Directed motion emerging from two coupled random processes:: translocation of a chain through a membrane nanopore driven by binding proteins

We investigate the translocation of a stiff polymer consisting of M monomers through a nanopore in a membrane, in the presence of binding particles (chaperones) that bind onto the polymer, and partially prevent backsliding of the polymer through the pore. The process is characterized by the rates: k for the polymer to make a diffusive jump through the pore, q for unbinding of a chaperone, and the rate q kappa for binding (with a binding strength K); except for the case of no binding kappa = 0 the presence of the chaperones gives rise to an effective force that drives the translocation process. In more detail, we develop a dynamical description of the process in terms of a (2 + 1)-variable master equation for the probability of having m monomers on the target side of the membrane with n bound chaperones at time t. Emphasis is put on the calculation of the mean first passage time T as a function of total chain length M. The transfer coefficients in the master equation are determined through detailed balance, and depend on the relative chaperone size k and binding strength K, as well as the two rate constants k and q. The ratio gamma = q/k between the two rates determines, together with K and;, three limiting cases, for which analytic results are derived: (i) for the case of slow binding (gamma kappa -> 0), the motion is purely diffusive, and T similar or equal to M-2 for large M; (ii) for fast binding (gamma kappa -> infinity) but slow unbinding (gamma -> 0), the motion is, for small chaperones = 1, ratchet-like, and T similar or equal to M; (iii) for the case of fast binding and unbinding dynamics (y -> infinity and gamma kappa -> infinity), we perform the adiabatic elimination of the fast variable n, and find that for a very long polymer T similar or equal to M, but with a smaller prefactor than for ratchet-like dynamics. We solve the general case numerically as a function of the dimensionless parameters lambda, kappa and gamma, and compare to the three limiting cases.