First passage time distribution for the 1D diffusion of particles with internal degrees of freedom

Proteins searching for their targets on DNA do so by non-specifically adsorbing and sliding along DNA. It became known recently that the proteins simultaneously undergo some internal conformational changes (partial folding and unfolding), such that in some of the sub-states their ability to recognize the target site is enhanced at the expense of diffusivity which is suppressed. We analyze the probability distribution of the first passage times of these proteins into the target taking into account the internal sub-states of proteins. We show that there are two distinct scaling regimes: one is diffusion controlled and the other is absorption controlled.