General relation between drift velocity and dispersion of a molecular motor

We model a processive linear molecular motor as a particle diffusing in a one-dimensional periodic lattice with arbitrary transition rates between its sites. We present a relatively simple proof of a theorem which states that the ratio of the drift velocity V to the diffusion coefficient D has the upper bound 2N/d, where N is the number of nodes in an elementary cell and d denotes its length. This relation can be used to estimate the minimal value of internal states of the motor and the maximal value of the so called Einstein force, which approximately equals to the maximal force exerted by a molecular motor.