A model of flagellar and ciliary functioning which uses the forces transverse to the axoneme as the regulator of dynein activation

The mechanics of cilia and flagella: What we know and what we need to know

In this review, we provide a condensed overview of what is currently known about the mechanical functioning of the flagellar/ciliary axoneme. We also present a list of 10 specific areas where our current knowledge is incomplete and explain the benefits of further experimental investigation. Many of the physical parameters of the axoneme and its component parts have not been determined. This limits our ability to understand how the axoneme structure contributes to its functioning in several regards. It restricts our ability to understand how the mechanics of the structure contribute to the regulation of motor function. It also confines our ability to understand the three-dimensional workings of the axoneme and how various beating modes are accomplished. Lastly, it prevents accurate computational modeling of the axoneme in three-dimensions.

Crystal structure of the stalk region of axonemal inner-arm dynein-d reveals unique features in the coiled-coil and microtubule-binding domain

Axonemal dynein is an ATP-dependent microtubular motor protein responsible for cilia and flagella beating, and its dysfunction can cause diseases such as primary ciliary dyskinesia and sperm dysmotility. Despite its biological importance, structure-based mechanisms underlying axonemal dynein motors remain unclear. Here, we determined the X-ray crystal structure of the human inner-arm dynein-d (DNAH1) stalk region, which contains a long antiparallel coiled-coil and a microtubule-binding domain (MTBD), at 2.7 Å resolution. Notably, differences in the relative orientation of the coiled-coil and MTBD in comparison with other dyneins, as well as the diverse orientations of the MTBD flap region among various isoforms, lead us to propose a 'spike shoe model' with an altered stepping angle for the interaction between IAD-d and microtubules. Based on these findings, we discuss isoform-specific functions of the axonemal dynein stalk MTBDs.

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.

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.

Rebound and scattering of motile Chlamydomonas algae in confined chambers

Motivated by recent experiments demonstrating that motile algae get trapped in draining foams, we study the trajectories of microorganisms confined in model foam channels (section of a Plateau border). We track single Chlamydomonas reinhardtii cells confined in a thin three-circle microfluidic chamber and show that their spatial distribution exhibits strong corner accumulation. Using empirical scattering laws observed in previous experiments (scattering with a constant scattering angle), we next develop a two-dimension geometrical model and compute the phase space of trapped and periodic trajectories of swimmers inside a three-circles billiard. We find that the majority of cell trajectories end up in a corner, providing a geometrical mechanism for corner accumulation. Incorporating the distribution of scattering angles observed in our experiments and including hydrodynamic interactions between the cells and the surfaces into the geometrical model enables us to reproduce the experimental probability density function of micro-swimmers in microfluidic chambers. Both our experiments and models demonstrate therefore that motility leads generically to trapping in complex geometries.

Flagellar energetics from high-resolution imaging of beating patterns in tethered mouse sperm

We demonstrate a technique for investigating the energetics of flagella or cilia. We record the planar beating of tethered mouse sperm at high resolution. Beating waveforms are reconstructed using proper orthogonal decomposition of the centerline tangent-angle profiles. Energy conservation is employed to obtain the mechanical power exerted by the dynein motors from the observed kinematics. A large proportion of the mechanical power exerted by the dynein motors is dissipated internally by the motors themselves. There could also be significant dissipation within the passive structures of the flagellum. The total internal dissipation is considerably greater than the hydrodynamic dissipation in the aqueous medium outside. The net power input from the dynein motors in sperm from Crisp2-knockout mice is significantly smaller than in wildtype samples, indicating that ion-channel regulation by cysteine-rich secretory proteins controls energy flows powering the axoneme.

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.

Mechanical induction of oscillatory movement in demembranated, immotile flagella of sea urchin sperm at very low ATP concentrations

Oscillation is a characteristic feature of eukaryotic flagellar movement. The mechanism involves the control of dynein-driven microtubule sliding under self-regulatory mechanical feedback within the axoneme. To define the essential factors determining the induction of oscillation, we developed a novel experiment by applying mechanical deformation of demembranated, immotile sea urchin sperm flagella at very low ATP concentrations, below the threshold of ATP required for spontaneous beating. Upon application of mechanical deformation at above 1.5 µmol l-1 ATP, a pair of bends could be induced and was accompanied by bend growth and propagation, followed by switching the bending direction. For an oscillatory, cyclical bending response to occur, the velocity of bend propagation towards the flagellar tip must be kept above certain levels. Continuous formation of new bends at the flagellar base was coupled with synchronized decay of the preceding paired bends. Induction of cyclical bends was initiated in a constant direction relative to the axis of the flagellar 9+2 structure, and resulted in the so-called principal bend. In addition, stoppage of the bending response occasionally occurred during development of a new principal bend, and in this situation, formation of a new reverse bend did not occur. This observation indicates that the reverse bend is always active, opposing the principal bend. The results show that mechanical strain of bending is a central component regulating the bend oscillation, and switching of the bend direction appears to be controlled, in part, by the velocity of wave propagation.

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

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

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.

High hydrostatic pressure induces vigorous flagellar beating in Chlamydomonas non-motile mutants lacking the central apparatus

The beating of eukaryotic flagella (also called cilia) depends on the sliding movements between microtubules powered by dynein. In cilia/flagella of most organisms, microtubule sliding is regulated by the internal structure of cilia comprising the central pair of microtubules (CP) and radial spokes (RS). Chlamydomonas paralyzed-flagella (pf) mutants lacking CP or RS are non-motile under physiological conditions. Here, we show that high hydrostatic pressure induces vigorous flagellar beating in pf mutants. The beating pattern at 40 MPa was similar to that of wild type at atmospheric pressure. In addition, at 80 MPa, flagella underwent an asymmetric-to-symmetric waveform conversion, similar to the one triggered by an increase in intra-flagella Ca2+ concentration during cell's response to strong light. Thus, our study establishes that neither beating nor waveform conversion of cilia/flagella requires the presence of CP/RS in the axoneme.

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'.

Instability-driven oscillations of elastic microfilaments

Cilia and flagella are highly conserved slender organelles that exhibit a variety of rhythmic beating patterns from non-planar cone-like motions to planar wave-like deformations. Although their internal structure, composed of a microtubule-based axoneme driven by dynein motors, is known, the mechanism responsible for these beating patterns remains elusive. Existing theories suggest that the dynein activity is dynamically regulated, via a geometric feedback from the cilium's mechanical deformation to the dynein force. An alternative, open-loop mechanism based on a 'flutter' instability was recently proven to lead to planar oscillations of elastic filaments under follower forces. Here, we show that an elastic filament in viscous fluid, clamped at one end and acted on by an external distribution of compressive axial forces, exhibits a Hopf bifurcation that leads to non-planar spinning of the buckled filament at a locked curvature. We also show the existence of a second bifurcation, at larger force values, that induces a transition from non-planar spinning to planar wave-like oscillations. We elucidate the nature of these instabilities using a combination of nonlinear numerical analysis, linear stability theory and low-order bead-spring models. Our results show that, away from the transition thresholds, these beating patterns are robust to perturbations in the distribution of axial forces and in the filament configuration. These findings support the theory that an open-loop, instability-driven mechanism could explain both the sustained oscillations and the wide variety of periodic beating patterns observed in cilia and flagella.

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.

The Kinetics of Nucleotide Binding to Isolated Chlamydomonas Axonemes Using UV-TIRF Microscopy

Cilia and flagella are long, slender organelles found in many eukaryotic cells, where they have sensory, developmental, and motile functions. All cilia and flagella contain a microtubule-based structure called the axoneme. In motile cilia and flagella, which drive cell locomotion and fluid transport, the axoneme contains, along most of its length, motor proteins from the axonemal dynein family. These motor proteins drive motility by using energy derived from the hydrolysis of ATP to generate a bending wave, which travels down the axoneme. As a first step toward visualizing the ATPase activity of the axonemal dyneins during bending, we have investigated the kinetics of nucleotide binding to axonemes. Using a specially built ultraviolet total internal reflection fluorescence microscope, we found that the fluorescent ATP analog methylanthraniloyl ATP (mantATP), which has been shown to support axonemal motility, binds all along isolated, immobilized axonemes. By studying the recovery of fluorescence after photobleaching, we found that there are three mantATP binding sites: one that bleaches rapidly (time constant ≈ 1.7 s) and recovers slowly (time constant ≈ 44 s), one that bleaches with the same time constant but does not recover, and one that does not bleach. By reducing the dynein content in the axoneme using mutants and salt extraction, we provide evidence that the slow-recovering component, but not the other components, corresponds to axonemal dyneins. The recovery rate of this component, however, is too slow to be consistent with the activation of beating observed at higher mantATP concentrations; this indicates that the dyneins may be inhibited due to their immobilization at the surface. The development of this method is a first step toward direct observation of the traveling wave of dynein activity.

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.

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.

Three-dimensional tracking of microbeads attached to the tip of single isolated tracheal cilia beating under external load

To study the properties of tracheal cilia beating under various conditions, we developed a method to monitor the movement of the ciliary tip. One end of a demembranated cilium was immobilized on the glass surface, while the other end was capped with a polystyrene bead and tracked in three dimensions. The cilium, when activated by ATP, stably repeated asymmetric beating as in vivo. The tip of a cilium in effective and recovery strokes moved in discrete trajectories that differed in height. The trajectory remained asymmetric in highly viscous solutions. Model calculation showed that cilia maintained a constant net flux during one beat cycle irrespective of the medium viscosity. When the bead attached to the end was trapped with optical tweezers, it came to display linear oscillation only in the longitudinal direction. Such a beating-mode transition may be an inherent nature of movement-restricted cilia.

Changed Expression of Cytoskeleton Proteins During Lung Injury in a Mouse Model of Streptococcus pneumoniae Infection

Infections by Streptococcus pneumoniae are a major cause of morbidity and mortality worldwide, often causing community-acquired pneumonia, otitis media and also bacteremia and meningitis. Studies on S. pneumoniae are mainly focused on its virulence or capacity to evade the host immune system, but little is known about the injury caused in lungs during a pneumococcal infection. Herein we investigated this issue comparing the proteome profile of lungs from S. pneumoniae-infected mice with control mice by means of difference gel electrophoresis (DIGE) technology. In order to obtain reliable results three biological replicas were used, and four technical replicas were carried out in each biological replica. Proteomic comparison was performed at two time points: 24 and 48 h post infection. A total of 91 proteins were identified with different abundance. We found important changes in the protein profiles during pneumococcal infection mainly associated with regulation of vesicle-mediated transport, wound healing, and cytoskeleton organization. In conclusion, the results obtained show that the cytoskeleton of the host cell is modified in S. pneumoniae infection.

Spontaneous oscillation and fluid-structure interaction of cilia

The exact mechanism to orchestrate the action of hundreds of dynein motor proteins to generate wave-like ciliary beating remains puzzling and has fascinated many scientists. We present a 3D model of a cilium and the simulation of its beating in a fluid environment. The model cilium obeys a simple geometric constraint that arises naturally from the microscopic structure of a real cilium. This constraint allows us to determine the whole 3D structure at any instant in terms of the configuration of a single space curve. The tensions of active links, which model the dynein motor proteins, follow a postulated dynamical law, and together with the passive elasticity of microtubules, this dynamical law is responsible for the ciliary motions. In particular, our postulated tension dynamics lead to the instability of a symmetrical steady state, in which the cilium is straight and its active links are under equal tensions. The result of this instability is a stable, wave-like, limit cycle oscillation. We have also investigated the fluid-structure interaction of cilia using the immersed boundary (IB) method. In this setting, we see not only coordination within a single cilium but also, coordinated motion, in which multiple cilia in an array organize their beating to pump fluid, in particular by breaking phase synchronization.

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.

Finite element models of flagella with sliding radial spokes and interdoublet links exhibit propagating waves under steady dynein loading

It remains unclear how flagella generate propulsive, oscillatory waveforms. While it is well known that dynein motors, in combination with passive cytoskeletal elements, drive the bending of the axoneme by applying shearing forces and bending moments to microtubule doublets, the origin of rhythmicity is still mysterious. Most conceptual models of flagellar oscillation involve dynein regulation or switching, so that dynein activity first on one side of the axoneme, then the other, drives bending. In contrast, a "viscoelastic flutter" mechanism has recently been proposed, based on a dynamic structural instability. Simple mathematical models of coupled elastic beams in viscous fluid, subjected to steady, axially distributed, dynein forces of sufficient magnitude, can exhibit oscillatory motion without any switching or dynamic regulation. Here we introduce more realistic finite element (FE) models of 6-doublet and 9-doublet flagella, with radial spokes and interdoublet links that slide along the central pair or corresponding doublet. These models demonstrate the viscoelastic flutter mechanism. Above a critical force threshold, these models exhibit an abrupt onset of propulsive, wavelike oscillations typical of flutter instability. Changes in the magnitude and spatial distribution of steady dynein force, or to viscous resistance, lead to behavior qualitatively consistent with experimental observations. This study demonstrates the ability of FE models to simulate nonlinear interactions between axonemal components during flagellar beating, and supports the plausibility of viscoelastic flutter as a mechanism of flagellar oscillation.

Developmental adaptations of trypanosome motility to the tsetse fly host environments unravel a multifaceted in vivo microswimmer system

The highly motile and versatile protozoan pathogen Trypanosoma brucei undergoes a complex life cycle in the tsetse fly. Here we introduce the host insect as an expedient model environment for microswimmer research, as it allows examination of microbial motion within a diversified, secluded and yet microscopically tractable space. During their week-long journey through the different microenvironments of the fly´s interior organs, the incessantly swimming trypanosomes cross various barriers and confined surroundings, with concurrently occurring major changes of parasite cell architecture. Multicolour light sheet fluorescence microscopy provided information about tsetse tissue topology with unprecedented resolution and allowed the first 3D analysis of the infection process. High-speed fluorescence microscopy illuminated the versatile behaviour of trypanosome developmental stages, ranging from solitary motion and near-wall swimming to collective motility in synchronised swarms and in confinement. We correlate the microenvironments and trypanosome morphologies to high-speed motility data, which paves the way for cross-disciplinary microswimmer research in a naturally evolved environment.

Evidence for a self-organized compliant mechanism for the spontaneous steady beating of cilia

Cilia or eukaryotic flagella are slender 200-nm-diameter organelles that move the immersing fluid relative to a cell and sense the environment. Their core structure is nine doublet microtubules (DMTs) arranged around a central-pair. When motile, thousands of tiny motors slide the DMTs relative to each other to facilitate traveling waves of bending along the cilium's length. These motors provide the energy to change the shape of the cilium and overcome the viscous forces of moving in the surrounding fluid. In planar beating, motors walk toward where the cilium is attached to the cell body. Traveling waves are initiated by motors bending the elastic cilium back and forth, a self-organized mechanical oscillator. We found remarkably that the energy in a wave is nearly constant over a wide range of (ATP) and medium viscosities and inter-doublet springs operate only in the central and not in the basal region. Since the energy in a wave does not depend on its rate of formation, the control mechanism is likely purely mechanical. Further the torque per length generated by the motors acting on the doublets is proportional to and nearly in phase with the microtubule sliding velocity with magnitude dependent on the medium. We determined the frequency-dependent elastic moduli and strain energies of beating cilia. Incorporation of these in an energy-based model explains the beating frequency, wavelength, limiting of the wave amplitude and the overall energy of the traveling wave. Our model describes the intricacies of the basal-wave initiation as well as the traveling wave.

The Central Apparatus of Cilia and Eukaryotic Flagella

The motile cilium is a complex organelle that is typically comprised of a 9+2 microtubule skeleton; nine doublet microtubules surrounding a pair of central singlet microtubules. Like the doublet microtubules, the central microtubules form a scaffold for the assembly of protein complexes forming an intricate network of interconnected projections. The central microtubules and associated structures are collectively referred to as the central apparatus (CA). Studies using a variety of experimental approaches and model organisms have led to the discovery of a number of highly conserved protein complexes, unprecedented high-resolution views of projection structure, and new insights into regulation of dynein-driven microtubule sliding. Here, we review recent progress in defining mechanisms for the assembly and function of the CA and include possible implications for the importance of the CA in human health.

Effects of external strain on the regulation of microtubule sliding induced by outer arm dynein of sea urchin sperm flagella

Oscillatory bending movement of eukaryotic flagella is powered by orchestrated activity of dynein motor proteins that hydrolyze ATP and produce microtubule sliding. Although the ATP concentration within a flagellum is kept uniform at a few mmol l−1 level, sliding activities of dyneins are dynamically coordinated along the flagellum in accordance with the phase of bending waves. Thus, at the organellar level the dynein not only generates force for bending but also modulates its motile activity by responding to bending of the flagellum. Single molecule analyses have suggested that dynein at the molecular level, even if isolated from the axoneme, could alter the modes of motility in response to mechanical strain. However, it still remains unknown whether the coordinated activities of multiple dyneins can be modulated directly by mechanical signals. Here, we studied the effects of externally applied strain on the sliding movement of microtubules interacted with ensemble of dynein molecules adsorbed on a glass surface. We found that by bending the microtubules with a glass microneedle, three modes of motility that have not been previously characterized without bending can be induced: those were, stoppage, backward sliding and dissociation. Modification in sliding velocities was also induced by imposed bending. These results suggest that the activities of dyneins interacted with a microtubule can be modified and coordinated through external strain in a quite flexible manner and that such regulatory mechanism may be the basis of flagellar oscillation.

Elastohydrodynamic Synchronization of Adjacent Beating Flagella

It is now well established that nearby beating pairs of eukaryotic flagella or cilia typically synchronize in phase. A substantial body of evidence supports the hypothesis that hydrodynamic coupling between the active filaments, combined with waveform compliance, provides a robust mechanism for synchrony. This elastohydrodynamic mechanism has been incorporated into 'bead-spring' models in which the beating flagella are represented by microspheres tethered by radial springs as they are driven about orbits by internal forces. While these low-dimensional models reproduce the phenomenon of synchrony, their parameters are not readily relatable to those of the filaments they represent. More realistic models which reflect the underlying elasticity of the axonemes and the active force generation, take the form of fourth-order nonlinear PDEs. While computational studies have shown the occurrence of synchrony, the effects of hydrodynamic coupling between nearby filaments governed by such continuum models have been theoretically examined only in the regime of interflagellar distances d large compared to flagellar length L. Yet, in many biological situations d/L ≪ 1. Here, we first present an asymptotic analysis of the hydrodynamic coupling between two extended filaments in the regime d/L ≪ 1, and find that the form of the coupling is independent of the microscopic details of the internal forces that govern the motion of the individual filaments. The analysis is analogous to that yielding the localized induction approximation for vortex filament motion, extended to the case of mutual induction. In order to understand how the elastohydrodynamic coupling mechanism leads to synchrony of extended objects, we introduce a heuristic model of flagellar beating. The model takes the form of a single fourth-order nonlinear PDE whose form is derived from symmetry considerations, the physics of elasticity, and the overdamped nature of the dynamics. Analytical and numerical studies of this model illustrate how synchrony between a pair of filaments is achieved through the asymptotic coupling.

Curvature regulation of the ciliary beat through axonemal twist

Cilia and flagella are hairlike organelles that propel cells through fluid. The active motion of the axoneme, the motile structure inside cilia and flagella, is powered by molecular motors of the axonemal dynein family. These motors generate forces and torques that slide and bend the microtubule doublets within the axoneme. To create regular waveforms, the activities of the dyneins must be coordinated. It is thought that coordination is mediated by stresses due to radial, transverse, or sliding deformations, and which build up within the moving axoneme and feed back on dynein activity. However, which particular components of the stress regulate the motors to produce the observed waveforms of the many different types of flagella remains an open question. To address this question, we describe the axoneme as a three-dimensional bundle of filaments and characterize its mechanics. We show that regulation of the motors by radial and transverse stresses can lead to a coordinated flagellar motion only in the presence of twist. We show that twist, which could arise from torque produced by the dyneins, couples curvature to transverse and radial stresses. We calculate emergent beating patterns in twisted axonemes resulting from regulation by transverse stresses. The resulting waveforms are similar to those observed in flagella of Chlamydomonas and sperm. Due to the twist, the waveform has nonplanar components, which result in swimming trajectories such as twisted ribbons and helices, which agree with observations.

Functional anatomy of the mammalian sperm flagellum

The eukaryotic flagellum is the organelle responsible for the propulsion of the male gamete in most animals. Without exception, sperm of all mammalian species use a flagellum for swimming. The mammalian sperm has a centrally located 9 + 2 arrangement of microtubule doublets and hundreds of accessory proteins that together constitute an axoneme. However, they also possess several characteristic peri-axonemal structures that make the mammalian sperm tail function differently. These modifications include nine outer dense fibers (ODFs) that are paired with the nine outer microtubule doublets of the axoneme, and are anchored in a structure called the connecting piece located at the base. The presence of the ODFs and connecting piece, and the absence of a basal body, dictate that physical forces generated by the dynein motors are transmitted to the base of the flagellum through the ODFs. Mammalian sperm flagella also possess a mitochondrial and a fibrous sheath that encircle most of the axoneme. These sheaths and the ODFs add mechanical rigidity to the flagellum creating the functional effect of increasing bend wavelength, which requires the entrainment of more dynein motors in the production of a single wave. The sheaths also act as a retinaculum and maintain the integrity of the central axoneme when large bending torques are generated by dynein. Large torque production is crucial to the process of hyperactivation and the unique motility transitions associated with effective fertilizing capacity. Consequently, these specialized anatomical features are essential for the effective interaction of sperm with the female reproductive tract and ovum. © 2016 Wiley Periodicals, Inc.

Steady dynein forces induce flutter instability and propagating waves in mathematical models of flagella

Cilia and flagella are highly conserved organelles that beat rhythmically with propulsive, oscillatory waveforms. The mechanism that produces these autonomous oscillations remains a mystery. It is widely believed that dynein activity must be dynamically regulated (switched on and off, or modulated) on opposite sides of the axoneme to produce oscillations. A variety of regulation mechanisms have been proposed based on feedback from mechanical deformation to dynein force. In this paper, we show that a much simpler interaction between dynein and the passive components of the axoneme can produce coordinated, propulsive oscillations. Steady, distributed axial forces, acting in opposite directions on coupled beams in viscous fluid, lead to dynamic structural instability and oscillatory, wave-like motion. This 'flutter' instability is a dynamic analogue to the well-known static instability, buckling. Flutter also occurs in slender beams subjected to tangential axial loads, in aircraft wings exposed to steady air flow and in flexible pipes conveying fluid. By analysis of the flagellar equations of motion and simulation of structural models of flagella, we demonstrate that dynein does not need to switch direction or inactivate to produce autonomous, propulsive oscillations, but must simply pull steadily above a critical threshold force.

Dynamic curvature regulation accounts for the symmetric and asymmetric beats of Chlamydomonas flagella

Cilia and flagella are model systems for studying how mechanical forces control morphology. The periodic bending motion of cilia and flagella is thought to arise from mechanical feedback: dynein motors generate sliding forces that bend the flagellum, and bending leads to deformations and stresses, which feed back and regulate the motors. Three alternative feedback mechanisms have been proposed: regulation by the sliding forces, regulation by the curvature of the flagellum, and regulation by the normal forces that deform the cross-section of the flagellum. In this work, we combined theoretical and experimental approaches to show that the curvature control mechanism is the one that accords best with the bending waveforms of Chlamydomonas flagella. We make the surprising prediction that the motors respond to the time derivative of curvature, rather than curvature itself, hinting at an adaptation mechanism controlling the flagellar beat.

DOI:http://dx.doi.org/10.7554/eLife.13258.001

Mechanics of the eukaryotic flagellar axoneme: Evidence for structural distortion during bending

The sliding doublet mechanism is the established explanation that allows us to understand the process of ciliary and flagellar bending. In this study, we apply the principles of the sliding doublet mechanism to analyze the mechanics of the counterbend phenomenon in sea urchin sperm flagella. When a passive, vanadate-treated, flagellum is forced into a bend with a glass microprobe, the portion of the flagellum distal to the probe exhibits a bend of opposite curvature (counterbend) to the imposed bend. This phenomenon was shown to be caused by the induction of inter-doublet shear and is dependent on the presence of an inter-doublet shear resistance. Here we report that in sea urchin flagella there is systematically less shear induced in the distal flagellum than is predicted by the sliding doublet mechanism, if we follow the assumption that the diameter of the flagellum is uniform. To account for the reduced shear that is observed, the likeliest and most direct interpretation is that the portion of the axoneme that is forced to bend undergoes substantial compression of the axoneme in the bending plane. A compression of 30-50 nm would be sufficient to account for the shear reduction from a bend of 2 radians. A compression of this magnitude would require considerable flexibility in the axoneme structure. This would necessitate that the radial spokes and/or the central pair apparatus are easily compressed by transverse stress. © 2016 Wiley Periodicals, Inc.

Analysis of unstable modes distinguishes mathematical models of flagellar motion

The mechanisms underlying the coordinated beating of cilia and flagella remain incompletely understood despite the fundamental importance of these organelles. The axoneme (the cytoskeletal structure of cilia and flagella) consists of microtubule doublets connected by passive and active elements. The motor protein dynein is known to drive active bending, but dynein activity must be regulated to generate oscillatory, propulsive waveforms. Mathematical models of flagellar motion generate quantitative predictions that can be analysed to test hypotheses concerning dynein regulation. One approach has been to seek periodic solutions to the linearized equations of motion. However, models may simultaneously exhibit both periodic and unstable modes. Here, we investigate the emergence and coexistence of unstable and periodic modes in three mathematical models of flagellar motion, each based on a different dynein regulation hypothesis: (i) sliding control; (ii) curvature control and (iii) control by interdoublet separation (the 'geometric clutch' (GC)). The unstable modes predicted by each model are used to critically evaluate the underlying hypothesis. In particular, models of flagella with 'sliding-controlled' dynein activity admit unstable modes with non-propulsive, retrograde (tip-to-base) propagation, sometimes at the same parameter values that lead to periodic, propulsive modes. In the presence of these retrograde unstable modes, stable or periodic modes have little influence. In contrast, unstable modes of the GC model exhibit switching at the base and propulsive base-to-tip propagation.

An ALE-based finite element model of flagellar motion driven by beating waves: A parametric study

A computational model of flagellar motility is presented using the finite element method. Two-dimensional traveling waves of finite amplitude are propagated down the flagellum and the swimmer is propelled through a viscous fluid according to Newto's second law of motion. Incompressible Navier-Stokes equations are solved on a triangular moving mesh and arbitrary Lagrangian-Eulerian formulation is employed to accommodate the deforming boundaries. The results from the present study are validated against the data available in the literature and close agreement with previous works is found. The effects of wave parameters as well as head morphology on the swimming characteristics are studied for different swimming conditions. We have found that the swimming velocities are linear functions of finite amplitudes and that the rate of work is independent of the channel height for large amplitudes. Furthermore, we have also demonstrated that for the range of wave parameters that are often encountered in human sperm motility studies, the propulsive velocity versus the wavelength exhibits dissimilar trends for different channel heights. Various head configurations were analyzed and it is also observed that wall proximity amplifies the effects induced by different head shapes. By taking non-Newtonian fluids into account, we present new efficiency analyzes through which we have found that the model microorganism swims much more efficiently in shear-thinning fluids.

Ultrastructural evidence that motility changes caused by variations in ATP, Mg2+ , and ADP correlate to conformational changes in reactivated bull sperm axonemes

We report the results of an ultrastructural study of Triton X-100-extracted, Mg-adenosine triphosphate (ATP)-reactivated bull sperm. We utilized a rapid fixation method to look for differences in the flagellar apparatus that correlate to the state of motility of reactivated sperm models. In a companion article, we examined the motility characteristics induced in bull sperm models by varying the concentration ratio of ATP and Mg(2+) and the stabilizing effect of adenosine diphosphate (ADP) on coordinated beating. Based on the results of that report, we selected four dissimilar states that appeared to represent extremes. One reactivation condition produces vigorous motility similar to live sperm, another produces large amplitude, low frequency beating while the remaining two conditions produce small amplitude vibrations of the flagellum with little coordinated beating. Morphometric analysis of transmission electron micrographs of sperm from these four treatment conditions revealed statistically significant differences between the samples in regard to axoneme diameter, inter-microtubule doublet spacing, and outer dense fiber (ODF) spacing. Our results show that Mg(2+) decreases the axoneme diameter and reduces interdoublet spacing, while ATP, uncomplexed with Mg(2+) , had the opposite effect. We also provide supporting evidence that this may be due to Mg(2+) increasing, and ATP decreasing, the interdoublet adhesion of dynein. We also found that 4 mM ADP significantly increases the separation between the ODFs and the space between the ODFs and the central axoneme within the middle piece. We present a hypothetical explanation that is consistent with our results to explain how ATP, ADP, and Mg(2+) act to regulate the beat cycle. © 2014 Wiley Periodicals, Inc.

The physiological role of ADP and Mg2+ in maintaining a stable beat cycle in bull sperm

Sperm flagella derive their motive power from the motor protein dynein. In this study, we show that maintenance of the flagellar beat cycle in detergent-extracted bull sperm models is highly dependent on the ratio of Mg(2+) to adenosine triphosphate (ATP). An excess of either ATP un-complexed with Mg(2+) , or an excess of Mg(2+) without an equivalent concentration of ATP, results in the loss of beat amplitude and a reduced curvature development in the beat cycle. In addition, we find that adenosine diphosphate (ADP) can stabilize the beat cycle and permit rhythmic beating across a broader range of ATP and Mg(2+) concentrations. We provide evidence that suggests that when ATP is un-complexed with Mg(2+) , it disrupts the beat cycle by reducing dynein adhesion and thereby, reduces the transmission of dynein-generated force between the doublets. Excess Mg(2+) does not act by the same mechanism and induces a condition where the flagellum is more resistant to bending. This is consistent with the idea that high Mg(2+) stabilizes rigor bridges, and ATP reduces the microtubule binding affinity of dynein. Our results may explain how intact sperm are able to sustain coordinated flagellar beating under a wide range of metabolic conditions, as intact sperm produce ADP in direct proportion to their consumption of ATP.

Dynein-deficient flagella respond to increased viscosity with contrasting changes in power and recovery strokes

Changes in the flagellar waveform in response to increased viscosity were investigated in uniflagellate mutants of Chlamydomonas reinhardtii. We hypothesized that the waveforms of mutants lacking different dynein arms would change in different ways as viscosity was increased, and that these variations would illuminate the feedback pathways from force to dynein activity. Previous studies have investigated the effects of viscosity on cell body motion, propulsive force, and power in different mutants, but the effect on waveform has not yet been fully characterized. Beat frequency decreases with viscosity in wild-type uniflagellate (uni1) cells, and outer dynein arm deficient (oda2) mutants. In contrast, the inner dynein arm mutant ida1 (lacking I1/f) maintains beat frequency at high viscosity but alters its flagellar waveform more than either wild-type or oda2. The ida1 waveform is narrower than wild-type, primarily due to an abbreviated recovery stroke; this difference is amplified at high viscosity. The oda2 mutant in contrast, maintains a consistent waveform at high and low viscosity with a slightly longer power stroke than wild-type. Analysis of the delays and shear displacements between bends suggest that direct force feedback in the outer dynein arm system may initiate switching of dynein activity. In contrast, I1/f dynein appears to delay switching, most markedly at the initiation of the power stroke, possibly by controlling inter-doublet separation.

Equations of interdoublet separation during flagella motion reveal mechanisms of wave propagation and instability

The motion of flagella and cilia arises from the coordinated activity of dynein motor protein molecules arrayed along microtubule doublets that span the length of axoneme (the flagellar cytoskeleton). Dynein activity causes relative sliding between the doublets, which generates propulsive bending of the flagellum. The mechanism of dynein coordination remains incompletely understood, although it has been the focus of many studies, both theoretical and experimental. In one leading hypothesis, known as the geometric clutch (GC) model, local dynein activity is thought to be controlled by interdoublet separation. The GC model has been implemented as a numerical simulation in which the behavior of a discrete set of rigid links in viscous fluid, driven by active elements, was approximated using a simplified time-marching scheme. A continuum mechanical model and associated partial differential equations of the GC model have remained lacking. Such equations would provide insight into the underlying biophysics, enable mathematical analysis of the behavior, and facilitate rigorous comparison to other models. In this article, the equations of motion for the flagellum and its doublets are derived from mechanical equilibrium principles and simple constitutive models. These equations are analyzed to reveal mechanisms of wave propagation and instability in the GC model. With parameter values in the range expected for Chlamydomonas flagella, solutions to the fully nonlinear equations closely resemble observed waveforms. These results support the ability of the GC hypothesis to explain dynein coordination in flagella and provide a mathematical foundation for comparison to other leading models.

The geometric clutch at 20: stripping gears or gaining traction?

It has been 20 years since the geometric clutch (GC) hypothesis was first proposed. The core principle of the GC mechanism is fairly simple. When the axoneme of a eukaryotic flagellum is bent, mechanical stress generates forces transverse to the outer doublets (t-forces). These t-forces can push doublets closer together or pry them apart. The GC hypothesis asserts that changes in the inter-doublet spacing caused by t-forces are responsible for the activation and deactivation of the dynein motors, that creates the beat cycle. A series of computer models utilizing the clutch mechanism has shown that it can simulate ciliary and flagellar beating. The objective of the present review is to assess where things stand with the GC hypothesis in the clarifying light of new information. There is considerable new evidence to support the hypothesis. However, it is also clear that it is necessary to modify some of the original conceptions of the hypothesis so that it can be consistent with the results of recent experimental and ultrastructural studies. In particular, dynein deactivation by t-forces must be able to occur with dyneins that remain attached to the B-subtubule of the adjacent doublet.

Motor regulation results in distal forces that bend partially disintegrated Chlamydomonas axonemes into circular arcs

The bending of cilia and flagella is driven by forces generated by dynein motor proteins. These forces slide adjacent microtubule doublets within the axoneme, the motile cytoskeletal structure. To create regular, oscillatory beating patterns, the activities of the axonemal dyneins must be coordinated both spatially and temporally. It is thought that coordination is mediated by stresses or strains, which build up within the moving axoneme, and somehow regulate dynein activity. During experimentation with axonemes subjected to mild proteolysis, we observed pairs of doublets associating with each other and forming bends with almost constant curvature. By modeling the statics of a pair of filaments, we show that the activity of the motors concentrates at the distal tips of the doublets. Furthermore, we show that this distribution of motor activity accords with models in which curvature, or curvature-induced normal forces, regulates the activity of the motors. These observations, together with our theoretical analysis, provide evidence that dynein activity can be regulated by curvature or normal forces, which may, therefore, play a role in coordinating the beating of cilia and flagella.

The effects of Ca2+ and ADP on dynein switching during the beat cycle of reactivated bull sperm models

Calcium regulation of flagellar motility is the basis for chemotaxis, phototaxis, and hyperactivation responses in eukaryotic flagellates and spermatozoa. Ca2+ is the internal messenger for these responses, but the coupling between Ca2+ and the motor mechanism that generates the flagellar beat is incompletely understood. We examined the effects of Ca2+ on the flagellar curvature at the switch-points of the beat cycle in bull sperm. The sperm were detergent extracted and reactivated with 0.1 mM adenosine triphosphate (ATP). With their heads immobilized and their tails beating freely it is possible to calculate the bending torque and the transverse force acting on the flagellum at the switch-points. An increase in the free Ca2+ concentration (pCa 8 to pCa 4) significantly decreased the development of torque and t-force in the principal bending direction, while having negligible effect on the reverse bend. The action of Ca2+ was more pronounced when the sperm were also treated with 4 mM adenosine diphosphate (ADP); it was sufficient to change the direction of bending that reaches the greater curvature. We also observed that the curvature of the distal half of the flagellum became locked in one direction in the presence of Ca2+ . This indicates that a subset of the dynein becomes continuously activated by Ca2+ and fails to switch with the beat cycle. Our evidence suggests this subset of dyneins is localized to doublets #1-4 of the axoneme.

Computer simulation of flagellar movement X: doublet pair splitting and bend propagation modeled using stochastic dynein kinetics

Experimental observations on cyclic splitting and bending by a flagellar doublet pair are modeled using forces obtained from a model for dynein mechanochemistry, based on ideas introduced by Andrew Huxley and Terrill Hill and extended previously for modeling flagellar movements. The new feature is elastic attachment of dynein to the A doublet, which allows movement perpendicular to the A doublet and provides adhesive force that can strain attached dyneins. This additional strain influences the kinetics of dynein attachment and detachment. Computations using this dynein model demonstrate that very simple and realistic ideas about dynein mechanochemistry are sufficient for explaining the separation and reattachment seen experimentally with flagellar doublet pairs. Additional simulations were performed after adding a "super-adhesion" elasticity. This elastic component is intended to mimic interdoublet connections, normally present in an intact axoneme, that would prevent visible splitting but allow sufficient separation to cause dynein detachment and cessation of shear force generation. This is the situation envisioned by Lindemann's "geometric clutch" hypothesis for control of dynein function in flagella and cilia. The simulations show abrupt disengagement of the "clutch" at one end of a bend, and abrupt reengagement of the "clutch" at the other end of a bend, ensuring that active sliding is only operating where it will cause bend propagation from base to tip.

Bending, twisting and beating trunk robot bioinspired from the '3 + 0' axoneme

The axoneme is the skeleton and motor axis of flagella and cilia in eukaryotic organisms. Basically it consists of a series of longitudinal fibers (outer doublets of microtubules) that design a cylinder and whose sliding, due to the coordinated activities of dedicated molecular motors (the dynein arms), is converted into a bending because outer doublets pairs are stabilized by elastic links (the nexine molecules). In spite of these interesting mechanical properties, mechanical and robotics engineers have never considered this amazing molecular machinery as a model. The aim of this paper is to propose the robotic design and the kinematic modeling of the '3 + 0' axoneme that makes motile the flagellum of Diplauxis hatti, the simplest that exists. The model that we propose bends and twists and combines the two movements. It is able to propagate wave trains that could be involved in the development of biomimetic actuators of various mechanisms such as (sub)aquatic robotic propellers as well as robotic trunks.

Human airway ciliary dynamics

Airway cilia depend on precise changes in shape to transport the mucus gel overlying mucosal surfaces. The ciliary motion can be recorded in several planes using video microscopy. However, cilia are densely packed, and automated computerized systems are not available to convert these ciliary shape changes into forms that are useful for testing theoretical models of ciliary function. We developed a system for converting planar ciliary motions recorded by video microscopy into an empirical quantitative model, which is easy to use in validating mathematical models, or in examining ciliary function, e.g., in primary ciliary dyskinesia (PCD). The system we developed allows the manipulation of a model cilium superimposed over a video of beating cilia. Data were analyzed to determine shear angles and velocity vectors of points along the cilium. Extracted waveforms were used to construct a composite waveform, which could be used as a standard. Variability was measured as the mean difference in position of points on individual waveforms and the standard. The shapes analyzed were the end-recovery, end-effective, and fastest moving effective and recovery with mean (± SE) differences of 0.31(0.04), 0.25(0.06), 0.50(0.12), 0.50(0.10), μm, respectively. In contrast, the same measures for three different PCD waveforms had values far outside this range.

Analyses of functional domains within the PF6 protein of the central apparatus reveal a role for PF6 sub-complex members in regulating flagellar beat frequency

Numerous studies have indicated that each of the seven projections associated with the central pair of microtubules plays a distinct role in regulating eukaryotic ciliary/flagellar motility. Mutants which lack specific projections have distinct motility phenotypes. For example, Chlamydomonas pf6 mutants lack the C1a projection and have twitchy, non-beating flagella. The C1a projection is a complex of proteins including PF6, C1a-86, C1a-34, C1a-32, C1a-18, and calmodulin. To define functional domains within PF6 and to potentially assign functions to specific C1a components, we generated deletion constructs of the PF6 gene and tested for their ability to assemble and rescue motility upon transformation of mutant pf6 cells. Our results demonstrate that domains near the carboxyl-terminus of PF6 are essential for motility and/or assembly of the projection. The amino terminal half of PF6 is not required for C1a assembly; however, this region is important for stability of the C1a-34, C1a-32, and C1a-18 sub-complex and wild-type beat frequency. Analysis of double mutants lacking the amino terminus of PF6 and outer dynein arms reveal that C1a may play a role in modulating both inner and outer dynein arm activity.

Propulsive forces on the flagellum during locomotion of Chlamydomonas reinhardtii

The distributed propulsive forces exerted on the flagellum of the swimming alga Chlamydomonas reinhardtii by surrounding fluid were estimated from experimental image data. Images of uniflagellate mutant Chlamydomonas cells were obtained at 350 frames/s with 125-nm spatial resolution, and the motion of the cell body and the flagellum were analyzed in the context of low-Reynolds-number fluid mechanics. Wild-type uniflagellate cells, as well as uniflagellate cells lacking inner dynein arms (ida3) or outer dynein arms (oda2) were studied. Ida3 cells exhibit stunted flagellar waveforms, whereas oda2 cells beat with lower frequency. Image registration and sorting algorithms provided high-resolution estimates of the motion of the cell body, as well as detailed kinematics of the flagellum. The swimming cell was modeled as an ellipsoid in Stokes flow, propelled by viscous forces on the flagellum. The normal and tangential components of force on the flagellum (f(N) and f(T)) were related by resistive coefficients (C(N) and C(T)) to the corresponding components of velocity (V(N) and V(T)).The values of these coefficients were estimated by satisfying equilibrium requirements for force and torque on the cell. The estimated values of the resistive coefficients are consistent among all three genotypes and similar to theoretical predictions.

How molecular motors shape the flagellar beat

Cilia and eukaryotic flagella are slender cellular appendages whose regular beating propels cells and microorganisms through aqueous media. The beat is an oscillating pattern of propagating bends generated by dynein motor proteins. A key open question is how the activity of the motors is coordinated in space and time. To elucidate the nature of this coordination we inferred the mechanical properties of the motors by analyzing the shape of beating sperm: Steadily beating bull sperm were imaged and their shapes were measured with high precision using a Fourier averaging technique. Comparing our experimental data with wave forms calculated for different scenarios of motor coordination we found that only the scenario of interdoublet sliding regulating motor activity gives rise to satisfactory fits. We propose that the microscopic origin of such "sliding control" is the load dependent detachment rate of motors. Agreement between observed and calculated wave forms was obtained only if significant sliding between microtubules occurred at the base. This suggests a novel mechanism by which changes in basal compliance could reverse the direction of beat propagation. We conclude that the flagellar beat patterns are determined by an interplay of the basal properties of the axoneme and the mechanical feedback of dynein motors.