Dynamical Boundary Following and Corner Trapping of Undulating Worms

We investigate the behavior of {\it Lumbriculus variegatus} in circular and polygonal chambers and show that the worms align with the boundaries as they move forward and then become dynamically trapped at the concave corners over prolonged periods. We model the worm as a self-propelled rod and derive analytical expressions for the evolution of its orientation when it encounters the flat and the circular boundaries of the chamber. By further incorporating translational and rotational diffusion, arising due to the undulatory and peristaltic body strokes, we demonstrate through numerical simulations that the self-propelled rod model can capture both the boundary aligning and the corner trapping behavior of the worm. The Péclet number $Pe$, representing the ratio of forward propulsion to rotational diffusion, is found to characterize the boundary alignment dynamics and trapping time distribution of the worm. Simulations show that the angle of the worm’s body with the boundary while entering a concave corner plays a key role in determining the trapping time, with shallow angles leading to faster escapes. Our study demonstrates that directed motion combined with limited angular diffusion can lead to spatial localization that mimics shelter seeking behavior in slender undulating limbless worms, even in the absence of thigmotaxis or contact seeking behavior.

💡 Research Summary

The paper investigates why the freshwater annelid Lumbriculus variegatus tends to follow walls and become trapped in concave corners when confined in water‑filled chambers of various shapes. Using transparent acrylic chambers (circular, square, and polygonal) with a height of 1.5 mm, the authors recorded the full‑body motion of individual worms (≈20 mm long, ≈0.1 mm diameter) at 24 fps for 30‑minute trials. By extracting head, centroid, and tail coordinates they quantified spatial distributions, root‑mean‑square displacements, and the orientation angle ψ_b between the worm’s body axis and the local wall normal. The experiments reveal that the head spends >80 % of the time in the corner regions, while the centroid and tail are more uniformly distributed. When the head contacts a flat wall, the body gradually aligns such that ψ_b → 0, i.e., the worm moves tangentially to the wall. Near a corner ψ_b fluctuates between 0 and ±π/2, and the time spent trapped depends strongly on the entry angle: shallow entry angles lead to rapid escape, whereas near‑perpendicular approaches result in long residence times.

To rationalize these observations the authors model the worm as a self‑propelled rigid rod of length ℓ ≈ 20 mm moving at a constant speed v₀ with rotational diffusion coefficient D_r. Contact with a wall generates a restoring torque τ = −κ ψ_b that tends to align the rod with the wall, while stochastic torque ξ(t) accounts for the undulatory and peristaltic strokes. The dynamics are described by a Langevin equation for the orientation θ(t) and translational motion for the center of mass. Two analytical cases are solved: (i) a flat wall where the torque simply drives ψ_b toward zero, and (ii) a circular wall where curvature introduces an additional deterministic rotation term proportional to v₀/R sin ψ_b (R is the chamber radius). A single dimensionless group, the Péclet number Pe = v₀/(D_r ℓ), quantifies the competition between directed propulsion and angular diffusion. In the experiments Pe≈10–30, indicating that propulsion dominates and alignment is rapid.

Numerical simulations integrate the Langevin equations using the experimentally measured v₀≈2 mm s⁻¹ and D_r≈0.05 rad² s⁻¹. The simulations reproduce (a) the high probability of head residence in corners, (b) the ψ_b versus arclength s scatter plot observed experimentally, and (c) the exponential tail of the corner‑trapping time distribution, which scales with Pe. By varying the entry angle ψ_entry in the simulations the authors confirm that shallow angles (|ψ_entry| < 30°) reduce the mean trapping time to <2 min, matching the empirical trend.

The key insight is that the observed “thigmotaxis” – the tendency to stay near surfaces – does not require an active sensory attraction to walls. Instead, simple mechanical contact combined with limited angular diffusion suffices to generate wall‑following and corner trapping. The Péclet number emerges as a universal predictor: high Pe yields strong alignment and long corner residence, low Pe leads to frequent detachment. This finding bridges biological observations with active‑matter theory, suggesting that many organisms may exploit purely physical mechanisms for shelter‑seeking behavior. The work also offers design principles for synthetic active rods or soft robots that need to navigate confined environments without explicit sensing, by tuning propulsion speed and rotational noise to achieve desired wall‑following or corner‑trapping characteristics.