Motor-Driven Bacterial Flagella and Buckling Instabilities

TL;DR

This paper models bacterial flagella as elastic rods driven by motors, revealing buckling instabilities under thrust forces, and provides analytical and numerical insights into the critical conditions for buckling during bacterial locomotion.

Contribution

It introduces a coarse-grained elastic-rod model for bacterial flagella driven by motors, analyzing buckling transitions and their relation to motor torque and cell size.

Findings

Buckling occurs at a critical motor torque, observable as a Hopf bifurcation.

A second buckling transition appears at higher torque levels.

The model predicts buckling in real bacterial flagella based on biological parameters.

Abstract

Many types of bacteria swim by rotating a bundle of helical filaments also called flagella. Each filament is driven by a rotary motor and a very flexible hook transmits the motor torque to the filament. We model it by discretizing Kirchhoff's elastic-rod theory and develop a coarse-grained approach for driving the helical filament by a motor torque. A rotating flagellum generates a thrust force, which pushes the cell body forward and which increases with the motor torque. We fix the rotating flagellum in space and show that it buckles under the thrust force at a critical motor torque. Buckling becomes visible as a supercritical Hopf bifurcation in the thrust force. A second buckling transition occurs at an even higher motor torque. We attach the flagellum to a spherical cell body and also observe the first buckling transition during locomotion. By changing the size of the cell body, we vary the necessary thrust force and thereby obtain a characteristic relation between the critical thrust force and motor torque. We present a sophisticated analytical model for the buckling transition based on a helical rod which quantitatively reproduces the critical force-torque relation. Real values for motor torque, cell body size, and the geometry of the helical filament suggest that buckling should occur in single bacterial flagella. We also find that the orientation of pulling flagella along the driving torque is not stable and comment on the biological relevance for marine bacteria.