Differential Bending Stiffness of the Bacterial Flagellar Hook under Counterclockwise and Clockwise Rotations

The bacterial hook, as a universal joint coupling rotation of the flagellar motor and the filament, is an important component of the flagellum that propels the bacteria to swim. The mechanical properties of the hook are essential for the flagellum to achieve normal functions. In multiflagellated bacteria such as Escherichia coli, the hook must be compliant so that it can bend for the filaments to form a coherently rotating bundle to generate the thrust when the motor rotates counterclockwise (CCW), yet it also must be rigid so that the bundle can disrupt for the bacteria to tumble to change swimming direction when the motor rotates clockwise (CW). Here, by combining an elastic rod model with high-resolution bead assay to accurately measure the bending stiffness of the hook under CCW or CW rotation in vivo, we elucidate how the hook accomplishes this dual functionality: the hook stiffens under CW rotation, with bending stiffness under CW rotation twice as large as that under CCW rotation. This enables a robust run-and-tumble swimming motility for multiflagellated bacteria.

Figure 1

(a) Microscopic geometry model of the bead assays in a constant flow field (not to scale). $XOY$ is the image focal plane, and $Y^{\prime }O{Z}^{\prime }$ is the hook bending plane where $O{Y}^{\prime }$ is its intersect with the plane $XOY$. (b) The hook centerline in the bending plane under a concentrated force $\mathit{f}$. The dashed lines represent the state of the hook (red) and filament stub (blue) before applying the force. The solid lines represent the equilibrium state of the hook (red) and filament stub (blue) under the force. (c) Comparison of the relation between the flow field force and ${\theta }_L$ for the elastic rod model (red circles) and linear torsional spring model (blue crosses).

Figure 2

Typical measurement of hook bending stiffness in 10% Ficoll 400. (a) Schematic view of the experimental setup. A living bacterium was adhered to a microscope slide with a latex bead attached to the filament stub, and the medium was controlled by a syringe pump to generate a variable flow field. (b) 2D trajectories in the focal plane $XOY$ (yellow) and the calculated original 3D trajectory (purple). (c)–(g) The 1-s-long projected trajectories on the $XOY$ plane of the rotating bead at pump speeds of 10 (red), 100 (blue), 140 (green), 180 (cyan), and $220\mathrm{\mu }\mathrm{L}/\min$ (magenta). The black dashed lines denote the fitting results with a function for an ellipse. (h) The time trace of the tilted angle $\theta$ of the rotation plane with respect to the focal plane, calculated from the fitted ellipses. Five time windows are highlighted (gray regions) and considered periods of steady state. Different colors represent the corresponding pump speeds, as denoted in (c)–(g). (i) The vectors ${\mathit{N}}_{\mathit{i}}^{\prime }$ obtained by projecting the normal vectors of the bead rotating planes to the bending plane ($Y^{\prime }O{Z}^{\prime }$) for the selected time windows in (h). (j) The angle ${\theta }_L$ between ${\mathit{N}}_{\mathit{i}}^{\prime }$ and the $Z^{\prime }$-axis as a function of the flow force $f$. The dots represent the experimental measurements, and the red solid line represents the fitted curve with the elastic rod model. The error bars denote SDs for the time periods.

Figure 3

(a) The hook bending stiffness for the CCW- and CW-rotating strains in 0%, 10%, and 15% Ficoll 400 solutions. The error bar denotes the standard error of the mean (SEM). (b) The motor torque for CCW- and CW-rotating strains in 0%, 10%, and 15% Ficoll 400 solutions. (c) The hook bending stiffness versus the motor torque under different conditions. The error bar denotes SEM.