The transport of cargoes within a cell takes place on Microtubules (MT) with the help of motor proteins like dynein and kinesin which pull the cargo in a certain direction. Starting from a discrete Asymmetric Simple Exclusion Process (ASEP), which models the individual interactions between these particles, we introduce a continuum model for the density of a fixed (large) amount of motors and one cargo on a microtubule with periodic boundary. This leads to a free boundary problem for a nonlinear transport-diffusion equation.
We are interested in the influence of the motors on the direction and velocity of the cargo and vice versa. Since the density of the motors depends on the position of the cargo, we transform the latter one to the left boundary and analyse the resulting problem for steady state solutions and features of transport, e.g. traffic jams visible in boundary layer effects.
Moreover, we compare our results with the ones from the discrete model and numerically investigate changes in density profiles for the motors when varying the jumping rates.