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

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.

Spontaneous oscillation of an active filament under viscosity gradients

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

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.

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.

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.

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

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

Oscillatory movement of a dynein-microtubule complex crosslinked with DNA origami

Bending of cilia and flagella occurs when axonemal dynein molecules on one side of the axoneme produce force and move toward the microtubule (MT) minus end. These dyneins are then pulled back when the axoneme bends in the other direction, meaning oscillatory back and forth movement of dynein during repetitive bending of cilia/flagella. There are various factors that may regulate the dynein activity, e.g. the nexin-dynein regulatory complex, radial spokes, and central apparatus. In order to understand the basic mechanism of dynein’s oscillatory movement, we constructed a simple model system composed of MTs, outer-arm dyneins, and crosslinks between the MTs made of DNA origami. Electron microscopy (EM) showed pairs of parallel MTs crossbridged by patches of regularly arranged dynein molecules bound in two different orientations, depending on which of the MTs their tails bind to. The oppositely oriented dyneins are expected to produce opposing forces when the pair of MTs have the same polarity. Optical trapping experiments showed that the dynein-MT-DNA-origami complex actually oscillates back and forth after photolysis of caged ATP. Intriguingly, the complex, when held at one end, showed repetitive bending motions. The results show that a simple system composed of ensembles of oppositely oriented dyneins, MTs, and inter-MT crosslinkers, without any additional regulatory structures, has an intrinsic ability to cause oscillation and repetitive bending motions.

Fluid-structure interaction of bio-inspired flexible slender structures: a review of selected topics

Flexible slender structures are ubiquitous in biological systems and engineering applications. Fluid-structure interaction (FSI) plays a key role in the dynamics of such structures immersed in fluids. Here, we survey recent studies on highly simplified bio-inspired models (either mathematical or mechanical) that aim to revealthe flow physics associated with FSI. Various models from different sources of biological inspiration are included, namely flexible flapping foil inspired by fish and insects, deformable membrane inspired by jellyfish and cephalopods, beating filaments inspired by flagella and cilia of microorganisms, and flexible wall-mounted filaments inspired by terrestrial and aquatic plants. Suggestions on directions for future research are also provided.

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 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.

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.

Swirling Instability of the Microtubule Cytoskeleton

In the cellular phenomena of cytoplasmic streaming, molecular motors carrying cargo along a network of microtubules entrain the surrounding fluid. The piconewton forces produced by individual motors are sufficient to deform long microtubules, as are the collective fluid flows generated by many moving motors. Studies of streaming during oocyte development in the fruit fly Drosophila melanogaster have shown a transition from a spatially disordered cytoskeleton, supporting flows with only short-ranged correlations, to an ordered state with a cell-spanning vortical flow. To test the hypothesis that this transition is driven by fluid-structure interactions, we study a discrete-filament model and a coarse-grained continuum theory for motors moving on a deformable cytoskeleton, both of which are shown to exhibit a swirling instability to spontaneous large-scale rotational motion, as observed.

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.

Multisynchrony in Active Microfilaments

Biological microfilaments exhibit a variety of synchronization modes. Recent experiments observed that a pair of isolated eukaryotic flagella, coupled solely via the fluid medium, display synchrony at nontrivial phase lags in addition to in-phase and antiphase synchrony. Using an elastohydrodynamic filament model in conjunction with numerical simulations and a Floquet-type theoretical analysis, we demonstrate that it is possible to reach multiple synchronization states by varying the intrinsic activity of the filament and the strength of hydrodynamic coupling between the two filaments. Then, we derive an evolution equation for the phase difference between the two filaments at weak coupling, and use a Kuramoto-style phase sensitivity analysis to reveal the nature of the bifurcations underlying the transitions between these different synchronized states.

Internal friction controls active ciliary oscillations near the instability threshold

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

Collective dynamics of sperm cells

While only a single sperm may fertilize the egg, getting to the egg can be facilitated, and possibly enhanced, by sperm group dynamics. Examples range from the trains formed by wood mouse sperm to the bundles exhibited by echidna sperm. In addition, observations of wave-like patterns exhibited by ram semen are used to score prospective sample fertility for artificial insemination in agriculture. In this review, we discuss these experimental observations of collective dynamics, as well as describe recent mechanistic models that link the motion of individual sperm cells and their flagella to observed collective dynamics. Establishing this link in models involves negotiating the disparate time- and length scales involved, typically separated by a factor of 1000, to capture the dynamics at the greatest length scales affected by mechanisms at the shortest time scales. Finally, we provide some outlook on the subject, in particular, the open questions regarding how collective dynamics impacts fertility. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Buckling instabilities and spatio-temporal dynamics of active elastic filaments

Biological filaments driven by molecular motors tend to experience tangential propulsive forces also known as active follower forces. When such a filament encounters an obstacle, it deforms, which reorients its follower forces and alters its entire motion. If the filament pushes a cargo, the friction on the cargo can be enough to deform the filament, thus affecting the transport properties of the cargo. Motivated by cytoskeletal filament motility assays, we study the dynamic buckling instabilities of a two-dimensional slender elastic filament driven through a dissipative medium by tangential propulsive forces in the presence of obstacles or cargo. We observe two distinct instabilities. When the filament's head is pinned or experiences significant translational but little rotational drag from its cargo, it buckles into a steadily rotating coiled state. When it is clamped or experiences both significant translational and rotational drag from its cargo, it buckles into a periodically beating, overall translating state. Using minimal analytically tractable models, linear stability theory and fully nonlinear computations, we study the onset of each buckling instability, characterize each buckled state, and map out the phase diagram of the system. Finally, we use particle-based Brownian dynamics simulations to show our main results are robust to moderate noise and steric repulsion. Overall, our results provide a unified framework to understand the dynamics of tangentially propelled filaments and filament-cargo assemblies.

Multiscale mechanics of mucociliary clearance in the lung

Mucociliary clearance (MCC) is one of the most important defence mechanisms of the human respiratory system. Its failure is implicated in many chronic and debilitating airway diseases. However, due to the complexity of lung organization, we currently lack full understanding on the relationship between these regional differences in anatomy and biology and MCC functioning. For example, it is unknown whether the regional variability of airway geometry, cell biology and ciliary mechanics play a functional role in MCC. It therefore remains unclear whether the regional preference seen in some airway diseases could originate from local MCC dysfunction. Though great insights have been gained into the genetic basis of cilia ultrastructural defects in airway ciliopathies, the scaling to regional MCC function and subsequent clinical phenotype remains unpredictable. Understanding the multiscale mechanics of MCC would help elucidate genotype-phenotype relationships and enable better diagnostic tools and treatment options. Here, we review the hierarchical and variable organization of ciliated airway epithelium in human lungs and discuss how this organization relates to MCC function. We then discuss the relevancy of these structure-function relationships to current topics in lung disease research. Finally, we examine how state-of-the-art computational approaches can help address existing open questions. This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.

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'.

On the unity and diversity of cilia

Cilia are specialized cellular organelles that are united in structure and implicated in diverse key life processes across eukaryotes. In both unicellular and multicellular organisms, variations on the same ancestral form mediate sensing, locomotion and the production of physiological flows. As we usher in a new, more interdisciplinary era, the way we study cilia is changing. This special theme issue brings together biologists, biophysicists and mathematicians to highlight the remarkable range of systems in which motile cilia fulfil vital functions, and to inspire and define novel strategies for future research. This article is part of the Theo Murphy meeting issue 'Unity and diversity of cilia in locomotion and transport'.

Cilia oscillations

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

Hydrodynamic Synchronization of Spontaneously Beating Filaments

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

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.

Morphological transitions of axially-driven microfilaments

The interactions of microtubules with motor proteins are ubiquitous in cellular and sub-cellular processes that involve motility and cargo transport. In vitro motility assays have demonstrated that motor-driven microtubules exhibit rich dynamical behaviors from straight to curved configurations. Here, we theoretically investigate the dynamic instabilities of elastic filaments, with free-ends, driven by single follower forces that emulate the action of molecular motors. Using the resistive force theory at low Reynolds number, and a combination of numerical techniques with linear stability analysis, we show the existence of four distinct regimes of filament behavior, including a novel buckled state with locked curvature. These successive instabilities recapitulate the full range of experimentally-observed microtubule behavior, implying that neither structural nor actuation asymmetry are needed to elicit this rich repertoire of motion.

How Does Cilium Length Affect Beating?

The effects of cilium length on the dynamics of cilia motion were investigated by high-speed video microscopy of uniciliated mutants of the swimming alga, Chlamydomonas reinhardtii. Cells with short cilia were obtained by deciliating cells via pH shock and allowing cilia to reassemble for limited times. The frequency of cilia beating was estimated from the motion of the cell body and of the cilium. Key features of the ciliary waveform were quantified from polynomial curves fitted to the cilium in each image frame. Most notably, periodic beating did not emerge until the cilium reached a critical length between 2 and 4 μm. Surprisingly, in cells that exhibited periodic beating, the frequency of beating was similar for all lengths with only a slight decrease in frequency as length increased from 4 μm to the normal length of 10-12 μm. The waveform average curvature (rad/μm) was also conserved as the cilium grew. The mechanical metrics of ciliary propulsion (force, torque, and power) all increased in proportion to length. The mechanical efficiency of beating appeared to be maximal at the normal wild-type length of 10-12 μm. These quantitative features of ciliary behavior illuminate the biophysics of cilia motion and, in future studies, may help distinguish competing hypotheses of the underlying mechanism of oscillation.

Fifty years of microtubule sliding in cilia

Motility of cilia (also known as flagella in some eukaryotes) is based on axonemal doublet microtubule sliding that is driven by the dynein molecular motors. Dyneins are organized into intricately patterned inner and outer rows of arms, whose collective activity is to produce inter-microtubule movement. However, to generate a ciliary bend, not all dyneins can be active simultaneously. The switch point model accounts, in part, for how dynein motors are regulated during ciliary movement. On the basis of this model, supported by key direct experimental observations as well as more recent theoretical and structural studies, we are now poised to understand the mechanics of how ciliary dynein coordination controls axonemal bend formation and propagation.

Coordination of eukaryotic cilia and flagella

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

Turning dyneins off bends cilia

Ciliary and flagellar motility is caused by the ensemble action of inner and outer dynein arm motors acting on axonemal doublet microtubules. The switch point or switching hypothesis, for which much experimental and computational evidence exists, requires that dyneins on only one side of the axoneme are actively working during bending, and that this active motor region propagate along the axonemal length. Generation of a reverse bend results from switching active sliding to the opposite side of the axoneme. However, the mechanochemical states of individual dynein arms within both straight and curved regions and how these change during beating has until now eluded experimental observation. Recently, Lin and Nicastro used high-resolution cryo-electron tomography to determine the power stroke state of dyneins along flagella of sea urchin sperm that were rapidly frozen while actively beating. The results reveal that axonemal dyneins are generally in a pre-power stroke conformation that is thought to yield a force-balanced state in straight regions; inhibition of this conformational state and microtubule release on specific doublets may then lead to a force imbalance across the axoneme allowing for microtubule sliding and consequently the initiation and formation of a ciliary bend. Propagation of this inhibitory signal from base-to-tip and switching the microtubule doublet subsets that are inhibited is proposed to result in oscillatory motion.

Spontaneous oscillations of elastic filaments induced by molecular motors

It is known from the wave-like motion of microtubules in motility assays that the piconewton forces that motors produce can be sufficient to bend the filaments. In cellular phenomena such as cytosplasmic streaming, molecular motors translocate along cytoskeletal filaments, carrying cargo which entrains fluid. When large numbers of such forced filaments interact through the surrounding fluid, as in particular stages of oocyte development in Drosophila melanogaster, complex dynamics are observed, but the detailed mechanics underlying them has remained unclear. Motivated by these observations, we study here perhaps the simplest model for these phenomena: an elastic filament, pinned at one end, acted on by a molecular motor treated as a point force. Because the force acts tangential to the filament, no matter what its shape, this 'follower-force' problem is intrinsically non-variational, and thereby differs fundamentally from Euler buckling, where the force has a fixed direction, and which, in the low-Reynolds-number regime, ultimately leads to a stationary, energy-minimizing shape. Through a combination of linear stability theory, analytical study of a solvable simplified 'two-link' model and numerical studies of the full elastohydrodynamic equations of motion, we elucidate the Hopf bifurcation that occurs with increasing forcing of a filament, leading to flapping motion analogous to the high-Reynolds-number oscillations of a garden hose with a free end.

Detecting singular weak-dissipation limit for flutter onset in reversible systems

A "flutter machine" is introduced for the investigation of a singular interface between the classical and reversible Hopf bifurcations that is theoretically predicted to be generic in nonconservative reversible systems with vanishing dissipation. In particular, such a singular interface exists for the Pflüger viscoelastic column moving in a resistive medium, which is proven by means of the perturbation theory of multiple eigenvalues with the Jordan block. The laboratory setup, consisting of a cantilevered viscoelastic rod loaded by a positional force with nonzero curl produced by dry friction, demonstrates high sensitivity of the classical Hopf bifurcation onset to the ratio between the weak air drag and Kelvin-Voigt damping in the Pflüger column. Thus, the Whitney umbrella singularity is experimentally confirmed, responsible for discontinuities accompanying dissipation-induced instabilities in a broad range of physical contexts.

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.

Human sperm steer with second harmonics of the flagellar beat

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