The effects of flagellar hook compliance on motility of monotrichous bacteria: A modeling study

A crucial structure in the motility of flagellated bacteria is the hook, which connects the flagellum filament to the motor in the cell body. Early mathematical models of swimming bacteria assume that the helically shaped flagellum rotates rigidly about its axis, which coincides with the axis of the cell body. Motivated by evidence that the hook is much more flexible than the rest of the flagellum, we develop a new model that allows a naturally straight hook to bend. Hook dynamics are based on the Kirchhoff rod model, which is combined with a boundary element method for solving viscous interactions between the bacterium and the surrounding fluid. For swimming in unbounded fluid, we find good support for using a rigid model since the hook reaches an equilibrium configuration within several revolutions of the motor. However, for effective swimming, there are constraints on the hook stiffness relative to the scale set by the product of the motor torque with the hook length. When the hook is too flexible, its shape cannot be maintained and large deformations and stresses build up. When the hook is too rigid, the flagellum does not align with the cell body axis and the cell “wobbles” with little net forward motion. We also examine the attraction of swimmers to no-slip surfaces and find that the tendency to swim steadily close to a surface can be very sensitive to the combination of the hook rigidity and the precise shape of the cell and flagellum.