Buckling instabilities and spatio-temporal dynamics of active elastic filaments

Abstract

<jats:p>Biological filaments driven by molecular motors tend to experience tangential propulsive forces also known as active follower forces. When such a filament encounters an obstacle, it deforms, which reorients its follower forces and alters its entire motion. If the filament pushes a cargo, the friction on the cargo can be enough to deform the filament, thus affecting the transport properties of the cargo. Motivated by cytoskeletal filament motility assays, we study the dynamic buckling instabilities of a two-dimensional slender elastic filament driven through a dissipative medium by tangential propulsive forces in the presence of obstacles or cargo. We observe two distinct instabilities. When the filament’s head is pinned or experiences significant translational but little rotational drag from its cargo, it buckles into a steadily rotating coiled state. When it is clamped or experiences both significant translational and rotational drag from its cargo, it buckles into a periodically beating, overall translating state. Using minimal analytically tractable models, linear stability theory and fully nonlinear computations, we study the onset of each buckling instability, characterize each buckled state, and map out the phase diagram of the system. Finally, we use particle-based Brownian dynamics simulations to show our main results are robust to moderate noise and steric repulsion. Overall, our results provide a unified framework to understand the dynamics of tangentially propelled filaments and filament-cargo assemblies.</jats:p>

Flapping, swirling and flipping: Non-linear dynamics of pre-stressed active filaments

“…0.05 1−L ee /L 0.5 our numerical analysis shows that beyond a critical value of the follower force, F cr , buckled shapes no longer maintain static equilibrium and planar oscillations -flapping oscillations -emerge when rod is subject to any infinitesimal planar perturbations. This result is consistent with the onset of Hopf bifurcation that is obtained by linear stability analysis in cantilever scenario [52][53][54]66] ESM Movie 3 in the included supplementary material demonstrates an example of flapping oscillations with the spatiotemporal distribution of curvature and angular velocity in the fixed-fixed scenario.…”

“…Here in the active context, fluid drag plays a crucial dual role -both dissipating the energy and providing a pathway to stabilize the system by forcing the emergence of oscillations with large amplitude and clear frequencies. The dependence of the frequencies on the active force density F follows power laws as shown in previous theoretical work by us [54,66]); the exact exponent depends (provided one is far from onset) on the form of the drag and is 5/6 for quadratic drag as shown here, and 4/3 for linear drag forms such as low Reynolds number Stokes drag.…”

“…In the absence of active forces, fluid drag competes with physical mechanisms that temporarily store energy and produced damped motions. One of us has previously worked on theoretical and experimental systems with solid-fluid interactions where bulk solid elasticity provides the storage mechanism [54], as well as in purely fluid-fluid contexts with surface tension serving to store energy temporarily [75,76]. Here in the active context, fluid drag plays a crucial dual role -both dissipating the energy and providing a pathway to stabilize the system by forcing the emergence of oscillations with large amplitude and clear frequencies.…”

“…This is due to the non-self-adjoint nature of the equations and boundary conditions [23][24][25][45][46][47][48][49][50][51]. Instead time dependent evolution equations (dynamical equations) derived using variants of the Kirchhoff-Love theory are appropriate and have been used to analyze the onset of instabilities and non-linear patterns in these systems [29,[52][53][54][55]. As part of this effort, studies have also focussed on elucidating material constitutive laws [56,57].…”

“…However, a significant fundamental gap in the literature exists. Most current theories relate to filaments/rods that are are not pre-stressed (equivalently not pre-strained) and further are only partially constrained; that is, in most cases the base state is an uncompressed rod with a straight shape [29,[52][53][54][55]. In many important contexts however, rods and filaments that are subject to deforming forces and torques start off from shapes that are neither planar nor stress-free.…”