Buckling Instabilities and Complex Trajectories in a Simple Model of Uniflagellar Bacteria

Abstract

Observations of uniflagellar bacteria show that buckling instabilities of the hook protein connecting the cell body and flagellum play a role in locomotion. To understand this phenomenon, we develop models at varying levels of description with a particular focus on the parameter dependence of the buckling instability. A key dimensionless group called the flexibility number measures the hook flexibility relative to the thrust exerted by the flagellum; this parameter and the geometric parameters of the cell determine the stability of straight swimming. Two very simple models amenable to analytical treatment are developed to examine buckling in stationary (pinned) and moving swimmers. We then consider a more detailed model incorporating a helical flagellum and the rotational degrees of freedom of the cell body and flagellum, and we use numerical simulations to map out the parameter dependence of the buckling instability. In all models, a bifurcation occurs as the flexibility number increases, separating equilibrium configurations into straight or bent, and for the full model, separating trajectories into straight or helical. More specifically for the latter, the critical flexibility marks the transition from periodicity to quasi-periodicity in the behavior of variables determining configuration. We also find that for a given body geometry, there is a specific flagellar geometry that minimizes the critical flexibility number at which buckling occurs. These results highlight the role of flexibility in the biology of real organisms and the engineering of artificial microswimmers.

Complex dynamics of long, flexible fibers in shear

“…In the simulations presented below, we will see the effects of the small fluctuations that occur due to the numerical perturbations that arise from time integration and from the finite discretization of the fiber surface. Figure 2 shows the periodic orbits of three fibers of increasing length that display the signature tumble, S-turn and snaking behavior reported by [6,13,22,16]. The first column of Figure 2 depicts the rotational orbit of a fiber of length L = 0.139 (μ = 5.35 × 10 2 ) that exhibits little deformation from its straight shape.…”

“…The fibers we consider need not be thin and are not represented purely by their centerlines, but by a discretization of their surface. We use a similar fiber model as that used to examine the dynamics of diatom chains in a non-zero Reynolds number environment [16]. However, here we assume that the length and velocity scales are small enough so that the fluid dynamics is well-described by the Stokes equations, and use a regularized Stokeslet formulation [2,1] of the fluid-fiber system.…”

“…While the diatom chain model of [16] was able to capture complex buckling behavior of long fibers, it was based on an adaptive mesh immersed boundary method that needed fine grid resolution on the fluid domain near the fiber. In contrast, the regularized Stokeslet formulation described here, while requiring fine resolution of the fiber surface to capture the complex behavior, is based on fundamental solutions of the Stokes equations and does not require a spatial grid on the surrounding fluid.…”

TheN-flagella problem: elastohydrodynamic motility transition of multi-flagellated bacteria

“…n . The first and second entries of the right-hand side of (22) give the effective force and torque for the entire body, corresponding to propulsive force and torques,…”

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

“…They did not address flicks and due to computational expense provided detailed results for only a few geome-try and hook stiffness scenarios. Modeling the hook as a torsional spring connected to a rigid flagellum, Nguyen et al [31] described a transition from straight hook and straight trajectories to bent hook and helical trajectories as hook stiffness decreased. Park et al [32] modeled flicks using a time-dependent decrease in hook stiffness and flagellum stiff enough not to bend, but their model could be at most qualitative since the bacteria were not freeswimming.…”