Flagellar and ciliary beating: the proven and the possible

Directional ciliary beats across epithelia require Ccdc57-mediated coupling between axonemal orientation and basal body polarity

Motile cilia unify their axonemal orientations (AOs), or beat directions, across epithelia to drive liquid flows. This planar polarity results from cytoskeleton-driven swiveling of basal foot (BF), a basal body (BB) appendage coincident with the AO, in response to regulatory cues. How and when the BF-AO relationship is established, however, are unaddressed. Here, we show that the BF-AO coupling occurs during rotational polarizations of BBs and requires Ccdc57. Ccdc57 localizes on BBs as a rotationally-asymmetric punctum, which polarizes away from the BF in BBs having achieved the rotational polarity to probably fix the BF-AO relationship. Consistently, Ccdc57-deficient ependymal multicilia lack the BF-AO coupling and display directional beats at only single cell level. Ccdc57 -/- tracheal multicilia also fail to fully align their BFs. Furthermore, Ccdc57 -/- mice manifest severe hydrocephalus, due to impaired cerebrospinal fluid flow, and high mortality. These findings unravel mechanisms governing the planar polarity of epithelial motile cilia.

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.

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.

CCDC176 stabilizes microtubule doublets 1 and 9 to ensure proper sperm movement

The molecular mechanism underlying asymmetric axonemal complexes in sperm flagella is still largely unknown. Here, we showed that the knockout of the coiled-coil domain-containing 176 (CCDC176) in mice led to male infertility due to decreased sperm motility. Ccdc176 knockout specifically destabilized microtubule doublets (MTDs) 1 and 9 during sperm maturation in the corpus epididymis. Single-sperm immunofluorescence showed that most CCDC176 was distributed along the axoneme, and further super-resolution imaging revealed that CCDC176 is asymmetrically localized in the sperm axoneme. CCDC176 could cooperate with microtubule and radial spoke proteins to stabilize MTDs 1 and 9, and its knockout results in the destabilization of some proteins in sperm flagella. Furthermore, as predicted by the sperm multibody dynamics (MBD) model, we found that MTDs 1 and 9 jutted out from the sperm flagellum annulus region in Ccdc176-/- spermatozoa, and these flagellar defects alter sperm flagellar beat patterns and swimming paths, potentially owing to the reduction and disequilibration of bending torque on the central pair. These results demonstrate that CCDC176 specifically stabilizes MTDs 1 and 9 in the sperm flagellum to ensure proper sperm movement for fertilization.

Bending Actuation of Hydrogels through Rotation of Light-Driven Molecular Motors

The unidirectional rotation of chemically crosslinked light-driven molecular motors is shown to progressively shift the swelling equilibrium of hydrogels. The concentration of molecular motors and the initial strand density of the polymer network are key parameters to modulate the macroscopic contraction of the material, and both parameters can be tuned using polymer chains of different molecular weights. These findings led to the design of optimized hydrogels revealing a half-time contraction of approximately 5 min. Furthermore, under inhomogeneous stimulation, the local contraction event was exploited to design useful bending actuators with an energy output 400 times higher than for previously reported self-assembled systems involving rotary motors. In the present configuration, we measure that a single molecular motor can lift up loads of 200 times its own molecular weight.

Calaxin is required for asymmetric bend initiation and propagation in sperm flagella

Regulation of waveform asymmetry in flagella is critical for changes in direction when sperm are swimming, as seen during the chemotaxis of sperm towards eggs. Ca2+ is an important regulator of asymmetry in flagellar waveforms. A calcium sensor protein, calaxin, is associated with the outer arm dynein and plays a key role in the regulation of flagellar motility in a Ca2+-dependent manner. However, the underlying mechanism of regulating asymmetric waves by means of Ca2+ and calaxin remains unclear. To clarify the calaxin-dependent mechanism for generating Ca2+-dependent asymmetric flagellar waveforms, we analyzed the initial step of flagellar bend formation and propagation in the sperm of the ascidian Ciona intestinalis. Our experiment used demembranated sperm cells, which were then reactivated by UV flash photolysis of caged ATP under both high and low Ca2+ concentrations. Here, we show that initial bends in the flagella are formed at the base of the sperm and propagate towards the tip during waveform generation. However, the direction of the initial bend differed between asymmetric and symmetric waves. When a calaxin inhibitor (repaglinide) was applied, it resulted in the failure of asymmetric wave formation and propagation. This was because repaglinide had no effect on initial bend formation, but it significantly inhibited the generation of the subsequent bend in the reverse direction. Switching of dynein sliding activity by mechanical feedback is crucial for flagellar oscillation. Our results suggest that the Ca2+/calaxin mechanism plays an important role in the switching of dynein activity from microtubule sliding in the principal bend into the suppressed sliding in the reverse bend, thereby allowing the sperm to successfully change direction.

Dynamic movement and turnover of extracellular matrices during tissue development and maintenance

Extracellular matrices (ECMs) are essential for the architecture and function of animal tissues. ECMs have been thought to be highly stable structures; however, too much stability of ECMs would hamper tissue remodelling required for organ development and maintenance. Regarding this conundrum, this article reviews multiple lines of evidence that ECMs are in fact rapidly moving and replacing components in diverse organisms including hydra, worms, flies, and vertebrates. Also discussed are how cells behave on/in such dynamic ECMs, how ECM dynamics contributes to embryogenesis and adult tissue homoeostasis, and what molecular mechanisms exist behind the dynamics. In addition, it is highlighted how cutting-edge technologies such as genome engineering, live imaging, and mathematical modelling have contributed to reveal the previously invisible dynamics of ECMs. The idea that ECMs are unchanging is to be changed, and ECM dynamics is emerging as a hitherto unrecognized critical factor for tissue development and maintenance.

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.

A Synthetic Minimal Beating Axoneme

Cilia and flagella are beating rod-like organelles that enable the directional movement of microorganisms in fluids and fluid transport along the surface of biological organisms or inside organs. The molecular motor axonemal dynein drives their beating by interacting with microtubules. Constructing synthetic beating systems with axonemal dynein capable of mimicking ciliary beating still represents a major challenge. Here, the bottom-up engineering of a sustained beating synthoneme consisting of a pair of microtubules connected by a series of periodic arrays of approximately eight axonemal dyneins is reported. A model leads to the understanding of the motion through the cooperative, cyclic association-dissociation of the molecular motor from the microtubules. The synthoneme represents a bottom-up self-organized bio-molecular machine at the nanoscale with cilia-like properties.

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.

Generation of ciliary beating by steady dynein activity: the effects of inter-filament coupling in multi-filament models

The structure of the axoneme in motile cilia and flagella is emerging with increasing detail from high-resolution imaging, but the mechanism by which the axoneme creates oscillatory, propulsive motion remains mysterious. It has recently been proposed that this motion may be caused by a dynamic 'flutter' instability that can occur under steady dynein loading, and not by switching or modulation of dynein motor activity (as commonly assumed). In the current work, we have built an improved multi-filament mathematical model of the axoneme and implemented it as a system of discrete equations using the finite-element method. The eigenvalues and eigenvectors of this model predict the emergence of oscillatory, wave-like solutions in the absence of dynein regulation and specify the associated frequencies and waveforms of beating. Time-domain simulations with this model illustrate the behaviour predicted by the system's eigenvalues. This model and analysis allow us to efficiently explore the potential effects of difficult to measure biophysical parameters, such as elasticity of radial spokes and inter-doublet links, on the ciliary waveform. These results support the idea that dynamic instability without dynamic dynein regulation is a plausible and robust mechanism for generating ciliary beating.

The Plasmodium falciparum CCCH Zinc Finger Protein ZNF4 Plays an Important Role in Gametocyte Exflagellation through the Regulation of Male Enriched Transcripts

CCCH zinc finger proteins (ZFPs) function mainly as RNA-binding proteins (RBPs) and play a central role in the mRNA metabolism. Over twenty seven CCCH-ZFPs are encoded in the genome of the human malaria parasite Plasmodium falciparum, the causative agent of malaria tropica. However, little is known about their functions. In this study, we characterize one member of the PfCCCH-ZFP named ZNF4. We show that ZNF4 is highly expressed in mature gametocytes, where it predominantly localizes to the cytoplasm. Targeted gene disruption of ZNF4 showed no significant effect in asexual blood stage replication and gametocyte development while male gametocyte exflagellation was significantly impaired, leading to reduced malaria transmission in the mosquito. Comparative transcriptomics between wildtype (WT) and the ZNF4-deficient line (ZNF4-KO) demonstrated the deregulation of about 473 genes (274 upregulated and 199 downregulated) in mature gametocytes. Most of the downregulated genes show peak expression in mature gametocyte with male enriched genes associated to the axonemal dynein complex formation, and cell projection organization is highly affected, pointing to the phenotype in male gametocyte exflagellation. Upregulated genes are associated to ATP synthesis. Our combined data therefore indicate that ZNF4 is a CCCH zinc finger protein which plays an important role in male gametocyte exflagellation through the regulation of male gametocyte-enriched genes.

Cryo-EM structure of an active central apparatus

Accurately regulated ciliary beating in time and space is critical for diverse cellular activities, which impact the survival and development of nearly all eukaryotic species. An essential beating regulator is the conserved central apparatus (CA) of motile cilia, composed of a pair of microtubules (C1 and C2) associated with hundreds of protein subunits per repeating unit. It is largely unclear how the CA plays its regulatory roles in ciliary motility. Here, we present high-resolution structures of Chlamydomonas reinhardtii CA by cryo-electron microscopy (cryo-EM) and its dynamic conformational behavior at multiple scales. The structures show how functionally related projection proteins of CA are clustered onto a spring-shaped scaffold of armadillo-repeat proteins, facilitated by elongated rachis-like proteins. The two halves of the CA are brought together by elastic chain-like bridge proteins to achieve coordinated activities. We captured an array of kinesin-like protein (KLP1) in two different stepping states, which are actively correlated with beating wave propagation of cilia. These findings establish a structural framework for understanding the role of the CA in cilia.

Prediction of Sperm Progression in Three Dimensions Using Rapid Optical Imaging and Dynamic Mechanical Modeling

We present a multidisciplinary approach for predicting how sperm cells with various morphologies swim in three-dimensions (3D), from milliseconds to much longer time scales at spatial resolutions of less than half a micron. We created the sperm 3D geometry and built a numerical mechanical model using the experimentally acquired dynamic 3D refractive-index profiles of sperm cells swimming in vitro as imaged by high-resolution optical diffraction tomography. By controlling parameters in the model, such as the size and shape of the sperm head and tail, we can then predict how different sperm cells, normal or abnormal, would swim in 3D, in the short or long term. We quantified various 3D structural factor effects on the sperm long-term motility. We found that some abnormal sperm cells swim faster than normal sperm cells, in contrast to the commonly used sperm selection assumption during in vitro fertilization (IVF), according to which sperm cells should mainly be chosen based on their progressive motion. We thus establish a new tool for sperm analysis and male-infertility diagnosis, as well as sperm selection criteria for fertility treatments.

High-Frequency Ultrasound Boosts Bull and Human Sperm Motility

Sperm motility is a significant predictor of male fertility potential and is directly linked to fertilization success in both natural and some forms of assisted reproduction. Sperm motility can be impaired by both genetic and environmental factors, with asthenozoospermia being a common clinical presentation. Moreover, in the setting of assisted reproductive technology clinics, there is a distinct absence of effective and noninvasive technology to increase sperm motility without detriment to the sperm cells. Here, a new method is presented to boost sperm motility by increasing the intracellular rate of metabolic activity using high frequency ultrasound. An increase of 34% in curvilinear velocity (VCL), 10% in linearity, and 32% in the number of motile sperm cells is shown by rendering immotile sperm motile, after just 20 s exposure. A similar effect with an increase of 15% in VCL treating human sperm with the same setting is also identified. This cell level mechanotherapy approach causes no significant change in cell viability or DNA fragmentation index, and, as such, has the potential to be applied to encourage natural fertilization or less invasive treatment choices such as in vitro fertilization rather than intracytoplasmic injection.

Decoding the Bell-Shaped Calcium Spikes in Phosphorylation Cycles of Flagella

We investigate the messenger role of calcium ions implicated in the regulation of wave-like bending dynamics of flagella. The emphasis is on microtubules of flagellar axoneme serving as nonlinear transmission lines for bell-shaped spikes of calcium ions. The calcium sensitive proteins, such as calmodulin, exhibit activation dependence on the spike train frequency and amplitude. Here, we analyze a Ca²⁺ decoding module IDA-I1 whose activity is controlled by Ca²⁺ activated kinase. We find that trains of Ca²⁺ spikes are advantageous compared to a constant rise in Ca²⁺ concentration as being more efficient and much less prone to noisy fluctuations.

Unraveling the Kinematics of Sperm Motion by Reconstructing the Flagellar Wave Motion in 3D

Sperm swim through the female reproductive tract by propagating a 3D flagellar wave that is self-regulatory in nature and driven by dynein motors. Traditional microscopy methods fail to capture the full dynamics of sperm flagellar activity as they only image and analyze sperm motility in 2D. Here, an automated platform to analyze sperm swimming behavior in 3D by using thin-lens approximation and high-speed dark field microscopy to reconstruct the flagellar waveform in 3D is presented. It is found that head-tethered mouse sperm exhibit a rolling beating behavior in 3D with the beating frequency of 6.2 Hz using spectral analysis. The flagellar waveform bends in 3D, particularly in the distal regions, but is only weakly nonplanar and ambidextrous in nature, with the local helicity along the flagellum fluctuating between clockwise and counterclockwise handedness. These findings suggest a nonpersistent flagellar helicity. This method provides new opportunities for the accurate measurement of the full motion of eukaryotic flagella and cilia which is essential for a biophysical understanding of their activation by dynein motors.

Active beating modes of two clamped filaments driven by molecular motors

Biological cilia pump the surrounding fluid by asymmetric beating that is driven by dynein motors between sliding microtubule doublets. The complexity of biological cilia raises the question about minimal systems that can re-create similar patterns of motion. One such system consists of a pair of microtubules that are clamped at the proximal end. They interact through dynein motors that cover one of the filaments and pull against the other one. Here, we study theoretically the static shapes and the active dynamics of such a system. Using the theory of elastica, we analyse the shapes of two filaments of different lengths with clamped ends. Starting from equal lengths, we observe a transition similar to Euler buckling leading to a planar shape. When further increasing the length ratio, the system assumes a non-planar shape with spontaneously broken chiral symmetry after a secondary bifurcation and then transitions to planar again. The predicted curves agree with experimentally observed shapes of microtubule pairs. The dynamical system can have a stable fixed point, with either bent or straight filaments, or limit cycle oscillations. The latter match many properties of ciliary motility, demonstrating that a two-filament system can serve as a minimal actively beating model.

The flagellar germ-line hypothesis: How flagellate and ciliate gametes significantly shaped the evolution of organismal complexity

This essay presents a hypothesis which contends that the development of organismic complexity in the eukaryotes depended extensively on propagation via flagellated and ciliated gametes. Organisms utilizing flagellate and ciliate gametes to propagate their germ line have contributed most of the organismic complexity found in the higher animals. The genes of the flagellum and the flagellar assembly system (intraflagellar transport) have played a disproportionately important role in the construction of complex tissues and organs. The hypothesis also proposes that competition between large numbers of haploid flagellated male gametes rigorously conserved the functionality of a key set of flagellar genes for more than 700 million years. This in turn has insured that a large set (>600) of highly functional cytoskeletal and signal pathway genes is always present in the lineage of organisms with flagellated or ciliated gametes to act as a dependable resource, or "toolkit," for organ elaboration.

Metachronal waves in concentrations of swimming Turbatrix aceti nematodes and an oscillator chain model for their coordinated motions

At high concentration, free swimming nematodes known as vinegar eels (Turbatrix aceti), collectively exhibit metachronal waves near a boundary. We find that the frequency of the collective traveling wave is lower than that of the freely swimming organisms. We explore models based on a chain of oscillators with nearest-neighbor interactions that inhibit oscillator phase velocity. The phase of each oscillator represents the phase of the motion of the eel's head back and forth about its mean position. A strongly interacting directed chain model mimicking steric repulsion between organisms robustly gives traveling wave states and can approximately match the observed wavelength and oscillation frequency of the observed traveling wave. We predict body shapes assuming that waves propagate down the eel body at a constant speed. The phase oscillator model that impedes eel head overlaps also reduces close interactions throughout the eel bodies.

Modelling Motility: The Mathematics of Spermatozoa

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

The biomechanical role of extra-axonemal structures in shaping the flagellar beat of Euglena gracilis

We propose and discuss a model for flagellar mechanics in Euglena gracilis. We show that the peculiar non-planar shapes of its beating flagellum, dubbed 'spinning lasso', arise from the mechanical interactions between two of its inner components, namely, the axoneme and the paraflagellar rod. The spontaneous shape of the axoneme and the resting shape of the paraflagellar rod are incompatible. Thus, the complex non-planar configurations of the coupled system emerge as the energetically optimal compromise between the two antagonistic components. The model is able to reproduce the experimentally observed flagellar beats and the characteristic geometric signature of spinning lasso, namely, traveling waves of torsion with alternating sign along the length of the flagellum.

EFFECT OF CHEMICAL REACTION AND THERMAL RADIATION ON AXISYMMETRIC MHD FLOW OF JEFFREY NANOFLUID THROUGH A CILIATED CHANNEL FILLED WITH POROUS MEDIUM

To prevent the respiratory diseases in an air ways, a defense mechanism based on mucus transport by the moving cilia plays an important role. The mucus transport is affected by the thermal radiation, chemical reaction that changes the physics of fluid due to nanoparticles and thickness of mucus, also different problems in respiratory tract may occur due to the mucus efficacy. In this study, it is observed that the mucus transport can be controlled by the magnetic field that is produced by the drug delivery of nanoparticles, thermal radiation due to temperature difference, porous medium due to respiratory infection, and diffusion of the nanoparticles (chemical reaction) due to the magnetic drug delivery. In this model, flow of Jeffrey nanofluid through the ciliated tube resembles with the mucus flow in a wind pipe. The movement of the mucus is observed by the momentum, energy and concentration equation in the presence of body forces due to magnetic field, heat source due to radiation, Darcy’s resistance due to infection and chemical reaction due to the concentration of nanoparticles. Mathematical model of this study forms a complex system of partial differential equations under the low Reynolds number and long wavelength approximation. The nonlinear set of partial differential equations is solved by the Homotopy perturbation method and software “Mathematica,” results are found for velocity, temperature and concentration profiles and concluded that the mucus flow decelerates due to magnetic field produced by the drug delivery of the nanoparticles but accelerates due to the viscoelastic parameter of Jeffrey fluid and Darcy’s resistance parameter due to infection. The heat transfer rate in the mucus flow rises by increasing the random motion and reduces by the radiation and energy loss. The diffusion of the nanoparticles in the mucus rises by the growing values of thermophoresis and chemical reaction parameter and reduces by the growing values of viscoelastic and Brownian motion parameter.

Thermal analysis of airway mucus clearance by ciliary activity in the presence of inertial forces

In this study heat transfer effects on cilia induced mucus flow in human airways is presented. The elliptic wave pattern of cilia tips produces metachronal wave which enables the transportation of highly viscous mucus with nonzero inertial forces. Upper Convective Maxwell model is considered as mucus. The governing partial differential equations are transformed from the fixed frame to the wave frame by using Galilean transformation and viscous dissipation is also incorporated in the energy equation. The non-linear governing equations are evaluated by the perturbation technique by using software “MATHEMATICA” and pressure rise is computed by numerical integration. The impact of interested parameters on temperature profile, velocity, pressure rise and pressure gradient are plotted by the graphs. The comparison of velocities due to symplectic and antiplectic metachronal wave are also achieved graphically.

The many modes of flagellar and ciliary beating: Insights from a physical analysis

The mechanism that allows the axoneme of eukaryotic cilia and flagella to produce both helical and planar beating is an enduring puzzle. The nine outer doublets of eukaryotic cilia and flagella are arranged in a circle. Therefore, each doublet pair with its associated dynein motors, should produce torque to bend the flagellum in a different direction. Sequential activation of each doublet pair should, therefore result in a helical bending wave. In reality, most cilia and flagella have a well‐defined bending plane and many exhibit an almost perfectly flat (planar) beating pattern. In this analysis we examine the physics that governs flagellar bending, and arrive at two distinct possibilities that could explain the mechanism of planar beating. Of these, the mechanism with the best observational support is that the flagellum behaves as two ribbons of doublets interacting with a central partition. We also examine the physics of torsion in flagella and conclude that torsion could play a role in transitioning from a planar to a helical beating modality in long flagella. Lastly, we suggest some tests that would provide theoretical and/or experimental evaluation of our proposals.

Three-dimensional nonlinear dynamics of prestressed active filaments: Flapping, swirling, and flipping

Initially straight slender elastic filaments or rods with constrained ends buckle and form stable two-dimensional shapes when prestressed by bringing the ends together. Beyond a critical value of this prestress, rods can also deform off plane and form twisted three-dimensional equilibrium shapes. Here, we analyze the three-dimensional instabilities and dynamics of such deformed filaments subject to nonconservative active follower forces and fluid drag. We find that softly constrained filaments that are clamped at one end and pinned at the other exhibit stable two-dimensional planar flapping oscillations when active forces are directed toward the clamped end. Reversing the directionality of the forces quenches the instability. For strongly constrained filaments with both ends clamped, computations reveal an instability arising from the twist-bend-activity coupling. Planar oscillations are destabilized by off-planar perturbations resulting in twisted three-dimensional swirling patterns interspersed with periodic flipping or reversal of the swirling direction. These striking swirl-flip transitions are characterized by two distinct timescales: the time period for a swirl (rotation) and the time between flipping events. We interpret these reversals as relaxation oscillation events driven by accumulation of torsional energy. Each cycle is initiated by a fast jump in torsional deformation with a subsequent slow decrease in net torsion until the next cycle. Our work reveals the rich tapestry of spatiotemporal patterns when weakly inertial strongly damped rods are deformed by nonconservative active forces. Taken together, our results suggest avenues by which prestress, elasticity, and activity may be used to design synthetic macroscale pumps or mixers.

RAC1 controls progressive movement and competitiveness of mammalian spermatozoa

Mammalian spermatozoa employ calcium (Ca2+) and cyclic adenosine monophosphate (cAMP) signaling in generating flagellar beat. However, how sperm direct their movement towards the egg cells has remained elusive. Here we show that the Rho small G protein RAC1 plays an important role in controlling progressive motility, in particular average path velocity and linearity. Upon RAC1 inhibition of wild type sperm with the drug NSC23766, progressive movement is impaired. Moreover, sperm from mice homozygous for the genetically variant t-haplotype region (tw5/tw32), which are sterile, show strongly enhanced RAC1 activity in comparison to wild type (+/+) controls, and quickly become immotile in vitro. Sperm from heterozygous (t/+) males, on the other hand, display intermediate RAC1 activity, impaired progressive motility and transmission ratio distortion (TRD) in favor of t-sperm. We show that t/+-derived sperm consist of two subpopulations, highly progressive and less progressive. The majority of highly progressive sperm carry the t-haplotype, while most less progressive sperm contain the wild type (+) chromosome. Dosage-controlled RAC1 inhibition in t/+ sperm by NSC23766 rescues progressive movement of (+)-sperm in vitro, directly demonstrating that impairment of progressive motility in the latter is caused by enhanced RAC1 activity. The combined data show that RAC1 plays a pivotal role in controlling progressive motility in sperm, and that inappropriate, enhanced or reduced RAC1 activity interferes with sperm progressive movement. Differential RAC1 activity within a sperm population impairs the competitiveness of sperm cells expressing suboptimal RAC1 activity and thus their fertilization success, as demonstrated by t/+-derived sperm. In conjunction with t-haplotype triggered TRD, we propose that Rho GTPase signaling is essential for directing sperm towards the egg cells.

Light-Powered Autonomous Flagella-Like Motion of Molecular Crystal Microwires

The ability to exhibit life-like oscillatory motion fueled by light represents a new capability for stimuli-responsive materials. Although this capability has been demonstrated in soft materials like polymers, it has never been observed in molecular crystals, which are not generally regarded as dynamic objects. In this work, it is shown that molecular crystalline microwires composed of (Z)-2-(3-(anthracen-9-yl)allylidene)malononitrile ((Z)-DVAM) can be continuously actuated when exposed to a combination of ultraviolet and visible light. The photo-induced motion mimics the oscillatory behavior of biological flagella and enables propagation of microwires across a surface and through liquids, with translational speeds up to 7 μm s^-1 . This is the first example of molecular crystals that show complex oscillatory behavior under continuous irradiation. A model that relates the rotation of the transition dipole moment between reversible E→Z photoisomerization to the microscopic torque can qualitatively reproduce how the rotational frequency depends on light intensity and polarization.

Synchronized oscillations, traveling waves, and jammed clusters induced by steric interactions in active filament arrays

Autonomous active, elastic filaments that interact with each other to achieve cooperation and synchrony underlie many critical functions in biology. The mechanisms underlying this collective response and the essential ingredients for stable synchronization remain a mystery. Inspired by how these biological entities integrate elasticity with molecular motor activity to generate sustained oscillations, a number of synthetic active filament systems have been developed that mimic oscillations of these biological active filaments. Here, we describe the collective dynamics and stable spatiotemporal patterns that emerge in such biomimetic multi-filament arrays, under conditions where steric interactions may impact or dominate the collective dynamics. To focus on the role of steric interactions, we study the system using Brownian dynamics, without considering long-ranged hydrodynamic interactions. The simulations treat each filament as a connected chain of self-propelling colloids. We demonstrate that short-range steric inter-filament interactions and filament roughness are sufficient - even in the absence of inter-filament hydrodynamic interactions - to generate a rich variety of collective spatiotemporal oscillatory, traveling and static patterns. We first analyze the collective dynamics of two- and three-filament clusters and identify parameter ranges in which steric interactions lead to synchronized oscillations and strongly occluded states. Generalizing these results to large one-dimensional arrays, we find rich emergent behaviors, including traveling metachronal waves, and modulated wavetrains that are controlled by the interplay between the array geometry, filament activity, and filament elasticity. Interestingly, the existence of metachronal waves is non-monotonic with respect to the inter-filament spacing. We also find that the degree of filament roughness significantly affects the dynamics - specifically, filament roughness generates a locking-mechanism that transforms traveling wave patterns into statically stuck and jammed configurations. Taken together, simulations suggest that short-ranged steric inter-filament interactions could combine with complementary hydrodynamic interactions to control the development and regulation of oscillatory collective patterns. Furthermore, roughness and steric interactions may be critical to the development of jammed spatially periodic states; a spatiotemporal feature not observed in purely hydrodynamically interacting systems.

Structure of the radial spoke head and insights into its role in mechanoregulation of ciliary beating

Motile cilia power cell locomotion and drive extracellular fluid flow by propagating bending waves from their base to tip. The coordinated bending of cilia requires mechanoregulation by the radial spoke (RS) protein complexes and the microtubule central pair (CP). Despite their importance for ciliary motility across eukaryotes, the molecular function of the RSs is unknown. Here, we reconstituted the Chlamydomonas reinhardtii RS head that abuts the CP and determined its structure using single-particle cryo-EM to 3.1-Å resolution, revealing a flat, negatively charged surface supported by a rigid core of tightly intertwined proteins. Mutations in this core, corresponding to those involved in human ciliopathies, compromised the stability of the recombinant complex, providing a molecular basis for disease. Partially reversing the negative charge on the RS surface impaired motility in C. reinhardtii. We propose that the RS-head architecture is well-suited for mechanoregulation of ciliary beating through physical collisions with the CP.

Detachment of Dunaliella tertiolecta Microalgae from a Glass Surface by a Near-Infrared Optical Trap

We report on the observation of the detachment in situ and in vivo of Dunaliella tertiolecta microalgae cells from a glass surface using a 1064 nm wavelength trapping laser beam. The principal bends of both flagella of Dunaliella were seen self-adhered to either the top or bottom coverslip surfaces of a 50 μm thick chamber. When a selected attached Dunaliella was placed in the trapping site, it photoresponded to the laser beam by moving its body and flagellar tips, which eventually resulted in its detachment. The dependence of the time required for detachment on the trapping power was measured. No significant difference was found in the detachment time for cells detached from the top or bottom coverslip, indicating that the induced detachment was not due solely to the optical forces applied to the cells. After detachment, the cells remained within the optical trap. Dunaliella detached from the bottom were seen rotating about their long axis in a counterclockwise direction, while those detached from the top did not rotate. The rotation frequency and the minimal force required to escape from the trap were also measured. The average rotation frequency was found to be independent of the trapping power, and the swimming force of a cell escaping the laser trap ranged from 4 to 10 picoNewtons. Our observations provide insight into the photostimulus produced when a near-infrared trapping beam encounters a Dunaliella. The microalgae frequently absorb more light than they can actually use in photosynthesis, which could cause genetic and molecular changes. Our findings may open new research directions into the study of photomovement in species of Dunaliella and other swimming microorganisms that could eventually help to solve technological problems currently confronting biomass production. In future work, studies of the response to excess light may uncover unrecognized mechanisms of photoprotection and photoacclimation.

Cfap97d1 is important for flagellar axoneme maintenance and male mouse fertility

The flagellum is essential for sperm motility and fertilization in vivo. The axoneme is the main component of the flagella, extending through its entire length. An axoneme is comprised of two central microtubules surrounded by nine doublets, the nexin-dynein regulatory complex, radial spokes, and dynein arms. Failure to properly assemble components of the axoneme in a sperm flagellum, leads to fertility alterations. To understand this process in detail, we have defined the function of an uncharacterized gene, Cfap97 domain containing 1 (Cfap97d1). This gene is evolutionarily conserved in mammals and multiple other species, including Chlamydomonas. We have used two independently generated Cfap97d1 knockout mouse models to study the gene function in vivo. Cfap97d1 is exclusively expressed in testes starting from post-natal day 20 and continuing throughout adulthood. Deletion of the Cfap97d1 gene in both mouse models leads to sperm motility defects (asthenozoospermia) and male subfertility. In vitro fertilization (IVF) of cumulus-intact oocytes with Cfap97d1 deficient sperm yielded few embryos whereas IVF with zona pellucida-free oocytes resulted in embryo numbers comparable to that of the control. Knockout spermatozoa showed abnormal motility characterized by frequent stalling in the anti-hook position. Uniquely, Cfap97d1 loss caused a phenotype associated with axonemal doublet heterogeneity linked with frequent loss of the fourth doublet in the sperm stored in the epididymis. This study demonstrates that Cfap97d1 is required for sperm flagellum ultra-structure maintenance, thereby playing a critical role in sperm function and male fertility in mice.

Identification and mapping of central pair proteins by proteomic analysis

Cilia or flagella of eukaryotes are small micro-hair like structures that are indispensable to single-cell motility and play an important role in mammalian biological processes. Cilia or flagella are composed of nine doublet microtubules surrounding a pair of singlet microtubules called the central pair (CP). Together, this arrangement forms a canonical and highly conserved 9+2 axonemal structure. The CP, which is a unique structure exclusive to motile cilia, is a pair of structurally dimorphic singlet microtubules decorated with numerous associated proteins. Mutations of CP-associated proteins cause several different physical symptoms termed as ciliopathies. Thus, it is crucial to understand the architecture of the CP. However, the protein composition of the CP was poorly understood. This was because the traditional method of identification of CP proteins was mostly limited by available Chlamydomonas mutants of CP proteins. Recently, more CP protein candidates were presented based on mass spectrometry results, but most of these proteins were not validated. In this study, we re-evaluated the CP proteins by conducting a similar comprehensive CP proteome analysis comparing the mass spectrometry results of the axoneme sample prepared from Chlamydomonas strains with and without CP complex. We identified a similar set of CP protein candidates and additional new 11 CP protein candidates. Furthermore, by using Chlamydomonas strains lacking specific CP sub-structures, we present a more complete model of localization for these CP proteins. This work has established a new foundation for understanding the function of the CP complex in future studies.

The FOXJ1 target Cfap206 is required for sperm motility, mucociliary clearance of the airways and brain development

Cilia are complex cellular protrusions consisting of hundreds of proteins. Defects in ciliary structure and function, many of which have not been characterised molecularly, cause ciliopathies: a heterogeneous group of human syndromes. Here, we report on the FOXJ1 target gene Cfap206, orthologues of which so far have only been studied in Chlamydomonas and Tetrahymena In mouse and Xenopus, Cfap206 was co-expressed with and dependent on Foxj1 CFAP206 protein localised to the basal body and to the axoneme of motile cilia. In Xenopus crispant larvae, the ciliary beat frequency of skin multiciliated cells was enhanced and bead transport across the epidermal mucociliary epithelium was reduced. Likewise, Cfap206 knockout mice revealed ciliary phenotypes. Electron tomography of immotile knockout mouse sperm flagella indicated a role in radial spoke formation reminiscent of FAP206 function in Tetrahymena Male infertility, hydrocephalus and impaired mucociliary clearance of the airways in the absence of laterality defects in Cfap206 mutant mice suggests that Cfap206 may represent a candidate for the subgroup of human primary ciliary dyskinesias caused by radial spoke defects.

Insights of the tubulin code in gametes and embryos: from basic research to potential clinical applications in humans†

Microtubules are intracellular filaments that define in space and in time a large number of essential cellular functions such as cell division, morphology and motility, intracellular transport and flagella and cilia assembly. They are therefore essential for spermatozoon and oocyte maturation and function, and for embryo development. The dynamic and functional properties of the microtubules are in large part defined by various classes of interacting proteins including MAPs (microtubule associated proteins), microtubule-dependent motors, and severing and modifying enzymes. Multiple mechanisms regulate these interactions. One of them is defined by the high diversity of the microtubules themselves generated by the combination of different tubulin isotypes and by several tubulin post-translational modifications (PTMs). This generates a so-called tubulin code that finely regulates the specific set of proteins that associates with a given microtubule thereby defining the properties and functions of the network. Here we provide an in depth review of the current knowledge on the tubulin isotypes and PTMs in spermatozoa, oocytes, and preimplantation embryos in various model systems and in the human species. We focus on functional implications of the tubulin code for cytoskeletal function, particularly in the field of human reproduction and development, with special emphasis on gamete quality and infertility. Finally, we discuss some of the knowledge gaps and propose future research directions.

Update on mammalian sperm capacitation: how much does the horse differ from other species?

In contrast to various other mammalian species, conventional in vitro fertilization (IVF) with horse gametes is not reliably successful. In particular, stallion spermatozoa fails to penetrate the zona pellucida, most likely due to incomplete activation of stallion spermatozoa (capacitation) under in vitro conditions. In other mammalian species, specific capacitation triggers have been described; unfortunately, none of these is able to induce full capacitation in stallion spermatozoa. Nevertheless, knowledge of capacitation pathways and their molecular triggers might improve our understanding of capacitation-related events observed in stallion sperm. When sperm cells are exposed to appropriate capacitation triggers, several molecular and biochemical changes should be induced in the sperm plasma membrane and cytoplasm. At the level of the sperm plasma membrane, (1) an increase in membrane fluidity, (2) cholesterol depletion and (3) lipid raft aggregation should occur consecutively; the cytoplasmic changes consist of protein tyrosine phosphorylation and elevated pH, cAMP and Ca2+ concentrations. These capacitation-related events enable the switch from progressive to hyperactivated motility of the sperm cells, and the induction of the acrosome reaction. These final capacitation triggers are indispensable for sperm cells to migrate through the viscous oviductal environment, penetrate the cumulus cells and zona pellucida and, finally, fuse with the oolemma. This review will focus on molecular aspects of sperm capacitation and known triggers in various mammalian species. Similarities and differences with the horse will be highlighted to improve our understanding of equine sperm capacitation/fertilizing events.

Force-Generating Mechanism of Axonemal Dynein in Solo and Ensemble

In eukaryotic cilia and flagella, various types of axonemal dyneins orchestrate their distinct functions to generate oscillatory bending of axonemes. The force-generating mechanism of dyneins has recently been well elucidated, mainly in cytoplasmic dyneins, thanks to progress in single-molecule measurements, X-ray crystallography, and advanced electron microscopy. These techniques have shed light on several important questions concerning what conformational changes accompany ATP hydrolysis and whether multiple motor domains are coordinated in the movements of dynein. However, due to the lack of a proper expression system for axonemal dyneins, no atomic coordinates of the entire motor domain of axonemal dynein have been reported. Therefore, a substantial amount of knowledge on the molecular architecture of axonemal dynein has been derived from electron microscopic observations on dynein arms in axonemes or on isolated axonemal dynein molecules. This review describes our current knowledge and perspectives of the force-generating mechanism of axonemal dyneins in solo and in ensemble.

Bioinspired reorientation strategies for application in micro/nanorobotic control

Engineers have recently been inspired by swimming methodologies of microorganisms in creating micro-/nanorobots for biomedical applications. Future medicine may be revolutionized by the application of these small machines in diagnosing, monitoring, and treating diseases. Studies over the past decade have often concentrated on propulsion generation. However, there are many other challenges to address before the practical use of robots at the micro-/nanoscale. The control and reorientation ability of such robots remain as some of these challenges. This paper reviews the strategies of swimming microorganisms for reorientation, including tumbling, reverse and flick, direction control of helical-path swimmers, by speed modulation, using complex flagella, and the help of mastigonemes. Then, inspired by direction change in microorganisms, methods for orientation control for microrobots and possible directions for future studies are discussed. Further, the effects of solid boundaries on the swimming trajectories of microorganisms and microrobots are examined. In addition to propulsion systems for artificial microswimmers, swimming microorganisms are promising sources of control methodologies at the micro-/nanoscale.

A framework for high-resolution phenotyping of candidate male infertility mutants: from human to mouse

Male infertility is a heterogeneous condition of largely unknown etiology that affects at least 7% of men worldwide. Classical genetic approaches and emerging next-generation sequencing studies support genetic variants as a frequent cause of male infertility. Meanwhile, the barriers to transmission of this disease mean that most individual genetic cases will be rare, but because of the large percentage of the genome required for spermatogenesis, the number of distinct causal mutations is potentially large. Identifying bona fide causes of male infertility thus requires advanced filtering techniques to select for high-probability candidates, including the ability to test causality in animal models. The mouse remains the gold standard for defining the genotype-phenotype connection in male fertility. Here, we present a best practice guide consisting of (a) major points to consider when interpreting next-generation sequencing data performed on infertile men, and, (b) a systematic strategy to categorize infertility types and how they relate to human male infertility. Phenotyping infertility in mice can involve investigating the function of multiple cell types across the testis and epididymis, as well as sperm function. These findings will feed into the diagnosis and treatment of male infertility as well as male health broadly.

Rsph4a is essential for the triplet radial spoke head assembly of the mouse motile cilia

Motile cilia/flagella are essential for swimming and generating extracellular fluid flow in eukaryotes. Motile cilia harbor a 9+2 arrangement consisting of nine doublet microtubules with dynein arms at the periphery and a pair of singlet microtubules at the center (central pair). In the central system, the radial spoke has a T-shaped architecture and regulates the motility and motion pattern of cilia. Recent cryoelectron tomography data reveal three types of radial spokes (RS1, RS2, and RS3) in the 96 nm axoneme repeat unit; however, the molecular composition of the third radial spoke, RS3 is unknown. In human pathology, it is well known mutation of the radial spoke head-related genes causes primary ciliary dyskinesia (PCD) including respiratory defect and infertility. Here, we describe the role of the primary ciliary dyskinesia protein Rsph4a in the mouse motile cilia. Cryoelectron tomography reveals that the mouse trachea cilia harbor three types of radial spoke as with the other vertebrates and that all triplet spoke heads are lacking in the trachea cilia of Rsph4a-deficient mice. Furthermore, observation of ciliary movement and immunofluorescence analysis indicates that Rsph4a contributes to the generation of the planar beating of motile cilia by building the distal architecture of radial spokes in the trachea, the ependymal tissues, and the oviduct. Although detailed mechanism of RSs assembly remains unknown, our results suggest Rsph4a is a generic component of radial spoke heads, and could explain the severe phenotype of human PCD patients with RSPH4A mutation.

Propulsive nanomachines: the convergent evolution of archaella, flagella and cilia

Echoing the repeated convergent evolution of flight and vision in large eukaryotes, propulsive swimming motility has evolved independently in microbes in each of the three domains of life. Filamentous appendages – archaella in Archaea, flagella in Bacteria and cilia in Eukaryotes – wave, whip or rotate to propel microbes, overcoming diffusion and enabling colonization of new environments. The implementations of the three propulsive nanomachines are distinct, however: archaella and flagella rotate, while cilia beat or wave; flagella and cilia assemble at their tips, while archaella assemble at their base; archaella and cilia use ATP for motility, while flagella use ion-motive force. These underlying differences reflect the tinkering required to evolve a molecular machine, in which pre-existing machines in the appropriate contexts were iteratively co-opted for new functions and whose origins are reflected in their resultant mechanisms. Contemporary homologies suggest that archaella evolved from a non-rotary pilus, flagella from a non-rotary appendage or secretion system, and cilia from a passive sensory structure. Here, we review the structure, assembly, mechanism and homologies of the three distinct solutions as a foundation to better understand how propulsive nanomachines evolved three times independently and to highlight principles of molecular evolution.

The use of biophysical approaches to understand ciliary beating

Motile cilia are a striking example of the functional cellular organelle, conserved across all the eukaryotic species. Motile cilia allow the swimming of cells and small organisms and transport of liquids across epithelial tissues. Whilst the molecular structure is now very well understood, the dynamics of cilia is not well established either at the single cilium level nor at the level of collective beating. Indeed, a full understanding of this requires connecting together behaviour across various lengthscales, from the molecular to the organelle, then at the cellular level and up to the tissue scale. Aside from the fundamental interest in this system, understanding beating is important to elucidate aspects of embryonic development and a variety of health conditions from fertility to genetic and infectious diseases of the airways.

Asymmetries in the cilia of Chlamydomonas

The generation of ciliary waveforms requires the spatial and temporal regulation of dyneins. This review catalogues many of the asymmetric structures and proteins in the cilia of Chlamydomonas, a unicellular alga with two cilia that are used for motility in liquid medium. These asymmetries, which have been identified through mutant analysis, cryo-EM tomography and proteomics, provide a wealth of information to use for modelling how waveforms are generated and propagated. This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.

LAX28 is required for the stable assembly of the inner dynein arm f complex, and the tether and tether head complex in Leishmania flagella

Motile eukaryotic flagella beat through coordinated activity of dynein motor proteins; however, the mechanisms of dynein coordination and regulation are incompletely understood. The inner dynein arm (IDA) f complex (also known as the I1 complex), and the tether and tether head (T/TH) complex are thought to be key regulators of dynein action but, unlike the IDA f complex, T/TH proteins remain poorly characterised. Here, we characterised T/TH-associated proteins in the protist Leishmania mexicana Proteome analysis of axonemes from null mutants for the CFAP44 T/TH protein showed that they lacked the IDA f protein IC140 and a novel 28-kDa axonemal protein, LAX28. Sequence analysis identified similarities between LAX28 and the uncharacterised human sperm tail protein TEX47, both sharing features with sensory BLUF-domain-containing proteins. Leishmania lacking LAX28, CFAP44 or IC140 retained some motility, albeit with reduced swimming speed and directionality and a propensity for flagellar curling. Expression of tagged proteins in different null mutant backgrounds showed that the axonemal localisation of LAX28 requires CFAP44 and IC140, and the axonemal localisations of CFAP44 and IC140 both depend on LAX28. These data demonstrate a role for LAX28 in motility and show mutual dependencies of IDA f and T/TH-associated proteins for axonemal assembly in Leishmania.

Dynamics of Swimmers in Fluids with Resistance

Micro-swimmers such as spermatozoa are able to efficiently navigate through viscous fluids that contain a sparse network of fibers or other macromolecules. We utilize the Brinkman equation to capture the fluid dynamics of sparse and stationary obstacles that are represented via a single resistance parameter. The method of regularized Brinkmanlets is utilized to solve for the fluid flow and motion of the swimmer in 2-dimensions when assuming the flagellum (tail) propagates a curvature wave. Extending previous studies, we investigate the dynamics of swimming when varying the resistance parameter, head or cell body radius, and preferred beat form parameters. For a single swimmer, we determine that increased swimming speed occurs for a smaller cell body radius and smaller fluid resistance. Progression of swimmers exhibits complex dynamics when considering hydrodynamic interactions; attraction of two swimmers is a robust phenomenon for smaller beat amplitude of the tail and smaller fluid resistance. Wall attraction is also observed, with a longer time scale of wall attraction with a larger resistance parameter.

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'.

Beating to rotational transition of a clamped active ribbon-like filament

We present a detailed study of a clamped ribbon-like filament under a compressive active force using Brownian dynamics simulations. We show that a clamped ribbon-like filament is able to capture beating as well as rotational motion under the compressive force. The nature of oscillation is governed by the torsional rigidity of the filament. The frequency of oscillation is almost independent of the torsional rigidity. The beating of the filament gives a butterfly-shaped trajectory of the free-end monomer, whereas rotational motion yields a circular trajectory on a plane. The binormal correlation and the principal component analysis reveal the butterfly, elliptical, and circular trajectories of the free end monomer. We present a phase diagram for different kinds of motion in the parameter regime of compressive force and torsional rigidity.