  Many types of bacteria swim by rotating a bundle of helical filaments also
called flagella. Each filament is driven by a rotary motor and a very flexible
hook transmits the motor torque to the filament. We model it by discretizing
Kirchhoff's elastic-rod theory and develop a coarse-grained approach for
driving the helical filament by a motor torque. A rotating flagellum generates
a thrust force, which pushes the cell body forward and which increases with the
motor torque. We fix the rotating flagellum in space and show that it buckles
under the thrust force at a critical motor torque. Buckling becomes visible as
a supercritical Hopf bifurcation in the thrust force. A second buckling
transition occurs at an even higher motor torque. We attach the flagellum to a
spherical cell body and also observe the first buckling transition during
locomotion. By changing the size of the cell body, we vary the necessary thrust
force and thereby obtain a characteristic relation between the critical thrust
force and motor torque. We present a sophisticated analytical model for the
buckling transition based on a helical rod which quantitatively reproduces the
critical force-torque relation. Real values for motor torque, cell body size,
and the geometry of the helical filament suggest that buckling should occur in
single bacterial flagella. We also find that the orientation of pulling
flagella along the driving torque is not stable and comment on the biological
relevance for marine bacteria.
