Helical trajectories of swimming cells with a flexible flagellar hook

TL;DR

This study models how the flexibility of the bacterial flagellar hook influences cell swimming paths, revealing that it can cause helical trajectories and bifurcations in movement patterns, with implications for understanding microorganism locomotion.

Contribution

The paper introduces a simplified model linking flagellar hook flexibility to helical swimming trajectories and identifies a bifurcation affecting cell motion.

Findings

Flagellar hook flexibility induces helical swimming paths.

A supercritical Hopf bifurcation affects trajectory stability.

Predicted helical amplitude matches P. aeruginosa observations.

Abstract

The flexibility of the bacterial flagellar hook is believed to have substantial consequences for microorganism locomotion. Using a simplified model of a rigid flagellum and a flexible hook, we show that the paths of axisymmetric cell bodies driven by a single flagellum in Stokes flow are generically helical. Phase-averaged resistance and mobility tensors are produced to describe the flagellar hydrodynamics, and a helical rod model which retains a coupling between translation and rotation is identified as a distinguished asymptotic limit. A supercritical Hopf bifurcation in the flagellar orientation beyond a critical ratio of flagellar motor torque to hook bending stiffness, which is set by the spontaneous curvature of the flexible hook, the shape of the cell body, and the flagellum geometry, can have a dramatic effect on the cell's trajectory through the fluid. Although the equilibrium hook angle can result in a wide variance in the trajectory's helical pitch, we find a very consistent prediction for the trajectory's helical amplitude using parameters relevant to swimming P. aeruginosa cells.