  Filamentous viruses like influenza and torovirus often display systematic
bends and arcs of mysterious physical origin. We propose that such viruses
undergo an instability from a cylindrically symmetric to a toroidally curved
state. This "toro-elastic" state emerges via a spontaneous symmetry breaking
under prestress, induced via short range spike protein interactions and
magnified by the filament's surface topography. Once surface stresses become
sufficiently large, the filament buckles and the toroidal, curved state
constitutes a soft mode that can propagate through the filament's material
frame around a "mexican-hat" potential. In the mucus of our airways, glycan
chains are omnipresent that influenza's spike proteins can bind to and cut. We
show that when coupled to such a non-equilibrium chemical reaction, the curved
toro-elastic state can attain a spontaneous rotation for sufficiently strong
enzymatic activity, leading to a whole body reshaping propulsion similar to --
but different from -- eukaryotic flagella and spirochetes.
