Egg-sperm interaction in sturgeon: role of ovarian fluid

Fertilization of freshwater fish occurs in the environment which negatively affects a lifespan of gametes mostly due to the osmotic shock; therefore, male gametes should reach the female gamete, as soon as possible. The existence of mechanisms controlling the encounter of gametes would be highly expedient in this case. By analogy with other species for which guidance was demonstrated, it is likely that this control may be performed by ovarian fluid or substances released by eggs. The aim was to study the effect of ovarian fluid and egg-released substances on spermatozoa behavior in sterlet. It was found that the presence of a particular concentration of ovarian fluid (30% solution in water) had an inhibiting effect on spermatozoa motility initiation. Lower concentrations of the ovarian fluid improved the longevity of spermatozoa and did not affect their trajectories. Test of chemotactic response (using a microcapillary injection of fluids into the suspension of motile spermatozoa) showed no effect of ovarian fluid on spermatozoa behavior, while at the same time, the attracting effect of the egg-conditioned medium was evident (i.e., due to some substances released from the eggs during their contact with freshwater). The results of the fertilization test showed that the presence of ovarian fluid prevented the eggs from losing the fertilizing ability due to the contact with water, as well as promoted the spermatozoa to fertilize the eggs during a longer period of time. Thus, the combined physicochemical action of "female factors" affects sterlet gametes during fertilization and may be involved in the guidance and selection mechanisms.

Long-term segmentation-free assessment of head-flagellum movement and intracellular calcium in swimming human sperm

Human spermatozoa are the archetype of long-term self-organizing transport in nature and are critical for reproductive success. They utilize coordinated head and flagellar movements to swim long distances within the female reproductive tract in order to find and fertilize the egg. However, to date, long-term analysis of the sperm head-flagellar movements, or indeed those of other flagellated microorganisms, remains elusive due to limitations in microscopy and flagellar-tracking techniques. Here, we present a novel methodology based on local orientation and isotropy of bio-images to obtain long-term kinematic and physiological parameters of individual free-swimming spermatozoa without requiring image segmentation (thresholding). This computer-assisted segmentation-free method evaluates, for the first time, characteristics of the head movement and flagellar beating for up to 9.2 min. We demonstrate its powerful use by showing how releasing Ca²⁺ from internal stores significantly alters long-term sperm behavior. The method allows for straightforward generalization to other bio-imaging applications, such as studies of bull sperm and Trypanosoma, or indeed of other flagellated microorganisms - appealing to communities other than those investigating sperm biology.

Expanding the Limits of Computer-Assisted Sperm Analysis through the Development of Open Software

Computer assisted sperm analysis (CASA) systems can reduce errors occurring in manual analysis. However, commercial CASA systems are frequently not applicable at the forefront of challenging research endeavors. The development of open source software may offer important solutions for researchers working in related areas. Here, we present an example of this, with the development of three new modules for the OpenCASA software (hosted at Github). The first is the Chemotactic Sperm Accumulation Module, a powerful tool for studying sperm chemotactic behavior, analyzing the sperm accumulation in the direct vicinity of the stimuli. This module was validated by comparing fish sperm accumulation, with or without the influence of an attractant. The analysis clearly indicated cell accumulation in the treatment group, while the distribution of sperm was random in the control group. The second is the Sperm Functionality Module, based on the ability to recognize five sperm subpopulations according to their fluorescence patterns associated with the plasma membrane and acrosomal status. The last module is the Sperm Concentration Module, which expands the utilities of OpenCASA. These last two modules were validated, using bull sperm, by comparing them with visual counting by an observer. A high level of correlation was achieved in almost all the data, and a good agreement between both methods was obtained. With these newly developed modules, OpenCASA is consolidated as a powerful free and open-source tool that allows different aspects of sperm quality to be evaluated, with many potential applications for researchers.

Human sperm uses asymmetric and anisotropic flagellar controls to regulate swimming symmetry and cell steering

Flagellar beating drives sperm through the female reproductive tract and is vital for reproduction. Flagellar waves are generated by thousands of asymmetric molecular components; yet, paradoxically, forward swimming arises via symmetric side-to-side flagellar movement. This led to the preponderance of symmetric flagellar control hypotheses. However, molecular asymmetries must still dictate the flagellum and be manifested in the beat. Here, we reconcile molecular and microscopic observations, reconnecting structure to function, by showing that human sperm uses asymmetric and anisotropic controls to swim. High-speed three-dimensional (3D) microscopy revealed two coactive transversal controls: An asymmetric traveling wave creates a one-sided stroke, and a pulsating standing wave rotates the sperm to move equally on all sides. Symmetry is thus achieved through asymmetry, creating the optical illusion of bilateral symmetry in 2D microscopy. This shows that the sperm flagellum is asymmetrically controlled and anisotropically regularized by fast-signal transduction. This enables the sperm to swim forward.

Flagellar ultrastructure suppresses buckling instabilities and enables mammalian sperm navigation in high-viscosity media

Eukaryotic flagellar swimming is driven by a slender motile unit, the axoneme, which possesses an internal structure that is essentially conserved in a tremendous diversity of sperm. Mammalian sperm, however, which are internal fertilizers, also exhibit distinctive accessory structures that further dress the axoneme and alter its mechanical response. This raises the following two fundamental questions. What is the functional significance of these structures? How do they affect the flagellar waveform and ultimately cell swimming? Hence we build on previous work to develop a mathematical mechanical model of a virtual human sperm to examine the impact of mammalian sperm accessory structures on flagellar dynamics and motility. Our findings demonstrate that the accessory structures reinforce the flagellum, preventing waveform compression and symmetry-breaking buckling instabilities when the viscosity of the surrounding medium is increased. This is in agreement with previous observations of internal and external fertilizers, such as human and sea urchin spermatozoa. In turn, possession of accessory structures entails that the progressive motion during a flagellar beat cycle can be enhanced as viscosity is increased within physiological bounds. Hence the flagella of internal fertilizers, complete with accessory structures, are predicted to be advantageous in viscous physiological media compared with watery media for the fundamental role of delivering a genetic payload to the egg.

Physical Biomarkers of Disease Progression: On-Chip Monitoring of Changes in Mechanobiology of Colorectal Cancer Cells

Disease can induce changes to subcellular components, altering cell phenotype and leading to measurable bulk-material mechanical properties. The mechanical phenotyping of single cells therefore offers many potential diagnostic applications. Cells are viscoelastic and their response to an applied stress is highly dependent on the magnitude and timescale of the actuation. Microfluidics can be used to measure cell deformability over a wide range of flow conditions, operating two distinct flow regimes (shear and inertial) which can expose subtle mechanical properties arising from subcellular components. Here, we investigate the deformability of three colorectal cancer (CRC) cell lines using a range of flow conditions. These cell lines offer a model for CRC metastatic progression; SW480 derived from primary adenocarcinoma, HT29 from a more advanced primary tumor and SW620 from lymph-node metastasis. HL60 (leukemia cells) were also studied as a model circulatory cell, offering a non-epithelial comparison. We demonstrate that microfluidic induced flow deformation can be used to robustly detect mechanical changes associated with CRC progression. We also show that single-cell multivariate analysis, utilising deformation and relaxation dynamics, offers potential to distinguish these different cell types. These results point to the benefit of multiparameter determination for improving detection and accuracy of disease stage diagnosis.

The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella

The inertialess fluid-structure interactions of active and passive inextensible filaments and slender-rods are ubiquitous in nature, from the dynamics of semi-flexible polymers and cytoskeletal filaments to cellular mechanics and flagella. The coupling between the geometry of deformation and the physical interaction governing the dynamics of bio-filaments is complex. Governing equations negotiate elastohydrodynamical interactions with non-holonomic constraints arising from the filament inextensibility. Such elastohydrodynamic systems are structurally convoluted, prone to numerical errors, thus requiring penalization methods and high-order spatio-temporal propagators. The asymptotic coarse-graining formulation presented here exploits the momentum balance in the asymptotic limit of small rod-like elements which are integrated semi-analytically. This greatly simplifies the elastohydrodynamic interactions and overcomes previous numerical instability. The resulting matricial system is straightforward and intuitive to implement, and allows for a fast and efficient computation, more than a hundred times faster than previous schemes. Only basic knowledge of systems of linear equations is required, and implementation achieved with any solver of choice. Generalizations for complex interaction of multiple rods, Brownian polymer dynamics, active filaments and non-local hydrodynamics are also straightforward. We demonstrate these in four examples commonly found in biological systems, including the dynamics of filaments and flagella. Three of these systems are novel in the literature. We additionally provide a Matlab code that can be used as a basis for further generalizations.

Multiparameter antibiotic resistance detection based on hydrodynamic trapping of individual E. coli

There is an urgent need to develop novel methods for assessing the response of bacteria to antibiotics in a timely manner. Antibiotics are traditionally assessed via their effect on bacteria in a culture medium, which takes 24-48 h and exploits only a single parameter, i.e. growth. Here, we present a multiparameter approach at the single-cell level that takes approximately an hour from spiking the culture to correctly classify susceptible and resistant strains. By hydrodynamically trapping hundreds of bacteria, we simultaneously monitor the evolution of motility and morphology of individual bacteria upon drug administration. We show how this combined detection method provides insights into the activity of antimicrobials at the onset of their action which single parameter and traditional tests cannot offer. Our observations complement the current growth-based methods and highlight the need for future antimicrobial susceptibility tests to take multiple parameters into account.

Cells Under Stress: An Inertial-Shear Microfluidic Determination of Cell Behavior

The deformability of a cell is the direct result of a complex interplay between the different constituent elements at the subcellular level, coupling a wide range of mechanical responses at different length scales. Changes to the structure of these components can also alter cell phenotype, which points to the critical importance of cell mechanoresponse for diagnostic applications. The response to mechanical stress depends strongly on the forces experienced by the cell. Here, we use cell deformability in both shear-dominant and inertia-dominant microfluidic flow regimes to probe different aspects of the cell structure. In the inertial regime, we follow cellular response from (visco-)elastic through plastic deformation to cell structural failure and show a significant drop in cell viability for shear stresses >11.8 kN/m2. Comparatively, a shear-dominant regime requires lower applied stresses to achieve higher cell strains. From this regime, deformation traces as a function of time contain a rich source of information including maximal strain, elastic modulus, and cell relaxation times and thus provide a number of markers for distinguishing cell types and potential disease progression. These results emphasize the benefit of multiple parameter determination for improving detection and will ultimately lead to improved accuracy for diagnosis. We present results for leukemia cells (HL60) as a model circulatory cell as well as for a colorectal cancer cell line, SW480, derived from primary adenocarcinoma (Dukes stage B). SW480 were also treated with the actin-disrupting drug latrunculin A to test the sensitivity of flow regimes to the cytoskeleton. We show that the shear regime is more sensitive to cytoskeletal changes and that large strains in the inertial regime cannot resolve changes to the actin cytoskeleton.

The counterbend dynamics of cross-linked filament bundles and flagella

Cross-linked filament bundles, such as in cilia and flagella, are ubiquitous in biology. They are considered in textbooks as simple filaments with larger stiffness. Recent observations of flagellar counterbend, however, show that induction of curvature in one section of a passive flagellum instigates a compensatory counter-curvature elsewhere, exposing the intricate role of the diminutive cross-linking proteins at large scales. We show that this effect, a material property of the cross-linking mechanics, modifies the bundle dynamics and induces a bimodal L² - L³ length-dependent material response that departs from the Euler-Bernoulli theory. Hence, the use of simpler theories to analyse experiments can result in paradoxical interpretations. Remarkably, the counterbend dynamics instigates counter-waves in opposition to driven oscillations in distant parts of the bundle, with potential impact on the regulation of flagellar bending waves. These results have a range of physical and biological applications, including the empirical disentanglement of material quantities via counterbend dynamics.

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.

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.

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.

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.

Development of tip-splitting and side-branching patterns in elastic fingering

Elastic fingering supplements the already interesting features of the traditional viscous fingering phenomena in Hele-Shaw cells with the consideration that the two-fluid separating boundary behaves like an elastic membrane. Sophisticated numerical simulations have shown that under maximum viscosity contrast the resulting patterned shapes can exhibit either finger tip-splitting or side-branching events. In this work, we employ a perturbative mode-coupling scheme to get important insights into the onset of these pattern formation processes. This is done at lowest nonlinear order and by considering the interplay of just three specific Fourier modes: a fundamental mode n and its harmonics 2n and 3n. Our approach further allows the construction of a morphology diagram for the system in a wide range of the parameter space without requiring expensive numerical simulations. The emerging interfacial patterns are conveniently described in terms of only two dimensionless controlling quantities: the rigidity fraction C and a parameter Γ that measures the relative strength between elastic and viscous effects. Visualization of the rigidity field for the various pattern-forming structures supports the idea of an elastic weakening mechanism that facilitates finger growth in regions of reduced interfacial bending rigidity.

Fock space, symbolic algebra, and analytical solutions for small stochastic systems

Randomness is ubiquitous in nature. From single-molecule biochemical reactions to macroscale biological systems, stochasticity permeates individual interactions and often regulates emergent properties of the system. While such systems are regularly studied from a modeling viewpoint using stochastic simulation algorithms, numerous potential analytical tools can be inherited from statistical and quantum physics, replacing randomness due to quantum fluctuations with low-copy-number stochasticity. Nevertheless, classical studies remained limited to the abstract level, demonstrating a more general applicability and equivalence between systems in physics and biology rather than exploiting the physics tools to study biological systems. Here the Fock space representation, used in quantum mechanics, is combined with the symbolic algebra of creation and annihilation operators to consider explicit solutions for the chemical master equations describing small, well-mixed, biochemical, or biological systems. This is illustrated with an exact solution for a Michaelis-Menten single enzyme interacting with limited substrate, including a consideration of very short time scales, which emphasizes when stiffness is present even for small copy numbers. Furthermore, we present a general matrix representation for Michaelis-Menten kinetics with an arbitrary number of enzymes and substrates that, following diagonalization, leads to the solution of this ubiquitous, nonlinear enzyme kinetics problem. For this, a flexible symbolic maple code is provided, demonstrating the prospective advantages of this framework compared to stochastic simulation algorithms. This further highlights the possibilities for analytically based studies of stochastic systems in biology and chemistry using tools from theoretical quantum physics.

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.

Stationary patterns in centrifugally driven interfacial elastic fingering

A vortex sheet formalism is used to search for equilibrium shapes in the centrifugally driven interfacial elastic fingering problem. We study the development of interfacial instabilities when a viscous fluid surrounded by another of smaller density flows in the confined environment of a rotating Hele-Shaw cell. The peculiarity of the situation is associated to the fact that, due to a chemical reaction, the two-fluid boundary becomes an elastic layer. The interplay between centrifugal and elastic forces leads to the formation of a rich variety of stationary shapes. Visually striking equilibrium morphologies are obtained from the numerical solution of a nonlinear differential equation for the interface curvature (the shape equation), determined by a zero vorticity condition. Classification of the various families of shapes is made via two dimensionless parameters: an effective bending rigidity (ratio of elastic to centrifugal effects) and a geometrical radius of gyration.

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.

Elastic fingering in rotating Hele-Shaw flows

The centrifugally driven viscous fingering problem arises when two immiscible fluids of different densities flow in a rotating Hele-Shaw cell. In this conventional setting an interplay between capillary and centrifugal forces makes the fluid-fluid interface unstable, leading to the formation of fingered structures that compete dynamically and reach different lengths. In this context, it is known that finger competition is very sensitive to changes in the viscosity contrast between the fluids. We study a variant of such a rotating flow problem where the fluids react and produce a gellike phase at their separating boundary. This interface is assumed to be elastic, presenting a curvature-dependent bending rigidity. A perturbative weakly nonlinear approach is used to investigate how the elastic nature of the interface affects finger competition events. Our results unveil a very different dynamic scenario, in which finger length variability is not regulated by the viscosity contrast, but rather determined by two controlling quantities: a characteristic radius and a rigidity fraction parameter. By properly tuning these quantities one can describe a whole range of finger competition behaviors even if the viscosity contrast is kept unchanged.

Interfacial elastic fingering in Hele-Shaw cells: a weakly nonlinear study

We study a variant of the classic viscous fingering instability in Hele-Shaw cells where the interface separating the fluids is elastic, and presents a curvature-dependent bending rigidity. By employing a second-order mode-coupling approach we investigate how the elastic nature of the interface influences the morphology of emerging interfacial patterns. This is done by focusing our attention on a conventionally stable situation in which the fluids involved have the same viscosity. In this framework, we show that the inclusion of nonlinear effects plays a crucial role in inducing sizable interfacial instabilities, as well as in determining the ultimate shape of the pattern-forming structures. Particularly, we have found that the emergence of either narrow or wide fingers can be regulated by tuning a rigidity fraction parameter. Our weakly nonlinear findings reinforce the importance of the so-called curvature weakening effect, which favors the development of fingers in regions of lower rigidity.

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.

Coriolis effects on rotating Hele-Shaw flows: a conformal-mapping approach

The zero surface tension fluid-fluid interface dynamics in a radial Hele-Shaw cell driven by both injection and rotation is studied by a conformal-mapping approach. The situation in which one of the fluids is inviscid and has negligible density is analyzed. When Coriolis force effects are ignored, exact solutions of the zero surface tension rotating Hele-Shaw problem with injection reveal suppression of cusp singularities for sufficiently high rotation rates. We study how the Coriolis force affects the time-dependent solutions of the problem, and the development of finite time singularities. By employing Richardson's harmonic moments approach we obtain conformal maps which describe the time evolution of the fluid boundary. Our results demonstrate that the inertial Coriolis contribution plays an important role in determining the time for cusp formation. Moreover, it introduces a phase drift that makes the evolving patterns rotate. The Coriolis force acts against centrifugal effects, promoting (inhibiting) cusp breakdown if the more viscous and dense fluid lies outside (inside) the interface. Despite the presence of Coriolis effects, the occurrence of finger bending events has not been detected in the exact solutions.

Nonlinear instability in flagellar dynamics: a novel modulation mechanism in sperm migration?

Throughout biology, cells and organisms use flagella and cilia to propel fluid and achieve motility. The beating of these organelles, and the corresponding ability to sense, respond to and modulate this beat is central to many processes in health and disease. While the mechanics of flagellum-fluid interaction has been the subject of extensive mathematical studies, these models have been restricted to being geometrically linear or weakly nonlinear, despite the high curvatures observed physiologically. We study the effect of geometrical nonlinearity, focusing on the spermatozoon flagellum. For a wide range of physiologically relevant parameters, the nonlinear model predicts that flagellar compression by the internal forces initiates an effective buckling behaviour, leading to a symmetry-breaking bifurcation that causes profound and complicated changes in the waveform and swimming trajectory, as well as the breakdown of the linear theory. The emergent waveform also induces curved swimming in an otherwise symmetric system, with the swimming trajectory being sensitive to head shape-no signalling or asymmetric forces are required. We conclude that nonlinear models are essential in understanding the flagellar waveform in migratory human sperm; these models will also be invaluable in understanding motile flagella and cilia in other systems.

Effects of normal viscous stresses on radial viscous fingering

We revisit the radial viscous fingering problem in a Hele-Shaw cell, and consider the action of viscous stresses originated from velocity gradients normal to the fluid-fluid interface. The evolution of the interface during linear and weakly nonlinear stages is described analytically through a mode-coupling approach. We find that the introduction of normal stresses influences the stability and the ultimate morphology of the emerging patterns. Although at early stages normal stresses tend to stabilize the interface, they act to favor the development of tip-splitting phenomena at the weakly nonlinear regime. We have also verified that finger competition events are only significantly affected by normal stresses for circumstances involving the development of a large number of interfacial fingers.

Coriolis effects on fingering patterns under rotation

The development of immiscible viscous fingering patterns in a rotating Hele-Shaw cell is investigated. We focus on understanding how the time evolution and the resulting morphologies are affected by the action of the Coriolis force. The problem is approached analytically and numerically by employing a vortex sheet formalism. The vortex sheet strength and a linear dispersion relation are derived analytically, revealing that the most relevant Coriolis force contribution comes from the normal component of the averaged interfacial velocity. It is shown that this normal velocity, uniquely due to the presence of the Coriolis force, is responsible for the complex-valued nature of the linear dispersion relation making the linear phases vary with time. Fully nonlinear stages are studied through intensive numerical simulations. A suggestive interplay between inertial and viscous effects is found, which modifies the dynamics, leading to different pattern-forming structures. The inertial Coriolis contribution plays a characteristic role: it generates a phase drift by deviating the fingers in the sense opposite to the actual rotation of the cell. However, the direction and intensity of finger bending is predominantly determined by viscous effects, being sensitive to changes in the magnitude and sign of the viscosity contrast. The finger competition behavior at advanced time stages is also discussed.

Radial viscous fingering in miscible Hele-Shaw flows: a numerical study

A modified version of the usual viscous fingering problem in a radial Hele-Shaw cell with immiscible fluids is studied by intensive numerical simulations. We consider the situation in which the fluids involved are miscible, so that the diffusing interface separating them can be driven unstable through the injection or suction of the inner fluid. The system is allowed to rotate in such a way that centrifugal and Coriolis forces come into play, imposing important changes on the morphology of the arising patterns. In order to bridge from miscible to immiscible pattern forming structures, we add the surface tensionlike effects due to Korteweg stresses. Our numerical experiments reveal a variety of interesting fingering behaviors, which depend on the interplay between injection (or suction), diffusive, rotational, and Korteweg stress effects. Whenever possible the features of the simulated miscible fronts are contrasted to existing experiments and other theoretical or numerical studies, usually resulting in close agreements. A number of additional complex morphologies, whose experimental realization is still not available, are predicted and discussed.

Dynamics of viscous fingers in rotating Hele-Shaw cells with Coriolis effects

A growing number of experimental and theoretical works have been addressing various aspects of the viscous fingering formation in rotating Hele-Shaw cells. However, only a few of them consider the influence of Coriolis forces. The studies including Coriolis effects are mostly restricted to the high-viscosity-contrast limit and rely on either purely linear stability analyses or intensive numerical simulations. We approach the problem analytically and use a modified Darcy's law including the exact form of the Coriolis effects to execute a mode-coupling analysis of the system. By imposing no restrictions on the viscosity contrast A (dimensionless viscosity difference) we go beyond linear stages and examine the onset of nonlinearities. Our results indicate that when Coriolis effects are taken into account, an interesting interplay between the Reynolds number Re and A arises. This leads to important changes in the stability and morphological features of the emerging interfacial patterns. We contrast our mode-coupling approach with previous theoretical models proposed in the literature.

Finger competition dynamics in rotating Hele-Shaw cells

We report analytical results for the development of interfacial instabilities in rotating Hele-Shaw cells. We execute a mode-coupling approach to the problem and examine the morphological features of the fluid-fluid interface at the onset of nonlinear effects. The impact of normal stresses is accounted for through a modified pressure jump boundary condition. A differential equation describing the early nonlinear evolution of the interface is derived, being conveniently written in terms of three relevant dimensionless parameters: viscosity contrast A , surface tension B , and gap spacing b . We focus our study on the influence of these parameters on finger competition dynamics. It is deduced that the link between finger competition and A , B , and b can be revealed by a mechanism based on the enhanced growth of subharmonic perturbations. Our results show good agreement with existing experimental and numerical investigations of the problem both in low and high A<0 limits. In particular, it is found that the condition of vanishing A suppresses the dynamic competition between fingers, regardless of the value of B and b . Moreover, our study enables one to extract analytical information about the problem by exploring the whole range of allowed values for A , B , and b . Specifically, it is verified that pattern morphology is significantly modified when the viscosity contrast -1< or =A< or =1 varies: increasingly larger values of A>0 (A<0) lead to enhanced competition of outward (inward) fingers. Within this context the role of B and b in determining different finger competition behaviors is also discussed.