  In the cellular phenomena of cytoplasmic streaming, molecular motors carrying
cargo along a network of microtubules entrain the surrounding fluid. The
piconewton forces produced by individual motors are sufficient to deform long
microtubules, as are the collective fluid flows generated by many moving
motors. Studies of streaming during oocyte development in the fruit fly
$D.~melanogaster$ have shown a transition from a spatially-disordered
cytoskeleton, supporting flows with only short-ranged correlations, to an
ordered state with a cell-spanning vortical flow. To test the hypothesis that
this transition is driven by fluid-structure interactions we study a
discrete-filament model and a coarse-grained continuum theory for motors moving
on a deformable cytoskeleton, both of which are shown to exhibit a
$swirling~instability$ to spontaneous large-scale rotational motion, as
observed.
