Fluid flow reconstruction around a free-swimming sperm in 3D

We investigate the dynamics and hydrodynamics of a human spermatozoa swimming freely in 3D. We simultaneously track the sperm flagellum and the sperm head orientation in the laboratory frame of reference via high-speed high-resolution 4D (3D+t) microscopy, and extract the flagellar waveform relative to the body frame of reference, as seen from a frame of reference that translates and rotates with the sperm in 3D. Numerical fluid flow reconstructions of sperm motility are performed utilizing the experimental 3D waveforms, with excellent accordance between predicted and observed 3D sperm kinematics. The reconstruction accuracy is validated by directly comparing the three linear and three angular sperm velocities with experimental measurements. Our microhydrodynamic analysis reveals a novel fluid flow pattern, characterized by a pair of vortices that circulate in opposition to each other along the sperm cell. Finally, we show that the observed sperm counter-vortices are not unique to the experimental beat, and can be reproduced by idealised waveform models, thus suggesting a fundamental flow structure for free-swimming sperm propelled by a 3D beating flagellum.

Modelling Motility: The Mathematics of Spermatozoa

In one of the first examples of how mechanics can inform axonemal mechanism, Machin's study in the 1950s highlighted that observations of sperm motility cannot be explained by molecular motors in the cell membrane, but would instead require motors distributed along the flagellum. Ever since, mechanics and hydrodynamics have been recognised as important in explaining the dynamics, regulation, and guidance of sperm. More recently, the digitisation of sperm videomicroscopy, coupled with numerous modelling and methodological advances, has been bringing forth a new era of scientific discovery in this field. In this review, we survey these advances before highlighting the opportunities that have been generated for both recent research and the development of further open questions, in terms of the detailed characterisation of the sperm flagellum beat and its mechanics, together with the associated impact on cell behaviour. In particular, diverse examples are explored within this theme, ranging from how collective behaviours emerge from individual cell responses, including how these responses are impacted by the local microenvironment, to the integration of separate advances in the fields of flagellar analysis and flagellar mechanics.

The art of coarse Stokes: Richardson extrapolation improves the accuracy and efficiency of the method of regularized stokeslets

The method of regularized stokeslets is widely used in microscale biological fluid dynamics due to its ease of implementation, natural treatment of complex moving geometries, and removal of singular functions to integrate. The standard implementation of the method is subject to high computational cost due to the coupling of the linear system size to the numerical resolution required to resolve the rapidly varying regularized stokeslet kernel. Here, we show how Richardson extrapolation with coarse values of the regularization parameter is ideally suited to reduce the quadrature error, hence dramatically reducing the storage and solution costs without loss of accuracy. Numerical experiments on the resistance and mobility problems in Stokes flow support the analysis, confirming several orders of magnitude improvement in accuracy and/or efficiency.

Numerical Modeling of Sperm Swimming

Due to rising human infertility, sperm motility has been an important subject. Among the hundreds of millions of sperms on the journey up the oviducts, only a few excellent travelers will reach the eggs. This journey is affected by many factors, some of which include sperm quality, sperm density, fluid rheology and chemotaxis. In addition, the sperm swimming through different body tracks and fluids involves complex sperm flagellar, complex fluid environment, and multi-sperm and sperm-wall interactions. Therefore, this topic has generated substantial research interest. In this paper, we present a review of computational studies on sperm swimming from an engineering perspective with focus on both simplified theoretical methods and fluid–structure interaction methods. Several open issues in this field are highlighted.

Passively parallel regularized stokeslets

Stokes flow, discussed by G.G. Stokes in 1851, describes many microscopic biological flow phenomena, including cilia-driven transport and flagellar motility; the need to quantify and understand these flows has motivated decades of mathematical and computational research. Regularized stokeslet methods, which have been used and refined over the past 20 years, offer significant advantages in simplicity of implementation, with a recent modification based on nearest-neighbour interpolation providing significant improvements in efficiency and accuracy. Moreover this method can be implemented with the majority of the computation taking place through built-in linear algebra, entailing that state-of-the-art hardware and software developments in the latter, in particular multicore and GPU computing, can be exploited through minimal modifications ('passive parallelism') to existing Matlab computer code. Hence, and with widely available GPU hardware, significant improvements in the efficiency of the regularized stokeslet method can be obtained. The approach is demonstrated through computational experiments on three model biological flows: undulatory propulsion of multiple Caenorhabditis elegans, simulation of progression and transport by multiple sperm in a geometrically confined region, and left-right symmetry breaking particle transport in the ventral node of the mouse embryo. In general an order-of-magnitude improvement in efficiency is observed. This development further widens the complexity of biological flow systems that are accessible without the need for extensive code development or specialist facilities. This article is part of the theme issue 'Stokes at 200 (part 2)'.

Effects of cell morphology and attachment to a surface on the hydrodynamic performance of unicellular choanoflagellates

Choanoflagellates, eukaryotes that are important predators on bacteria in aquatic ecosystems, are closely related to animals and are used as a model system to study the evolution of animals from protozoan ancestors. The choanoflagellate Salpingoeca rosetta has a complex life cycle with different morphotypes, some unicellular and some multicellular. Here we use computational fluid dynamics to study the hydrodynamics of swimming and feeding by different unicellular stages of S. rosetta: a swimming cell with a collar of prey-capturing microvilli surrounding a single flagellum, a thecate cell attached to a surface and a dispersal-stage cell with a slender body, long flagellum and short collar. We show that a longer flagellum increases swimming speed, longer microvilli reduce speed and cell shape only affects speed when the collar is very short. The flux of prey-carrying water into the collar capture zone is greater for swimming than sessile cells, but this advantage decreases with collar size. Stalk length has little effect on flux for sessile cells. We show that ignoring the collar, as earlier models have done, overestimates flux and greatly overestimates the benefit to feeding performance of swimming versus being attached, and of a longer stalk for attached cells.

Simulations of particle tracking in the oligociliated mouse node and implications for left-right symmetry-breaking mechanics

The concept of internal anatomical asymmetry is familiar-usually in humans the heart is on the left and the liver is on the right; however, how does the developing embryo know to produce this consistent laterality? Symmetry-breaking initiates with left-right asymmetric cilia-driven fluid mechanics in a small fluid-filled structure called the ventral node in mice. However, the question of what converts this flow into left-right asymmetric development remains unanswered. A leading hypothesis is that flow transports morphogen-containing vesicles within the node, the absorption of which results in asymmetrical gene expression. To investigate how vesicle transport might result in the situs patterns observe in wild-type and mutant experiments, we extend the open-source Stokes flow package, NEAREST, to consider the hydrodynamic and Brownian motion of particles in a mouse model with flow driven by one, two and 112 beating cilia. Three models for morphogen-containing particle released are simulated to assess their compatibility with observed results in oligociliated and wild-type mouse embryos: uniformly random release, localized cilium stress-induced release and localized release from motile cilia themselves. Only the uniformly random release model appears consistent with the data, with neither localized release model resulting in significant transport in the oligociliated embryo. This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.

From flagellar undulations to collective motion: predicting the dynamics of sperm suspensions

Swimming cells and microorganisms are as diverse in their collective dynamics as they are in their individual shapes and propulsion mechanisms. Even for sperm cells, which have a stereotyped shape consisting of a cell body connected to a flexible flagellum, a wide range of collective dynamics is observed spanning from the formation of tightly packed groups to the display of larger-scale, turbulence-like motion. Using a detailed mathematical model that resolves flagellum dynamics, we perform simulations of sperm suspensions containing up to 1000 cells and explore the connection between individual and collective dynamics. We find that depending on the level of variation in individual dynamics from one swimmer to another, the sperm exhibit either a strong tendency to aggregate, or the suspension exhibits large-scale swirling. Hydrodynamic interactions govern the formation and evolution of both states. In addition, a quantitative analysis of the states reveals that the flows generated at the time scale of flagellum undulations contribute significantly to the overall energy in the surrounding fluid, highlighting the importance of resolving these flows.

Clustering-induced self-propulsion of isotropic autophoretic particles

Self-diffusiophoretic particles exploit local concentration gradients of a solute species in order to self-propel at the micron scale. While an isolated chemically- and geometrically-isotropic particle cannot swim, we show that it can achieve self-propulsion through interactions with other individually-non-motile particles by forming geometrically-anisotropic clusters via phoretic and hydrodynamic interactions. This result identifies a new route to symmetry-breaking for the concentration field and to self-propulsion, that is not based on an anisotropic design, but on the collective dynamics of identical and homogeneous active particles. Using full numerical simulations as well as theoretical modelling of the clustering process, the statistics of the propulsion properties are obtained for arbitrary initial arrangement of the particles. The robustness of these results to thermal noise, and more generally the effect of Brownian motion of the particles, is also discussed.

Modeling polymorphic transformation of rotating bacterial flagella in a viscous fluid

The helical flagella that are attached to the cell body of bacteria such as Escherichia coli and Salmonella typhimurium allow the cell to swim in a fluid environment. These flagella are capable of polymorphic transformation in that they take on various helical shapes that differ in helical pitch, radius, and chirality. We present a mathematical model of a single flagellum described by Kirchhoff rod theory that is immersed in a fluid governed by Stokes equations. We perform numerical simulations to demonstrate two mechanisms by which polymorphic transformation can occur, as observed in experiments. First, we consider a flagellar filament attached to a rotary motor in which transformations are triggered by a reversal of the direction of motor rotation [L. Turner et al., J. Bacteriol. 182, 2793 (2000)10.1128/JB.182.10.2793-2801.2000]. We then consider a filament that is fixed on one end and immersed in an external fluid flow [H. Hotani, J. Mol. Biol. 156, 791 (1982)10.1016/0022-2836(82)90142-5]. The detailed dynamics of the helical flagellum interacting with a viscous fluid is discussed and comparisons with experimental and theoretical results are provided.

Coarse-Graining the Fluid Flow around a Human Sperm

The flagellar beat is extracted from human sperm digital imaging microscopy and used to determine the flow around the cell and its trajectory, via boundary element simulation. Comparison of the predicted cell trajectory with observation demonstrates that simulation can predict fine-scale sperm dynamics at the qualitative level. The flow field is also observed to reduce to a time-dependent summation of regularized Stokes flow singularities, approximated at leading order by a blinking force triplet. Such regularized singularity decompositions may be used to upscale cell level detail into population models of human sperm motility.

Dynamics of cilia length in left-right development

Reduction in the length of motile cilia in the zebrafish left-right organizer (LRO), also known as Kupffer's vesicle, has a large impact on left-right development. Here we demonstrate through genetic overexpression in zebrafish embryos and mathematical modelling that the impact of increased motile cilia length in embryonic LRO fluid flow is milder than that of short cilia. Through Arl13b overexpression, which increases cilia length without impacting cilia beat frequency, we show that the increase in cilium length is associated with a decrease in beat amplitude, resulting in similar flow strengths for Arl13b overexpression and wild-type (WT) embryos, which were not predicted by current theory. Longer cilia exhibit pronounced helical beat patterns and, consequently, lower beat amplitudes relative to WT, a result of an elastohydrodynamic shape transition. For long helical cilia, fluid dynamics modelling predicts a mild (approx. 12%) reduction in the torque exerted on the fluid relative to the WT, resulting in a proportional reduction in flow generation. This mild reduction is corroborated by experiments, providing a mechanism for the mild impact on organ situs.

Microscale flow dynamics of ribbons and sheets

Numerical study of the hydrodynamics of thin sheets and ribbons presents difficulties associated with resolving multiple length scales. To circumvent these difficulties, asymptotic methods have been developed to describe the dynamics of slender fibres and ribbons. However, such theories entail restrictions on the shapes that can be studied, and often break down in regions where standard boundary element methods are still impractical. In this paper we develop a regularised stokeslet method for ribbons and sheets in order to bridge the gap between asymptotic and boundary element methods. The method is validated against the analytical solution for plate ellipsoids, as well as the dynamics of ribbon helices and an experimental microswimmer. We then demonstrate the versatility of this method by calculating the flow around a double helix, and the swimming dynamics of a microscale "magic carpet".

Hydrodynamic evolution of sperm swimming: Optimal flagella by a genetic algorithm

Swimming performance of spermatozoa is an important index for the success of fertilization. For many years, numerous studies have reported the optimal swimming of flagellar organisms. Nevertheless, there is still a question as to which is optimal among planar, circular helical and ellipsoidal helical beating. In this paper, we use a genetic algorithm to investigate the beat pattern with the best swimming efficiency based on hydrodynamic dissipation and internal torque exertion. For the parameters considered, our results show that the planar beat is optimal for small heads and the helical flagellum is optimum for a larger heads, while the ellipsoidal beat is never optimal. Also, the genetic optimization reveals that the wavenumber and shape of wave envelope are relevant parameters, whereas the wave shape and head geometry have relatively minor effects on efficiency. The optimal beat with respect to the efficiency based on the internal torque exertion of an active elastic flagellum is characterized by a small-wavenumber and large-amplitude wave in a lower-viscosity medium. The obtained results on the optimal waveform are consistent with observations for planar waveforms, but in many respects, the results suggest the necessity of a detailed flagellar structure-fluid interaction to address whether real spermatozoa exhibit hydrodynamically efficient swimming. The evolutional optimization approach used in this study has distinguished biologically important parameters, and the methodology can potentially be applicable to various swimmers.

Three-dimensional flow in Kupffer's Vesicle

Whilst many vertebrates appear externally left-right symmetric, the arrangement of internal organs is asymmetric. In zebrafish, the breaking of left-right symmetry is organised by Kupffer’s Vesicle (KV): an approximately spherical, fluid-filled structure that begins to form in the embryo 10 hours post fertilisation. A crucial component of zebrafish symmetry breaking is the establishment of a cilia-driven fluid flow within KV. However, it is still unclear (a) how dorsal, ventral and equatorial cilia contribute to the global vortical flow, and (b) if this flow breaks left-right symmetry through mechanical transduction or morphogen transport. Fully answering these questions requires knowledge of the three-dimensional flow patterns within KV, which have not been quantified in previous work. In this study, we calculate and analyse the three-dimensional flow in KV. We consider flow from both individual and groups of cilia, and (a) find anticlockwise flow can arise purely from excess of cilia on the dorsal roof over the ventral floor, showing how this vortical flow is stabilised by dorsal tilt of equatorial cilia, and (b) show that anterior clustering of dorsal cilia leads to around 40 % faster flow in the anterior over the posterior corner. We argue that these flow features are supportive of symmetry breaking through mechano-sensory cilia, and suggest a novel experiment to test this hypothesis. From our new understanding of the flow, we propose a further experiment to reverse the flow within KV to potentially induce situs inversus.

A regularised singularity approach to phoretic problems

An efficient, accurate, and flexible numerical method is proposed for the solution of the swimming problem of one or more autophoretic particles in the purely diffusive limit. The method relies on successive boundary element solutions of the Laplacian and the Stokes flow equations using regularised Green's functions for swift, simple implementations, an extension of the well-known method of "regularised stokeslets" for Stokes flow problems. The boundary element method is particularly suitable for phoretic problems, since no quantities in the domain bulk are required to compute the swimming velocity. For time-dependent problems, the method requires no re-meshing and simple boundaries such as a plane wall may be added at no increase to the size of the linear system through the method of images. The method is validated against two classical examples for which an analytical or semi-analytical solution is known, a two-sphere system and a Janus particle, and provides a rigorous computational pipeline to address further problems with complex geometry and multiple bodies.

Spermatozoa scattering by a microchannel feature: an elastohydrodynamic model

Sperm traverse their microenvironment through viscous fluid by propagating flagellar waves; the waveform emerges as a consequence of elastic structure, internal active moments and low Reynolds number fluid dynamics. Engineered microchannels have recently been proposed as a method of sorting and manipulating motile cells; the interaction of cells with these artificial environments therefore warrants investigation. A numerical method is presented for large-amplitude elastohydrodynamic interaction of active swimmers with domain features. This method is employed to examine hydrodynamic scattering by a model microchannel backstep feature. Scattering is shown to depend on backstep height and the relative strength of viscous and elastic forces in the flagellum. In a 'high viscosity' parameter regime corresponding to human sperm in cervical mucus analogue, this hydrodynamic contribution to scattering is comparable in magnitude to recent data on contact effects, being of the order of 5°-10°. Scattering can be positive or negative depending on the relative strength of viscous and elastic effects, emphasizing the importance of viscosity on the interaction of sperm with their microenvironment. The modulation of scattering angle by viscosity is associated with variations in flagellar asymmetry induced by the elastohydrodynamic interaction with the boundary feature.

Optimal swimming of a sheet

Propulsion at microscopic scales is often achieved through propagating traveling waves along hairlike organelles called flagella. Taylor's two-dimensional swimming sheet model is frequently used to provide insight into problems of flagellar propulsion. We derive numerically the large-amplitude wave form of the two-dimensional swimming sheet that yields optimum hydrodynamic efficiency: the ratio of the squared swimming speed to the rate-of-working of the sheet against the fluid. Using the boundary element method, we show that the optimal wave form is a front-back symmetric regularized cusp that is 25% more efficient than the optimal sine wave. This optimal two-dimensional shape is smooth, qualitatively different from the kinked form of Lighthill's optimal three-dimensional flagellum, not predicted by small-amplitude theory, and different from the smooth circular-arc-like shape of active elastic filaments.

Modelling the fluid mechanics of cilia and flagella in reproduction and development

Cilia and flagella are actively bending slender organelles, performing functions such as motility, feeding and embryonic symmetry breaking. We review the mechanics of viscous-dominated microscale flow, including time-reversal symmetry, drag anisotropy of slender bodies, and wall effects. We focus on the fundamental force singularity, higher-order multipoles, and the method of images, providing physical insight and forming a basis for computational approaches. Two biological problems are then considered in more detail: 1) left-right symmetry breaking flow in the node, a microscopic structure in developing vertebrate embryos, and 2) motility of microswimmers through non-Newtonian fluids. Our model of the embryonic node reveals how particle transport associated with morphogenesis is modulated by the gradual emergence of cilium posterior tilt. Our model of swimming makes use of force distributions within a body-conforming finite-element framework, allowing the solution of nonlinear inertialess Carreau flow. We find that a three-sphere model swimmer and a model sperm are similarly affected by shear-thinning; in both cases swimming due to a prescribed beat is enhanced by shear-thinning, with optimal Deborah number around 0.8. The sperm exhibits an almost perfect linear relationship between velocity and the logarithm of the ratio of zero to infinite shear viscosity, with shear-thickening hindering cell progress.

Generic flow profiles induced by a beating cilium

We describe a multipole expansion for the low-Reynolds-number fluid flows generated by a localized source embedded in a plane with a no-slip boundary condition. It contains 3 independent terms that fall quadratically with the distance and 6 terms that fall with the third power. Within this framework we discuss the flows induced by a beating cilium described in different ways: a small particle circling on an elliptical trajectory, a thin rod and a general ciliary beating pattern. We identify the flow modes present based on the symmetry properties of the ciliary beat.