Dancing volvox: hydrodynamic bound states of swimming algae

Abstract

The spherical alga Volvox swims by means of flagella on thousands of surface somatic cells. This geometry and its large size make it a model organism for studying the fluid dynamics of multicellularity. Remarkably, when two nearby Volvox colonies swim close to a solid surface, they attract one another and can form stable bound states in which they "waltz" or "minuet" around each other. A surface-mediated hydrodynamic attraction combined with lubrication forces between spinning, bottom-heavy Volvox explains the formation, stability, and dynamics of the bound states. These phenomena are suggested to underlie observed clustering of Volvox at surfaces.

FIG. 1

(color online). Waltzing of V. carteri. (a) Top view. Superimposed images taken 4 s apart, graded in intensity. (b) Side, and (c) top views of a colony swimming against a cover slip, with fluid streamlines. Scales are 200 μm. (d) A linear Volvox cluster viewed from above (scale is 1 mm).

FIG. 2

(color online). Dual-view apparatus.

FIG. 3

(color online). Swimming properties of V. carteri as a function of radius. (a) upswimming speed, (b) rotational frequency, (c) sedimentation speed, (d) bottom-heaviness reorientation time, (e) density offset, and (f) components of average flagellar force density.

FIG. 4

(color online). Waltzing dynamics. Geometry of (a) two interacting Stokeslets (side view) and (b) nearby spinning colonies (top view). (c) Radial separation r, normalized by mean colony radius, as a function of rescaled time for 60 events (black). Running average (green) compares well with predictions of the singularity model (red). Inset shows orbiting frequency Ω as a function of mean spinning frequency $\stackrel{‒}{\omega }$, and linear fit.

FIG. 5

(color online). “Minuet” bound state. (a) Side views 3 s apart of two colonies near the chamber bottom. Arrows indicate the anterior-posterior axes p_i at angles θ_i to vertical. Scale bar is 600 μm. (b) Bifurcation diagram, and phase portrait (inset), showing a limit cycle, with realistic model parameters F = 6πηRV, R = 300 μm, h₁ = 450 μm, h₂ = 1050 μm.