Random walks of molecular motors arising from diffusional encounters with immobilized filaments

Movements of molecular motors on cytoskeletal filaments are described by directed walks on a line. Detachment from this line is allowed to occur with a small probability. Motion in the surrounding fluid is described by symmetric random walks. Effects of detachment and reattachment are calculated by an analytical solution of the master equation in two and three dimensions. Results are obtained for the fraction of bound motors, their average velocity, displacement, and dispersion. The analytical results are in good agreement with results from Monte Carlo simulations and confirm the behavior predicted by scaling arguments. The diffusion coefficient parallel to the filament becomes anomalously large since detachment and subsequent reattachment, in the presence of directed motion of the bound motors, leads to a broadening of the density distribution. The occurrence of protofilaments on a microtubule is modeled by internal states of the binding sites. After a transient time, all protofilaments become equally populated.

Figure 1

Random walk of a molecular motor: The motor performs directed movement along a filament (gray rod) and unbinds from it after a certain walking distance. The unbound motor diffuses in the surrounding fluid until it rebinds to the filament and resumes directed motion.

Figure 2

The random walks of molecular motors are modeled as random walks on a lattice. A line of lattice sites, which is indicated here as a black line, represents a filament. Motors at filament sites perform directed random walks, while motors at nonfilament sites undergo symmetric random walks. For the movement in two dimensions, the jump probabilities at nonfilament sites are $1∕4$ for each of the four neighbor sites. At filament sites, a motor steps forward with probability $1-\gamma -\delta ∕2-ϵ∕2$ and backward with probability $\delta ∕2$; jumps to each of the adjacent nonfilament sites that lead to unbinding occur with probability $ϵ∕4$. The dwell probability is $\gamma$.

Figure 3

Fraction $N_0$ of motors bound to the filament as a function of time $t$ for the two-dimensional case. The three curves correspond to $ϵ=0.03$ (circles), $ϵ=0.05$ (squares), and $ϵ=0.08$ (diamonds). Lines are obtained from the exact integral (15), the data points from MC simulation. In the simulations, the other parameters have been chosen as $\gamma =0$, $\delta =0.6$, but the results shown here depend only on $ϵ$.

Figure 4

Displacement of motors as a function of time $t$ in the two-dimensional case, as obtained from the exact integrals (lines) and MC simulations (data points). Open circles show the average position ${\overline{n}}_{\mathrm{b}}$ of the bound motors, diamonds the average position ${\overline{n}}_{\mathrm{ub}}$ of the unbound motors, and full circles indicate the displacement $\overline{n}$ averaged over all motors, which interpolates between the curve for the bound motors at small times and the curve for the unbound motors at large times. The parameters are $ϵ=0.05$, $\gamma =0$, and $\delta =0.6$.

Figure 5

Density profile $P_{\mathrm{b}}(n,t)=P_{n,0}\left(t\right)$ of motors bound to the filament as a function of the spatial coordinate $n$ parallel to the filament for different times $t$. The profiles are normalized with respect to the probability $N_0$ to be bound at the filament. The lines indicate the analytical result from Eq. (45); data points are from MC simulations. The parameters are $ϵ=0.05$ and $\gamma =\delta =0$.

Figure 6

Diffusion of motors parallel to the filament in $d=2$ and $d=3$. Dispersion of bound motors parallel to the filament, $\mathrm{\Delta }{n}_{\mathrm{b}}^2$, as a function of time $t$. The dotted lines indicate the linear behavior described by the large-time diffusion coefficient $D_{\mathrm{b}}(t=\infty )$ as given by Eqs. (41, 99). In two dimensions, this diffusion coefficient is anomalously high; in three dimensions, it is given by the diffusion away from the filament but exhibits large logarithmic corrections. The parameters are the same as in Fig. 4.

Figure 7

Diffusion of motors perpendicular to the filament in $d=2$. Transverse dispersion $\mathrm{\Delta }{m}^2$ as a function of time $t$. The parameters are the same as in Fig. 4.

Figure 8

Random walk probabilities for movements in three dimensions: At a nonfilament site, a motor jumps with probability $1∕6$ to each of the six neighbor sites; at a filament site, a forward step has probability $1-\gamma -\delta ∕2-2ϵ∕3$, a backward step $\delta ∕2$, and a jump to each of the six adjacent nonfilament sites has probability $ϵ∕6$. As in two dimensions, a motor does not step at all with probability $\gamma$. As in Fig. 2, the line of black lattice sites represents the filament. The shaded areas with white lattice sites indicate the planes spanned by the $n$ and $m_1$ axes of the lattice perpendicular to the paper plane.

Figure 9

Fraction $N_0$ of bound motors as a function of time for the three-dimensional case. The two curves are for $ϵ=0.03$ (circles) and $ϵ=0.05$ (diamonds). The other parameters are $\gamma =0$ and $\delta =0.6$.

Figure 10

Average position ${\overline{n}}_{\mathrm{b}}$ of the motors bound to the filament as a function of time $t$ in three dimensions. The parameters are the same as in Fig. 9.

Figure 11

Density profile $P_{\mathrm{b}}\left(n\right)=P_{n,0}$ of motors bound to the filament for the three-dimensional case as a function of the spatial coordinate $n$ parallel to the filament at times $t=2000$ (solid line, circles) and $t={10}^4$ (dashed line, diamonds). Lines are obtained by numerical evaluation of the approximate analytical expression (100); data points are from MC simulations. The parameters are the same as in Fig. 4.