Instability of hooks during bacterial flagellar swimming

In bacteria, a flexible hook transmits torque from the rotary motor at the cell body to the flagellum. Previously, the hook has been modeled as a Kirchhoff rod between the cell body and rotating flagellum. To study effects of the hook's flexibility on the bacteria's swimming speed and trajectory for wide range hook stiffnesses and flagellum configurations, we develop an efficient simplified spring model for the hook by linearizing the Kirchhoff rod. We treat the hydrodynamics of the cell body and helical flagellum using resistance matrices calculated by the method of regularized Stokeslets. We investigate flagellar and swimming dynamics for a range of hook flexibilities and flagellar orientations relative to the cell body and compare the results to models without hook flexibility. We investigate in detail parameters corresponding to E. coli and Vibrio alginolyticus. Generally, the flagellum changes orientation relative to the cell body, undergoing an orbit with the period of the motor rotation. We find that as the hook stiffness decreases, steady-state orbits of the flagellum first become unstable before the hook buckles, which may suggest a new mechanism of flick initiation in run-reverse-flick motility. We also find that for some parameter ranges, there are multiple stable steady state orbits, which may have implications for the tumbling and turning of bacteria.