Boundary curvature-dependent dynamical trapping of undulating worms

TL;DR

This work analyzes boundary-driven localization of slender, undulatory worms in chambers with circular, square, and polygonal geometries. Using full-body tracking, it demonstrates that alignment to boundaries and trapping at concave corners arise from purely contact-based steric interactions, not thigmotaxis or long-range hydrodynamics. The authors develop a two-tier modeling approach: a kinematic boundary-alignment description and a self-propelled rigid-rod (SPR) model with translational/rotational diffusion and optional end-to-end length fluctuations, capturing boundary-following, corner trapping, and escape. A key finding is that the Péclet number $Pe$ governs both the extent of boundary alignment and the characteristic trapping times, linking locomotion details to spatial organization. The results provide a minimal, predictive framework for boundary-induced localization in confined, slender organisms and inform design principles for biomimetic systems operating under confinement.

Abstract

We investigate the behavior of {\it Lumbriculus variegatus} in polygonal chambers and show that the worms align with the boundaries as they move forward and get dynamically trapped at concave corners over prolonged periods of time before escaping. We develop a kinematic model to calculate and describe the evolution of the worm's mean body orientation angle relative to the boundary. Performing simulations with a minimal active elastic dumbbell model, we then show that both the boundary aligning and corner trapping behavior of the worm are captured by steric interactions with the boundaries. The dimensionless ratio of the strength of forward motion and diffusion caused by the worm's undulatory and peristaltic strokes is shown to determine the boundary alignment dynamics and trapping time scales of the worm. The simulations show that that the body angle with the boundary while entering the concave corner is important to the trapping time distributions with shallow angles leading to faster escapes. Our study demonstrates that directed motion and limited angular diffusion can give rise to aggregation which can mimic shelter seeking behavior in slender undulating limbless worms even when thigmotaxis or contact seeking behavior is absent.

Figures (12)

• Figure 1: (a) An image of Lumbriculus variegatus, also known as California blackworm being studied. (b-d) Superposed snapshots of the worm at $\Delta t = 10$ seconds time intervals moving inside a circular chamber with diameter $d = 4$ cm (b), a square chamber with side length $l= 4 \textrm{ cm}$ (c), and a polygon chamber with long sides $l = 4 \textrm{ cm}$ (d), over total observation time $T = 18$ min, 21 min, and 25 min, respectively. The snapshots correspond to time $t$ denoted by the color map. For reference, the snapshot at $t = 0$ is highlighted in bold. The location of the head is indicated by a solid blue marker and is observed to stay close to the boundary in the circular chamber, and predominantly near the concave corners in the square chamber. The head remains close to the concave corner and does not follow the boundary around the convex corner in the polygonal chamber. The arrows labeled 1, 2, 3 in the color bar indicate three spontaneous departures of the worm from the polygonal chamber boundary in (d).
• Figure 2: (a) The trajectory of the head (blue), centroid (red), and tail (green) of a worm moving in a square chamber over time $T = 10$ minutes ($l_w = 23$ mm; l = 4 cm). (b) The corner, boundary, and center regions in the chamber. (c) The probability of finding the head, centroid, and tail in various parts of the chamber shows the localization of the head near the corners. (d) The probability of finding the head in square chambers with worm to side length aspect ratio, $l_w/l$ = 2, 1 and 0.5 all show localization of the head at the corners ($l_w = 20 \pm 3$ mm). The error bars denote root mean square deviations of a trial from the mean ($n_T =10$ trails; $T=30$ minutes each).
• Figure 3: (a) Snapshots of a worm as it aligns with a boundary and gets trapped in a corner. The time corresponds to the color map. (b) A scatter plot of body orientation $\psi_{b}$ versus distance along perimeter $s$. The dashed (black) lines correspond to Eq. (\ref{['eq:psib']}) calculated with the kinematic model, and the vertical dotted (red) lines correspond to the worms trapped at the corners. (c) The scatter plot of $v_h^{||}$ as a function of $\sin\psi_b$ recorded each time a worm contacts a boundary, shows increasing trend which is well captured by a linear fit (solid black line) based on Eq. (\ref{['eq:linear_eqn']}) with $v_h = 1.74$ mm s$^{-1}$. (d) The scatter plot of the measured worm centroid velocity $v_c^{||}$ as a function of $\sin(\psi_b)$ is described by Eq. (\ref{['eq:v_c']}) assuming $v_r$ is the same as $v_h$ obtained by fitting Eq. (\ref{['eq:linear_eqn']}).
• Figure 4: (a-c) Schematics of a self propelled rigid rod of length $l_r$ interacting with the boundary. The unit vectors $\mathbf{e}_\parallel$ and $\mathbf{e}_\perp$ denote directions parallel and perpendicular to the rod axis, respectively. $v_c$ and $v_h$ are the velocities of the rod center and head, respectively. (a) The rod head velocity components before and after contact with a flat boundary. (b) The velocity components of the rod center parallel and perpendicular to the flat boundary. (c) A rod on a curved boundary, illustrating the variation $d\psi_b$ in its orientation resulting from a parallel displacement by distance $ds$ along the curve while the slope angle changes by $d\phi$. (d) $\psi_b(t)$ obtained by numerical integration of Eq. (\ref{['eq:psit']}) for three different initial conditions ($R = 20$ mm; $l_r=15$ mm). Each example converges to the stable value $\psi_o$ consistent with Eq. (\ref{['eq:Stable angle']}). (e) The evolution of the stable orientation of a rigid rod moving in a circular chamber as a function of $R/l_r$.
• Figure 5: The mean trapping time of the rod $\langle \tau_\textrm{sim} \rangle$ at a corner of square chamber ($l = 40$ mm) increases with increasing $Pe$. At $Pe = 13$, $\langle \tau_\textrm{sim} \rangle$ matches the trapping time scale observed in experiments within the deviation (mean - dashed line, deviation - shaded region).