Amoeboid motion in confined geometry

Many eukaryotic cells undergo frequent shape changes (described as amoeboid motion) that enable them to move forward. We investigate the effect of confinement on a minimal model of amoeboid swimmer. Complex pictures emerge: (i) The swimmer’s nature (i.e., either pusher or puller) can be modified by confinement, thus suggesting that this is not an intrinsic property of the swimmer. This swimming nature transition stems from intricate internal degrees of freedom of membrane deformation. (ii) The swimming speed might increase with increasing confinement before decreasing again for stronger confinements. (iii) A straight amoeoboid swimmer’s trajectory in the channel can become unstable, and ample lateral excursions of the swimmer prevail. This happens for both pusher- and puller-type swimmers. For weak confinement, these excursions are symmetric, while they become asymmetric at stronger confinement, whereby the swimmer is located closer to one of the two walls. In this study, we combine numerical and theoretical analyses.

💡 Research Summary

In this paper the authors investigate how confinement influences the dynamics of a minimal model of an amoeboid swimmer. The swimmer is represented as a two‑dimensional, inextensible membrane that encloses a fluid of viscosity η and is itself immersed in a fluid of the same viscosity. Active forces are applied normal to the membrane and are decomposed into Fourier modes; only the second and third harmonics (k = 2, 3) are retained, producing a periodic stroke with amplitude A, frequency ω, and a dimensionless stroke strength S = A/(ωη) fixed at 10. Two dimensionless control parameters are introduced: Γ, which measures the excess perimeter relative to a circle (i.e., the degree of deformability), and C = 2R₀/W, the confinement ratio defined by the swimmer’s effective radius R₀ and the channel width W.

The governing Stokes equations with appropriate boundary conditions are solved numerically using both a boundary integral method (BIM) and an immersed boundary method (IBM), ensuring that the results are robust with respect to the numerical scheme. The authors first examine swimmers that move axially (along the channel centreline). They find a non‑monotonic dependence of the time‑averaged swimming speed V̄ on the confinement C: as C increases, V̄ initially rises, reaches a maximum at an optimal confinement C₀, and then declines for stronger confinement. This behaviour reflects a competition between wall‑induced lubrication friction (which slows the swimmer at very low C) and the beneficial hydrodynamic “squeezing” effect of the walls that enhances propulsion at intermediate C. At very strong confinement the swimmer’s shape deformations are suppressed, leading to a sharp drop in speed, a phenomenon also reported for helical flagella and other models.

A second key result concerns the swimmer’s stresslet Σ, the symmetric part of the far‑field force dipole. By averaging Σ over a swimming cycle the authors obtain a mean stresslet Σ̄ that changes sign as C varies: for C < 0.5 the swimmer behaves as a pusher (Σ̄ > 0), while for C > 0.5 it behaves as a puller (Σ̄ < 0). Thus the classic pusher/puller classification is not an intrinsic property of the swimmer but can be switched by confinement. The sign change originates from the way the walls modify the membrane deformation modes and consequently the distribution of active stresses.

Linear stability analysis of the centre‑line trajectory reveals that a positive stresslet (pusher) makes the straight path unstable, whereas a negative stresslet (puller) stabilises it. Numerical simulations, however, show that both pushers and pullers can experience an instability of the central trajectory when the confinement is moderate. The swimmer then performs large lateral excursions between the two walls, a behaviour the authors term a “navigating swimmer” (NS). For weak confinement the NS displays symmetric zig‑zag motion, while for stronger confinement the motion becomes asymmetric, with the swimmer spending most of its time near one wall. This symmetry‑breaking is identified as a supercritical bifurcation at a critical confinement C*; the average transverse position ⟨Y⟩/W jumps abruptly from zero to a finite value as C crosses C*.

The authors also explore the effect of varying the active force distribution up to the sixth Fourier harmonic and different amplitudes. The overall phenomenology—non‑monotonic speed, stresslet sign reversal, central‑line instability, and symmetry‑breaking bifurcation—remains unchanged, indicating that these features are generic to amoeboid swimming rather than artefacts of a particular force spectrum.

A further parameter, β, controls the strength of the stresslet component of the active force. When β is large (β > β_c ≈ 1), the classic behaviours reappear: strong pushers crash into the wall, while strong pullers settle into a straight trajectory. For weaker β, both types can adopt the NS mode, highlighting the importance of the internal degrees of freedom in determining the swimmer’s macroscopic behaviour.

The navigation period T, defined as the time needed for the swimmer to complete a full lateral cycle, exhibits three scaling regimes as a function of confinement. At very low C, the swimmer moves essentially straight toward the opposite wall, so T scales as the travel distance (∝ W) divided by the speed, giving T ∝ C⁻¹. In the intermediate regime the speed grows linearly with C, leading to T ∝ C⁻². After the symmetry‑breaking bifurcation, the swimmer remains close to one wall and its centre of mass oscillates with the intrinsic stroke period T_s, making T independent of C. These scalings are supported by heuristic arguments and agree with the numerical data.

The paper concludes by discussing the relevance of the findings to three‑dimensional systems and real cells such as leukocytes. While extending the model to 3D and incorporating cytoskeletal dynamics will be challenging, the authors argue that the qualitative features uncovered—optimal confinement, stresslet‑driven pusher/puller transition, central‑line instability, and symmetry‑breaking navigation—are likely to persist. The work therefore provides a comprehensive framework for understanding how confinement and internal shape dynamics together shape the locomotion of amoeboid cells.