``Burnt Bridge'' Mechanism of Molecular Motor Motion

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.