Random Walk by Majority Rule and L\'evy walk

We have studied a random walk model based on majority rule. At a given instant, the moving direction of a cargo is determined by motor coordination mediated by a tug-of-war mechanism between two kinds of competing motor proteins. We have demonstrated that the probability distribution $P(t)$ for unidirectional run time $t$ of a cargo can be remarkably described by Levy walk for $t<{\gamma }_u^{-1}$ as $P(t)\propto t^{-3\mathrm{/}2}{e}^{-{\gamma }_{u}t}$ with ${\gamma }_u$ being the unbinding rate of a motor protein from microtubule. The mean squared displacement of a cargo changes from super-diffusive behavior $\langle X^2\rangle\propto t^2$for$t<\gamma_u^{-1}$tonormaldiffusion$\langle X^2\rangle\propto t$for$t>\gamma_u^{-1}$. By considering the correlation effect in binding of a motor protein to microtubule, we have shown that Levy walk behavior of $P(t)\propto t^{-3\mathrm{/}2}$ persists robustly against correlations only adding an effective cutoff time ${\gamma }_b\mathrm{/}{\gamma }_c^2$ with ${\gamma }_c$ representing the amount of correlations.