Hermes Gadêlha

Bend Propagation in the Flagella of Migrating Human Sperm, and Its Modulation by Viscosity

A pre-requisite for sexual reproduction is successful unification of the male and female gametes; in externally-fertilising echinoderms the male gamete is brought into close proximity to the female gamete through chemotaxis, the associated signalling and flagellar beat changes being elegantly characterised in several species. In the human, sperm traverse a relatively high-viscosity mucus coating the tract surfaces, there being a tantalising possible role for chemotaxis. To understand human sperm migration and guidance, studies must therefore employ similar viscous in vitro environments. High frame rate digital imaging is used for the first time to characterise the flagellar movement of migrating sperm in low and high viscosities. While qualitative features have been reported previously, we show in precise spatial and temporal detail waveform evolution along the flagellum. In low viscosity the flagellum continuously moves out of the focal plane, compromising the measurement of true curvature, nonetheless the presence of torsion can be inferred. In high viscosities curvature can be accurately determined and we show how waves propagate at approximately constant speed. Progressing waves increase in curvature approximately linearly except for a sharper increase over a distance approximately 20-27 microm from the head/midpiece junction. Curvature modulation, likely influenced by the outer dense fibres, creates the characteristic waveforms of high viscosity swimming, with remarkably effective cell progression against greatly increased resistance, even in high viscosity liquids. Assessment of motility in physiological viscosities will be essential in future basic research, studies of chemotaxis and novel diagnostics.

The counterbend phenomenon in flagellar axonemes and cross-linked filament bundles

Recent observations of flagellar counterbend in sea urchin sperm show that the mechanical induction of curvature in one part of a passive flagellum induces a compensatory countercurvature elsewhere. This apparent paradoxical effect cannot be explained using the standard elastic rod theory of Euler and Bernoulli, or even the more general Cosserat theory of rods. Here, we develop a geometrically exact mechanical model to describe the statics of microtubule bundles that is capable of predicting the curvature reversal events observed in eukaryotic flagella. This is achieved by allowing the interaction of deformations in different material directions, by accounting not only for structural bending, but also for the elastic forces originating from the internal cross-linking mechanics. Large-amplitude static configurations can be described analytically, and an excellent match between the model and the observed counterbend deformation was found. This allowed a simultaneous estimation of multiple sperm flagellum material parameters, namely the cross-linking sliding resistance, the bending stiffness, and the sperm head junction compliance ratio. We further show that small variations on the empirical conditions may induce discrepancies for the evaluation of the flagellar material quantities, so that caution is required when interpreting experiments. Finally, our analysis demonstrates that the counterbend emerges as a fundamental property of sliding resistance in cross-linked filamentous polymer bundles, which also suggests that cross-linking proteins may contribute to the regulation of the flagellar waveform in swimming sperm via counterbend mechanics.

Comment on the article by J. Elgeti, U. B. Kaupp, and G. Gompper: hydrodynamics of sperm cells near surfaces.

A recent study by Elgeti et al. used multiparticle collision dynamics to simulate a long-standing problem: the approach of sperm to surfaces, and subsequent accumulation. The authors highlight differences in their predictions with those of the earlier Stokes flow simulations of Smith et al. attributing the differences to methodological flaws in the earlier article. In this Comment, we discuss the criticisms leveled in detail, and review some recently published work that shows how species-specific details of cell morphology provides a more likely explanation for the differing predictions of the two studies. We also highlight experimental work that supports the study of Smith et al.

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.

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.

The mechanics of hyperactivation in adhered human sperm

Hyperactivation is an important phenomenon exhibited by mammalian sperm during the process of acquiring fertilization capacity. The majority of studies have focused on incubation-induced hyperactivation in non-human species, which typically differ in size, shape, and are more homogeneous than human sperm. We develop an alternative approach via drug-induction, using high-speed imaging and analysis of same-cell changes in the flagellar movement of adhered cells. Following stimulation with 4-aminopyridine, approximately two-thirds (21 of 34) of the cells analysed exhibited a waveform with a single characteristic frequency; in all cases, the frequency was lower than before stimulation. The remaining cells (13 of 34) exhibited a more complex motility with multiple-frequency modes. The lowest mode in all cases was lower than the frequency prior to stimulation. Flagellar bending increased in all cells following stimulation and was significantly greater in the multiple-frequency responders. Despite the increased bending, time-averaged hydrodynamic power dissipation decreased significantly when assessed across all cells, the effect being significantly greater in the multiple-frequency responders than single frequency. These results reveal the heterogeneity of responses of human sperm to a hyperactivating stimulus, the methodology being potentially useful for assessing dynamic responses to stimuli in human sperm, and physiological selection of cells for assisted reproduction.

On the optimal shape of magnetic swimmers

Magnetic actuation of elasto-magnetic devices has long been proposed as a simple way to propel fluid and achieve locomotion in environments dominated by viscous forces. Under the action of an oscillating magnetic field, a permanent magnet, when attached to an elastic tail, is able to generate bending waves and sufficient thrust for propulsion. We study the hydrodynamical effects of the magnetic head geometry using a geometrically exact formulation for the elastic tail elastohydrodynamics.We show that the spherical head geometry fails to take full advantage of the propulsive potential from the flexible tail. Nevertheless, while elongated prolate spheroids demonstrate a superior swimming performance, this is still regulated by the nature of the imposed magnetic field. Interestingly, the highest swimming speed is observed when the magnitude of the magnetic field is weak due to delays between the orientation of the magnetic moment and the oscillating magnetic field. This allows the stored elastic energy from the deformed tail to relax during the phase lag between the imposed magnetic field and the swimmer’s magnetic moment, favouring in this way the net propulsion. In particular, this result suggests the existence of optimal magnetic actuations that are non-smooth, and even discontinuous in time, in order to fully explore the propulsive potential associated with the relaxation dynamics of periodically deformed elastic filaments.

Nonlinear amplitude dynamics in flagellar beating

The physical basis of flagellar and ciliary beating is a major problem in biology which is still far from completely understood. The fundamental cytoskeleton structure of cilia and flagella is the axoneme, a cylindrical array of microtubule doublets connected by passive cross-linkers and dynein motor proteins. The complex interplay of these elements leads to the generation of self-organized bending waves. Although many mathematical models have been proposed to understand this process, few attempts have been made to assess the role of dyneins on the nonlinear nature of the axoneme. Here, we investigate the nonlinear dynamics of flagella by considering an axonemal sliding control mechanism for dynein activity. This approach unveils the nonlinear selection of the oscillation amplitudes, which are typically either missed or prescribed in mathematical models. The explicit set of nonlinear equations are derived and solved numerically. Our analysis reveals the spatio-temporal dynamics of dynein populations and flagellum shape for different regimes of motor activity, medium viscosity and flagellum elasticity. Unstable modes saturate via the coupling of dynein kinetics and flagellum shape without the need of invoking a nonlinear axonemal response. Hence, our work reveals a novel mechanism for the saturation of unstable modes in axonemal beating.

Subharmonic oscillations of collective molecular motors

We study a generic two-state model for an assembly of molecular motors which is described by means of a pair of integro-partial differential equations and leads to oscillatory motion in the presence of an elastic coupling to its environment. We discuss a reduction of the system to a minimal set of three ordinary differential equations that successfully capture the complex nonlinear dynamics of the full system. In the limit of high mobility and large elastic modulus, we report on the emergence of subharmonics in the power spectrum of the oscillations. This provides a rationale for the unexplained observation of secondary peaks in a minimal actomyosin system in vitro (Placais P.-Y.et al., Phys. Rev. Lett. 103, (2009) 158102), showing that the phenomenon is robust and genuine. editors choice Copyright c EPLA, 2014

Human sperm swimming in a high viscosity mucus analogue

Remarkably, mammalian sperm maintain a substantive proportion of their progressive swimming speed within highly viscous fluids, including those of the female reproductive tract. Here, we analyse the digital microscopy of a human sperm swimming in a highly viscous, weakly elastic mucus analogue. We exploit principal component analysis to simplify its flagellar beat pattern, from which boundary element calculations are used to determine the time-dependent flow field around the sperm cell. The sperm flow field is further approximated in terms of regularised point forces, and estimates of the mechanical power consumption are determined, for comparison with analogous low viscosity media studies. This highlights extensive differences in the structure of the flows surrounding human sperm in different media, indicating how the cell-cell and cell-boundary hydrodynamic interactions significantly differ with the physical microenvironment. The regularised point force decomposition also provides cell-level information that may ultimately be incorporated into sperm population models. We further observe indications that the core feature in explaining the effectiveness of sperm swimming in high viscosity media is the loss of cell yawing, which is related with a greater density of regularised point force singularities along the axis of symmetry of the flagellar beat to represent the flow field. In turn this implicates a reduction of the wavelength of the distal beat pattern - and hence dynamical wavelength selection of the flagellar beat - as the dominant feature governing the effectiveness of sperm swimming in highly viscous media.