Metachronal waves in the flagellar beating of Volvox and their hydrodynamic origin

Groups of eukaryotic cilia and flagella are capable of coordinating their beating over large scales, routinely exhibiting collective dynamics in the form of metachronal waves. The origin of this behaviour—possibly influenced by both mechanical interactions and direct biological regulation—is poorly understood, in large part due to a lack of quantitative experimental studies. Here we characterize in detail flagellar coordination on the surface of the multicellular alga Volvox carteri, an emerging model organism for flagellar dynamics. Our studies reveal for the first time that the average metachronal coordination observed is punctuated by periodic phase defects during which synchrony is partial and limited to specific groups of cells. A minimal model of hydrodynamically coupled oscillators can reproduce semi-quantitatively the characteristics of the average metachronal dynamics, and the emergence of defects. We systematically study the model''s behaviour by assessing the effect of changing intrinsic rotor characteristics, including oscillator stiffness and the nature of their internal driving force, as well as their geometric properties and spatial arrangement. Our results suggest that metachronal coordination follows from deformations in the oscillators'' limit cycles induced by hydrodynamic stresses, and that defects result from sufficiently steep local biases in the oscillators'' intrinsic frequencies. Additionally, we find that random variations in the intrinsic rotor frequencies increase the robustness of the average properties of the emergent metachronal waves.     

Parallel computational microhydrodynamics: scalable load-balancing strategies

The hydrodynamic interactions in a suspension of small particles in a viscous fluid is computed by a boundary integral velocity representation featuring a completed double-layer potential (completed double-layer boundary integral equation method or CDL = BIEM). A multiple expansion is used to represent interactions between distance particles, leading to a considerable improvement in computational speed. The resulting large linear system of equations provides an ideal setting for asynchronous iterative solvers (block Gauss-Seidel) on a message-passing MIMD parallel computer (Intel iPSC/860 Hypercube) using a supervisor-workers load-balancing strategy. Our benchmark results as a function of problem size and number of processors suggest that our algorithm will scale successfully to the massively parallel computers of the future.     

Stokes flow past an ellipsoid in a special ambient flow field: A marvelous cubic equation for a quadratic ambient field

A three-dimensional eigenspace appears in the analysis of an ellipsoid immersed in quadratic ambient viscous flow fields. To celebrate the mathematical achievements of Babatunde Ogunnaike, we focus on mathematical details of the eigenvalues and eigenfunctions of the double-layer operator that appears in the integral representation of velocity fields as a function of the shape parameters of the ellipsoid. Three special quadratic ambient fields that lack an ambient pressure gradient (toroidal fields) form a decoupled subsystem with eigenvalues associated with three real roots of a cubic equation. The discriminant of this cubic vanishes at discrete points in the parameter space of the ellipsoidal shape revealing repeated roots (and thus repeated eigenvalues) of the cubic. The discrete nature of these points is illustrated with explicit derivation of the discriminant's level set contours in the region near the repeated eigenvalues for the sphere.