Force-extension curves of bacterial flagella

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.

13 pages, 23 figures; The final publication is vailable at http://www.epj.org; Supplementary material in the form of two movies available from the Journal web page

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• A numerical method for inextensible elastic filaments in viscous fluids
• An Element of Determinism in a Stochastic Flagellar Motor Switch
• Propulsion by stiff elastic filaments in viscous fluids
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• How flagellated bacteria wobble