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, 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.
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Movie 1: The movie shows a filament that is stretched at zero temperature T = 0 . The left end of the filament is attached to a wall and we pull with a constant velocity vp at its right end. At a certain extension a small segment of the filament close to the fixed end switches into the normal state. Then the filament is stretched again. Further adjacent segments transform suddenly until the filament is nearly completely in the normal state. From here we invert the velocity vp and move both ends together. However, the filament does not transform back into the coiled state but remains in the normal state and starts to buckle. Movie 2: The movie shows a filament that is stretched at finite temperature T > 0 . The left end of the filament is attached to a wall and we pull with a constant velocity vp at its right end. At a certain extension a small segment of the filament close to the fixed end switches into the normal state. This transition occurs at a smaller extension compared to the deterministic case at T = 0 . Then the filament is further stretched and adjacent segments transform suddenly until the filament is nearly completely in the normal state. From here we invert the velocity vp and move both ends together. The filament completely transforms back to the coiled state and buckling is not observed.
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Vogel, R., Stark, H. Force-extension curves of bacterial flagella. Eur. Phys. J. E 33, 259–271 (2010). https://doi.org/10.1140/epje/i2010-10664-5
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DOI: https://doi.org/10.1140/epje/i2010-10664-5