The Periodic Three-state Hopping Model for Molecular Motors

Motivated by recent applications to experiments on molecular motors, the directed motion of molecular motor based on a periodic one-dimensional three-states hopping model is studied. The model combines the biochemical cycle o nucleotide hydrolysis with the motor′s translation. An explicit solution is obtained for the probability distribution as function of the time for any initial distribution with all the transients included, and the drift velocity v, the diffusion constant D and the randomness parameter can also be obtained at any time from the probability distribution. Meanwhile the characteristic time for the motor to reach steady state has been calculated. Lastly, several possible applications arproposed: the pure asymmetric case, the random symmetric case and the random asymmetric case. In the long-time limit, the drift velocity v and the diffusion constant D are obtained in terms of microscopic transition rates that are parameters in the three-state stochastic model for the pure asymmetric case. By comparison with experiments (drift velocity and randomness parameter rversus ATP), it is shown that the model presented here can rather satisfactorily explain the available data. The theoretical model provides a conceptual framework for realistic studies of molecular motor.