Periodic Chirality Transformations Propagating On Bacterial Flagella

When a helical bacterial flagellum, clamped at one end, is placed in an external flow, it has been observed that regions of the flagellum transform to the opposite chirality, and travel as pulses down the length of the filament, the process repeating periodically [H. Hotani, J. Mol. Biol. 156, 791 (1982)]. We propose a theory for this phenomenon based on a treatment of the flagellum as an elastic object with multiple stable configurations. The simplest possible implementation of the model accurately reproduces key features seen in experiment.

Figure 1

Cyclic transformation between normal (grey) and curly (black) forms on a salmonella flagellum (length $23\mu \mathrm{m}$). Time runs left-to-right at intervals of 0.55 s. Courtesy of Hotani.

Figure 2

Initial front velocity versus flow speed, comparing numerical results (dots) and Eq. (5) (solid line). The minimal flow speed for nucleation of fronts in the computations is $10\mu \mathrm{m}/\mathrm{s}$.

Figure 3

Filament configuration time series for flow speed close to critical ($U=10\mu \mathrm{m}/\mathrm{s}$) with corresponding twist-rate ($\Omega$) plots. Time runs from top ($t=0\mathrm{s}$) to bottom ($t=11.7\mathrm{s}$). Arrows indicate points at which the handedness changes. The oscillations seen in the twist, while reminiscent of computational artifacts (Gibbs phenomena), are real; they arise from a phase mismatch between the actual and locally preferred pitch of the filament.

Figure 4

Numerical results for inversion frequency and number of domains vs fluid speed. Dots are numerically determined inversion frequencies; lines separate regimes in which different numbers of fronts exist simultaneously on the filament.