Stepping molecular motor amid Lévy white noise

We consider a model of a stepping molecular motor consisting of two connected heads. Directional motion of the stepper takes place along a one-dimensional track. Each head is subject to a periodic potential without spatial reflection symmetry. When the potential for one head is switched on, it is switched off for the other head. Additionally, the system is subject to the influence of symmetric, white Lévy noise that mimics the action of external random forcing. The stepper exhibits motion with a preferred direction which is examined by analyzing the median of the displacement of a midpoint between the positions of the two heads. We study the modified dynamics of the stepper by numerical simulations. We find flux reversals as noise parameters are changed. Speed and direction appear to very sensitively depend on characteristics of the noise.

Figure 1

Model system: The heads are subject to an external potential (a) from the filamentous track (e.g., microtubule) and an internal potential (b) due to sterical interactions within the motor. The internal potential is the sum of a harmonic and a Lennard-Jones potential [cf. Eqs. (2) and (3)]. In the absence of a constraint on the distance $|y-x|$ and in the presence of “bursting fluctuations,” the two heads can be pulled apart to an unphysically large distance, i.e., a catastrophe as shown in (c) can occur. Parameter values are $a=0.25,\epsilon =0.1,\tau =5,k=0.8,L=1,dt=0.01$.

Figure 2

A sketch illustrating the model for the system's response to bursting fluctuations. If there were no constraint, the leading head would have been in the dotted position. For explanation, see the main text.

Figure 3

Simulated trajectories [cf. Eq. (2)] showing the displacement of the median $q_{0.5}$ of the midpoint between $x$ and $y$ for 1000 motors. The distance between the heads $x$ and $y$ has been limited to $|y-x|=d_{\max }=2a$. The noise parameter is $\sigma =0.1$ in panel (a) and $\sigma =0.5$ in panel (b). The other parameter values are $a=0.25,\epsilon =0.1,\tau =5.0$, and $k=0.8$. The time step is $dt=0.01$.

Figure 4

Velocity of the median $q_{0.5}$ of 1000 independent walkers after 100 time units as a function of noise intensity $\sigma$. The parameter values are $a=0.25,d_{\max }=0.5,\epsilon =0.1,\tau =5.0,k=0.8$, and $dt=0.01$.

Figure 5

Velocity of the median $q_{0.5}$ of 1000 independent walkers after 100 time units as a function of $d_{\max }$ for different values of the stability index $\alpha =1.5$, 1.8, 2.0 and for different lengths of the fully relaxed linker: $a=0.25$ (a) and $a=0.1$ (b). The variable $d_{\max }$ represents the maximum allowable distance $|y-x|$ between the heads. The parameter values are $\epsilon =0.1,\tau =5,k=0.8,dt=0.01$, and $\sigma =0.5$. The duration of the individual motor trajectories has been set to $t=100$. Lines are drawn to guide the eye.

Figure 6

Trajectories of the median $q_{0.5}$ of 1000 independent walkers. Graphs have been presented for different values of the linker extension $d_{\max }$ close to and beyond the minimum depicted in Fig. 5. Top (bottom) panels refer to linkers characterized by natural (relaxed) lengths $a=0.25(a=0.1)$. Stability index is $\alpha =2.0$ and $\alpha =1.5$ for left and right panels, respectively.

Figure 7

The velocity of the median $q_{0.5}$ of 1000 independent walkers as a function of the period $\tau$, for different values of the stability index $\alpha$, namely $\alpha =1.5,1.8,2$. The parameter values are $a=0.25,\epsilon =0.1,d_{\max }=0.5,k=0.8,\sigma =0.5$, and $dt=0.01$. Lines, obtained by curve fitting, are drawn to guide the eye.