alpha}_{L,n}=0$$

(A.60)

$$\left({\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P,n}\right)+{\beta}_{L}\left({\theta}_{P},{\varphi}_{P},{\beta}_{P,n+1}\right)\right)-\pi =0$$

(A.61)

Optimum values for *θ _{P}* and

We thank Dr. Robert Dale for useful comments on the manuscript and Drs. Zev Bryant and David Parker for coordinates of an actomyosin-V model.

This work was supported by National Institutes of Health grant R01GM086352.

Richard L. Moss served as editor.

- BR
- bifunctional rhodamine
- CaM
- calmodulin
- DOPI
- defocused orientation and position imaging
- EM
- electron microscopy
- LCD
- light chain domain
- myoV
- myosin V
- PFI
- polarized fluorescence intensity
- polTIRF
- polarized total internal reflection microscopy

- Ali M.Y., Uemura S., Adachi K., Itoh H., Kinosita K., Jr, Ishiwata S. 2002. Myosin V is a left-handed spiral motor on the right-handed actin helix. Nat. Struct. Biol. 9:464–467 10.1038/nsb803 [PubMed] [Cross Ref]
- Ali M.Y., Krementsova E.B., Kennedy G.G., Mahaffy R., Pollard T.D., Trybus K.M., Warshaw D.M. 2007. Myosin Va maneuvers through actin intersections and diffuses along microtubules. Proc. Natl. Acad. Sci. USA. 104:4332–4336 10.1073/pnas.0611471104 [PubMed] [Cross Ref]
- Amos, L.A., and W.B. Amos. 1991. Molecules of the Cytoskeleton. Guilford Press. 253 pp.
- Beausang J.F., Schroeder H.W., III, Nelson P.C., Goldman Y.E. 2008a. Twirling of actin by myosins II and V observed via polarized TIRF in a modified gliding assay. Biophys. J. 95:5820–5831 10.1529/biophysj.108.140319 [PubMed] [Cross Ref]
- Beausang, J.F., Y. Sun, and Y.E. Goldman. 2008b. Single molecule fluorescence polarization via polarized total internal reflection fluorescent microscopy.
*In*Laboratory Manual for Single Molecule Studies. P.R. Selvin and T.J. Ha, editors. Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY. 121–148. - Block S.M. 2007. Kinesin motor mechanics: binding, stepping, tracking, gating, and limping. Biophys. J. 92:2986–2995 10.1529/biophysj.106.100677 [PubMed] [Cross Ref]
- Bustamante C., Cheng W., Mejia Y.X. 2011. Revisiting the central dogma one molecule at a time. Cell. 144:480–497 10.1016/j.cell.2011.01.033 [PMC free article] [PubMed] [Cross Ref]
- Cheney R.E., O’Shea M.K., Heuser J.E., Coelho M.V., Wolenski J.S., Espreafico E.M., Forscher P., Larson R.E., Mooseker M.S. 1993. Brain myosin-V is a two-headed unconventional myosin with motor activity. Cell. 75:13–23 [PubMed]
- Cooke R. 1986. The mechanism of muscle contraction. CRC Crit. Rev. Biochem. 21:53–118 10.3109/10409238609113609 [PubMed] [Cross Ref]
- Corrie J.E., Craik J.S. 1994. Synthesis and characterization of iodacetamidotetramethylrhodamine. J. Chem. Soc. Perkin Trans. 1:2967–2973 10.1039/p19940002967 [Cross Ref]
- Corrie J.E., Craik J.S., Munasinghe V.R. 1998. A homobifunctional rhodamine for labeling proteins with defined orientations of a fluorophore. Bioconjug. Chem. 9:160–167 10.1021/bc970174e [PubMed] [Cross Ref]
- Corrie J.E., Brandmeier B.D., Ferguson R.E., Trentham D.R., Kendrick-Jones J., Hopkins S.C., van der Heide U.A., Goldman Y.E., Sabido-David C., Dale R.E., et al. 1999. Dynamic measurement of myosin light-chain-domain tilt and twist in muscle contraction. Nature. 400:425–430 10.1038/22704 [PubMed] [Cross Ref]
- Dobbie I., Linari M., Piazzesi G., Reconditi M., Koubassova N., Ferenczi M.A., Lombardi V., Irving M. 1998. Elastic bending and active tilting of myosin heads during muscle contraction. Nature. 396:383–387 10.1038/24647 [PubMed] [Cross Ref]
- Egelman E.H. 1997. New angles on actin dynamics. Structure. 5:1135–1137 10.1016/S0969-2126(97)00264-5 [PubMed] [Cross Ref]
- Endow S.A., Higuchi H. 2000. A mutant of the motor protein kinesin that moves in both directions on microtubules. Nature. 406:913–916 10.1038/35022617 [PubMed] [Cross Ref]
- Forkey J.N., Quinlan M.E., Goldman Y.E. 2000. Protein structural dynamics by single-molecule fluorescence polarization. Prog. Biophys. Mol. Biol. 74:1–35 10.1016/S0079-6107(00)00015-8 [PubMed] [Cross Ref]
- Forkey J.N., Quinlan M.E., Shaw M.A., Corrie J.E., Goldman Y.E. 2003. Three-dimensional structural dynamics of myosin V by single-molecule fluorescence polarization. Nature. 422:399–404 10.1038/nature01529 [PubMed] [Cross Ref]
- Forkey J.N., Quinlan M.E., Goldman Y.E. 2005. Measurement of single macromolecule orientation by total internal reflection fluorescence polarization microscopy. Biophys. J. 89:1261–1271 10.1529/biophysj.104.053470 [PubMed] [Cross Ref]
- Gennerich A., Vale R.D. 2009. Walking the walk: how kinesin and dynein coordinate their steps. Curr. Opin. Cell Biol. 21:59–67 10.1016/j.ceb.2008.12.002 [PMC free article] [PubMed] [Cross Ref]
- Hecht, E. 2001. Optics. Fourth edition. Addison Wesley, New York. 680 pp.
- Klostermeier D. 2011. Single-molecule FRET reveals nucleotide-driven conformational changes in molecular machines and their link to RNA unwinding and DNA supercoiling. Biochem. Soc. Trans. 39:611–616 10.1042/BST0390611 [PubMed] [Cross Ref]
- Kodera N., Yamamoto D., Ishikawa R., Ando T. 2010. Video imaging of walking myosin V by high-speed atomic force microscopy. Nature. 468:72–76 10.1038/nature09450 [PubMed] [Cross Ref]
- Mehta A.D., Rock R.S., Rief M., Spudich J.A., Mooseker M.S., Cheney R.E. 1999. Myosin-V is a processive actin-based motor. Nature. 400:590–593 10.1038/23072 [PubMed] [Cross Ref]
- Moffitt J.R., Chemla Y.R., Aathavan K., Grimes S., Jardine P.J., Anderson D.L., Bustamante C. 2009. Intersubunit coordination in a homomeric ring ATPase. Nature. 457:446–450 10.1038/nature07637 [PMC free article] [PubMed] [Cross Ref]
- Nakanishi-Matsui M., Sekiya M., Nakamoto R.K., Futai M. 2010. The mechanism of rotating proton pumping ATPases. Biochim. Biophys. Acta. 1797:1343–1352 10.1016/j.bbabio.2010.02.014 [PubMed] [Cross Ref]
- Numata N., Kon T., Shima T., Imamula K., Mogami T., Ohkura R., Sutoh K., Sutoh K. 2008. Molecular mechanism of force generation by dynein, a molecular motor belonging to the AAA+ family. Biochem. Soc. Trans. 36:131–135 10.1042/BST0360131 [PubMed] [Cross Ref]
- Oke O.A., Burgess S.A., Forgacs E., Knight P.J., Sakamoto T., Sellers J.R., White H., Trinick J. 2010. Influence of lever structure on myosin 5a walking. Proc. Natl. Acad. Sci. USA. 107:2509–2514 10.1073/pnas.0906907107 [PubMed] [Cross Ref]
- Okuno D., Iino R., Noji H. 2011. Rotation and structure of FoF1-ATP synthase. J. Biochem. 149:655–664 10.1093/jb/mvr049 [PubMed] [Cross Ref]
- Pardee J.D., Spudich J.A. 1982. Purification of muscle actin. Methods Cell Biol. 24:271–289 10.1016/S0091-679X(08)60661-5 [PubMed] [Cross Ref]
- Parker D., Bryant Z., Delp S.L. 2009. Coarse-grained structural modeling of molecular motors using multibody dynamics. Cell Mol Bioeng. 2:366–374 10.1007/s12195-009-0084-4 [PMC free article] [PubMed] [Cross Ref]
- Purcell T.J., Morris C., Spudich J.A., Sweeney H.L. 2002. Role of the lever arm in the processive stepping of myosin V. Proc. Natl. Acad. Sci. USA. 99:14159–14164 10.1073/pnas.182539599 [PubMed] [Cross Ref]
- Putkey J.A., Slaughter G.R., Means A.R. 1985. Bacterial expression and characterization of proteins derived from the chicken calmodulin cDNA and a calmodulin processed gene. J. Biol. Chem. 260:4704–4712 [PubMed]
- Reck-Peterson S.L., Provance D.W., Jr, Mooseker M.S., Mercer J.A. 2000. Class V myosins. Biochim. Biophys. Acta. 1496:36–51 10.1016/S0167-4889(00)00007-0 [PubMed] [Cross Ref]
- Rief M., Rock R.S., Mehta A.D., Mooseker M.S., Cheney R.E., Spudich J.A. 2000. Myosin-V stepping kinetics: a molecular model for processivity. Proc. Natl. Acad. Sci. USA. 97:9482–9486 10.1073/pnas.97.17.9482 [PubMed] [Cross Ref]
- Sakamoto T., Amitani I., Yokota E., Ando T. 2000. Direct observation of processive movement by individual myosin V molecules. Biochem. Biophys. Res. Commun. 272:586–590 10.1006/bbrc.2000.2819 [PubMed] [Cross Ref]
- Sakamoto T., Yildez A., Selvin P.R., Sellers J.R. 2005. Step-size is determined by neck length in myosin V. Biochemistry. 44:16203–16210 10.1021/bi0512086 [PubMed] [Cross Ref]
- Schmeing T.M., Ramakrishnan V. 2009. What recent ribosome structures have revealed about the mechanism of translation. Nature. 461:1234–1242 10.1038/nature08403 [PubMed] [Cross Ref]
- Snyder G.E., Sakamoto T., Hammer J.A., III, Sellers J.R., Selvin P.R. 2004. Nanometer localization of single green fluorescent proteins: evidence that myosin V walks hand-over-hand via telemark configuration. Biophys. J. 87:1776–1783 10.1529/biophysj.103.036897 [PubMed] [Cross Ref]
- Sun Y., Goldman Y.E. 2011. Lever-arm mechanics of processive myosins. Biophys. J. 101:1–11 10.1016/j.bpj.2011.05.026 [PubMed] [Cross Ref]
- Sun Y., Schroeder H.W., III, Beausang J.F., Homma K., Ikebe M., Goldman Y.E. 2007. Myosin VI walks “wiggly” on actin with large and variable tilting. Mol. Cell. 28:954–964 10.1016/j.molcel.2007.10.029 [PMC free article] [PubMed] [Cross Ref]
- Sweeney H.L., Houdusse A. 2010. Structural and functional insights into the myosin motor mechanism. Annu Rev Biophys. 39:539–557 10.1146/annurev.biophys.050708.133751 [PubMed] [Cross Ref]
- Syed S., Snyder G.E., Franzini-Armstrong C., Selvin P.R., Goldman Y.E. 2006. Adaptability of myosin V studied by simultaneous detection of position and orientation. EMBO J. 25:1795–1803 10.1038/sj.emboj.7601060 [PubMed] [Cross Ref]
- Terrak M., Rebowski G., Lu R.C., Grabarek Z., Dominguez R. 2005. Structure of the light chain-binding domain of myosin V. Proc. Natl. Acad. Sci. USA. 102:12718–12723 10.1073/pnas.0503899102 [PubMed] [Cross Ref]
- Toprak E., Enderlein J., Syed S., McKinney S.A., Petschek R.G., Ha T., Goldman Y.E., Selvin P.R. 2006. Defocused orientation and position imaging (DOPI) of myosin V. Proc. Natl. Acad. Sci. USA. 103:6495–6499 10.1073/pnas.0507134103 [PubMed] [Cross Ref]
- Trybus K.M., Gushchin M.I., Lui H., Hazelwood L., Krementsova E.B., Volkmann N., Hanein D. 2007. Effect of calcium on calmodulin bound to the IQ motifs of myosin V. J. Biol. Chem. 282:23316–23325 10.1074/jbc.M701636200 [PubMed] [Cross Ref]
- Veigel C., Wang F., Bartoo M.L., Sellers J.R., Molloy J.E. 2002. The gated gait of the processive molecular motor, myosin V. Nat. Cell Biol. 4:59–65 10.1038/ncb732 [PubMed] [Cross Ref]
- Vilfan A. 2005a. Elastic lever-arm model for myosin V. Biophys. J. 88:3792–3805 10.1529/biophysj.104.046763 [PubMed] [Cross Ref]
- Vilfan A. 2005b. Influence of fluctuations in actin structure on myosin V step size. J. Chem. Inf. Model. 45:1672–1675 10.1021/ci050182m [PubMed] [Cross Ref]
- Walker M.L., Burgess S.A., Sellers J.R., Wang F., Hammer J.A., III, Trinick J., Knight P.J. 2000. Two-headed binding of a processive myosin to F-actin. Nature. 405:804–807 10.1038/35015592 [PubMed] [Cross Ref]

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