Bundling of elastic filaments induced by hydrodynamic interactions
Peritrichous bacteria swim in viscous fluids by rotating multiple helical flagellar filaments. As the bacterium swims forward, all its flagella rotate in synchrony behind the cell in a tight helical bundle. When the bacterium changes its direction, the flagellar filaments unbundle and randomly reorient the cell for a short period of time before returning to their bundled state and resuming swimming. This rapid bundling and unbundling is, at its heart, a mechanical process whereby hydrodynamic interactions balance with elasticity to determine the time-varying deformation of the filaments. Inspired by this biophysical problem, we present in this paper what is perhaps the simplest model of bundling whereby two, or more, straight elastic filaments immersed in a viscous fluid rotate about their centreline, inducing rotational flows which tend to bend the filaments around each other. We derive an integro-differential equation governing the shape of the filaments resulting from mechanical balance in a viscous fluid at low Reynolds number. We show that such equation may be evaluated asymptotically analytically in the long-wavelength limit, leading to a local partial differential equation governed by a single dimensionless Bundling number. A numerical study of the dynamics predicted by the model reveals the presence of two configuration instabilities with increasing Bundling numbers: first to a crossing state where filaments touch at one point and then to a bundled state where filaments wrap along each other in a helical fashion. We also consider the case of multiple filaments, and the unbundling dynamics. We next provide an intuitive physical model for the crossing instability and show that it may be used to predict analytically its threshold, and adapted to address the transition to a bundling state. We then use a macro-scale experimental implementation of the two-filament configuration in order to validate our theoretical predictions and obtain excellent agreement. This long-wavelength model of bundling will be applicable to other problems in biological physics and provides the groundwork for further, more realistic, models of flagellar bundling.