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.

STUDY QUESTION Can flagellar analyses be scaled up to provide automated tracking of motile sperm, and does knowledge of the flagellar waveform provide new insight not provided by routine head tracking? SUMMARY ANSWER High-throughput flagellar waveform tracking and analysis enable measurement of experimentally intractable quantities such as energy dissipation, disturbance of the surrounding medium and viscous stresses, which are not possible by tracking the sperm head alone. WHAT IS KNOWN ALREADY The clinical gold standard for sperm motility analysis comprises a manual analysis by a trained professional, with existing automated sperm diagnostics [computer-aided sperm analysis (CASA)] relying on tracking the sperm head and extrapolating measures. It is not currently possible with either of these approaches to track the sperm flagellar waveform for large numbers of cells in order to unlock the potential wealth of information enclosed within. STUDY DESIGN, SIZE, DURATION The software tool in this manuscript has been developed to enable high-throughput, repeatable, accurate and verifiable analysis of the sperm flagellar beat. PARTICIPANTS/MATERIALS, SETTING, METHODS Using the software tool [Flagellar Analysis and Sperm Tracking (FAST)] described in this manuscript, we have analysed 176 experimental microscopy videos and have tracked the head and flagellum of 205 progressive cells in diluted semen (DSM), 119 progressive cells in a high-viscosity medium (HVM) and 42 stuck cells in a low-viscosity medium. Unscreened donors were recruited at Birmingham Women’s and Children’s NHS Foundation Trust after giving informed consent. MAIN RESULTS AND THE ROLE OF CHANCE We describe fully automated tracking and analysis of flagellar movement for large cell numbers. The analysis is demonstrated on freely motile cells in low- and high-viscosity fluids and validated on published data of tethered cells undergoing pharmacological hyperactivation. Direct analysis of the flagellar beat reveals that the CASA measure ‘beat cross frequency’ does not measure beat frequency; attempting to fit a straight line between the two measures gives \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}${\mathrm{R}}^2$\end{document} values of 0.042 and 0.00054 for cells in DSM and HVM, respectively. A new measurement, track centroid speed, is validated as an accurate differentiator of progressive motility. Coupled with fluid mechanics codes, waveform data enable extraction of experimentally intractable quantities such as energy dissipation, disturbance of the surrounding medium and viscous stresses. We provide a powerful and accessible research tool, enabling connection ...

The dynamics of geometrically non-linear flexible filaments play an important role in a host of biological processes, from flagella-driven cell transport to the polymeric structure of complex fluids. Such problems have historically been computationally expensive due to numerical stiffness associated with the inextensibility constraint, as well as the often non-trivial boundary conditions on the governing high-order PDEs. Formulating the problem for the evolving shape of a filament via an integral equation in the tangent angle has recently been found to greatly alleviate this numerical stiffness. The contribution of the present manuscript is to enable the simulation of nonlocal interactions of multiple filaments in a computationally efficient manner using the method of regularized stokeslets within this framework. The proposed method is benchmarked against a nonlocal bead and link model, and recent code utilizing a local drag velocity law. Systems of multiple filaments (1) in a background fluid flow, (2) under a constant body force, and (3) undergoing active self-motility are modeled efficiently. Buckling instabilities are analyzed by examining the evolving filament curvature, as well as by coarse-graining the body frame tangent angles using a Chebyshev approximation for various choices of the relevant non-dimensional parameters. From these experiments, insight is gained into how filament-filament interactions can promote buckling, and further reveal the complex fluid dynamics resulting from arrays of these interacting fibers. By examining active moment-driven filaments, we investigate the speed of worm-and sperm-like swimmers for different governing parameters. The MATLAB R implementation is made available as an open-source library, enabling flexible extension for alternate discretizations and different surrounding flows.