Hydrodynamics of helical bacteria with polar flagella

Amphitrichous bacteria possessing a helical cell body propel themselves by coordinating the rotation of two helical flagella anchored at opposite poles of a spirillum-shaped body. The motility modes of these cells depend on the relative rotational directions of the two polar flagella and the magnitude of the motor torque. Here, we develop a mathematical model of an amphitrichous bacterium to elucidate its swimming mechanisms in a viscous fluid. The hydrodynamics of the system is governed by the incompressible Stokes equations, subject to instantaneous force- and torque-free constraints, with flagellar motor rotation providing the sole actuation. For a helical bacterium with a single polar flagellum, we identify four dynamically stable swimming modes—pull, push, wrapping, and overwhirling—consistent with those observed for rod-shaped cells. We further show that swimming efficiency is greater when the cell body and flagellum have opposite helical handedness than when they share the same handedness. Extending the model to two oppositely anchored flagella, we systematically characterize the swimming states arising from different combinations of motor rotation and torque magnitude and delineate the coordination regimes that yield stable propulsion. Finally, we incorporate external magnetic fields through an applied magnetic torque and demonstrate that magnetic and hydrodynamic torques compete to control cell orientation, with stronger fields increasingly aligning swimming trajectories with the applied field.