Nonlinear amplitude dynamics in flagellar beating

Spontaneous oscillation of an active filament under viscosity gradients

We investigate the effects of uniform viscosity gradients on the spontaneous oscillations of an elastic, active filament in viscous fluids. Combining numerical simulations and linear stability analysis, we demonstrate that a viscosity gradient increasing from the filament's base to tip destabilises the system, facilitating its self-oscillation. This effect is elucidated through a reduced-order model, highlighting the delicate balance between destabilising active forces and stabilising viscous forces. Additionally, we reveal that while a perpendicular viscosity gradient to the filament's orientation minimally affects instability, it induces asymmetric ciliary beating, thus generating a net flow along the gradient. Our findings offer new insights into the complex behaviours of biological and artificial filaments in complex fluid environments, contributing to the broader understanding of filament dynamics in heterogeneous viscous media.

Human sperm rotate with a conserved direction during free swimming in four dimensions

Head rotation in human spermatozoa is essential for different swimming modes and fertilisation, as it links the molecular workings of the flagellar beat with sperm motion in three-dimensional (3D) space over time. Determining the direction of head rotation has been hindered by the symmetry and translucent nature of the sperm head, and by the fast 3D motion driven by the helical flagellar beat. Analysis has been mostly restricted to two-dimensional (2D) single focal plane image analysis, which enables tracking of head centre position but not tracking of head rotation. Despite the conserved helical beating of the human sperm flagellum, human sperm head rotation has been reported to be uni- or bi-directional, and even to intermittently change direction in a given cell. Here, we directly measure the head rotation of freely swimming human sperm using multi-plane 4D (3D+t) microscopy and show that: (1) 2D microscopy is unable to distinguish head rotation direction in human spermatozoa; (2) head rotation direction in non-capacitating and capacitating solutions, for both aqueous and viscous media, is counterclockwise (CCW), as seen from head to tail, in all rotating spermatozoa, regardless of the experimental conditions; and (3) head rotation is suppressed in 36% of spermatozoa swimming in non-capacitating viscous medium, although CCW rotation is recovered after incubation in capacitating conditions within the same viscous medium, possibly unveiling an unexplored aspect of the essential need of capacitation for fertilisation. Our observations show that the CCW head rotation in human sperm is conserved. It constitutes a robust and persistent helical driving mechanism that influences sperm navigation in 3D space over time, and thus is of critical importance in cell motility, propulsion of flagellated microorganisms, sperm motility assessments, human reproduction research, and self-organisation of flagellar beating patterns and swimming in 3D space.

The reaction-diffusion basis of animated patterns in eukaryotic flagella

The flagellar beat of bull spermatozoa and C. Reinhardtii are modelled by a minimal, geometrically exact, reaction-diffusion system. Spatio-temporal animated patterns describe flagellar waves, analogous to chemical-patterns from classical reaction-diffusion systems, with sliding-controlled molecular motor reaction-kinetics. The reaction-diffusion system is derived from first principles as a consequence of the high-internal dissipation by the flagellum relative to the external hydrodynamic dissipation. Quantitative comparison with nonlinear, large-amplitude simulations shows that animated reaction-diffusion patterns account for the experimental beating of both bull sperm and C. Reinhardtii. Our results suggest that a unified mechanism may exist for motors controlled by sliding, without requiring curvature-sensing, and uninfluenced by hydrodynamics. High-internal dissipation instigates autonomous travelling waves independently of the external fluid, enabling progressive swimming, otherwise not possible, in low viscosity environments, potentially critical for external fertilizers and aquatic microorganisms. The reaction-diffusion system may prove a powerful tool for studying pattern formation of movement on animated structures.

Flagella-like beating of actin bundles driven by self-organized myosin waves

Wave-like beating of eukaryotic cilia and flagella-threadlike protrusions found in many cells and microorganisms-is a classic example of spontaneous mechanical oscillations in biology. This type of self-organized active matter raises the question of the coordination mechanism between molecular motor activity and cytoskeletal filament bending. Here we show that in the presence of myosin motors, polymerizing actin filaments self-assemble into polar bundles that exhibit wave-like beating. Importantly, filament beating is associated with myosin density waves initiated at twice the frequency of the actin-bending waves. A theoretical description based on curvature control of motor binding to the filaments and of motor activity explains our observations in a regime of high internal friction. Overall, our results indicate that the binding of myosin to actin depends on the actin bundle shape, providing a feedback mechanism between the myosin activity and filament deformations for the self-organization of large motor filament assemblies.

Self-sustained three-dimensional beating of a model eukaryotic flagellum

Flagella and cilia are common features of a wide variety of biological cells and play important roles in locomotion and feeding at the microscale. The beating of flagella is controlled by molecular motors that exert forces along the length of the flagellum and are regulated by a feedback mechanism coupled to the flagella deformation. We develop a three-dimensional (3D) flagellum beating model based on sliding-controlled motor feedback, accounting for both bending and twist, as well as differential bending resistances along and orthogonal to the major bending plane of the flagellum. We show that beating is generated and sustained spontaneously for a sufficiently high motor activity through an instability mechanism. Isotropic bending rigidities in the flagellum lead to 3D helical beating patterns. By contrast, anisotropic flagella present a rich variety of wave-like beating dynamics, including both 3D beating patterns as well as planar beating patterns. We show that the ability to generate nearly planar beating despite the 3D beating machinery requires only a modest degree of bending anisotropy, and is a feature observed in many eukaryotic flagella such as mammalian spermatozoa.

Self-sustaining oscillation of two axonemal microtubules based on a stochastic bonding model between microtubules and dynein

The motility of cilia and flagella plays important physiological roles, and there has been a great deal of research on the mechanisms underlying the motility of molecular motors. Although recent molecular structural analyses have revealed the components of the ciliary axoneme, the mechanisms involved in the regulation of dynein activity are still unknown, and how multiple dyneins coordinate their movements remains unclear. In particular, the mode of binding for axonemal dynein has not been elucidated. In this study, we constructed a thermodynamic stochastic model of microtubule-dynein coupling and reproduced the experiments of Aoyama and Kamiya on the minimal component of axonemal microtubule-dynein. We then identified the binding mode of axonemal dynein and clarified the relationship between dynein activity distribution and axonemal movement. Based on our numerical results, the slip-bond mechanism agrees quantitatively with the experimental results in terms of amplitude, frequency, and propagation velocity, implying that axial microtubule-dynein coupling may follow a slip-bond mechanism. Moreover, the frequency and propagation velocity decayed in proportion to the fourth power of microtubule length, and the critical load of the trigger for the oscillation agreed well with Euler's critical load.

A multiscale biophysical model gives quantized metachronal waves in a lattice of beating cilia

Motile cilia are slender, hair-like cellular appendages that spontaneously oscillate under the action of internal molecular motors and are typically found in dense arrays. These active filaments coordinate their beating to generate metachronal waves that drive long-range fluid transport and locomotion. Until now, our understanding of their collective behavior largely comes from the study of minimal models that coarse grain the relevant biophysics and the hydrodynamics of slender structures. Here we build on a detailed biophysical model to elucidate the emergence of metachronal waves on millimeter scales from nanometer-scale motor activity inside individual cilia. Our study of a one-dimensional lattice of cilia in the presence of hydrodynamic and steric interactions reveals how metachronal waves are formed and maintained. We find that, in homogeneous beds of cilia, these interactions lead to multiple attracting states, all of which are characterized by an integer charge that is conserved. This even allows us to design initial conditions that lead to predictable emergent states. Finally, and very importantly, we show that, in nonuniform ciliary tissues, boundaries and inhomogeneities provide a robust route to metachronal waves.

Symmetry Breaking of Tail-Clamped Filaments in Stokes Flow

Symmetry breaking (SB) of fluid-structure interaction problems plays an important role in our understanding of animals' locomotive and sensing behaviors. In this Letter, we study the SB of flexible filaments clamped at one end and placed in a spanwise periodic array in Stokes flow. The equilibrium state of the filament along the streamwise direction loses stability and experiences two-dimensional and then three-dimensional SBs as the spanwise distance increases, or as the filament rigidity reduces. For slightly deformed filaments, the viscous and pressure forces are commensurate, while for extremely deformed filaments the viscous force becomes dominant.

Minimal model of the hydrodynamical coupling of flagella on a spherical body with application to Volvox

Flagella are hairlike appendages attached to microorganisms that allow the organisms to traverse their fluid environment. The algae Volvox are spherical swimmers with thousands of individual flagella on their surface, and their coordination is not fully understood. In this work, a previously developed minimal model of flagella synchronization is extended to the outer surface of a sphere submerged in a fluid. Each beating flagellum tip is modeled as a small sphere, elastically bound to a circular orbit just above the spherical surface and a regularized image system for Stokes flow outside of a sphere is used to enforce the no-slip condition. Biologically relevant distributions of rotors results in a rapidly developing and robust symplectic metachronal wave traveling from the anterior to the posterior of the spherical Volvox body.

Internal friction controls active ciliary oscillations near the instability threshold

Ciliary oscillations driven by molecular motors cause fluid motion at micron scale. Stable oscillations require a substantial source of dissipation to balance the energy input of motors. Conventionally, it stems from external fluid. We show, in contrast, that external fluid friction is negligible compared to internal elastic stress through a simultaneous measurement of motion and flow field of an isolated and active Chlamydomonas cilium beating near the instability threshold. Consequently, internal friction emerges as the sole source of dissipation for ciliary oscillations. We combine these experimental insights with theoretical modeling of active filaments to show that an instability to oscillations takes place when active stresses are strain softening and shear thinning. Together, our results reveal a counterintuitive mechanism of ciliary beating and provide a general experimental and theoretical methodology to analyze other active filaments, both biological and synthetic ones.

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.

Cilia oscillations

Cilia, or eukaryotic flagella, are microscopic active filaments expressed on the surface of many eukaryotic cells, from single-celled protozoa to mammalian epithelial surfaces. Cilia are characterized by a highly conserved and intricate internal structure in which molecular motors exert forces on microtubule doublets causing cilia oscillations. The spatial and temporal regulations of this molecular machinery are not well understood. Several theories suggest that geometric feedback control from cilium deformations to molecular activity is needed. Here, we implement a recent sliding control model, where the unbinding of molecular motors is dictated by the sliding motion between microtubule doublets. We investigate the waveforms exhibited by the model cilium, as well as the associated molecular motor dynamics, for hinged and clamped boundary conditions. Hinged filaments exhibit base-to-tip oscillations while clamped filaments exhibit both base-to-tip and tip-to-base oscillations. We report the change in oscillation frequencies and amplitudes as a function of motor activity and sperm number, and we discuss the validity of these results in the context of experimental observations of cilia behaviour. This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.

Hydrodynamic Synchronization of Spontaneously Beating Filaments

Using a geometric feedback model of the flagellar axoneme accounting for dynein motor kinetics, we study elastohydrodynamic phase synchronization in a pair of spontaneously beating filaments with waveforms ranging from sperm to cilia and Chlamydomonas. Our computations reveal that both in-phase and antiphase synchrony can emerge for asymmetric beats while symmetric waveforms go in phase, and elucidate the mechanism for phase slips due to biochemical noise. Model predictions agree with recent experiments and illuminate the crucial roles of hydrodynamics and mechanochemical feedback in synchronization.

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.

Mathematical Modeling of Mucociliary Clearance: A Mini-Review

Mucociliary clearance is an important innate host defense of the mammalian respiratory system, as it traps foreign substances, including pollutants, pathogens, and allergens, and transports them out of the airway. The underlying mechanism of the actuation and coordination of cilia, the interplay between the cilia and mucus, and the formation of the metachronal wave have been explored extensively both experimentally and mathematically. In this mini-review, we provide a survey of the mathematical models of mucociliary clearance, from the motion of one single cilium to the emergence of the metachronal wave in a group of them, from the fundamental theoretical study to the state-of-the-art three-dimensional simulations. The mechanism of cilium actuation is discussed, together with the mathematical simplification and the implications or caveats of the results.

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.

Coordination of eukaryotic cilia and flagella

Propulsion by slender cellular appendages called cilia and flagella is an ancient means of locomotion. Unicellular organisms evolved myriad strategies to propel themselves in fluid environments, often involving significant differences in flagella number, localisation and modes of actuation. Remarkably, these appendages are highly conserved, occurring in many complex organisms such as humans, where they may be found generating physiological flows when attached to surfaces (e.g. airway epithelial cilia), or else conferring motility to male gametes (e.g. undulations of sperm flagella). Where multiple cilia arise, their movements are often observed to be highly coordinated. Here I review the two main mechanisms for motile cilia coordination, namely, intracellular and hydrodynamic, and discuss their relative importance in different ciliary systems.

Hydrodynamic Clustering of Human Sperm in Viscoelastic Fluids

We have numerically investigated sperm clustering behaviours, modelling cells as superpositions of regularised flow singularities, coarse-grained from experimentally obtained digital microscopy of human sperm, both in watery medium and a highly viscous-weakly elastic, methylcellulose medium. We find that the cell yaw and cell pulling dynamics inhibit clustering in low viscosity media. In contrast clustering is readily visible in simulations modelling sperm within a methylcellulose medium, in line with previous observations that bovine sperm clustering is much more prominent in a rheological polyacrylamide medium. Furthermore, the fine-scale details of sperm flagellar movement substantially impact large-scale collective behaviours, further motivating the need for the digital microscopy and characterization of sperm to understand their dynamics.

3-D Human Sperm Flagellum Tracing in Low SNR Fluorescence Images

Tracing tubular structures from biomedical images is important for a wide range of applications. Particularly, the spermatozoon is an essential cell whose flagella have a tubular form. Its main function is to fertilize the egg, and the flagellum is fundamental to achieve this task which depends importantly on the dynamics of intracellular calcium ([Ca²⁺]_i). Measuring [Ca²⁺]_i along the flagellum in 3-D is not a simple matter since it requires: 1) sophisticated fluorescence imaging techniques dealing with low intensity and signal to noise ratio (SNR) and 2) tracing the flagellum's centerline. Most of the algorithms proposed to trace tubular structures have been developed for multi-branch structures not being adequate for single tubular structures with low SNR. Taking into account the prior knowledge that the flagellum is constituted by a single tubular structure, we propose an automatic method to trace and track multiple single tubular structures from 3-D images. First, an algorithm based on one-class classification allows enhancement of the flagellum. This enhanced 3-D image permits guiding an iterative centerline algorithm toward the flagellum's centerline. Each sperm is assigned an ID to keep track of it in 3-D . Our algorithm was quantitatively evaluated using a ground truth 564 semi-manual traces (six 3-D image stacks) comparing them to those obtained from state-of-the-art tubular structure centerline extraction algorithms. The qualitative and quantitative results show that our algorithm is extracting similar traces as compared with ground truth, and it is more robust and accurate to trace the flagellum's centerline than multi-branch algorithms.

Time Irreversibility and Criticality in the Motility of a Flagellate Microorganism

Active living organisms exhibit behavioral variability, partitioning between fast and slow dynamics. Such variability may be key to generating rapid responses in a heterogeneous, unpredictable environment wherein cellular activity effects continual exchanges of energy fluxes. We demonstrate a novel, noninvasive strategy for revealing nonequilibrium control of swimming-specifically, in an octoflagellate microalga. These organisms exhibit surprising features of flagellar excitability and mechanosensitivity, which characterize a novel, time-irreversible "run-stop-shock" motility comprising forward runs, knee-jerk shocks with dramatic beat reversal, and long stops during which cells are quiescent yet continue to exhibit submicron flagellar vibrations. Entropy production, associated with flux cycles arising in a reaction graph representation of the gait-switching dynamics, provides a direct measure of detailed balance violation in this primitive alga.

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.