Asymptotics of a Brownian ratchet for Protein Translocation

Abstract

Protein translocation in cells has been modelled by \emph{Brownian ratchets}. In such models, the protein diffuses through a nanopore. On one side of the pore, ratcheting molecules bind to the protein and hinder it to diffuse out of the pore. We study a Brownian ratchet by means of a reflected Brownian motion $(X_t)_{t\geq 0}$ with a changing reflection point $(R_t)_{t\geq 0}$. The rate of change of $R_t$ is $\gamma(X_t-R_t)$ and the new reflection boundary is distributed uniformly between $R_{t-}$ and $X_t$. The asymptotic speed of the ratchet scales with $\gamma^{1/3}$ and the asymptotic variance is independent of $\gamma$.