  We use a two-state ratchet model to study the cooperative bidirectional
motion of molecular motors on cytoskeletal tracks with randomly alternating
polarities. Our model is based on a previously proposed model [Badoual et al.,
{\em Proc. Natl. Acad. Sci. USA} {\bf 99}, 6696 (2002)] for collective motor
dynamics and, in addition, takes into account the cooperativity effect arising
from the elastic tension that develops in the cytoskeletal track due to the
joint action of the walking motors. We show, both computationally and
analytically, that this additional cooperativity effect leads to a dramatic
reduction in the characteristic reversal time of the bidirectional motion,
especially in systems with a large number of motors. We also find that
bidirectional motion takes place only on (almost) a-polar tracks, while on even
slightly polar tracks the motion is unidirectional. We argue that the origin of
these observations is the sensitive dependence of the cooperative dynamics on
the difference between the number of motors typically working in and against
the instantaneous direction of motion.
