Meshfree and efficient modeling of swimming cells

Locomotion in Stokes flow is an intensively studied problem because it describes important biological phenomena such as the motility of many species' sperm, bacteria, algae, and protozoa. Numerical computations can be challenging, particularly in three dimensions, due to the presence of moving boundaries and complex geometries; methods which combine ease of implementation and computational efficiency are therefore needed. A recently proposed method to discretize the regularized Stokeslet boundary integral equation without the need for a connected mesh is applied to the inertialess locomotion problem in Stokes flow. The mathematical formulation and key aspects of the computational implementation in matlab or GNU Octave are described, followed by numerical experiments with biflagellate algae and multiple uniflagellate sperm swimming between no-slip surfaces, for which both swimming trajectories and flow fields are calculated. These computational experiments required minutes of time on modest hardware; an extensible implementation is provided in a GitHub repository. The nearest-neighbor discretization dramatically improves convergence and robustness, a key challenge in extending the regularized Stokeslet method to complicated three-dimensional biological fluid problems.

Figure 1

Computational results for a free-swimming model biflagellate in an unbounded fluid, implemented with the script GenerateSwimmingFigureChlamy.m. Model biflagellate show the beat pattern, visualized via (a) force discretization and (b) quadrature discretization. (c) The $x_2$ coordinate of the free-swimming cell over five beat cycles, where positive $x_2$ is the overall swimming direction. The computed flow fields are shown at (d) $t=2\pi /3$, (e) $t=4\pi /3$, and (f) $t=2\pi$ (three points during the beat cycle). The computed velocity profile with streamlines is shown at $t=0$ for cross sections of the (g) $(x_1,x_2)$ and (h) $(x_2,x_3)$ planes.

Figure 2

Computational results for a group of free-swimming model sperm swimming midway between two opposed no-slip surfaces separated by $0.4$ flagellar lengths, implemented with the script GenerateSwimmingFigureSperm.m. Model sperm show the beat pattern of Dresdner and Katz [33], visualized via (a) force discretization and (b) quadrature discretization. (c) Visualization of sperm placed between the discretized boundaries (note that the sperm heads appear rounder than they actually are due to the aspect ratio chosen for plotting). (d) Trajectories of free-swimming cells over five beat cycles. The computed cell positions (with five randomly perturbed beat cycles and head dimensions) and flow fields are shown at (e) $t=0$ and (f) $t=2.5$ cycles. The computed velocity profile with streamlines of a single sperm is shown at $t=0$ for cross sections of the (g) $(x_1,x_2)$ and (h) $(x_1,x_3)$ planes.

Figure 3

Visual comparison of how the swimming distance for the biflagellate swimmer converges with (a) the nearest-neighbor and (b) the classic (Nyström) discretizations. The swimming distance is shown with increasing number of points for both the head $\left(N^H\right)$ and flagellum $\left(N^F\right)$ discretizations. In (a) the quadrature discretization is chosen to be twice as fine as the force discretization.