Dynamic instability in the hook-flagellum system that triggers bacterial flicks

Dynamical bending, buckling, and polymorphic transformations of the flagellum are known to affect bacterial motility, but run-reverse-flick motility of monotrichous bacteria also involves the even more flexible hook connecting the flagellum to its rotary motor. Although flick initiation has been hypothesized to involve either static Euler buckling or dynamic bending of the hook, the precise mechanism of flick initiation remains unknown. Here, we find that flicks initiate via a dynamic instability requiring flexibility in both the hook and flagellum. We obtain accurate estimates of forces and torques on the hook that suggest that flicks occur for stresses below the (static) Euler buckling criterion, then provide a mechanistic model for flick initiation that requires combined bending of the hook and flagellum. We calculate the triggering torque-stiffness ratio and find that our predicted onset of dynamic instability corresponds well with experimental observations.

Figure 1

Axial torques and forces on hook during preflick (flick-triggering, red circles, $E{I}_u)$ and postflick (non-flick-triggering, blue triangles, $E{I}_w)$ runs of different bacteria, estimated by Son et al.'s [29] 1D model (open symbols) and our full hydrodynamic model (filled symbols). The solid line is the static Euler buckling criterion, while the dashed line is the Euler buckling criterion if $EI=1\times {10}^{-26}\mathrm{N}{\mathrm{m}}^2$ (see text). Inset: After swimming backwards (A) with clockwise flagellar rotation, which unwinds the hook and decreases its stiffness to $E{I}_u$, reversal of the flagellar motor leads to a short preflick forward run (B) that increases loading on the hook and causes a flick (C) in which the flagellum deforms off-axis and turns the bacteria. Postflick, the wound hook with increased bending stiffness $E{I}_w$ returns the flagellum on-axis for a forward run (D).

Figure 2

Forces $\mathbf{F}$, motor $\left(M_M\right)$, and perpendicular $\left(M_{\mathrm{app}}\right)$ torques at one end of a hook deform it from a straight rest position in the $x$ direction so that the tangent direction at the other end is ${\mathbf{d}}_3\left(L_H\right)$, described by polar angles $\left({\theta }_M,{\varphi }_M\right)$.

Figure 3

(a) Orientation of flagellum along $\widehat{\mathbf{r}}$ described by angles $\left(\theta ,\varphi \right)$ traces a nearly circular trajectory (dashed line) around average orientation ${\widehat{\mathbf{r}}}^{*}$. (b),(c) Examples of the flagellar orientation trajectories during free swimming with linearized hook dynamics. The flagellum is initially oriented in the $x$ direction and, after a transient motion, settles into either orbits or precession. The blue trace is the motion of the flagellum endpoint relative to the cell body for motor moment of $M_M=2000$ pN nm. (b) For $k_H=1>k_H^{*}$, the long-time motion is a small orbit, shown as a red circle. One orbit traversal takes one flagellar rotation. (c) For $k_H=0.36<k_H^{*}$, the long-time motion is precession closely following the largest circular paths. One traversal around the path takes multiple flagellar rotations. (d) Trajectory size measured by minimum $\left({\theta }_p^{\min }\right)$ and maximum $\left({\theta }_p^{\max }\right)$ deflection angles (red lines) or radius of endpoint motion $(R$, blue line). In the model with rigid flagellum, as stiffness parameter $k_H$ decreases, flagellum trajectories transition from small orbits to large precession at $k_H^{*}$.

Figure 4

Adding flagellum flexibility, for $k_H>k_H^{*}$ flagellum remains nearly unbent and on-axis, while for $k_H<k_H^{*}$ stresses from precession lead to large deformations from rigid flagellum (straight gray line), as measured by endpoint displacement between bent and rigid flagellum, $|\mathbf{r}-{\mathbf{r}}_0{|}_{(s=L)}$.

Figure 5

Calculated $1/k_H$ values of postflick (blue triangles) runs lie in the stable regime to the left of the critical value (dashed line) predicted by the dynamic buckling instability, while those of preflick runs (red circles) lie in the unstable regime. Nearly all preflick runs do not exceed the static Euler buckling criterion (solid line).

Figure 6

The linear response of hook deflection angles to applied moments. (a),(b) Dots are the response of deflection angles $\left(\theta ,\varphi \right)$ for the exact Kirchhoff-rod model, while lines are the linear interpolations for different relative stiffnesses $k_H$. The coefficients plotted in (c) are the intercepts and slopes of these linear interpolations. (c) The linearization coefficients for the orientation angles $\left(\theta ,\varphi \right)$ and associated errors. The corresponding relations are Eqs. (4) and (5).

Figure 7

The effects of applied force on the deflection angles. (a) ${\theta }_{M,F}$ is the deflection angle due to applying a typical moment and upper bound for the force, while ${\theta }_M$ and ${\theta }_F$ are the deflection angles when applying only the moment or force, respectively. The dashed line represents the percentage change in the deflection due to the force compared to the moment. (b) Similar quantities for the torsional deflection $\varphi$ due to applied moment and force.

Figure 8

The regularized Stokeslet method is coupled with an extensible Kirchhoff-rod model to model a flexible flagellum. The flagellum is discretized into cylindrical segments of length $\mathrm{\Delta }s$. Cross sections between the segments are labeled by integers $p$, while segment centers are labeled by half integers $p+1/2$. Each segment is treated as a rigid body with translational and angular velocities $\mathbf{V}$ and $\mathrm{\Omega }$. The elastic forces from deformation are calculated from the relative motion of segments. Points show regularized Stokeslets on the surface of each segment. A prescribed velocity and rotation rate are applied to the cell end of the flagellum $\left(s=0\right)$ and then the time-dependent deflections of the flagellum are calculated.