Motor-driven dynamics of cytoskeletal filaments in motility assays

We model analytically the dynamics of a cytoskeletal filament in a motility assay. The filament is described as rigid rod free to slide in two dimensions. The motor proteins consist of polymeric tails tethered to the plane and modeled as linear springs and motor heads that bind to the filament. As in related models of rigid and soft two-state motors, the binding and unbinding dynamics of the motor heads and the dependence of the transition rates on the load exerted by the motor tails play a crucial role in controlling the filament’s dynamics. Our work shows that the filament effectively behaves as a self-propelled rod at long times, but with non-Markovian noise sources arising from the coupling to the motor binding and unbinding dynamics. The effective propulsion force of the filament and the active renormalization of the various friction and diffusion constants are calculated in terms of microscopic motor and filament parameters. These quantities could be probed by optical force microscopy.

Figure 1

The figure shows the four steps of a motor cycle. In (a) a filament is sliding with velocity $v$ over a uniform density of unbound motors with tails tethered to the substrate. In (b) a motor attaches to the filament at a position $s_0$ from the filament’s midpoint. The stretch of the motor tails at the time of attachment is neglected. In (c) the motor has walked toward the polar head of the filament, stretching the tails by an amount $\Delta$. Finally, in (d) the bound motor detaches and relaxes instantaneously to its unstretched state. The filament has undergone a net displacement in the direction opposite to that of motor motion.

Figure 2

The velocity $-v_m$ of a loaded motor head as a function of the load $f_{\Vert }=\stackrel{\wedge }{\mathbf{u}}·\mathit{\Delta }$. The figure shows the stall force $f_s$ where $v_m=0$ and the detachment force $-f_d$.

Figure 3

Force-velocity curves for $\zeta =0.002\hspace{0.5em}\mathrm{pN}\hspace{0.5em}{\mathrm{nm}}^{-1}\hspace{0.5em}\mathrm{s}$ and various values of the motor stiffness $k$, showing the transition to nonmonotonicity as $k$ increases. The values of the stiffness $k$ (in pN/nm) and the corresponding values for ${\alpha }^{-1}$ (in nm) and $\epsilon$ are as follows: $k=0$, ${\alpha }^{-1}=0$, $\epsilon =0$ (black dotted line); $k=1$, ${\alpha }^{-1}=0.75$, $\epsilon =0.5$ (red dashed line); $k=2$, ${\alpha }^{-1}=1.5$, $\epsilon =1$ (blue dashed-dotted line); $k=8$, ${\alpha }^{-1}=6$, $\epsilon =4$ (black solid line). At high velocities the curves merge into the linear curve $F_{\mathrm{ext}}=\zeta v$ (black dotted line), corresponding to the case where no motors are present. The remaining parameters have the following values: $N={\rho }_m\mathit{Lb}=100$, $v_0=1000\hspace{0.5em}\mathrm{nm}/\mathrm{s}$, $f_s=4\hspace{0.5em}\mathrm{pN}$, ${\omega }_0=0.5\hspace{0.5em}(\mathrm{m}\mathrm{\hspace{0.5em}}\mathrm{s}){}^{-1}$, $r=0.06$.

Figure 4

“Phase diagram” in $k$-$\zeta$ plane showing the region where the $F_{\mathrm{ext}}$-$v$ curves exhibit nonmonotonic behavior (blue shaded region) for $N={\rho }_m\mathit{Lb}=100$ and $v_0=1\hspace{0.5em}\mu \mathrm{m}\mathrm{\hspace{0.5em}}{\mathrm{s}}^{-1}$, $f_s=4\hspace{0.5em}\mathrm{pN}$, $\alpha /k=1.33\hspace{0.5em}\mathrm{pN}$, ${\omega }_0=0.5\hspace{0.5em}(\mathrm{m}\mathrm{\hspace{0.5em}}\mathrm{s}){}^{-1}$, $r=0.06$.

Figure 5

The figure sketches the hysteretic behavior that may be obtained in an experiment where an external force $F_{\mathrm{ext}}$ is applied to a filament in a motility assay. The response of the filament will generally display two regions of hysteresis, at positive and negative forces.

Figure 6

The motor-induced sliding velocity $v_s$ of an actin filament in the absence of external force is shown as a function of $\epsilon ={\ell }_0/{\delta }_s$ for $N=10$ (dotted line), $N=25$ (dashed line), $N=100$ (dashed-dotted line), and $N=500$ (solid line). We observe that $v_s\to v_0$ for stiff motors as $N$ is increased. Parameter values: $\zeta =0.002\hspace{0.5em}\mathrm{pN}(\mathrm{nm}){}^{-1}\hspace{0.5em}\mathrm{s}$, $r=0.06$, $\alpha /k=1.33\hspace{0.5em}\mathrm{pN}$.

Figure 7

Mean-field attachment time ${\tau }_{\mathrm{MF}}$ as a function of $v$ for parameter values appropriate for acto-myosin systems: $v_0=1000\hspace{0.5em}\mathrm{nm}\mathrm{\hspace{0.5em}}{\mathrm{s}}^{-1}$, $k=10\hspace{0.5em}\mathrm{pN}\mathrm{\hspace{0.5em}}{\mathrm{nm}}^{-1}$, $f_s=4\hspace{0.5em}\mathrm{pN}$, ${\alpha }^{-1}=7.5\hspace{0.5em}\mathrm{nm}$, ${\omega }_0=0.5\hspace{0.5em}(\mathrm{m}\mathrm{\hspace{0.5em}}\mathrm{s}{)}^{-1}$, $r=0.06$, corresponding to $\epsilon =5$. The dashed line is the numerical solution of Eq. (A1) obtained using the exponential dependence of the unbinding rate on the stretch. The solid line is obtained using the parabolic ansatz given in Eq. (A3).

Figure 8

Stretch $\Delta$ as a function of velocity $v$ obtained using the mean-field value of the attachment time displayed in Fig. 7. The parameter values are the same as in Fig. 7. The dashed line is obtained using the exponential dependence of the unbinding rate on the stretch. The solid line is obtained using the parabolic ansatz given in Eq. (A3).