Morphological and dynamical properties of semiflexible filaments driven by molecular motors

We consider an explicit model of a semiflexible filament moving in two dimensions on a gliding assay of motor proteins, which attach to and detach from filament segments stochastically, with a detachment rate that depends on the local load experienced. Attached motor proteins move along the filament to one of its ends with a velocity that varies nonlinearly with the motor protein extension. The resultant force on the filament drives it out of equilibrium. The distance from equilibrium is reflected in the end-to-end distribution, modified bending stiffness, and a transition to spiral morphology of the polymer. The local stress dependence of activity results in correlated fluctuations in the speed and direction of the center of mass leading to a series of ballistic-diffusive crossovers in its dynamics.

Figure 1

(a) Schematic of the system showing the molecular motors arranged on a square grid. The semiflexible polymer glides on the bed of molecular motors. (b), (c) Simulation snapshots of the polymer in an open and spiral state for a polymer with persistence ratio $u=3.33$, under the influence of MP activity $\text{Pe}=100$, and bare persistence ratio $\mathrm{\Omega }=5/6$.

Figure 2

Activity dependence of end-to-end distribution functions and their difference from equilibrium for a filament with $N=64$ having persistence ratio $u=3.33$. The MP activity is controlled by turnover with a bare processivity $\mathrm{\Omega }=5/6$, and nonzero active velocity $v_0$ set by $\text{Pe}=1$. (a) The logarithmic ratio of probabilities of filament under active drive with respect to that of the equilibrium polymer, $\mathrm{\Delta }\mathrm{\Sigma }$, provides a measure of the difference in distributions. The legends denote parameter values (detachment rate, MP velocity), where, in this figure, all data sets correspond to a constant detachment rate ${\omega }_0$ and MP velocity varies between constant values zero, $v_0$, and stretching dependent active velocity $v_t^a$ as denoted by Eq. (3). (b) The end-to-end distribution of stretchable semiflexible polymer $p\left(\tilde{r}\right)$, with local strain dependent detachment ${\omega }_{\mathrm{off}}$ as in Eq. (2). (c) The end-to-end distribution of the stretchable semiflexible polymer at equilibrium $p_{\mathrm{eq}}\left(\tilde{r}\right)$.

Figure 3

End-to-end distribution for three different values of Pe using stretching dependent turnover ${\omega }_{\mathrm{off}}$. All other parameters are as in Fig. 2.

Figure 4

End-to-end distribution functions. We use constant detachment rate ${\omega }_0$ with $\mathrm{\Omega }=5/6$ for all the figures. The variation of MP active velocities are as in Fig. 2, with the nonzero active velocities set by $\text{Pe}=1$. The three graphs show results for (a) $N=64,u=3.33,\lambda =18.92\sigma$, (b) $N=128,u=3.33,\lambda =38.14\sigma$, and (c) $N=128,u=6.66,\lambda =18.92\sigma$.

Figure 5

Tangent-tangent correlation function for a chain of $N=64,u=3.33$, and activity $v_t^a$ set by $\text{Pe}=1,10$, and 100 and load dependent detachment rate ${\omega }_{\mathrm{off}}$ with $\mathrm{\Omega }=5/6$. The data set passive denotes equilibrium result. The solid line shows a single exponential fit to $\text{Pe}=10$ data used to extract the effective persistence length ${\lambda }_{\mathrm{eff}}=(15.99\pm 0.24)\sigma$.

Figure 6

Analysis of the turning number $\psi \left(s\right)$ using $N=64,u=3.33$ and load dependent detachment ${\omega }_{\mathrm{off}}$ with $\mathrm{\Omega }=5/6$ with activity $v_t^a$ set by Pe. (a) Plot of $\psi \left(s\right)$ for three different configurations with $\text{Pe}=100$. It shows that $\psi \left(s\right)$ is an effective order parameter, distinguishing between the open state (green), clockwise spiral (blue), and counterclockwise spiral (red). (b) Probability distributions for $\psi \left(L\right)$ at $\text{Pe}=1,10$, and 100.

Figure 7

Dynamics of center of mass for a chain of $N=64,u=3.33$, with load dependent MP activity $v_t^a$ controlled by Pe and detachment rate ${\omega }_{\mathrm{off}}$ determined by $\mathrm{\Omega }=20/21$. (a) Mean squared displacement of the center of mass at different Péclet $\left(\text{Pe}=1,10,100\right)$. Numerical analysis of the dynamics at $\text{Pe}=100$ is presented in (b) and (c). (b) The gray line shows a center of mass trajectory. Structure of polymer corresponding to the blue, red, and green points indicated on the trajectory are shown in the respective colors. (c) The end-to-end length $r_{\mathrm{ee}}$ (red line), end-to-end orientation $\varphi$ (green line), and root mean squared fluctuations of the center of mass position (blue line) for a single trajectory are shown as a function of time at $\text{Pe}=100$.

Figure 8

End-to-end distribution at $\text{Pe}=1,10,100$ for different values of bare processivity $\mathrm{\Omega }$, using stress dependent active velocity and detachment rate with $N=64,u=3.33$. The graphs correspond to (a) $\mathrm{\Omega }=2/3$, (b) $\mathrm{\Omega }=5/6$, and (c) $\mathrm{\Omega }=20/21$.

Figure 9

Speed autocorrelation of the center of mass of the polymer. Single exponential decays are observed for both $\text{Pe}=1,10$, with correlation times $t_s\approx 0.1\tau ,1.0\tau$, respectively. (Inset) In the log-log plot, for $\text{Pe}=100$, multiple exponential decays with $t_s\approx 1\tau ,250\tau ,2000\tau$ are shown. The three exponential fits are indicated by black points.

Figure 10

Orientational autocorrelation of the center of mass velocity vector. This shows multiple exponential decays for all Pe. (Inset) For $\text{Pe}=1$, the log-log plot shows multiple exponential decays with time scales $t_{\theta }\approx 1\tau ,1500\tau$. The two exponential fits are indicated by black points.

Figure 11

Cross correlation of the orientation and speed of the center of mass velocity for $\text{Pe}=1$ (red), 10 (green), and 100 (blue, and values correspond to right ordinate).