  Molecular motors are involved in key transport processes inside actin-based
cellular protrusions. The motors carry cargo proteins to the protrusion tip
which participate in regulating the actin polymerization, and play a key role
in facilitating the growth and formation of such protrusions. It is observed
that the motors accumulate at the tips of cellular protrusions, and in addition
form aggregates that are found to drift towards the protrusion base at the rate
of actin treadmilling. We present a one-dimensional driven lattice model, where
motors become inactive after delivering their cargo at the tip, or by loosing
their cargo to a cargo-less neighbor. The results suggest that the experimental
observations may be explained by the formation of traffic jams that form at the
tip. The model is solved using a novel application of mean-field and shock
analysis. We find a new class of shocks that undergo intermittent collapses,
and on average do not obey the Rankine-Hugoniot relation.
