Extended Data Fig. 6: Schematics showing the 3D configuration of a swimming bacterium and its helical trajectory.

a, The top view of the configuration. The left-handed helical trajectory encloses a cylindrical space of radius R_w (the grey region). The angular velocity of the body and the flagellar bundle ω_b and ω_t are shown. The velocity of the bacterium tangential to the helical trajectory V_b is indicated. Note that ω_b and V_b tilt above the paper (the solid arrows), whereas ω_t tilts into the paper (the dashed arrow). The average swimming speed of bacteria measured in experiments V is normal to and points out of the paper. The ω plane is normal to the paper as indicated by the purple dashed line. The cross-section of the plane with the helical cylinder has a rectangular shape with the width w < R_w. The viewpoint of b is indicated at the lower right corner. b, The side view of the ω plane. The axis of the helical trajectory is indicated by the vertical dashed line, which is in front of the ω plane above the paper. The projection of V_b along the direction of –ω_cm gives V, whereas the projection of V_b along the direction of the bacterial flagellar bundle (–ω_t) gives V_b^z. The angles α, β and θ are the same as those defined in Extended Data Fig. 4b. The coordinate defined in Extended Data Fig. 4b is reproduced on the lower right. As colloids are depleted from the cylindrical space due to the long-range hydrodynamics (Extended Data Fig. 7), colloids that exert torque on the bacterium via the short-range lubrication interaction are outside the cylindrical space. The effect breaks the symmetric role of colloids around the bacterium, which preferably reduces α and therefore suppresses bacterial wobbling.