Hook flexibility and body trajectories of swimming microorganisms

The flexibility of the hook connecting the bacterial flagellum to the cell body, and an associated buckling instability, is believed to play an important role in microorganism locomotion. We consider a simplified model for the flagellum-cell dynamics and solve analytically for the flagellum orientation and cell trajectories through space. To better understand how hook flexibility affects the swimming pathway, we consider a sequence of problems, from fixed flagellar orientation, to specified orientation, to free, flexible motion dictated by force and torque balance. Exact helical trajectories yield to nearly-helical and then more complex paths. Other geometrical features are also explored, including baseline flagellum orientation. Numerical simulations reveal the regions of accuracy of our analytical predictions.