Spontaneous oscillations of elastic filaments induced by molecular motors

Abstract

It is known from the wave-like motion of microtubules in motility assays that the piconewton forces that motors produce can be sufficient to bend the filaments. In cellular phenomena such as cytosplasmic streaming, molecular motors translocate along cytoskeletal filaments, carrying cargo which entrains fluid. When large numbers of such forced filaments interact through the surrounding fluid, as in particular stages of oocyte development in , complex dynamics are observed, but the detailed mechanics underlying them has remained unclear. Motivated by these observations, we study here perhaps the simplest model for these phenomena: an elastic filament, pinned at one end, acted on by a molecular motor treated as a point force. Because the force acts tangential to the filament, no matter what its shape, this 'follower-force' problem is intrinsically non-variational, and thereby differs fundamentally from Euler buckling, where the force has a fixed direction, and which, in the low-Reynolds-number regime, ultimately leads to a stationary, energy-minimizing shape. Through a combination of linear stability theory, analytical study of a solvable simplified 'two-link' model and numerical studies of the full elastohydrodynamic equations of motion, we elucidate the Hopf bifurcation that occurs with increasing forcing of a filament, leading to flapping motion analogous to the high-Reynolds-number oscillations of a garden hose with a free end.

Instabilities and Spatiotemporal Dynamics of Active Elastic Filaments

“…The small discrepancy in β hb may be due to a different discretization method. 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.…”

“…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 .…”

“…The clamped case was previously studied by De Canio et al 25 . Using a method similar to that of section 3.1, they showed that the most unstable eigenvalue exhibits the same qualitative behavior seen in the green curves of Figure 3 (23) Imposition of boundary conditions at s = 1 yields two equations for the two unknowns a 1 and a 2 : a 1 (λ 2 1 coshλ 1 + λ 2 2 cosλ 2 ) + a 2 λ 1 (λ 1 sinhλ 1 + λ 2 sinλ 2 ) = 0, (24)…”

Harnessing elasticity to generate self-oscillation via an electrohydrodynamic instability

“…On the other hand, whenĒ ≤ 1, the extra anchored filament will further stabilise the original QR particle, hence the composite object will be stationary. We further note that the optimal translational velocity ≈ 6 × 10 −3 is in the range (1,15) By analogy to the results in figure 9a-c, we showΩ mag versusĒ as the bifurcation parameter for three EEV numbersμ = 500, 2000 and 8000. A similar bifurcation diagram is identified: increasingĒ from zero, the stationary fixed point solution transits to a limitcycle solution through a supercritical Hopf bifurcation atĒ cri 1 (green star); that solution then jumps to a second fixed point solution (steady spinning) via a secondary bifurcation at E cri 2 (magenta star).…”

Structures loaded with a force acting along a fixed straight line, or the “Reut’s column problem”

“…The introduced mechanical set-up shows for the first time after the 80-years-old Reut's paper, that a load acting on a fixed line can be made a real and usable condition, thus opening the way to new and unexpected applications, for instance in mechanical actuation or energy harvesting [18], but also in models for locomotion [19], biomechanics [20], and fluid-structure interaction [21], research arenas where flutter instability may play an important role. 2 Elastic double pendulum subject to four different loads of conservative and nonconservative nature A double pendulum is considered, namely, the two-degree-of-freedom rigid and heavy (the rods have unit mass densities ρ 1 , ρ 2 , and ρ 3 ) rods system shown in Fig.…”