Force-extension curves of bacterial flagella

TL;DR

This paper develops an elastic model based on Kirchhoff's theory to simulate and analyze the force-extension behavior of bacterial flagella, capturing polymorphic transformations and environmental effects.

Contribution

It introduces a comprehensive elastic model and Brownian dynamics simulations that accurately reproduce experimental force-extension curves of bacterial flagella.

Findings

The model quantitatively matches experimental data.

The ratio of torsional to bending rigidity influences flagella response.

Extension depends logarithmically on the extensional rate.

Abstract

Bacterial flagella assume different helical shapes during the tumbling phase of a bacterium but also in response to varying environmental conditions. Force-extension measurements by Darnton and Berg explicitly demonstrate a transformation from the coiled to the normal helical state [N.C. Darnton and H.C. Berg, Biophys. J. {92}, 2230 (2007)]. We here develop an elastic model for the flagellum based on Kirchhoff's theory of an elastic rod that describes such a polymorphic transformation and use resistive force theory to couple the flagellum to the aqueous environment. We present Brownian dynamics simulations that quantitatively reproduce the force-extension curves and study how the ratio $\Gamma$ of torsional to bending rigidity and the extensional rate influence the response of the flagellum. An upper bound for $\Gamma$ is given. Using clamped flagella, we show in an adiabatic approximation that the mean extension, where a local coiled-to-normal transition occurs first, depends on the logarithm of the extensional rate.