Asymptotics of a Brownian ratchet for Protein Translocation

TL;DR

This paper analyzes the long-term behavior of a Brownian ratchet model for protein translocation, revealing how its speed scales with the ratcheting rate and showing the variance remains unaffected by it.

Contribution

It introduces a novel mathematical model of the Brownian ratchet with a dynamic reflection boundary and derives its asymptotic properties.

Findings

Asymptotic speed scales with b3^{1/3}.

Asymptotic variance is independent of b3.

Provides a rigorous mathematical framework for protein translocation modeling.

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 $\gamma$.