  The cytoskeleton in eukaryotic cells plays several crucial roles. In terms of
intracellular transport, motor proteins use the cytoskeletal filaments as a
backbone along which they can actively transport biological cargos such as
vesicles carrying biochemical reactants. Crossings between such filaments
constitute a key element, as they may serve to alter the destination of such
payload. Although motor proteins are known to display a rich behaviour at such
crossings, the latter have so far only been modelled as simple branching
points. Here we explore a model for a crossing between two microtubules which
retains the individual tracks consisting of protofilaments, and we construct a
schematic representation of the transport paths. We study collective transport
exemplified by the Totally Asymmetric Simple Exclusion Process (TASEP), and
provide a full analysis of the transport features and the associated phase
diagram, by a generic mean-field approach which we confirm through
particle-based stochastic simulations. In particular we show that transport
through such a compound crossing cannot be approximated from a coarse-grained
structure with a simple branching point. Instead, it gives rise to entirely new
and counterintuitive features: the fundamental current-density relation for
traffic flow is no longer a single-valued function, and it furthermore differs
according to whether it is observed upstream or downstream from the crossing.
We argue that these novel features may be directly relevant for interpreting
experimental measurements.
