A "Burnt Bridge'' Brownian Ratchet

TL;DR

This paper analyzes a one-dimensional biased random walk model inspired by molecular motors, where the destruction of weak links influences directed motion, with exact solutions for velocity and diffusion in specific cases.

Contribution

It introduces a novel model of a Brownian ratchet with state-dependent bridge destruction and provides analytical solutions for velocity and diffusion constants.

Findings

Velocity and diffusion constants are derived analytically for equidistant bridges.

Exact solutions are obtained for the case where p=1 with both periodic and random bridge distributions.

The model captures key features of biased diffusion influenced by substrate state changes.

Abstract

Motivated by a biased diffusion of molecular motors with the bias dependent on the state of the substrate, we investigate a random walk on a one-dimensional lattice that contains weak links (called "bridges'') which are affected by the walker. Namely, a bridge is destroyed with probability p when the walker crosses it; the walker is not allowed to cross it again and this leads to a directed motion. The velocity of the walker is determined analytically for equidistant bridges. The special case of p=1 is more tractable -- both the velocity and the diffusion constant are calculated for uncorrelated locations of bridges, including periodic and random distributions.