Growing, Buckling, and Swirling: motility from polymerization

Technical Summary: Growing, Buckling, and Swirling: Motility from Polymerization

Problem Statement
The paper addresses the physical mechanisms underlying a distinct class of microswimmer propulsion driven by the growth of surface-anchored cytoskeletal filaments. While traditional models of microswimming (e.g., flagellar rotation or ciliary beating) often prescribe boundary conditions or surface slip velocities, this work investigates how active forces emerge self-consistently from the microscopic kinetics of polymerization, filament reorientation, and hydrodynamic interactions. The study is motivated by biological systems such as Acetobacter xylinum, which extrudes cellulose fibers that buckle to generate motility, and synthetic colloids propelled by actin polymerization. The central challenge is to formulate a theoretical framework that simultaneously resolves filament kinetics, orientational dynamics, and fluid-structure interactions in a low-Reynolds-number (Stokesian) environment, where the resulting locomotion is determined by the self-organized dynamics of the filament carpet rather than pre-defined boundary conditions.

Methodology
The authors develop a self-consistent continuum framework that couples microscopic polymerization kinetics with macroscopic fluid dynamics:

1. Mean-Field Kinetics: The system is modeled as a dense carpet of filaments undergoing nucleation, constant-velocity growth ($V_g$), and stochastic depolymerization (catastrophe). A distribution function $\Psi(\ell, t)$ describes filament lengths, while a generalized distribution $\psi(y, \ell, p, t)$ accounts for spatial position, length, and orientation $p$.
2. Separation of Time Scales: The model assumes that polymerization and depolymerization occur on faster time scales than filament reorientation. This allows for the adiabatic elimination of length dynamics, reducing the problem to the evolution of an orientational distribution $\tilde{g}(y, p, t)$ and a polarization field $n(y, t)$.
3. Filament Dynamics: Individual filaments are modeled as rigid rods anchored to a surface by torsional springs (stiffness $k_\alpha$). Their reorientation is governed by Jeffery's equation (fluid shear) and restoring torques, balanced against the active forces generated by polymerization against the viscous fluid.
4. Hydrodynamic Coupling: The active forces are coarse-grained into an effective traction jump acting on an offset surface $\partial S$ (located at the mean filament length $\bar{\ell}$ above the rigid body). This traction drives the Stokes flow. The rigid body is treated as force-free and torque-free, with its translational ($U$) and angular ($\Omega$) velocities determined self-consistently.
5. Numerical Implementation:
   • Closure: A polar Bingham closure is used to approximate higher-order orientational moments ($\langle pp \rangle, \langle ppp \rangle$) in terms of the polarization vector $n$, avoiding the computational cost of solving the full Fokker-Planck equation.
   • Solver: The Stokes equations are solved using a Boundary Integral Method (BIM) with a fifth-order quadrature scheme and Quadrature by Expansion (QBX) for near-singular interactions. The system is solved iteratively using GMRES to handle the implicit coupling between surface tractions and velocity gradients.

Key Results

• Linear Stability Analysis: In a half-space geometry, the authors demonstrate that a uniform carpet of vertically growing filaments is unstable to long-wavelength perturbations. Polymerization-induced compressive forces drive a buckling instability that breaks symmetry, leading to spontaneous tilting of filaments and the generation of large-scale flows. The instability is governed by the competition between destabilizing active stresses and stabilizing elastic restoring torques and depolymerization rates.
• Fixed Sphere Dynamics: For a rigid sphere with uniform nucleation, the instability leads to a swirling pattern characterized by two antipodal $+1$ topological defects. This configuration generates coherent rotational flows exterior to the sphere, analogous to cytoplasmic flows in Drosophila oocytes.
• Free Sphere Dynamics (Spontaneous Rotation): When the sphere is free to move, the symmetry-broken swirling flow generates a net hydrodynamic torque, causing the sphere to undergo sustained rigid-body rotation about a spontaneously selected axis.
• Patterned Nucleation and Motility: By breaking the spatial symmetry of the nucleation sites, the authors demonstrate control over the balance of forces and torques:
  • Equatorial Band: Generates a net torque but zero net force, resulting in pure spinning.
  • Polar Patch: Generates both net force and torque, resulting in circular trajectories (combined translation and rotation).
  • Four-Fold Symmetry: An arrangement of four patches cancels net torque while summing forces, producing directed, persistent translation.
  • Chiral Patterns: A helical arrangement of nucleation sites couples rotation to translation, producing straight trajectories with simultaneous spinning, mimicking chiral propulsion mechanisms found in nature (e.g., Ginkgo sperm).
• Role of Geometry: Extending the analysis to prolate spheroids reveals that body shape significantly alters trajectories. Due to drag anisotropy, a spheroid with a chiral pattern traces a circular helix rather than a straight line, as the misalignment between the net traction and the principal axis causes continuous precession of the propulsion direction.

Significance and Claims
The paper establishes a general theoretical foundation for motility driven by the collective dynamics of polymerizing filaments. It demonstrates that polymerization itself is a generic route to self-propulsion, capable of generating diverse behaviors (spinning, directed motility, chiral swimming) purely through the self-organization of filament orientation and growth.

The authors claim that their framework provides a new design principle for synthetic micron-scale swimmers, where the "surface code" (the spatial patterning of nucleation sites) dictates the emergent locomotion strategy. The work bridges the gap between microscopic polymerization kinetics and macroscopic hydrodynamics, offering a unified continuum description applicable to both polymerization-driven and motor-driven active carpets. Furthermore, the results suggest that swimming speeds on the order of $0.1\text{--}1 \, \mu\text{m/s}$ are achievable through these mechanisms. The paper concludes by noting that while the current model assumes a "thin carpet" approximation, future work is needed to address regimes where filaments grow to lengths exceeding the body size, requiring hybrid discrete-continuum approaches.