Instability-driven oscillations of elastic microfilaments

Abstract

Cilia and flagella are highly conserved slender organelles that exhibit a variety of rhythmic beating patterns from non-planar cone-like motions to planar wave-like deformations. Although their internal structure, composed of a microtubule-based axoneme driven by dynein motors, is known, the mechanism responsible for these beating patterns remains elusive. Existing theories suggest that the dynein activity is dynamically regulated, via a geometric feedback from the cilium's mechanical deformation to the dynein force. An alternative, open-loop mechanism based on a 'flutter' instability was recently proven to lead to planar oscillations of elastic filaments under follower forces. Here, we show that an elastic filament in viscous fluid, clamped at one end and acted on by an external distribution of compressive axial forces, exhibits a Hopf bifurcation that leads to non-planar spinning of the buckled filament at a locked curvature. We also show the existence of a second bifurcation, at larger force values, that induces a transition from non-planar spinning to planar wave-like oscillations. We elucidate the nature of these instabilities using a combination of nonlinear numerical analysis, linear stability theory, and low-order bead-spring models. Our results show that away from the transition thresholds, these beating patterns are robust to perturbations in the distribution of axial forces and in the filament configuration. These findings support the theory that an open-loop, instability-driven mechanism could explain both the sustained oscillations and the wide variety of periodic beating patterns observed in cilia and flagella. arXiv:1808.01583v2 [physics.flu-dyn]

Instabilities and Spatiotemporal Dynamics of Active Elastic Filaments

“…De Canio et al 25 studied the linear stability and nonlinear dynamics of the clamped point follower load model of section 3.2.2 and found the same qualitative behavior we see in the clamped distributed follower force case. Ling et al 26 studied the linear stability and nonlinear dynamics of the 3D clamped case and found two distinct types of beating, a planar one similar to the one we observe and a helical one, each having its own instability threshold and a range of β in which it dominates. Fatehiboroujeni et al 46 studied the double-clamped 3D case, in which both ends of the filament are clamped and the base state is pre-buckled.…”

“…Second, the 2D constraint may be relaxed to allow motion in 3D. Ling et al 26 found two distinct beating instabilities, one of them helical, in the constrained problem, however we are not aware of any such study for filament-cargo assemblies. Third, long-range hydrodynamic interactions may be included, e.g., by assuming the head is spherical and adding stresslets distributed along the filament [48][49][50] .…”

“…Experiments reveal the resulting buckled shape is determined by the strength and type of surface defect or pinning 17,22 . Buckling instabilities have also been observed in simulations of a variety of models of pinned or clamped motility assays filaments 22,[24][25][26][27] . Recent Brownian dynamics simulations also found buckling instabilities in follower-force propelled filaments pushing a cargo 28 .…”

Harnessing elasticity to generate self-oscillation via an electrohydrodynamic instability

“…To achieve persistent motions, cyclic or oscillatory motions are needed, yet, the mechanism underlying the emergence of such oscillations remains unclear. Two major hypotheses, geometric feedback [8][9][10][11][12][13] and "flutter" or buckling instability [14][15][16][17][18] , have been raised based on theory and/or simulations: the first hypothesis assumes that a time-dependent dynein activity (switching on/off or modulation) is necessary to trigger the oscillations; the second one suggests that a steady point force or force distributions acting along the axial direction of a flexible filament can trigger its oscillatory motion through a "flutter" or buckling instability.…”

Synchronized oscillations, metachronal waves, and jammed clusters in sterically interacting active filament arrays

“…These instabilities have been the subject of recent theoretical and computational inquiries. Continuum as well as discrete agent-based models have been used to investigate the emergence of oscillations in single filaments, and coupling-induced synchrony in systems of two rotating filaments [29][30][31][32][33][34][35][36][37][38][39][40] .…”