  Transport of molecular motors along protein filaments in a half-closed
geometry is a common feature of biologically relevant processes in cellular
protrusions. Using a lattice gas model we study how the interplay between
active and diffusive transport and mass conservation leads to localised domain
walls and tip localisation of the motors. We identify a mechanism for task
sharing between the active motors (maintaining a gradient) and the diffusive
motion (transport to the tip), which ensures that energy consumption is low and
motor exchange mostly happens at the tip. These features are attributed to
strong nearest-neighbour correlations that lead to a strong reduction of active
currents, which we calculate analytically using an exact moment-identity, and
might prove useful for the understanding of correlations and active transport
also in more elaborate systems.
