A homogenization approach for the motion of motor proteins

TL;DR

This paper analyzes the asymptotic behavior of a coupled Fokker-Planck system modeling motor proteins, showing that in the small-period limit, proteins either move with a fixed velocity or stay immobile.

Contribution

It introduces a homogenization approach for a coupled Fokker-Planck system, revealing the limiting motion behavior of motor proteins in periodic environments.

Findings

Proteins either propagate with a fixed velocity or remain stationary as the period tends to zero.

The system's limiting behavior depends on the asymptotic scaling of diffusion and coupling.

The results provide insight into the movement patterns of motor proteins in cellular environments.

Abstract

We consider the asymptotic behavior of an evolving weakly coupled Fokker-Planck system of two equations set in a periodic environment. The magnitudes of the diffusion and the coupling are respectively proportional and inversely proportional to the size of the period. We prove that, as the period tends to zero, the solutions of the system either propagate (concentrate) with a fixed constant velocity (determined by the data) or do not move at all. The system arises in the modeling of motor proteins which can take two different states. Our result implies that, in the limit, the molecules either move along a filament with a fixed direction and constant speed or remain immobile.