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 spontaneous symmetry breaking under prestress due to short range spike protein interactions magnified by surface topography. Once surface stresses are sufficiently large, the filament buckles and the curved state constitutes …