1. Physics of Living Systems
  2. Structural Biology and Molecular Biophysics
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Myosin V executes steps of variable length via structurally constrained diffusion

  1. David Hathcock  Is a corresponding author
  2. Riina Tehver
  3. Michael Hinczewski  Is a corresponding author
  4. D Thirumalai
  1. Department of Physics, Cornell University, United States
  2. Department of Physics and Astronomy, Denison University, United States
  3. Department of Physics, Case Western Reserve University, United States
  4. Department of Chemistry, University of Texas, United States
Research Article
Cite this article as: eLife 2020;9:e51569 doi: 10.7554/eLife.51569
9 figures, 2 tables and 2 additional files

Figures

Myosin V geometry.

(A) Side view, with the actin filament plus end oriented toward the 𝐳^ direction. Small circles on the actin monomers denote the binding sites 𝐫n, described by Equation 1. The site n=0 corresponds to the position of the bound head. The bound polymer leg has a preferred post-power stroke direction in the x-z plane defined by a constraint angle θc relative to the 𝐳^ axis. Due to the hypothesized structural constraint at the joint, the preferred angle between the lever arms is θp. The force transmitted through the tail domain has a polar angle θF relative to the -𝐳^ direction. (B) Front view, with the actin plus end pointing out of the page. Each binding site has an associated outward pointing normal direction with azimuthal angle ϕn. As an example, one such angle is shown for the red-colored site. All azimuthal angles are measured counter-clockwise with respect to the 𝐱^ direction. For binding to occur, the head has to be in the vicinity of the site, and oriented approximately along the normal. We approximately capture this condition by a binding criterion that requires the azimuthal angle of the free leg, ϕf, to be anti-parallel to ϕn within a cutoff range ±δϕac, highlighted in light red. The load force may have an off-axis component with azimuthal angle ϕF.

Myosin V kinetic pathways.
Figure 3 with 7 supplements
Contours of the myosin V free head position distribution 𝒫(𝐫) projected onto the z-x plane.

Top row: theoretical predictions for (A) free diffusion (μc=0) and (B–D) constrained diffusion with inter-leg constraint strength (B) μc=3, (C) μc=5, and (D) μc=12. Bottom row: the corresponding contours measured from Brownian dynamics simulations, with inter-leg constraint strength (E) μc=0, (F) μc=3, (G) μc=5, and (H) μc=12. (I) Experimental measurements of the diffusion by Andrecka et al. (2015). Adding an inter-leg constraint potential produces a multi-peaked diffusion pattern. The heights of the peaks are similar to the experimental measurements for 3μc12. Note that the x=0 axis in the experimental data corresponds to the position of the gold bead attached to the myosin head when the head is bound to actin. Given the ∼5 nm size of the head and ∼10 nm radius of the bead, this accounts for the approximately 15 nm vertical shift between the theoretical/simulation distributions and experiment. In the former the x=0 axis corresponds to the top of the actin filament (where the bound head is attached).

Figure 3—figure supplement 1
Alternative projections of the constrained diffusion (μc=5).

Top row: theoretical calculations of the diffusion projected onto the (A) z-x, (B) y-x, and (C) z-y planes, and (D) the cylindrical plane z-ρ. Bottom row: diffusion contours measured from Brownian dynamics simulations, again projected onto the (E) z-x, (F) y-x, and (G) z-y planes, and (H) the cylindrical plane z-ρ. Agreement between theory and simulations is excellent.

Figure 3—figure supplement 2
Alternative projections of the free diffusion (μc=0).

Top row: theoretical calculations of the diffusion projected onto the (A) z-x, (B) y-x, and (C) z-y planes, and (D) the cylindrical plane z-ρ. Bottom row: diffusion contours measured from Brownian dynamics simulations, again projected onto the (E) z-x, (F) y-x, and (G) z-y planes, and (H) the cylindrical plane z-ρ. In the analytical theory we include a joint Hamiltonian that penalizes small inter-leg angles. This mimics the effects of steric repulsion between the legs, which are explicitly included in the BD simulations. With this adjustment to the theory, agreement between the theory and simulations is excellent.

Figure 3—figure supplement 3
Constrained diffusion of myosin V under 2 pN backward force.

The constraint strength was set to μc=5. The force F = 2 pN is in the stall regime, just above the stall force. Top row: theoretical calculations of the diffusion projected onto the (A) z-x, (B) y-x, and (C) z-y planes, and (D) the cylindrical plane z-ρ. Bottom row: diffusion contours measured from Brownian dynamics simulations, again projected onto the (E) z-x, (F) y-x, and (G) z-y planes, and (H) the cylindrical plane z-ρ. The theory and simulations agree qualitatively, showing the diffusion is rotated toward the minus end of the actin compared to that at zero load (Figure 3—figure supplement 1). Quantitative discrepancies are primarily due to a small difference in the stall force between the theoretical and numerical models.

Figure 3—figure supplement 4
Constrained diffusion of the 4IQ myosin mutant.

The constraint strength was set to μc=5. Top row: theoretical calculations of the diffusion projected onto the (A) z-x, (B) y-x, and (C) z-y planes, and (D) the cylindrical plane z-ρ. Bottom row: diffusion contours measured from Brownian dynamics simulations, again projected onto the (E) z-x, (F) y-x, and (G) z-y planes, and (H) the cylindrical plane z-ρ. Notice the difference in probability and length scales in this figure versus those preceding. The 4IQ mutant has a nearly identical diffusion pattern to the wild-type myosin, scaled down due to the decreased lever arm length.

Figure 3—figure supplement 5
Constrained diffusion of the 8IQ myosin mutant.

The constraint strength was set to μc=5. Top row: theoretical calculations of the diffusion projected onto the (A) z-x, (B) y-x, and (C) z-y planes, and (D) the cylindrical plane z-ρ. Bottom row: diffusion contours measured from Brownian dynamics simulations, again projected onto the (E) z-x, (F) y-x, and (G) z-y planes, and (H) the cylindrical plane z-ρ. Notice the difference in probability and length scales in this figure versus those preceding. The 8IQ mutant has a nearly identical diffusion pattern to the wild-type myosin, scaled up due to the increased lever arm length.

Figure 3—figure supplement 6
Effect of the form of the joint potential on free head diffusion.

(A) The log of the KL divergence DKL(𝒫cos|𝒫) between the free head spatial probability densities 𝒫cos(𝐫) and 𝒫(𝐫) corresponding to the cosine potential used in the main text and HJ=μckBT[(Δθ)2/2+h3(Δθ)3/3!+h4(Δθ)4/4!] respectively. The minimum KL divergence occurs at (h3,h4)=(0,1), which is the quartic expansion of the cosine potential. The stars indicate the potentials used to compute diffusion contours in panels B–D. The x-z diffusion contours are shown for (B) the harmonic potential (h3=h4=0), as well as potentials with (C) (h3,h4)=(2,4), and (D) (h3,h4)=(-2,-2). The harmonic potential produces diffusion that is nearly identical to the cosine potential (Figure 3C), while the contour shown in C has only subtle differences. The diffusion shown in D is qualitatively different because the potential has an additional energy minimum giving rise to a new peak in the distribution. In all cases we set μc=5, as in the main text.

Figure 3—figure supplement 7
Effects of cover-slip volume exclusion on diffusion.

Data was taken using Brownian Dynamics simulations with a glass cover-slip cutting off half the 3-dimensional space, modeled with a soft-core repulsive potential as described in Appendix 2. The cover-slip is at y = —5.5 nm, parallel to the x-z plane and is indicated by the dashed line in the relevant diffusion projections. Top row: BD simulations of free diffusion (μc=0), projected onto the (A) z-x, (B) y-x, and (C) z-y planes, and (D) the cylindrical plane z-ρ. Bottom row: BD simulations of free diffusion (μc=5), again projected onto the (E) z-x, (F) y-x, and (G) z-y planes, and (H) the cylindrical plane z-ρ. The z-x diffusion contour is not considerably altered by the cover-slip in both the free and constrained diffusion models. Therefore, the multi-peaked contour measured by Andrecka et al. (2015) is not an artifact of entropic forces due to volume exclusion.

Figure 4 with 7 supplements
Step size distributions for myosin V and mutants with altered leg length.

Top row: raw step distributions for (A) the 4IQ mutant, (B) the 6IQ wild-type, and (C) the 8IQ mutant. Bottom row: full (convolved) step distributions for (D) the 4IQ mutant, (E) the 6IQ wild-type, and (F) the 8IQ mutant, with three theoretical peak locations indicated by arrows. Theoretical distributions are shown as histograms with Brownian dynamics simulations and experimental data from Oke et al. (2010), Sakamoto et al. (2005), and Yildiz et al. (2003) indicated by symbols. The raw data from Oke et al. (2010) is convolved and binned in the bottom row. Since the imaging methods used in this experiment did not resolve large steps taken by the 8IQ mutant, in panel C we show an alternative theory (in red) with a cutoff where only small steps are allowed. The actin monomers drawn below the top row are shaded according to the analytical theory results, with the darkest color normalized to the peak of the distribution.

Figure 4—figure supplement 1
Step distributions for myosin V and mutants with a freely rotating inter-leg joint.

Same as Figure 4, but analytical theory and Brownian dynamics simulations used the free diffusion model. The analytical model was fit to data using the procedure described in the main text. We used parameters: lp=320nm, θc=59.7, νc=140, μc=0, a=0.3nm, b=0.07, and δϕac=57.0, with all other parameters identical to Table 2. Agreement with experimental measurements is comparable to that for the constrained diffusion model.

Figure 4—video 1
BD simulation trajectory illustrating stepping for the 6IQ wild-type.

The myosin dimer is depicted in blue, and the locations of potential myosin binding sites on the two actin filaments are shown in light grey and red. The BD parameters are listed in Appendix 2. This video shows the case for a motor with a freely rotating inter-leg joint making a step where the final head separation is is 36 nm (13 actin subunits).

Figure 4—video 2
BD simulation of stepping for the 6IQ wild-type with a freely rotating joint, but with a longer step (final head separation > 36 nm).
Figure 4—video 3
BD simulation of stepping for the 6IQ wild-type with a freely rotating joint, but with a shorter step (final head separation < 36 nm).
Figure 4—video 4
Similar to Figure 4—video 1, but with an inter-leg joint constraint (μc=5).
Figure 4—video 5
Similar to Figure 4—video 2, but with an inter-leg joint constraint (μc=5).
Figure 4—video 6
Similar to Figure 4—video 3, but with an inter-leg joint constraint (μc=5).
Figure 5 with 1 supplement
Changes in the full step distribution, including leading and trailing leg contributions, under backward load.

(A) Distributions for zero force F = 0 pN (solid line), sub-stall force F = 1 pN (dashed line), stall force F = 1.9 pN (dot-dashed line), and super-stall force F = 2.5 pN (dotted line). The peaks near 72 nm, 0 nm and –72 nm correspond to forward steps, stomps, and backward steps respectively. Applying force shifts the forward step distribution backward slightly (by about 1 actin subunit) and increases the probability of stomps and backward steps. (B) Normalized forward step distributions for F = 0 pN, F = 1 pN, and F = 1.9 pN. Even when other kinetic pathways are dominant the shape of the forward step distribution remains robust to load force.

Figure 5—figure supplement 1
Robust forward step distributions from Brownian dynamics simulations.

Shown are the normalized forward step distribution measured from BD trajectories with zero force and under backward load of 1 pN and 2 pN. The distribution is fairly robust, shifting back about one actin subunit per piconewton of applied force, in qualitative agreement with the analytical model (Figure 5B).

Myosin V timescales as a function of F, the backward load force.

tdiff is the mean timescale for the detached head to diffuse within radius a of any of the actin binding sites. tT and tL are the mean times for the trailing and leading heads to bind after detachment. For comparison, th is the mean timescale of ATP hydrolysis.

Forward step distribution width (solid lines) and mean binding time after trailing leg detachment (dashed lines) for F=0 as a function of the inter-leg constraint strength μc.

We carried out this calculation for θp=83 (the value used throughout this paper) as well as θp=65 and 95. As the constraint is increased the step distribution narrows, while changes in the binding time are relatively small.

Figure 8 with 1 supplement
Load-dependent aspects of myosin V dynamics.

(A) Backward-to-forward step ratio 𝒫b/𝒫f; (B) mean run length zrun; (C) mean run velocity vrun. Analytical theory results are drawn as curves, experimental results as symbols. The legend symbols are the same as those in Hinczewski et al. (2013), for ease of comparison, but the theory curves have been updated.

Figure 8—figure supplement 1
Load-dependence of step ratio, run length, and run velocity is captured by the free diffusion model.

Same as Figure 8, but using the analytical free diffusion model, fit to experiments as described in the main text. We used parameters: lp=320nm, θc=59.7, νc=140, μc=0, a=0.3nm, b=0.07, and δϕac=57.0, with all other parameters identical to Table 2. Agreement with experimental measurements is comparable to that for the constrained diffusion model.

Myosin V run length under off-axis forces.

Shown is the percent change in run length from that under backward force zrun(θF,ϕF)/zrun(0,0)-1 computed using Equation 9. In the worst case (θF20,ϕF=0) the run length is decreased by 15%. The run length most dramatically increases under fully off-axis forces (θF>0,ϕF=±90).

Tables

Table 1
Summary of main analytical results.
QuantityMeaningDefinition
𝐫nposition of actin subunitsEquation 1
tfpnfirst passage time to subunit nEquation 3
𝒫(𝐫)equilibrium distribution of the free head positionfollowing Equation 3
𝒫Tnbinding probabilities for trailing legEquation 4
𝒫Lnbinding probabilities for leading legfollowing Equation 4
𝒫distndistribution of head-to-head distancesEquation 5
𝒫T(zn)convolved trailing leg step distributionfollowing Equation 5
𝒫L(zn)convolved leading leg step distributionpreceding Equation 6
𝒫(zn)full convolved step distributionEquation 6
μ^cconstraint direction (under force)Equation 7
𝒯power stroke effectiveness (under force)Equation 7
𝒫b/𝒫fbackward-to-backward step ratioStep ratio section
zrunmean run lengthEquation 10
vrunmean run velocitypreceding Equation 11
trunmean run timeEquation 11
Table 2
Summary of myosin V model parameters.

For the parameters identified as fit to experiments, the following approach was used: as described in the text, θc, θp, and δϕac were varied to fit the step distributions, while b, a, and 𝒯=1+20νc/(20+7κνc) were varied to fit the force response data. Parameters lp and νc were also allowed to vary along curves of constant 𝒯 while fitting the step distributions.

ParameterValueSource
Mechanical Parameters
Leg contour length, L35 nmCraig and Linke, 2009
Head diffusivity, Dh5.7 × 10—7 cm2/sOrtega et al., 2011; Coureux et al., 2004
Leg persistence length, lp350 nmFit to experiment*, Howard and Spudich, 1996; Vilfan, 2005a
Bound leg constraint angle, θc65.0°Fit to experiment*, Lewis et al., 2012
Bound leg constraint strength, νc261Fit to experiment
Inter-leg preferred angle, θp83.0°Fit to experiment*, Takagi et al., 2014
Inter-leg constraint strength, μc5Fit to experiment
Binding Parameters
Actin radius, R5.5 nmLan and Sun, 2006
Actin monomer size, Δz72/13 nmLan and Sun, 2006
Actin rotation angles , ϕn—12πn/13Lan and Sun, 2006
Capture radius, a0.4 nmFit to experiment*, Craig and Linke, 2009
Binding penalty, b0.045Fit to experiment
Acceptance region, δϕac55.6°Fit to experiment
Chemical Rates
Hydrolysis rate, th-1750 s—1De La Cruz et al., 1999
TH detachment rate, td1-112 s—1De La Cruz et al., 1999
LH detachement rate, td2-11.5 s—1Purcell et al., 2005
Gating ratio, g=td2/td18
  1. *Fits restricted to physically plausible parameter ranges as determined from the indicated literature.

    The TH detachment rate assumes saturating ATP conditions. This is used throughout the paper except for the low ATP run velocity calculation (see Run velocity).

Additional files

Source data 1

This ZIP file contains both the numerical data and Python scripts used to produce Figure 3 through Figure 9, with individual directories corresponding to the materials for each figure.

https://cdn.elifesciences.org/articles/51569/elife-51569-data1-v2.zip
Transparent reporting form
https://cdn.elifesciences.org/articles/51569/elife-51569-transrepform-v2.docx

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