  The deformation of thin rods in a viscous liquid is central to the mechanics
of motility in cells ranging from \textit{Escherichia coli} to sperm. Here we
use experiments and theory to study the shape transition of a flexible rod
rotating in a viscous fluid driven either by constant torque or at constant
speed. The rod is tilted relative to the rotation axis. At low applied torque,
the rod bends gently and generates small propulsive force. At a critical
torque, the rotation speed increases abruptly and the rod forms a helical shape
with much greater propulsive force. We find good agreement between theory and
experiment.
