  Many bacteria use rotating helical flagellar filaments to swim. The filaments
undergo polymorphic transformations in which the helical pitch and radius
change abruptly. These transformations arise in response to mechanical loading,
changes in solution temperature and ionic strength, and point substitutions in
the amino acid sequence of the protein subunits that make up the filament. To
explain polymorphism, we propose a new coarse-grained continuum rod theory
based on the quaternary structure of the filament. The model has two molecular
switches. The first is a double-well potential for the extension of a
protofilament, which is one of the eleven almost longitudinal columns of
subunits. Curved filament shapes occur in the model when there is a mismatch
strain, i.e. when inter-subunit bonds in the inner core of the filament prefer
a subunit spacing which is intermediate between the two spacings favored by the
double-well potential. The second switch is a double-well potential for twist,
due to lateral interactions between neighboring protofilaments. Cooperative
interactions between neighboring subunits within a protofilament are necessary
to ensure the uniqueness of helical ground states. We calculate a phase diagram
for filament shapes and the response of a filament to external moment and
force.
