Elastohydrodynamic Synchronization of Rotating Bacterial Flagella

To rotate continuously without jamming, the flagellar filaments of bacteria need to be locked in phase. While several models have been proposed for eukaryotic flagella, the synchronization of bacterial flagella is less well understood. Starting from a reduced model of flexible and hydrodynamically coupled bacterial flagella, we rigorously coarse grain the equations of motion using the method of multiple scales, and hence show that bacterial flagella generically synchronize to zero phase difference via an elastohydrodynamic mechanism. Remarkably, the far-field rate of synchronization is maximized at an intermediate value of elastic compliance, with surprising implications for bacteria.

Figure 1

Reduced model of interacting bacterial flagella. Two parallel helical filaments have phase angles ${\varphi }_j$. A constant torque, $T_0{\mathbf{e}}_z$, rotates each filament about its axis, while an elastic restoring force, $-k{x}_j{\mathbf{e}}_x$, pulls the axis back to a reference position. The tethering points are separated along $x$ by a distance much greater than the length of the filaments, $d\gg L$.

Figure 2

Interacting bacterial flagella generically synchronize in-phase via an elastohydrodynamic mechanism. (a) Evolution of phase difference on the slow time scale of synchronization and (inset) on the fast time scale of rotation, for inter-filament distance $d=2L$. The timescale for synchronization is (b) proportional to the hydrodynamic coupling between the filaments and (c) minimized at intermediate elastic compliance. Data points circled in blue represent identical input parameters (pitch angle $\psi =0.446$ and filament thickness $\epsilon =0.0038$ correspond to a “normal” flagellar filament [29]; full list of parameters available in the Supplemental Material [41]).

Figure 3

Physical mechanism for elastohydrodynamic synchronization. (a) The viscous drag on a rotating helix with phase angle $\varphi$ is balanced out primarily by (b) the viscous drag due to translation ($k\ll 1$) or (c) by the elastic restoring force ($k\gg 1$). (d) For general values of $k$, the axis of the helix, $\mathbf{x}$, lags behind the phase angle $\varphi$ by an angle between $\pi /2$ and $\pi$. (e),(f) Filament $i$ speeds up only if the flow induced by filament $j\ne i$ has a positive ${\mathbf{e}}_{\varphi }({\varphi }_i)$ component. (g) Hydrodynamically coupled helices synchronize in phase due to the flows that each filament $j$ produces along the positive ${\mathbf{e}}_r({\varphi }_j)$ direction and the resulting hydrodynamic stresses on filament $i\ne j$.