Dynamics of individual active elastic filaments with chiral self-propulsion

TL;DR

The paper develops a planar, overdamped continuum model for one-dimensional elastic filaments subjected to a chiral active force at angle $\alpha$ to the tangent, leading to a pair of coupled sixth-order nonlinear PDEs for the intrinsic stretch $u(s)$ and scaled curvature $w(s)$. It identifies stationary constant-curvature solutions, derives their rotation rates, and performs a linear stability analysis to map stable regions in parameter space, complemented by nonlinear simulations. The results show coexisting straight and circular-arc states and reveal nonuniform, time-periodic dynamics under certain conditions, illustrating rich single-filament behavior that can underpin observed multi-stability in gliding assays. The framework provides a foundation for including stochastic forces or external potentials and for extending to more complex active-matter systems, offering a mechanistic lens on chiral active filaments and their potential switching between conformations.

Abstract

We study the over-damped dynamics of individual one-dimensional elastic filaments subjected to a chiral active force which propels each point of the filament at a fixed angle relative to the tangent vector of the filament at that point. Such a model is a reasonable starting point for describing the behavior of polymers such as microtubules in gliding assay experiments. We derive sixth-order nonlinear coupled partial differential equations for the intrinsic properties of the filament, namely, its curvature and metric, and show that these equations are capable of supporting multiple different stationary solutions in a co-moving frame, i.e. that chiral active elastic filaments exhibit dynamic multi-stability in their shapes. A linear stability analysis of these solutions is carried out to determine which solutions are stable and a brief analysis of the time-dependent approach to stationary shape is considered. Finally, simulations are presented which confirm many of our predictions while also revealing additional complexity.

Figures (7)

• Figure 1: (a) Schematic illustration of a one-dimensional filament, such as a microtubule, experiencing a chiral active force $\vec{F}_A$. At each point along the curved filament, the active force $\vec{F}_A$ points at an angle $\alpha$ counter-clockwise from the unit tangent vector $\hat{T}$ at that point. This active force can then be decomposed into its tangential component $\vec{F}_{A,T}$ and its normal component $\vec{F}_{A,N}$. (b) Schematic representation of a microtubule in a straight conformation. Such a microtubule will simply translate across the surface. (c) Schematic representation of a microtubule in a curved conformation. Such a microtubule will rotate as it travels across the surface (see inset).
• Figure 2: Schematic solutions to Eq. (\ref{['eq:polynomial']}) for the curvature $w_0$ of uniformly curved solutions to the equations of motion. The points where the dashed lines intersect the solid curve (if any) constitute the roots of Eq. (\ref{['eq:polynomial']}). There are two, one, or zero such roots if $g_b / \sin(\alpha)$ is, respectively, less than (blue line), equal to (red line) or greater than (purple line) $(3^3/4^4)(g_s / \sin(\alpha))^3$. $w_{0,c}$ marks the value of $w_0$ for which Eq. (\ref{['eq:polynomial']}) has only a single root.
• Figure 3: Phase space of non-trivial constant curvature solutions. The solid line corresponds to the critical line given by Eq. (\ref{['eq:g_B-g_S critical line']}). On this line, Eq. (\ref{['eq:polynomial']}) has exactly one real root, above it, there are no real roots, and beneath it, there are two real roots. The region with two real roots is then further divided by the blue dashed line, $w_0 = 2\pi$, which separates regions where both, one or neither solution fully wraps around and self-intersects.
• Figure 4: The theoretically predicted uniform solutions, as a function of $g_S/\sin(\alpha)$, described in terms of their (a) uniform curvature $w_0$, (b) uniform stretching $u_0$, and (c) rotation rate $\omega/\cos(\alpha)$. The value of $g_B/\sin(\alpha)$ is given for each curve in the legends. The plots show that for small $g_S/\sin(\alpha)$, the only available constant-curvature solution is the uncurved ($w_0 = 0$), unstretched ($u_0 = 1$), non-rotating ($\omega = 0$) state. Above a critical value which depends on $g_B/\sin(\alpha)$, additional constant curvature solutions become possible.
• Figure 5: Linear stability of the curved solutions $(u_0,w_0)$ for (a)$\alpha = 0.1$, (b)$\alpha = 1.0$. The green regions show where both roots of Eq. (\ref{['eq:polynomial']}) are stable and the red regions show where both roots are unstable. In the blue regions, the root with smaller curvature is stable while the larger-curvature root is unstable. The opposite is the case for the cyan regions. The dashed line is the same as that in Fig. \ref{['fig:phase space 1']} and separates the regions where one or both solutions self-intersect by wrapping around.