Instabilities of a rotating helical rod in a viscous fluid

Discovery of oscillations in rotational speed of body-tethered Caulobacter crescentus

Swarmer cells of Caulobacter crescentus have been found to tether to glass at a point on the cell body. The rolling of the freely rotating flagellum near the glass surface causes the cell body to rotate. We describe the discovery of damped oscillations in the rotational speed of these cell bodies. We show that the damped oscillations are robust over multiple cells and that they depend more on the cell's accumulated rotation angle than on time. We also find that their phase is determined by the moment the flagellar motor changes the direction of its rotation. The oscillations occur only for one direction of cell rotation, when the flagellum is in pulling mode. We discuss possible explanations for these oscillations, including fluctuations in flagellar motor torque and periodic changes in flagellar orientation, and illustrate both of these cases using simplified computer models. Finally, we present the hypothesis that the oscillations are the result of fluctuations in the proton motive force, initiated by a sudden change in proton current that occurs when the motor switches rotation direction.

Large deformations of the hook affect free-swimming singly flagellated bacteria during flick motility

Hook dynamics are important in the motility of singly flagellated bacteria during flick motility. Although the hook is relatively short, during reorientation events it may undergo large deformations, leading to nonlinear behavior. Here, we explore when these nonlinear and large deformations are important for the swimming dynamics in different ranges of hook flexibilities and flagellar motor torques. For this purpose, we investigate progressively more faithful models for the hook, starting with linear springs, then models that incorporate nonlinearities due to larger hook deformations. We also employ these models both with and without hydrodynamic interactions between the flagellum and cell body to test the importance of those hydrodynamic interactions. We show that for stiff hooks, bacteria swim with a flagellum rotating on-axis in orbits and hydrodynamic interactions between the cell body and flagellum change swimming speeds by about 40%. As the hook stiffness decreases, there is a critical hook stiffness that predicts the initiation of the dynamic instability causing flicks. We compare the transition value of stiffnesses predicted by our models to experiments and show that nonlinearity and large deflections do not significantly affect critical transition values, while hydrodynamic interactions can change transition values by up to 13%. Below the transition value, we observe precession of the flagellum, in which it deflects off-axis to undergo nearly circular stable trajectories. However, only slightly below the transition stiffness, nonlinearity in hook response destabilizes precession, leading to unstable deflections of the flagellum. We conclude that while the linear hook response can qualitatively predict transition stiffnesses, nonlinear models are necessary to capture the behavior of hooks for stiffnesses below transition. Furthermore, we show that for the lower range of hook stiffnesses observed in actual bacteria, models which capture the full deformations of hooks are necessary. Inclusion of the hydrodynamic interactions of the cell body, hook, and flagellum is required to quantitatively simulate nonlinear dynamics of soft hooks during flick motility.

Emergent three-dimensional sperm motility: coupling calcium dynamics and preferred curvature in a Kirchhoff rod model

Changes in calcium concentration along the sperm flagellum regulate sperm motility and hyperactivation, characterized by an increased flagellar bend amplitude and beat asymmetry, enabling the sperm to reach and penetrate the ovum (egg). The signalling pathways by which calcium increases within the flagellum are well established. However, the exact mechanisms of how calcium regulates flagellar bending are still under investigation. We extend our previous model of planar flagellar bending by developing a fluid-structure interaction model that couples the 3D motion of the flagellum in a viscous Newtonian fluid with the evolving calcium concentration. The flagellum is modelled as a Kirchhoff rod: an elastic rod with preferred curvature and twist. The calcium dynamics are represented as a 1D reaction-diffusion model on a moving domain, the flagellum. The two models are coupled assuming that the preferred curvature and twist of the sperm flagellum depend on the local calcium concentration. To investigate the effect of calcium on sperm motility, we compare model results of flagellar bend amplitude and swimming speed for three cases: planar, helical (spiral with equal amplitude in both directions), and quasi-planar (spiral with small amplitude in one direction). We observe that for the same parameters, the planar swimmer is faster and a turning motion is more clearly observed when calcium coupling is accounted for in the model. In the case of flagellar bending coupled to the calcium concentration, we observe emergent trajectories that can be characterized as a hypotrochoid for both quasi-planar and helical bending.

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.