  It is known from the wave-like motion of microtubules in motility assays that
the piconewton forces that motors produce can be sufficient to bend the
filaments. In cellular phenomena such as cytosplasmic streaming, molecular
motors translocate along cytoskeletal filaments, carrying cargo which entrains
fluid. When large numbers of such forced filaments interact through the
surrounding fluid, as in particular stages of oocyte development in
$Drosophila~melanogaster$, complex dynamics are observed, but the detailed
mechanics underlying them has remained unclear. Motivated by these
observations, we study here perhaps the simplest model for these phenomena: an
elastic filament, pinned at one end, acted on by a molecular motor treated as a
point force. Because the force acts tangential to the filament, no matter what
its shape, this "follower-force" problem is intrinsically non-variational, and
thereby differs fundamentally from Euler buckling, where the force has a fixed
direction, and which, in the low Reynolds number regime, ultimately leads to a
stationary, energy-minimizing shape. Through a combination of linear stability
theory, analytical study of a solvable simplified "two-link" model, and
numerical studies of the full elastohydrodynamic equations of motion we
elucidate the Hopf bifurcation that occurs with increasing forcing of a
filament, leading to flapping motion analogous to the high Reynolds number
oscillations of a garden hose with a free end.
