Force-extension curves of bacterial flagella

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, 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 Γ of torsional to bending rigidity and the extensional rate influence the response of the flagellum. An upper bound for Γ 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.