Tug-of-war in motor proteins and the emergence of Lévy walk

In recent experiment, transport of mRNA–protein complex in neurons has been observed to follow a truncated Lévy walk behavior. We have theoretically 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 run-time distribution P(t) for unidirectional transport of a cargo can be described by a truncated Lévy walk P(t)∝t−3∕2e−γut with γu being the unbinding rate of a motor protein from microtubule. The mean squared displacement of a cargo changes from super-diffusive behavior 〈X2〉∝t2 for t<γu−1 to normal diffusion 〈X2〉∝t for t>γu−1. By considering the correlation effect in binding of a motor protein to microtubule, we have shown that Lévy walk behavior of P(t)∝t−3∕2 persists robustly against correlation simply adding a finite cutoff time γb∕γc2 with γc representing the amount of correlation.