  Microtubules are filamentous tubular protein polymers which are essential for
a range of cellular behaviour, and are generally straight over micron length
scales. However, in some gliding assays, where microtubules move over a carpet
of molecular motors, individual microtubules can also form tight arcs or rings,
even in the absence of crosslinking proteins. Understanding this phenomenon may
provide important explanations for similar highly curved microtubules which can
be found in nerve cells undergoing neurodegeneration. We propose a model for
gliding assays where the kinesins moving the microtubules over the surface
induce ring formation through differential binding, substantiated by recent
findings that a mutant version of the motor protein kinesin applied in solution
is able to lock-in microtubule curvature. For certain parameter regimes, our
model predicts that both straight and curved microtubules can exist
simultaneously as stable steady-states, as has been seen experimentally.
Additionally, unsteady solutions are found, where a wave of differential
binding propagates down the microtubule as it glides across the surface, which
can lead to chaotic motion. Whilst this model explains two-dimensional
microtubule behaviour in an experimental gliding assay, it has the potential to
be adapted to explain pathological curling in nerve cells.
