Asymptotics of a Brownian ratchet for protein translocation

Protein translocation in cells has been modelled by 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 (Xt)t≥0(Xt)t≥0 with a changing reflection point (Rt)t≥0(Rt)t≥0. The rate of change of RtRt is γ(Xt−Rt)γ(Xt−Rt) and the new reflection boundary is distributed uniformly between Rt−Rt− and XtXt. The asymptotic speed of the ratchet scales with γ1/3γ1/3 and the asymptotic variance is independent of γγ.