Growth, Bending, and Swirling: How Polymerizing Filaments Generate Motion ✛ ✦
Context
Movement in the world of low Reynolds numbers (typical for microorganisms) is fundamentally different from our macroscopic experience: here, inertial forces are negligible compared to viscous forces. In this regime, the Stokes equations are linear, and the superposition principle holds: the total velocity field is the sum of fields from each force source. However, this same linearity makes the flow time-reversible—Purcell's famous 'scallop theorem' states that symmetric motions do not produce net translation. Therefore, microorganisms must use irreversible deformations, and filament polymerization is one evolutionarily tested way to circumvent this constraint. Understanding how the collective behavior of many growing filaments generates directed motion is important not only for biology but also for creating autonomous microdevices.
Methods
The authors constructed a continuum model in which active forces arise self-consistently from polymerization kinetics, orientational dynamics, and hydrodynamic interactions. Each filament is treated as a rigid rod growing from an anchor on the surface; its orientation changes under the local fluid flow and an elastic restoring moment. Then an equation for the average orientation field (polarization) of the filaments is derived, closed through a moment approximation based on a kinetic model. The hydrodynamic problem is solved using a boundary integral method that relies on the superposition principle and requires a nontrivial self-consistency procedure, since forces depend on velocity gradients at the surface. For numerical simulations, a boundary integral method with fast multipole acceleration and high-order quadratures was used, along with an implicit scheme to account for feedback.
Results
Linear stability analysis in a half-space showed that a uniformly oriented carpet becomes unstable to long-wavelength bending perturbations when the dimensionless activity exceeds a critical threshold: ¯σ > χ + (¯λ+1)/¯ρ. In the nonlinear regime on a fixed sphere, symmetry spontaneously breaks, and the polarity swirls, forming two antisymmetric vortices with two +1 topological defects at the poles—a structure reminiscent of cytoplasmic flows in Drosophila oocytes. If the sphere is allowed to move freely, this swirling creates a constant torque, and the sphere starts rotating with a steady angular velocity. Since the sphere is free, it must satisfy conditions of zero net force and torque—consequences of the momentum conservation law and the angular momentum conservation law. With nonuniform nucleation distributions, much richer regimes emerge. An equatorial belt produces only rotation without translation, because forces balance each other; a polar patch creates both force and torque, causing the sphere to move along a circular trajectory. A four-point symmetric arrangement cancels the torque, and the sphere swims straight. Finally, a chiral (spiral) nucleation pattern leads to coupled motion: the sphere simultaneously rotates and translates along the spiral axis, with the direction determined by the 'handedness' of the spiral. In an oblate spheroid, the center-of-mass trajectory becomes helical—an effect caused by the anisotropy of hydrodynamic drag. Typical swimming speeds are 0.1–1 μm/s, comparable to speeds of real microorganisms.
Implications
This work provides a unified theoretical framework for describing motility driven by polymer growth and demonstrates that the spatial pattern of nucleation centers acts as a 'program code': its symmetry determines whether the body rotates, swims straight, or moves along a helical path. Unlike the classical squirmmer model, where the surface slip field is prescribed externally, here active forces and filament orientations are computed self-consistently. This approach is also applicable to other systems, such as cortical microtubules driven by molecular motors, allowing a unified description of motor and polymerization activity in cells. For astrobiology, it is important that the polymerization-based motility mechanism broadens the range of possible biosignatures: if microorganisms using filament growth for movement are found on exoplanets, their swimming trajectories could be recognized by Earth-based instruments. This is a direct consequence of Emmy Noether's theorem linking space symmetries to conservation laws. In nonequilibrium systems, the production of entropy determines the direction of processes, and our polymerization carpet is an example of an active system with continuous energy dissipation.
Future development
The next steps will include studying hydrodynamic interactions between several polymerization-driven swimmers, which could lead to collective effects such as schooling behavior or the formation of active crystals. Extending the model to long filaments (like those of Acetobacter xylinum), where the thin-layer approximation breaks down, will require accounting for steric interactions and large bending deformations—here hybrid discrete-continuum methods may be useful. In the long term, such models will allow engineers to design synthetic micro-robots with prescribed motility: by modifying the chemical pattern on a particle's surface, one could program its trajectory. The described phenomena, where reconnection of topological defects governs collective dynamics, find analogies at vastly different scales—from active galactic nuclei to turbulent flows—making this class of models interdisciplinary valuable.
Impact
The results will influence cellular biophysics (understanding motility of Listeria, Acetobacter), materials science (creation of active colloids), and microscale robotics.
Next steps
Experimental verification of predictions on synthetic colloids with tunable actin density and extension of the model to collectives of interacting swimmers.
Key open problems
This work directly addresses the fundamental problem of self-organization in nonequilibrium systems: how do local interaction rules (filament growth and reorientation) generate global motion? This resonates with open questions in the physics of active matter about spontaneous symmetry breaking and the emergence of directed transport in the absence of external fields.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
This work presents a continuum model of the collective dynamics of polymerizing filaments on a surface, where active forces emerge self-consistently from growth kinetics, orientational dynamics, and hydrodynamic interactions. Linear stability analysis and numerical simulations reveal spontaneous symmetry breaking, leading to vortex structures and the rotation of a free sphere under inhomogeneous nucleation.
Where its strength is
The strength of this work lies in coupling polymerization kinetics with hydrodynamics through a self-consistent continuum model, enabling the description of directed motion arising from the collective behavior of growing filaments without phenomenological assumptions about forces.
Where it could go next
- It might be interesting to see how the self-organization principles found here for polymerizing filaments relate to the collective dynamics of active particles without explicit growth. In our archive, there is a nearby work on the formation of swarms in living particles without centralized control—there, ordered motion also emerges from local interactions.based on: Living Particles Form Swarms Without a Leader arXiv:2608.03317v1
- The observed vortex structure with topological defects resembles flows in the cytoplasm of Drosophila oocytes. A nearby work discusses how cells regulate adhesion and motility, which could suggest a biological context for such vortices and their role in morphogenesis.based on: How Cells Stick Together or Scatter arXiv:2604.26928
- It is intriguing to consider how inhomogeneities in nucleation and filament growth might interact with wave processes in polymer systems. The archive contains a work on the motion of long polymers on waves, discussing transport and orientation under external fields—this could provide ideas for controlling motion through growth modulation.based on: Long Polymers Catch the Wave arXiv:2603.02777
Worth trying
- Test adaptive control of rotation via dynamic redistribution of nucleation zones, drawing on self-organization principles from the work on swarms.
- Investigate how introducing variable adhesion between filaments and the surface (analogous to regulation of cell adhesion) affects stability and types of motion.
- Explore the system's response to external oscillatory flows, using ideas about polymer interactions with waves to achieve directed transport.
Why this may matter
Such models could serve as a foundation for autonomous microdevices that harness polymerization as a source of motion, and deepen our understanding of cell motility and morphogenesis, where filament growth plays a key role.
Nearby works in our archive
- How Cells Stick Together or Scattersimilarity 0.64
- How a cell quickly assembles DNA copiessimilarity 0.62
- Living Particles Form Swarms Without a Leadersimilarity 0.62
- Mail Inside a Neuron: How Molecules Find Their Recipientssimilarity 0.61
- The Gold-Digger Cellsimilarity 0.61
- Long Polymers Catch the Wavesimilarity 0.61
The closest work in the archive sits at 0.64 by meaning — the usual level for our corpus.
Original abstract
as written by the authors arXiv:2607.10446
Locomotion in low-Reynolds-number environments is achieved through a remarkable diversity of strategies, from flagellar rotation and ciliary beating to large-scale body deformations. A distinct and biologically important class of propulsion arises when surface-anchored filaments grow and collectively reorient - as seen in the cellulose-extruding bacterium Acetobacter xylinum and in recent experiments on actin-propelled synthetic colloids inspired by the motility of Listeria monocytogenes - suggesting that polymerization itself is a generic route to self-propulsion. Developing a theoretical framework for this class of problems requires simultaneously resolving filament kinetics, their orientational dynamics, and fluid-structure interactions - all self-consistently coupled to the resulting locomotion. To address this, we formulate a continuum framework in which the active forces driving locomotion emerge self-consistently from filament nucleation, growth, catastrophe, and hydrodynamic interactions. We show analytically that polymerization-induced compressive forces drive a long-wavelength buckling instability, leading to spontaneous symmetry breaking of the filament carpet and large-scale flows. In coupling this framework to a force- and torque-free motile spheroidal particle, a wide variety of behaviors emerge - this includes spontaneous spinning, directed motility, and chiral swimming - whose selection is governed by the spatial patterning of polymerizing filaments. These results establish a general theoretical foundation for motility, driven by collective dynamics of polymerizing filaments and point towards new design principles for synthetic micron-scale swimmers.
The bacterium Acetobacter xylinum secretes cellulose fibers that can be several times longer than the cell itself; if placed in a microfluidic channel, the bacterium turns into a fuzzy ball rolling along the walls due to reactive forces from the growing filaments.
The idea of programmable microswimmers whose motion is dictated by a chemical pattern on the surface echoes the concept of 'smart dust' from science fiction — swarms of microscopic robots capable of collective actions, as in Stanisław Lem's novel 'The Invincible'.
Key numbers
- typical swimmer speed: 0.1–1 μm/s
- average microtubule length: ~20 μm
- rotational relaxation time of a filament: ~1000 s
- characteristic polymerization time: 20–30 s
- critical activity threshold for instability: determined by the inequality given χ, λ̄, ρ̄