The Transient Characters of the Directed Motion of Protein Motors

A periodic one-dimensional four-state hopping model of protein motors is studied through using the master equation. In the model, the substeps between arbitrary adjacent states are proposed to be equal, and an explicit solution of the master equation is first obtained for the probability distribution as a function of the time and position for any initial distribution with all the transients included. In particularly, the transient behaviors in the initial period of time and the characteristic time to reach the steady state for the protein motors are discussed. Finally, the results are obtained that the time evolution of the protein motors´ probability is determined by the transition rates and the initial condition, while the characteristic time to reach the steady state is only determined by the transition rates.