A Brownian ratchet for protein translocation including dissociation of ratcheting sites

We study a model for the translocation of proteins across membranes through a nanopore using a ratcheting mechanism. When the protein enters the nanopore it diffuses in and out of the pore according to a Brownian motion. Moreover, it is bound by ratcheting molecules which hinder the diffusion of the protein out of the nanopore, i.e. the Brownian motion is reflected such that no ratcheting molecule exits the pore. New ratcheting molecules bind at rate gamma. Extending our previous approach (Depperschmidt and Pfaffelhuber in Stoch Processes Appl 120:901-925, 2010) we allow the ratcheting molecules to dissociate (at rate delta) from the protein (Model I). We also provide an approximate model (Model II) which assumes a Poisson equilibrium of ratcheting molecules on one side of the current reflection boundary. Using analytical methods and simulations we show that the speeds of both models are approximately the same. Our analytical results on Model II give the speed of translocation by means of a solution of an ordinary differential equation. This speed gives an approximation for the time it takes to translocate a protein of given length.