The computational sperm cell

Reviewing mathematical models of sperm signaling networks

Dave Garbers' work significantly contributed to our understanding of sperm's regulated motility, capacitation, and the acrosome reaction. These key sperm functions involve complex multistep signaling pathways engaging numerous finely orchestrated elements. Despite significant progress, many parameters and interactions among these elements remain elusive. Mathematical modeling emerges as a potent tool to study sperm physiology, providing a framework to integrate experimental results and capture functional dynamics considering biochemical, biophysical, and cellular elements. Depending on research objectives, different modeling strategies, broadly categorized into continuous and discrete approaches, reveal valuable insights into cell function. These models allow the exploration of hypotheses regarding molecules, conditions, and pathways, whenever they become challenging to evaluate experimentally. This review presents an overview of current theoretical and experimental efforts to understand sperm motility regulation, capacitation, and the acrosome reaction. We discuss the strengths and weaknesses of different modeling strategies and highlight key findings and unresolved questions. Notable discoveries include the importance of specific ion channels, the role of intracellular molecular heterogeneity in capacitation and the acrosome reaction, and the impact of pH changes on acrosomal exocytosis. Ultimately, this review underscores the crucial importance of mathematical frameworks in advancing our understanding of sperm physiology and guiding future experimental investigations.

Viscous Loading Regulates the Flagellar Energetics of Human and Bull Sperm

The viscoelastic properties of the female reproductive tract influence sperm swimming behavior, but the exact role of these rheological changes in regulating sperm energetics remains unknown. Using high-speed dark-field microscopy, the flagellar dynamics of free-swimming sperm across a physiologically relevant range of viscosities is resolved. A transition from 3D to 2D slither swimming under an increased viscous loading is revealed, in the absence of any geometrical or chemical stimuli. This transition is species-specific, aligning with viscosity variations within each species' reproductive tract. Despite substantial drag increase, 2D slithering sperm maintain a steady swimming speed across a wide viscosity range (20-250 and 75-1000 mPa s for bull and human sperm) by dissipating over sixfold more energy into the fluid without elevating metabolic activity, potentially by altering the mechanisms of dynein motor activity. This energy-efficient motility mode is ideally suited for the viscous environment of the female reproductive tract.

Learning optimal integration of spatial and temporal information in noisy chemotaxis

We investigate the boundary between chemotaxis driven by spatial estimation of gradients and chemotaxis driven by temporal estimation. While it is well known that spatial chemotaxis becomes disadvantageous for small organisms at high noise levels, it is unclear whether there is a discontinuous switch of optimal strategies or a continuous transition exists. Here, we employ deep reinforcement learning to study the possible integration of spatial and temporal information in an a priori unconstrained manner. We parameterize such a combined chemotactic policy by a recurrent neural network and evaluate it using a minimal theoretical model of a chemotactic cell. By comparing with constrained variants of the policy, we show that it converges to purely temporal and spatial strategies at small and large cell sizes, respectively. We find that the transition between the regimes is continuous, with the combined strategy outperforming in the transition region both the constrained variants as well as models that explicitly integrate spatial and temporal information. Finally, by utilizing the attribution method of integrated gradients, we show that the policy relies on a nontrivial combination of spatially and temporally derived gradient information in a ratio that varies dynamically during the chemotactic trajectories.

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.

Human fertilization in vivo and in vitro requires the CatSper channel to initiate sperm hyperactivation

The infertility of many couples rests on an enigmatic dysfunction of the man's sperm. To gain insight into the underlying pathomechanisms, we assessed the function of the sperm-specific multisubunit CatSper-channel complex in the sperm of almost 2,300 men undergoing a fertility workup, using a simple motility-based test. We identified a group of men with normal semen parameters but defective CatSper function. These men or couples failed to conceive naturally and upon medically assisted reproduction via intrauterine insemination and in vitro fertilization. Intracytoplasmic sperm injection (ICSI) was, ultimately, required to conceive a child. We revealed that the defective CatSper function was caused by variations in CATSPER genes. Moreover, we unveiled that CatSper-deficient human sperm were unable to undergo hyperactive motility and, therefore, failed to penetrate the egg coat. Thus, our study provides the experimental evidence that sperm hyperactivation is required for human fertilization, explaining the infertility of CatSper-deficient men and the need of ICSI for medically assisted reproduction. Finally, our study also revealed that defective CatSper function and ensuing failure to hyperactivate represents the most common cause of unexplained male infertility known thus far and that this sperm channelopathy can readily be diagnosed, enabling future evidence-based treatment of affected couples.

Inverted Sedimentation of Active Particles in Unbiased ac Fields

Gaining control over the motion of active particles is crucial for applications ranging from targeted cargo delivery to nanomedicine. While much progress has been made recently to control active motion based on external forces, flows, or gradients in concentration or light intensity, which all have a well-defined direction or bias, little is known about how to steer active particles in situations where no permanent bias can be realized. Here, we show that ac fields with a vanishing time average provide an alternative route to steering active particles. We exemplify this route for inertial active particles in a gravitational field, observing that a substantial fraction of them persistently travels in the upward direction upon switching on the ac field, resulting in an inverted sedimentation profile at the top wall of a confining container. Our results offer a generic control principle that could be used in the future to steer active motion, direct collective behaviors, and purify mixtures.

Efficient Preparation of a Magnetic Helical Carbon Nanomotor for Targeted Anticancer Drug Delivery

The applications of nanomotors in the biomedical field have been attracting extensive attention. However, it remains a challenge to fabricate nanomotors in a facile way and effectively load drugs for active targeted therapy. In this work, we combine the microwave heating method and chemical vapor deposition (CVD) to fabricate magnetic helical nanomotors efficiently. The microwave heating method can accelerate intermolecular movement, which converts kinetic energy into heat energy and shortens the preparation time of the catalyst used for carbon nanocoil (CNC) synthesis by 15 times. Fe₃O₄ nanoparticles are in situ nucleated on the CNC surface by the microwave heating method to fabricate magnetically driven CNC/Fe₃O₄ nanomotors. In addition, we achieved precise control of the magnetically driven CNC/Fe₃O₄ nanomotors through remote manipulation of magnetic fields. Anticancer drug doxorubicin (DOX) is then efficiently loaded onto the nanomotors via π-π stacking interactions. Finally, the drug-loaded CNC/Fe₃O₄@DOX nanomotor can accurately accomplish cell targeting under external magnetic field control. Under short-time irradiation of near-infrared light, DOX can be quickly released onto target cells to effectively kill the cells. More importantly, CNC/Fe₃O₄@DOX nanomotors allow for single-cell or cell-cluster-targeted anticancer drug delivery, providing a dexterous platform to potentially perform many medically relevant tasks in vivo. The efficient preparation method and application in drug delivery are beneficial for future industrial production and provide inspiration for advanced micro/nanorobotic systems using the CNC as a carrier for a wide range of biomedical applications.

Synthesis and Functional Characterization of Novel RU1968-Derived CatSper Inhibitors with Reduced Stereochemical Complexity

The sperm-specific Ca2+ channel CatSper (cation channel of sperm) controls the intracellular Ca2+ concentration and, thereby, the swimming behavior of sperm from many species. The steroidal ethylenediamine RU1968 (1) represents a well-characterized, potent, and fairly selective cross-species inhibitor of CatSper. Due to its two additional centers of chirality in the amine-bearing side chain, RU1968 is a mixture of diastereomeric pairs of enantiomers and, thus, difficult to synthesize. This has hampered the use of this commercially not available inhibitor as a powerful tool for research. Here, simplifying both structure and synthesis, we introduced novel stereochemically less complex and enantiomerically pure aminomethyl RU1968 analogues lacking the C-21 CH3 moiety. Starting from (+)-estrone, a five-step synthesis was developed comprising a Wittig reaction as the key step, leading to a diastereomerically pure 17β-configured aldehyde. Subsequent reductive amination yielded diastereomerically and enantiomerically pure amines. Compared to RU1968, the novel ethylenediamine 2d and homologous trimethylenediamine derivative 2e inhibited CatSper with similar and even twofold enhanced potency, respectively. Considering that these aminomethyl analogues are enantiomerically pure and much easier to synthesize than RU1968, we envisage their common use in future studies investigating the physiology of CatSper in sperm.

Sea urchin sperm exploit extremum seeking control to find the egg

Sperm cells perform extremely demanding tasks with minimal capabilities. The cells must quickly navigate in a noisy environment to find an egg within a short time window for successful fertilization without any global positioning information. Many research efforts have been dedicated to derive mathematical principles that explain their superb navigation strategy. Here we show that the navigation strategy of sea urchin sperm, also known as helical klinotaxis, is a natural implementation of a well-established adaptive control paradigm known as extremum seeking. This bridge between control theory and the biology of taxis in microorganisms is expected to deepen our understanding of the process. For example, the formulation leads to a coarse-grained model of the signaling pathway that offers new insights on the peculiar switching-like behavior between high- and low-gain steering modes observed in sea urchin sperm. Moreover, it may guide engineers in developing bioinspired miniaturized robots with minimal sensors.

Biophysical Determinants and Constraints on Sperm Swimming Velocity

Over the last 50 years, sperm competition has become increasingly recognised as a potent evolutionary force shaping male ejaculate traits. One such trait is sperm swimming speed, with faster sperm associated with increased fertilisation success in some species. Consequently, sperm are often thought to have evolved to be longer in order to facilitate faster movement. However, despite the intrinsic appeal of this argument, sperm operate in a different biophysical environment than we are used to, and instead increasing length may not necessarily be associated with higher velocity. Here, we test four predictive models (ConstantPower Density, Constant Speed, Constant Power Transfer, Constant Force) of the relationship between sperm length and speed. We collated published data on sperm morphology and velocity from 141 animal species, tested for structural clustering of sperm morphology and then compared the model predictions across all morphologically similar sperm clusters. Within four of five morphological clusters of sperm, we did not find a significant positive relationship between total sperm length and velocity. Instead, in four morphological sperm clusters we found evidence for the Constant Speed model, which predicts that power output is determined by the flagellum and so is proportional to flagellum length. Our results show the relationship between sperm morphology (size, width) and swimming speed is complex and that traditional models do not capture the biophysical interactions involved. Future work therefore needs to incorporate not only a better understanding of how sperm operate in the microfluid environment, but also the importance of fertilising environment, i.e., internal and external fertilisers. The microenvironment in which sperm operate is of critical importance in shaping the relationship between sperm length and form and sperm swimming speed.

A Stable Chemokine Gradient Controls Directional Persistence of Migrating Dendritic Cells

Navigation of dendritic cells (DCs) from the site of infection to lymphoid organs is guided by concentration gradients of CCR7 ligands. How cells interpret chemokine gradients and how they couple directional sensing to polarization and persistent chemotaxis has remained largely elusive. Previous experimental systems were limited in the ability to control fast de novo formation of the final gradient slope, long-lasting stability of the gradient and to expose cells to dynamic stimulation. Here, we used a combination of microfluidics and quantitative in vitro live cell imaging to elucidate the chemotactic sensing strategy of DCs. The microfluidic approach allows us to generate soluble gradients with high spatio-temporal precision and to analyze actin dynamics, cell polarization, and persistent directional migration in both static and dynamic environments. We demonstrate that directional persistence of DC migration requires steady-state characteristics of the soluble gradient instead of temporally rising CCL19 concentration, implying that spatial sensing mechanisms control chemotaxis of DCs. Kymograph analysis of actin dynamics revealed that the presence of the CCL19 gradient is essential to stabilize leading edge protrusions in DCs and to determine directionality, since both cytoskeletal polarization and persistent chemotaxis are abrogated in the range of seconds when steady-state gradients are perturbed. In contrast to Dictyostelium amoeba, DCs are unable to decode oscillatory stimulation of soluble chemokine traveling waves into a directional response toward the wave source. These findings are consistent with the notion that DCs do not employ adaptive temporal sensing strategies that discriminate temporally increasing and decreasing chemoattractant concentrations in our setting. Taken together, in our experimental system DCs do not depend on increasing absolute chemokine concentration over time to induce persistent migration and do not integrate oscillatory stimulation. The observed capability of DCs to migrate with high directional persistence in stable gradients but not when subjected to periodic temporal cues, identifies spatial sensing as a key requirement for persistent chemotaxis of DCs.

Bistability in orbital trajectories of a chiral self-propelled particle interacting with an external field

In this paper, the dynamics of a self-propelled stochastic particle under the influence of an axisymmetric light field is experimentally studied. The particle under consideration has the main characteristic of carrying a light sensor in an eccentric location. For the chosen experimental conditions, the emerging trajectories are orbital, and, more interestingly, they suggest the existence of bistability. A mathematical model incorporating the key experimental components is introduced. By means of numerical simulations and theoretical analysis, it is found that, in addition to the orbiting behavior, the sensor location could produce trapped or diffusive behaviors. Furthermore, the study reveals that stochastic perturbation and the eccentric location of the sensor are responsible for inducing bistability in the orbital trajectories, supporting experimental observations.

Pheromone Guidance of Polarity Site Movement in Yeast

Cells' ability to track chemical gradients is integral to many biological phenomena, including fertilization, development, accessing nutrients, and combating infection. Mating of the yeast Saccharomyces cerevisiae provides a tractable model to understand how cells interpret the spatial information in chemical gradients. Mating yeast of the two different mating types secrete distinct peptide pheromones, called a-factor and α-factor, to communicate with potential partners. Spatial gradients of pheromones are decoded to guide mobile polarity sites so that polarity sites in mating partners align towards each other, as a prerequisite for cell-cell fusion and zygote formation. In ascomycetes including S. cerevisiae, one pheromone is prenylated (a-factor) while the other is not (α-factor). The difference in physical properties between the pheromones, combined with associated differences in mechanisms of secretion and extracellular pheromone metabolism, suggested that the pheromones might differ in the spatial information that they convey to potential mating partners. However, as mating appears to be isogamous in this species, it is not clear why any such signaling difference would be advantageous. Here we report assays that directly track movement of the polarity site in each partner as a way to understand the spatial information conveyed by each pheromone. Our findings suggest that both pheromones convey very similar information. We speculate that the different pheromones were advantageous in ancestral species with asymmetric mating systems and may represent an evolutionary vestige in yeasts that mate isogamously.

In Silico Reconstruction of Sperm Chemotaxis

In echinoderms, sperm swims in random circles and turns in response to a chemoattractant. The chemoattractant evokes transient Ca²⁺ influx in the sperm flagellum and induces turning behavior. Recently, the molecular mechanisms and biophysical properties of this sperm response have been clarified. Based on these experimental findings, in this study, we reconstructed a sperm model in silico to demonstrate an algorithm for sperm chemotaxis. We also focused on the importance of desensitizing the chemoattractant receptor in long-range chemotaxis because sperm approach distantly located eggs, and they must sense the chemoattractant concentration over a broad range. Using parameters of the sea urchin, simulations showed that a number of sperm could reach the egg from millimeter-order distances with desensitization, indicating that we could organize a functional sperm model, and that desensitization of the receptor is essential for sperm chemotaxis. Then, we compared the model with starfish sperm, which has a different desensitization scheme and analyzed the properties of the model against various disturbances. Our approach can be applied as a novel tool in chemotaxis research.

Multifocal imaging for precise, label-free tracking of fast biological processes in 3D

Many biological processes happen on a nano- to millimeter scale and within milliseconds. Established methods such as confocal microscopy are suitable for precise 3D recordings but lack the temporal or spatial resolution to resolve fast 3D processes and require labeled samples. Multifocal imaging (MFI) allows high-speed 3D imaging but is limited by the compromise between high spatial resolution and large field-of-view (FOV), and the requirement for bright fluorescent labels. Here, we provide an open-source 3D reconstruction algorithm for multi-focal images that allows using MFI for fast, precise, label-free tracking spherical and filamentous structures in a large FOV and across a high depth. We characterize fluid flow and flagellar beating of human and sea urchin sperm with a z-precision of 0.15 µm, in a volume of 240 × 260 × 21 µm, and at high speed (500 Hz). The sampling volume allowed to follow sperm trajectories while simultaneously recording their flagellar beat. Our MFI concept is cost-effective, can be easily implemented, and does not rely on object labeling, which renders it broadly applicable.

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.

Reconstruction of the three-dimensional beat pattern underlying swimming behaviors of sperm

 The eukaryotic flagellum propels sperm cells and simultaneously detects physical and chemical cues that modulate the waveform of the flagellar beat. Most previous studies have characterized the flagellar beat and swimming trajectories in two space dimensions (2D) at a water/glass interface. Here, using refined holographic imaging methods, we report high-quality recordings of three-dimensional (3D) flagellar bending waves. As predicted by theory, we observed that an asymmetric and planar flagellar beat results in a circular swimming path, whereas a symmetric and non-planar flagellar beat results in a twisted-ribbon swimming path. During swimming in 3D, human sperm flagella exhibit torsion waves characterized by maxima at the low curvature regions of the flagellar wave. We suggest that these torsion waves are common in nature and that they are an intrinsic property of beating axonemes. We discuss how 3D beat patterns result in twisted-ribbon swimming paths. This study provides new insight into the axoneme dynamics, the 3D flagellar beat, and the resulting swimming behavior.

Sperm migration in the genital tract-In silico experiments identify key factors for reproductive success

Sperm migration in the female genital tract controls sperm selection and, therefore, reproductive success as male gametes are conditioned for fertilization while their number is dramatically reduced. Mechanisms underlying sperm migration are mostly unknown, since in vivo investigations are mostly unfeasible for ethical or practical reasons. By presenting a spatio-temporal model of the mammalian female genital tract combined with agent-based description of sperm motion and interaction as well as parameterizing it with bovine data, we offer an alternative possibility for studying sperm migration in silico. The model incorporates genital tract geometry as well as biophysical principles of sperm motion observed in vitro such as positive rheotaxis and thigmotaxis. This model for sperm migration from vagina to oviducts was successfully tested against in vivo data from literature. We found that physical sperm characteristics such as velocity and directional stability as well as sperm-fluid interactions and wall alignment are critical for success, i.e. sperms reaching the oviducts. Therefore, we propose that these identified sperm parameters should be considered in detail for conditioning sperm in artificial selection procedures since the natural processes are normally bypassed in reproductive in vitro technologies. The tremendous impact of mucus flow to support sperm accumulation in the oviduct highlights the importance of a species-specific optimum time window for artificial insemination regarding ovulation. Predictions from our extendable in silico experimental system will improve assisted reproduction in humans, endangered species, and livestock.

Microgel that swims to the beat of light

Complementary to the quickly advancing understanding of the swimming of microorganisms, we demonstrate rather simple design principles for systems that can mimic swimming by body shape deformation. For this purpose, we developed a microswimmer that could be actuated and controlled by fast temperature changes through pulsed infrared light irradiation. The construction of the microswimmer has the following features: (i) it is a bilayer ribbon with a length of 80 or 120 [Formula: see text]m, consisting of a thermo-responsive hydrogel of poly-N-isopropylamide coated with a 2-nm layer of gold and equipped with homogeneously dispersed gold nanorods; (ii) the width of the ribbon is linearly tapered with a wider end of 5 [Formula: see text]m and a tip of 0.5 [Formula: see text]m; (iii) a thickness of only 1 and 2 [Formula: see text]m that ensures a maximum variation of the cross section of the ribbon along its length from square to rectangular. These wedge-shaped ribbons form conical helices when the hydrogel is swollen in cold water and extend to a filament-like object when the temperature is raised above the volume phase transition of the hydrogel at [Formula: see text]. The two ends of these ribbons undergo different but coupled modes of motion upon fast temperature cycling through plasmonic heating of the gel-objects from inside. Proper choice of the IR-light pulse sequence caused the ribbons to move at a rate of 6 body length/s (500 [Formula: see text]m/s) with the wider end ahead. Within the confinement of rectangular container of 30 [Formula: see text]m height and 300 [Formula: see text]m width, the different modes can be actuated in a way that the movement is directed by the energy input between spinning on the spot and fast forward locomotion.

Multi-ciliated microswimmers-metachronal coordination and helical swimming

The dynamics and motion of multi-ciliated microswimmers with a spherical body and a small number N (with [Formula: see text]) of cilia with length comparable to the body radius, is investigated by mesoscale hydrodynamics simulations. A metachronal wave is imposed for the cilia beat, for which the wave vector has both a longitudinal and a latitudinal component. The dynamics and motion is characterized by the swimming velocity, its variation over the beat cycle, the spinning velocity around the main body axis, as well as the parameters of the helical trajectory. Our simulation results show that the microswimmer motion strongly depends on the latitudinal wave number and the longitudinal phase lag. The microswimmers are found to swim smoothly and usually spin around their own axis. Chirality of the metachronal beat pattern generically generates helical trajectories. In most cases, the helices are thin and stretched, i.e., the helix radius is about an order of magnitude smaller than the pitch. The rotational diffusion of the microswimmer is significantly smaller than the passive rotational diffusion of the body alone, which indicates that the extended cilia contribute strongly to the hydrodynamic radius. The swimming velocity is found to increase with the cilia number N with a slightly sublinear power law, consistent with the behavior expected from the dependence of the transport velocity of planar cilia arrays on the cilia separation.

Sperm chemotaxis in marine species is optimal at physiological flow rates according theory of filament surfing

Sperm of marine invertebrates have to find eggs cells in the ocean. Turbulent flows mix sperm and egg cells up to the millimeter scale; below this, active swimming and chemotaxis become important. Previous work addressed either turbulent mixing or chemotaxis in still water. Here, we present a general theory of sperm chemotaxis inside the smallest eddies of turbulent flow, where signaling molecules released by egg cells are spread into thin concentration filaments. Sperm cells ‘surf’ along these filaments towards the egg. External flows make filaments longer, but also thinner. These opposing effects set an optimal flow strength. The optimum predicted by our theory matches flow measurements in shallow coastal waters. Our theory quantitatively agrees with two previous fertilization experiments in Taylor-Couette chambers and provides a mechanistic understanding of these early experiments. ‘Surfing along concentration filaments’ could be a paradigm for navigation in complex environments in the presence of turbulent flow.

Many motile cells navigate in complex environments along concentration gradients of signaling molecules. This chemotaxis has been studied extensively both experimentally and theoretically, yet mostly for idealized conditions of perfect chemical gradients. But under physiological conditions, concentration fields are subject to distortions, e.g., by turbulent flows in the ocean. Pioneering experiments suggest that in species with external fertilization, chemotaxis of sperm cells towards the egg may even work better at an optimal flow strength compared to conditions of still water. Yet to date, the mechanistic cause for this optimum is not known. We present a general theory of chemotactic navigation in external flow. We characterize how external flow distorts concentration fields into long filaments, and show how chemotaxing cells can subsequently ‘surf’ along these filaments towards a chemoattractant source. Stronger flows make concentration filaments longer, but also thinner; together, these two counter-acting effects set an optimal flow strength. Beyond fertilization of marine invertebrates, we believe that ‘surfing along concentration filaments’ could be a more general paradigm, relevant also for the ecology of marine bacteria feeding on organic marine snow in the ocean, or chemotaxis inside multi-cellular organisms with internal flows.

Origins of eukaryotic excitability

All living cells interact dynamically with a constantly changing world. Eukaryotes, in particular, evolved radically new ways to sense and react to their environment. These advances enabled new and more complex forms of cellular behaviour in eukaryotes, including directional movement, active feeding, mating, and responses to predation. But what are the key events and innovations during eukaryogenesis that made all of this possible? Here we describe the ancestral repertoire of eukaryotic excitability and discuss five major cellular innovations that enabled its evolutionary origin. The innovations include a vastly expanded repertoire of ion channels, the emergence of cilia and pseudopodia, endomembranes as intracellular capacitors, a flexible plasma membrane and the relocation of chemiosmotic ATP synthesis to mitochondria, which liberated the plasma membrane for more complex electrical signalling involved in sensing and reacting. We conjecture that together with an increase in cell size, these new forms of excitability greatly amplified the degrees of freedom associated with cellular responses, allowing eukaryotes to vastly outperform prokaryotes in terms of both speed and accuracy. This comprehensive new perspective on the evolution of excitability enriches our view of eukaryogenesis and emphasizes behaviour and sensing as major contributors to the success of eukaryotes. This article is part of the theme issue 'Basal cognition: conceptual tools and the view from the single cell'.

CiliaQ: a simple, open-source software for automated quantification of ciliary morphology and fluorescence in 2D, 3D, and 4D images

Cilia are hair-like membrane protrusions that emanate from the surface of most vertebrate cells and are classified into motile and primary cilia. Motile cilia move fluid flow or propel cells, while also fulfill sensory functions. Primary cilia are immotile and act as a cellular antenna, translating environmental cues into cellular responses. Ciliary dysfunction leads to severe diseases, commonly termed ciliopathies. The molecular details underlying ciliopathies and ciliary function are, however, not well understood. Since cilia are small subcellular compartments, imaging-based approaches have been used to study them. However, tools to comprehensively analyze images are lacking. Automatic analysis approaches require commercial software and are limited to 2D analysis and only a few parameters. The widely used manual analysis approaches are time consuming, user-biased, and difficult to compare. Here, we present CiliaQ, a package of open-source, freely available, and easy-to-use ImageJ plugins. CiliaQ allows high-throughput analysis of 2D and 3D, static or time-lapse images from fluorescence microscopy of cilia in cell culture or tissues, and outputs a comprehensive list of parameters for ciliary morphology, length, bending, orientation, and fluorescence intensity, making it broadly applicable. We envision CiliaQ as a resource and platform for reproducible and comprehensive analysis of ciliary function in health and disease.

Biomechanics of Early Life in the Female Reproductive Tract

Early human life that starts at the onset of fertilization and ends with implantation of the embryo in the uterine wall is the foundation for a successful pregnancy. The different stages during this period require biomechanical mechanisms, which are mostly unknown due to difficulties to conduct in vivo studies in humans.

Propulsion Mechanism of Flexible Microbead Swimmers in the Low Reynolds Number Regime

A propulsion mechanism for a flexible microswimmer constructed from superparamagnetic microbeads with different diameters and subjected to an oscillating field was studied experimentally and theoretically herein. Various types of artificial swimmers with different bending patterns were fabricated to determine the flexibility and an effective waveform for a planar beating flagellum. Waveform evolutions for various swimmer configurations were studied to determine the flexible mechanism of the swimmers. A one-armed microswimmer can propel itself only if the friction of its wavelike body is anisotropic. A swimmer with a larger head and a stronger magnetic dipole moment with a flexible tail allows the bending wave to propagate from the head toward the tail to generate forward thrust. The oscillating head and tail do not simultaneously generate positive thrust all the time within a period of oscillation. To increase the propulsion for a bending swimmer, this study proposes a novel configuration for a microbead swimmer that ensures better swimming efficiency. The ratio of the oscillation amplitude of the head to the length of the swimmer (from 0.26 to 0.28) produces a faster swimmer. On the other hand, the swimmer is propelled more effectively if the ratio of the oscillation amplitude of the tail to the length of the swimmer is from 0.29 to 0.33. This study determined the optimal configuration for a flexible microbead swimmer that generates the greatest propulsion in a low Reynolds number environment.

The anatomy, movement, and functions of human sperm tail: an evolving mystery

Sperms have attracted attention of many researchers since it was discovered by Antonie van Leeuwenhoek in 1677. Though a small cell, its every part has complex structure and different function to play in carrying life. Sperm tail is most complicated structure with more than 1000 proteins involved in its functioning. With the advent of three-dimensional microscopes, many studies are undergoing to understand exact mechanism of sperm tail movement. Most recent studies have shown that sperms move by spinning rather than swimming. Each subunit of tail, including axonemal, peri-axonemal structures, plays essential roles in sperm motility, capacitation, hyperactivation, fertilization. Furthermore, over 2300 genes are involved in spermatogenesis. A number of genetic mutations have been linked with abnormal sperm flagellar development leading to motility defects and male infertility. It was found that 6% of male infertility cases are related to genetic causes, and 4% of couples undergoing intracytoplasmic sperm injection for male subfertility have chromosomal abnormalities. Hence, an understanding of sperm tail development and genes associated with its normal functioning can help in better diagnosis of male infertility and its management. There is still a lot that needs to be discovered about genes, proteins contributing to normal human sperm tail development, movement, and role in male fertility. Sperm tail has complex anatomy, with surrounding axoneme having 9 + 2 microtubules arrangement along its entire length and peri-axonemal structures that contribute in sperm motility and fertilization. In future sperm tail-associated genes, proteins and subunits can be used as markers of male fertility.

Triangulation Sensing to Determine the Gradient Source from Diffusing Particles to Small Cell Receptors

How does a cell locate the source of molecular guidance cues from within a concentration gradient? We present a computational approach to recover the source from the absorbed fluxes at narrow receptor windows located on the surface of the cell. In the limit of fast binding, we solve the steady-state diffusion equation using an asymptotic approach and hybrid stochastic-analytical simulations. We show that the sensitivity to the gradient direction decays too rapidly to enable long-distance sensing. We illustrate how this constraint can be alleviated when triangulating the source with an increasing number of receptor windows and quantify the susceptibility of this process to flux perturbations.

Active dipolar spheroids in shear flow and transverse field: Population splitting, cross-stream migration, and orientational pinning

We study the steady-state behavior of active, dipolar, Brownian spheroids in a planar channel subjected to an imposed Couette flow and an external transverse field, applied in the "downward" normal-to-flow direction. The field-induced torque on active spheroids (swimmers) is taken to be of magnetic form by assuming that they have a permanent magnetic dipole moment, pointing along their self-propulsion (swim) direction. Using a continuum approach, we show that a host of behaviors emerges over the parameter space spanned by the particle aspect ratio, self-propulsion and shear/field strengths, and the channel width. The cross-stream migration of the model swimmers is shown to involve a regime of linear response (quantified by a linear-response factor) in weak fields. For prolate swimmers, the weak-field behavior crosses over to a regime of full swimmer migration to the bottom half of the channel in strong fields. For oblate swimmers, a counterintuitive regime of reverse migration arises in intermediate fields, where a macroscopic fraction of swimmers reorient and swim to the top channel half at an acute "upward" angle relative to the field axis. The diverse behaviors reported here are analyzed based on the shear-induced population splitting (bimodality) of the swim orientation, giving two distinct, oppositely polarized, swimmer subpopulations (albeit very differently for prolate/oblate swimmers) in each channel half. In strong fields, swimmers of both types exhibit net upstream currents relative to the laboratory frame. The onsets of full migration and net upstream current depend on the aspect ratio, enabling efficient particle separation strategies in microfluidic setups.

The 2020 motile active matter roadmap

Activity and autonomous motion are fundamental in living and engineering systems. This has stimulated the new field of 'active matter' in recent years, which focuses on the physical aspects of propulsion mechanisms, and on motility-induced emergent collective behavior of a larger number of identical agents. The scale of agents ranges from nanomotors and microswimmers, to cells, fish, birds, and people. Inspired by biological microswimmers, various designs of autonomous synthetic nano- and micromachines have been proposed. Such machines provide the basis for multifunctional, highly responsive, intelligent (artificial) active materials, which exhibit emergent behavior and the ability to perform tasks in response to external stimuli. A major challenge for understanding and designing active matter is their inherent nonequilibrium nature due to persistent energy consumption, which invalidates equilibrium concepts such as free energy, detailed balance, and time-reversal symmetry. Unraveling, predicting, and controlling the behavior of active matter is a truly interdisciplinary endeavor at the interface of biology, chemistry, ecology, engineering, mathematics, and physics. The vast complexity of phenomena and mechanisms involved in the self-organization and dynamics of motile active matter comprises a major challenge. Hence, to advance, and eventually reach a comprehensive understanding, this important research area requires a concerted, synergetic approach of the various disciplines. The 2020 motile active matter roadmap of Journal of Physics: Condensed Matter addresses the current state of the art of the field and provides guidance for both students as well as established scientists in their efforts to advance this fascinating area.

Chemokinetic Scattering, Trapping, and Avoidance of Active Brownian Particles

We present a theory of chemokinetic search agents that regulate directional fluctuations according to distance from a target. A dynamic scattering effect reduces the probability to penetrate regions with high fluctuations and thus reduces search success for agents that respond instantaneously to positional cues. In contrast, agents with internal states that initially suppress chemokinesis can exploit scattering to increase their probability to find the target. Using matched asymptotics between the case of diffusive and ballistic search, we obtain analytic results beyond Fox colored noise approximation.

The steering gaits of sperm

Sperm are highly specialized cells, which have been subject to substantial evolutionary pressure. Whereas some sperm features are highly conserved, others have undergone major modifications. Some of these variations are driven by adaptation to mating behaviours or fitness at the organismic level. Others represent alternative solutions to the same task. Sperm must find the egg for fertilization. During this task, sperm rely on long slender appendages termed flagella that serve as sensory antennas, propellers and steering rudders. The beat of the flagellum is periodic. The resulting travelling wave generates the necessary thrust for propulsion in the fluid. Recent studies reveal that, for steering, different species rely on different fundamental features of the beat wave. Here, we discuss some examples of unity and diversity across sperm from different species with a particular emphasis on the steering mechanisms. This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.

Quorum-sensing active particles with discontinuous motility

We develop a dynamic mean-field theory for polar active particles that interact through a self-generated field, in particular one generated through emitting a chemical signal. While being a form of chemotactic response, it is different from conventional chemotaxis in that particles discontinuously change their motility when the local concentration surpasses a threshold. The resulting coupled equations for density and polarization are linear and can be solved analytically for simple geometries, yielding inhomogeneous density profiles. Specifically, here we consider a planar and circular interface. Our theory thus explains the observed coexistence of dense aggregates with an active gas. There are, however, differences from the more conventional picture of liquid-gas coexistence based on a free energy, most notably the absence of a critical point. We corroborate our analytical predictions by numerical simulations of active particles under confinement and interacting through volume exclusion. Excellent quantitative agreement is reached through an effective translational diffusion coefficient. We finally show that an additional response to the chemical gradient direction is sufficient to induce vortex clusters. Our results pave the way to engineer motility responses in order to achieve aggregation and collective behavior even at unfavorable conditions.

Ratiometric GPCR signaling enables directional sensing in yeast

Accurate detection of extracellular chemical gradients is essential for many cellular behaviors. Gradient sensing is challenging for small cells, which can experience little difference in ligand concentrations on the up-gradient and down-gradient sides of the cell. Nevertheless, the tiny cells of the yeast Saccharomyces cerevisiae reliably decode gradients of extracellular pheromones to find their mates. By imaging the behavior of polarity factors and pheromone receptors, we quantified the accuracy of initial polarization during mating encounters. We found that cells bias the orientation of initial polarity up-gradient, even though they have unevenly distributed receptors. Uneven receptor density means that the gradient of ligand-bound receptors does not accurately reflect the external pheromone gradient. Nevertheless, yeast cells appear to avoid being misled by responding to the fraction of occupied receptors rather than simply the concentration of ligand-bound receptors. Such ratiometric sensing also serves to amplify the gradient of active G protein. However, this process is quite error-prone, and initial errors are corrected during a subsequent indecisive phase in which polarity clusters exhibit erratic mobile behavior.

Cells use surface receptors to decode spatial information from chemical gradients, but accurate decoding is hampered by small cell size and the presence of molecular noise. This study shows that yeast cells decode pheromone gradients by measuring the local ratio of bound to unbound receptors. This mechanism corrects for uneven receptor density at the surface and amplifies the gradient transmitted to downstream components.

The flagellar protein Enkurin is required for mouse sperm motility and for transport through the female reproductive tract

Enkurin was identified initially in mouse sperm where it was suggested to act as an intracellular adaptor protein linking membrane calcium influx to intracellular signaling pathways. In order to examine the function of this protein, a targeted mutation was introduced into the mouse Enkurin gene. Males that were homozygous for this mutated allele were subfertile. This was associated with lower rates of sperm transport in the female reproductive tract, including reduced entry into the oviduct and slower migration to the site of fertilization in the distal oviduct, and with poor progressive motility in vitro. Flagella from wild-type animals exhibited symmetrical bending and progressive motility in culture medium, and demembranated flagella exhibited the "curlicue" response to Ca2+ in vitro. In contrast, flagella of mice homozygous for the mutated allele displayed only asymmetric bending, nonprogressive motility, and a loss of Ca2+-responsiveness following demembrantion. We propose that Enkurin is part of a flagellar Ca2+-sensor that regulates bending and that the motility defects following mutation of the locus are the proximate cause of subfertility.

Hydrodynamic mobility reversal of squirmers near flat and curved surfaces

Self-propelled particles have been experimentally shown to orbit spherical obstacles and move along surfaces. Here, we theoretically and numerically investigate this behavior for a hydrodynamic squirmer interacting with spherical objects and flat walls using three different methods of approximately solving the Stokes equations: The method of reflections, which is accurate in the far field; lubrication theory, which describes the close-to-contact behavior; and a lattice Boltzmann solver that accurately accounts for near-field flows. The method of reflections predicts three distinct behaviors: orbiting/sliding, scattering, and hovering, with orbiting being favored for lower curvature as in the literature. Surprisingly, it also shows backward orbiting/sliding for sufficiently strong pushers, caused by fluid recirculation in the gap between the squirmer and the obstacle leading to strong forces opposing forward motion. Lubrication theory instead suggests that only hovering is a stable point for the dynamics. We therefore employ lattice Boltzmann to resolve this discrepancy and we qualitatively reproduce the richer far-field predictions. Our results thus provide insight into a possible mechanism of mobility reversal mediated solely through hydrodynamic interactions with a surface.

Cyclic Nucleotide-Specific Optogenetics Highlights Compartmentalization of the Sperm Flagellum into cAMP Microdomains

Inside the female genital tract, mammalian sperm undergo a maturation process called capacitation, which primes the sperm to navigate across the oviduct and fertilize the egg. Sperm capacitation and motility are controlled by 3′,5′-cyclic adenosine monophosphate (cAMP). Here, we show that optogenetics, the control of cellular signaling by genetically encoded light-activated proteins, allows to manipulate cAMP dynamics in sperm flagella and, thereby, sperm capacitation and motility by light. To this end, we used sperm that express the light-activated phosphodiesterase LAPD or the photo-activated adenylate cyclase bPAC. The control of cAMP by LAPD or bPAC combined with pharmacological interventions provides spatiotemporal precision and allows to probe the physiological function of cAMP compartmentalization in mammalian sperm.

A hydrodynamic-stochastic model of chemotactic ciliated microorganisms

Biological systems like ciliated microorganisms are capable of responding to the external chemical gradients, a process known as chemotaxis. In this process, the internal signaling network of the microorganism is triggered due to binding of the chemoattractant molecules with the receptors on the surface of the body. This can alter the activity at the surface of the microorganism. We study the chemotaxis of ciliated microorganisms using the chiral squirmer model, a spherical body with a surface slip velocity. In the presence of a chemical gradient, the coefficients of the slip velocity get modified resulting in a change in the path followed by the body. We observe that the strength of the gradient is not the only parameter which controls the dynamics of the body but also the adaptation time plays a very significant role in the success of chemotaxis. The trajectory of the body is smooth if we ignore the discreteness in the ligand-receptor binding which is stochastic in nature. In the presence of the latter, the path is not only irregular but the whole dynamics of the body changes. We calculate the mean first passage time, by varying the strength of the chemical gradient and the adaptation time, to determine the success rate of chemotaxis.

Strictures of a microchannel impose fierce competition to select for highly motile sperm

Investigating sperm locomotion in the presence of external fluid flow and geometries simulating the female reproductive tract can lead to a better understanding of sperm motion during fertilization. Using a microfluidic device featuring a stricture that simulates the fluid mechanical properties of narrow junctions inside the female reproductive tract, we documented the gate-like role played by the stricture in preventing sperm with motilities below a certain threshold from advancing through the stricture to the other side (i.e., fertilization site). All the slower sperm accumulate below (i.e., in front of) the stricture and swim in a butterfly-shaped path between the channel walls, thus maintaining the potential for penetrating the stricture and ultimately advancing toward the fertilization site. Accumulation below the stricture occurs in a hierarchical manner so that dense concentrations of sperm with higher velocities remain closer to the stricture, with more sparsely distributed arrays of lower-velocity sperm lagging behind.

SpermQ⁻A Simple Analysis Software to Comprehensively Study Flagellar Beating and Sperm Steering

Motile cilia, also called flagella, are found across a broad range of species; some cilia propel prokaryotes and eukaryotic cells like sperm, while cilia on epithelial surfaces create complex fluid patterns e.g., in the brain or lung. For sperm, the picture has emerged that the flagellum is not only a motor but also a sensor that detects stimuli from the environment, computing the beat pattern according to the sensory input. Thereby, the flagellum navigates sperm through the complex environment in the female genital tract. However, we know very little about how environmental signals change the flagellar beat and, thereby, the swimming behavior of sperm. It has been proposed that distinct signaling domains in the flagellum control the flagellar beat. However, a detailed analysis has been mainly hampered by the fact that current comprehensive analysis approaches rely on complex microscopy and analysis systems. Thus, knowledge on sperm signaling regulating the flagellar beat is based on custom quantification approaches that are limited to only a few aspects of the beat pattern, do not resolve the kinetics of the entire flagellum, rely on manual, qualitative descriptions, and are only a little comparable among each other. Here, we present SpermQ, a ready-to-use and comprehensive analysis software to quantify sperm motility. SpermQ provides a detailed quantification of the flagellar beat based on common time-lapse images acquired by dark-field or epi-fluorescence microscopy, making SpermQ widely applicable. We envision SpermQ becoming a standard tool in flagellar and motile cilia research that allows to readily link studies on individual signaling components in sperm and distinct flagellar beat patterns.

A novel cross-species inhibitor to study the function of CatSper Ca2+ channels in sperm

BACKGROUND AND PURPOSE

Sperm from many species share the sperm-specific Ca2+ channel CatSper that controls the intracellular Ca2+ concentration and, thereby, the swimming behaviour. A growing body of evidence suggests that the mechanisms controlling the activity of CatSper and its role during fertilization differ among species. A lack of suitable pharmacological tools has hampered the elucidation of the function of CatSper. Known inhibitors of CatSper exhibit considerable side effects and also inhibit Slo3, the principal K+ channel of mammalian sperm. The compound RU1968 was reported to suppress Ca2+ signaling in human sperm by an unknown mechanism. Here, we examined the action of RU1968 on CatSper in sperm from humans, mice, and sea urchins.

EXPERIMENTAL APPROACH

We resynthesized RU1968 and studied its action on sperm from humans, mice, and the sea urchin Arbacia punctulata by Ca2+ fluorimetry, single-cell Ca2+ imaging, electrophysiology, opto-chemistry, and motility analysis.

KEY RESULTS

RU1968 inhibited CatSper in sperm from invertebrates and mammals. The compound lacked toxic side effects in human sperm, did not affect mouse Slo3, and inhibited human Slo3 with about 15-fold lower potency than CatSper. Moreover, in human sperm, RU1968 mimicked CatSper dysfunction and suppressed motility responses evoked by progesterone, an oviductal steroid known to activate CatSper. Finally, RU1968 abolished CatSper-mediated chemotactic navigation in sea urchin sperm.

CONCLUSION AND IMPLICATIONS

We propose RU1968 as a novel tool to elucidate the function of CatSper channels in sperm across species.

The escape problem for mortal walkers

We introduce and investigate the escape problem for random walkers that may eventually die, decay, bleach, or lose activity during their diffusion towards an escape or reactive region on the boundary of a confining domain. In the case of a first-order kinetics (i.e., exponentially distributed lifetimes), we study the effect of the associated death rate onto the survival probability, the exit probability, and the mean first passage time. We derive the upper and lower bounds and some approximations for these quantities. We reveal three asymptotic regimes of small, intermediate, and large death rates. General estimates and asymptotics are compared to several explicit solutions for simple domains and to numerical simulations. These results allow one to account for stochastic photobleaching of fluorescent tracers in bio-imaging, degradation of mRNA molecules in genetic translation mechanisms, or high mortality rates of spermatozoa in the fertilization process. Our findings provide a mathematical ground for optimizing storage containers and materials to reduce the risk of leakage of dangerous chemicals or nuclear wastes.

Biohybrid and Bioinspired Magnetic Microswimmers

Many motile microorganisms swim and navigate in chemically and mechanically complex environments. These organisms can be functionalized and directly used for applications (biohybrid approach), but also inspire designs for fully synthetic microbots. The most promising designs of biohybrids and bioinspired microswimmers include one or several magnetic components, which lead to sustainable propulsion mechanisms and external controllability. This Review addresses such magnetic microswimmers, which are often studied in view of certain applications, mostly in the biomedical area, but also in the environmental field. First, propulsion systems at the microscale are reviewed and the magnetism of microswimmers is introduced. The review of the magnetic biohybrids and bioinspired microswimmers is structured gradually from mostly biological systems toward purely synthetic approaches. Finally, currently less explored parts of this field ranging from in situ imaging to swarm control are discussed.

Re-entrant bimodality in spheroidal chiral swimmers in shear flow

We use a continuum model to report on the behavior of a dilute suspension of chiral swimmers subject to externally imposed shear in a planar channel. Swimmer orientation in response to the imposed shear can be characterized by two distinct phases of behavior, corresponding to unimodal or bimodal distribution functions for swimmer orientation along the channel. These phases indicate the occurrence (or not) of a population splitting phenomenon changing the swimming direction of a macroscopic fraction of active particles to the exact opposite of that dictated by the imposed flow. We present a detailed quantitative analysis elucidating the complexities added to the population splitting behavior of swimmers when they are chiral. In particular, the transition from unimodal to bimodal and vice versa are shown to display a re-entrant behavior across the parameter space spanned by varying the chiral angular speed. We also present the notable effects of particle aspect ratio and self-propulsion speed on system phase behavior and discuss potential implications of our results in applications such as swimmer separation/sorting.

Decision making improves sperm chemotaxis in the presence of noise

To navigate their surroundings, cells rely on sensory input that is corrupted by noise. In cells performing chemotaxis, such noise arises from the stochastic binding of signalling molecules at low chemoattractant concentrations. We reveal a fundamental relationship between the speed of chemotactic steering and the strength of directional fluctuations that result from the amplification of noise in a chemical input signal. This relation implies a trade-off between steering that is slow and reliable, and steering that is fast but less reliable. We show that dynamic switching between these two modes of steering can substantially increase the probability to find a target, such as an egg to be found by sperm cells. This decision making confers no advantage in the absence of noise, but is beneficial when chemical signals are detectable, yet characterized by low signal-to-noise ratios. The latter applies at intermediate distances from a target, where signalling molecules are diluted, thus defining a ‘noise zone’ that cells have to cross. Our results explain decision making observed in recent experiments on sea urchin sperm chemotaxis. More generally, our theory demonstrates how decision making enables chemotactic agents to cope with high levels of noise in gradient sensing by dynamically adjusting the persistence length of a biased random walk.

Many cells can navigate upwards a concentration gradient of signalling molecules, a process called chemotaxis. Chemotaxis is used e.g. by sperm cells to find the egg. To measure and compare concentrations, cells count stochastic binding events of signalling molecules that diffuse to cellular receptors. Efficient chemotaxis strategies must be adapted to this molecular shot noise of concentration measurements. We reveal a fundamental relationship between the speed of chemotactic steering and the strength of directional fluctuations that result from the amplification of noise. This implies a trade-off between steering fast and steering reliable. Inspired by recent experiments on chemotaxis of sperm cells of marine invertebrates, we develop a theory that allows to efficiently compute optimal chemotaxis strategies. We show that dynamic switching between either fast or reliable steering can substantially increase the probability for a sperm cell to find the egg. Furthermore, the optimal strategy requires only minimal computational capacities of the chemotactic agent, a key constraint for biological cells. More generally, our work demonstrates a benefit of decision making for chemotaxis in the presence of noise, which could inspire optimal control designs for artificial microswimmers.

Chemotactic droplet swimmers in complex geometries

Chemotaxis¹ and auto-chemotaxis are key mechanisms in the dynamics of micro-organisms, e.g. in the acquisition of nutrients and in the communication between individuals, influencing the collective behaviour. However, chemical signalling and the natural environment of biological swimmers are generally complex, making them hard to access analytically. We present a well-controlled, tunable artificial model to study chemotaxis and autochemotaxis in complex geometries, using microfluidic assays of self-propelling oil droplets in an aqueous surfactant solution (Herminghaus et al 2014 Soft Matter 10 7008-22; Krüger et al 2016 Phys. Rev. Lett. 117). Droplets propel via interfacial Marangoni stresses powered by micellar solubilisation. Moreover, filled micelles act as a chemical repellent by diffusive phoretic gradient forces. We have studied these chemotactic effects in a series of microfluidic geometries, as published in Jin et al (2017 Proc. Natl Acad. Sci. 114 5089-94): first, droplets are guided along the shortest path through a maze by surfactant diffusing into the maze from the exit. Second, we let auto-chemotactic droplet swimmers pass through bifurcating microfluidic channels and record anticorrelations between the branch choices of consecutive droplets. We present an analytical Langevin model matching the experimental data. In a previously unpublished experiment, pillar arrays of variable sizes and shapes provide a convex wall interacting with the swimmer and, in the case of attachment, bending its trajectory and forcing it to revert to its own trail. We observe different behaviours based on the interplay of wall curvature and negative autochemotaxis, i.e. no attachment for highly curved interfaces, stable trapping at large pillars, and a narrow transition region where negative autochemotaxis makes the swimmers detach after a single orbit.

Reconstructing the gradient source position from steady-state fluxes to small receptors

Recovering the position of a source from the fluxes of diffusing particles through small receptors allows a biological cell to determine its relative position, spatial localization and guide it to a final target. However, how a source can be recovered from point fluxes remains unclear. Using the Narrow Escape approach for an open domain, we compute the diffusion fluxes of Brownian particles generated by a steady-state gradient from a single source through small holes distributed on a surface in two dimensions. We find that the location of a source can be recovered when there are at least 3 receptors and the source is positioned no further than 10 cell radii away, but this condition is not necessary in a narrow strip. The present approach provides a computational basis for the first step of direction sensing of a gradient at a single cell level.

Egg chemoattractants moderate intraspecific sperm competition

Interactions among eggs and sperm are often assumed to generate intraspecific variation in reproductive fitness, but the specific gamete-level mechanisms underlying competitive fertilization success remain elusive in most species. Sperm chemotaxis-the attraction of sperm by egg-derived chemicals-is a ubiquitous form of gamete signaling, occurring throughout the animal and plant kingdoms. The chemical cues released by eggs are known to act at the interspecific level (e.g., facilitating species recognition), but recent studies have suggested that they could have roles at the intraspecific level by moderating sperm competition. Here, we exploit the experimental tractability of a broadcast spawning marine invertebrate to test this putative mechanism of gamete-level sexual selection. We use a fluorescently labeled mitochondrial dye in mussels to track the real-time success of sperm as they compete to fertilize eggs, and provide the first direct evidence in any species that competitive fertilization success is moderated by differential sperm chemotaxis. Furthermore, our data are consistent with the idea that egg chemoattractants selectively attract ejaculates from genetically compatible males, based on relationships inferred from both nuclear and mitochondrial genetic markers. These findings for a species that exhibits the ancestral reproductive strategy of broadcast spawning have important implications for the numerous species that also rely on egg chemoattractants to attract sperm, including humans, and have potentially important implications for our understanding of the evolutionary cascade of sexual selection.

Human sperm steer with second harmonics of the flagellar beat

Sperm are propelled by bending waves traveling along their flagellum. For steering in gradients of sensory cues, sperm adjust the flagellar waveform. Symmetric and asymmetric waveforms result in straight and curved swimming paths, respectively. Two mechanisms causing spatially asymmetric waveforms have been proposed: an average flagellar curvature and buckling. We image flagella of human sperm tethered with the head to a surface. The waveform is characterized by a fundamental beat frequency and its second harmonic. The superposition of harmonics breaks the beat symmetry temporally rather than spatially. As a result, sperm rotate around the tethering point. The rotation velocity is determined by the second-harmonic amplitude and phase. Stimulation with the female sex hormone progesterone enhances the second-harmonic contribution and, thereby, modulates sperm rotation. Higher beat frequency components exist in other flagellated cells; therefore, this steering mechanism might be widespread and could inspire the design of synthetic microswimmers.