We consider in this eighth chapter bacterial locomotion powered by the rotation of helical filaments. We focus on the canonical case of a cell body rotating a single filament. Bacterial locomotion features two differences from the swimming of spermatozoa. First, bacterial swimming does not involve time-varying shape changes but may be understood physically as due to the relative rotation of rigid bodies. Second, in order to balance hydrodynamic torques, a bacterium needs a cell body of finite size to swim. We derive the Stokes resistance matrix of a rigid helix as predicted by resistive-force theory. We use it to compute the velocity of a bacterium moving along a straight line and compare our results with experimental measurements on E. coli. We employ our theoretical estimates to address the energy expended by the motors powering the rotation of filaments. Extending the theory to the 3D motion of finite-size bacteria, we obtain helical trajectories and compare to measurements for B. subtilis. We also show that there exists an optimal size of the cell body and that we can define an intrinsic efficiency for the helical propeller, allowing us to rationalise the shapes of natural flagellar filaments.

Motivated by the propulsion of spermatozoa, the seventh chapter focuses on the planar waving of eukaryotic flagella. We first show how to apply resistive-force theory to compute the swimming speeds of simple flagellar waves and detail how wave geometry influences locomotion. Next we introduce a measure of swimming efficiency and show how to use it in order to derive the shape of the optimal waving motion. Finally we model the eukaryotic flagellum as an active filament where actuation from molecular motors is accounted for in a continuum manner and the waving motion is obtained as a mechanical balance between molecular forcing, fluid dynamics and passive elasticity. We close with a demonstration of how that model allows us to quantify cellular energy consumption.