  Non-equilibrium processes which convert chemical energy into mechanical
motion enable the motility of organisms. Bundles of inextensible filaments
driven by energy transduction of molecular motors form essential components of
micron-scale motility engines like cilia and flagella. The mimicry of
cilia-like motion in recent experiments on synthetic active filaments supports
the idea that generic physical mechanisms may be sufficient to generate such
motion. Here we show, theoretically, that the competition between the
destabilising effect of hydrodynamic interactions induced by force-free and
torque-free chemomechanically active flows, and the stabilising effect of
nonlinear elasticity, provides a generic route to spontaneous oscillations in
active filaments. These oscillations, reminiscent of prokaryotic and eukaryotic
flagellar motion, are obtained without having to invoke structural complexity
or biochemical regulation. This minimality implies that biomimetic
oscillations, previously observed only in complex bundles of active filaments,
can be replicated in simple chains of generic chemomechanically active beads.
