A simplified "ratchet" model of molecular motors

A limiting case of one of the ratchet models of Ajdari, Frost, rt al. is analyzed. An explicit solution is obtained for the probability distribution as a function of the time for any initial distribution with all the transients included. In the longtime limit the drift velocity and diffusion coefficient are obtained in terms of the microscopic transition rates that are the parameters in the model. In spite of its extreme simplicity, with realistic values of its kinetic parameters the model yields values of the drift velocity and effective force that are of the right magnitude for a molecular motor. The model proves to be a simple special case of Derrida's periodic one-dimensional hopping model, for which he found a solution in the long-time limit.