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

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.

Time Irreversibility and Criticality in the Motility of a Flagellate Microorganism

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

Asymmetric distribution and spatial switching of dynein activity generates ciliary motility

Motile cilia and flagella are essential, highly conserved organelles, and their motility is driven by the coordinated activities of multiple dynein isoforms. The prevailing "switch-point" hypothesis posits that dyneins are asymmetrically activated to drive flagellar bending. To test this model, we applied cryo-electron tomography to visualize activity states of individual dyneins relative to their locations along beating flagella of sea urchin sperm cells. As predicted, bending was generated by the asymmetric distribution of dynein activity on opposite sides of the flagellum. However, contrary to predictions, most dyneins were in their active state, and the smaller population of conformationally inactive dyneins switched flagellar sides relative to the bending direction. Thus, our data suggest a "switch-inhibition" mechanism in which force imbalance is generated by inhibiting, rather than activating, dyneins on alternating sides of the flagellum.

Microtubules soften due to cross-sectional flattening

We use optical trapping to continuously bend an isolated microtubule while simultaneously measuring the applied force and the resulting filament strain, thus allowing us to determine its elastic properties over a wide range of applied strains. We find that, while in the low-strain regime, microtubules may be quantitatively described in terms of the classical Euler-Bernoulli elastic filament, above a critical strain they deviate from this simple elastic model, showing a softening response with increasingdeformations. A three-dimensional thin-shell model, in which the increased mechanical compliance is caused by flattening and eventual buckling of the filament cross-section, captures this softening effect in the high strain regime and yields quantitative values of the effective mechanical properties of microtubules. Our results demonstrate that properties of microtubules are highly dependent on the magnitude of the applied strain and offer a new interpretation for the large variety in microtubule mechanical data measured by different methods.

Sperm morphology and the evolution of intracellular sperm-egg interactions

Sperm morphology is incredibly diverse, even among closely related species, yet the coevolution between males and females of fertilization recognition systems is necessary for successful karyogamy (male and female pronuclear fusion). In most species, the entire sperm enters the egg during fertilization so sperm morphological diversity may impact the intracellular sperm-egg interactions necessary for karyogamy. We quantified morphological variation of sperm inside eggs prior to and following karyogamy in several species of Drosophila to understand whether evolution of sperm morphology could influence intracellular sperm-egg interactions (ISEIs). We measured seven parameters that describe ISEIs among species to determine whether these parameters varied both within a species across development and across species at the same developmental stage. We used heterospecific crosses to test the relative role of male origin, female origin, and interaction between the male and female in determining ISEIs. We found that sperm shape changed within a species as development proceeded and, at particular development stages, species varied in some ISEIs. Parental origin had an effect on some ISEIs, with a general trend for a stronger female effect. Overall, our findings identify conserved and variable ISEIs among species and demonstrate the potential to contribute understanding to gamete evolution and development.

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.

The counterbend dynamics of cross-linked filament bundles and flagella

Cross-linked filament bundles, such as in cilia and flagella, are ubiquitous in biology. They are considered in textbooks as simple filaments with larger stiffness. Recent observations of flagellar counterbend, however, show that induction of curvature in one section of a passive flagellum instigates a compensatory counter-curvature elsewhere, exposing the intricate role of the diminutive cross-linking proteins at large scales. We show that this effect, a material property of the cross-linking mechanics, modifies the bundle dynamics and induces a bimodal L² - L³ length-dependent material response that departs from the Euler-Bernoulli theory. Hence, the use of simpler theories to analyse experiments can result in paradoxical interpretations. Remarkably, the counterbend dynamics instigates counter-waves in opposition to driven oscillations in distant parts of the bundle, with potential impact on the regulation of flagellar bending waves. These results have a range of physical and biological applications, including the empirical disentanglement of material quantities via counterbend dynamics.

Phase reduction approach to elastohydrodynamic synchronization of beating flagella

We formulate a theory for the phase reduction of a beating flagellum. The theory enables us to describe the dynamics of a beating flagellum in a systematic manner using a single variable called the phase. The theory can also be considered as a phase reduction method for the limit-cycle solutions in infinite-dimensional dynamical systems, namely, the limit-cycle solutions to partial differential equations representing beating flagella. We derive the phase sensitivity function, which quantifies the phase response of a beating flagellum to weak perturbations applied at each point and at each time. Using the phase sensitivity function, we analyze the phase synchronization between a pair of beating flagella through hydrodynamic interactions at a low Reynolds number.

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.

Regulatory mechanisms of sperm flagellar motility by metachronal and synchronous sliding of doublet microtubules

STUDY QUESTION

What is the role of metachronal and synchronous sliding in sperm flagellar motility?

SUMMARY ANSWER

Both metachronal and oscillatory synchronous sliding are essential for sperm flagellar motility, while the change in mode of synchronous sliding between the non-oscillatory synchronous sliding of a specific pair of the doublet microtubules and the oscillatory synchronous sliding between most pairs of doublet microtubules modulates the sperm flagellar motility.

WHAT IS KNOWN ALREADY

Metachronal and synchronous sliding of doublet microtubules are involved in sperm flagellar motility and regulation of these sliding movements controls flagellar bend formation.

STUDY DESIGN, SIZE, DURATION

To study the regulatory mechanisms of metachronal and synchronous sliding in flagellar movement of golden hamster spermatozoa, changes in these sliding movements during hyperactivation were examined by measuring the angle of the tangent to the flagellar shaft with reference to the central axis of the sperm head (the shear angle) along the flagellum. Golden hamster spermatozoa were obtained from the caudal epididymis of five sexually mature golden hamsters. Results from three experiments were averaged. The number of spermatozoa analyzed is 15 activated sperm, 22 hyperactivated sperm and 20 acrosome-reacted sperm.

PARTICIPANTS/MATERIALS, SETTING, METHODS

For detailed field-by-field analysis, an individual flagellar image was tracked automatically using the Autotrace module of image analysis software. The coordinate values of the flagellar shaft were used to calculate the shear angle, which is proportional to the amount of microtubule sliding at any given position along the flagellum. The maximum shear angles of metachronal and synchronous sliding were obtained from the mean shear angles between the maximum shear angles of pro-hook bends and the absolute values of the minimum shear angles of anti-hook bends, which represent the amplitude of a set of successive shear angle curves, with 3-12 shear curves covering one beat cycle of sperm flagellar movement. Asymmetry of flagellar waves was expressed by the mean shear angle between the maximum shear angle of pro-hook bends and the minimum shear angle of anti-hook bends at 100 μm from the head-midpiece junction.

MAIN RESULTS AND THE ROLE OF CHANCE

The asymmetrical flagellar movements observed in the activated (non-hyperactivated) and hyperactivated spermatozoa were characterized by the non-oscillatory synchronous sliding of a specific pair of the doublets; the large asymmetrical flagellar movement in the hyperactivated spermatozoa was generated by the large non-oscillatory synchronous sliding. Both the metachronal and synchronous sliding increased during the hyperactivation; however, the large symmetrical flagellar movement of the acrosome-reacted spermatozoa was characterized by the oscillatory synchronous sliding between most pairs of doublets. These results demonstrated that the metachronal and synchronous sliding are involved in generation and modulation of sperm flagellar motility; however, two types of synchronous sliding, non-oscillatory and oscillatory sliding, modulate the sperm flagellar motility by enhancing the sliding of a specific pair of the doublets or the sliding between most pairs of the doublets.

LARGE SCALE DATA

None.

LIMITATIONS, REASONS FOR CAUTION

This is an indirect study of the metachronal and synchronous sliding of doublet microtubules. Studies based on the direct observation of behavior of dynein are needed to clarify the sliding microtubule theory of flagellar movement of spermatozoa.

WIDER IMPLICATIONS OF THE FINDINGS

Both the metachronal and oscillatory synchronous sliding of doublet microtubule generate and modulate sperm flagellar motility, while the change in mode of synchronous sliding between the non-oscillatory synchronous sliding and oscillatory synchronous sliding modulates the sperm flagellar motility. The coordination between these sliding leads to various types of flagellar and ciliary motility, including the asymmetrical beating in flagellar and ciliary movement and planar or helical beating in sea urchin spermatozoa. Moreover, the finding that the metachronal sliding and two types of synchronous sliding generate and modulate the flagellar motility will open a new avenue for quantitative analysis of flagellar and ciliary motility.

STUDY FUNDING AND COMPETING INTEREST(S)

The authors have no conflict of interest and no funding to declare.

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.

Nonlinear amplitude dynamics in flagellar beating

The physical basis of flagellar and ciliary beating is a major problem in biology which is still far from completely understood. The fundamental cytoskeleton structure of cilia and flagella is the axoneme, a cylindrical array of microtubule doublets connected by passive cross-linkers and dynein motor proteins. The complex interplay of these elements leads to the generation of self-organized bending waves. Although many mathematical models have been proposed to understand this process, few attempts have been made to assess the role of dyneins on the nonlinear nature of the axoneme. Here, we investigate the nonlinear dynamics of flagella by considering an axonemal sliding control mechanism for dynein activity. This approach unveils the nonlinear selection of the oscillation amplitudes, which are typically either missed or prescribed in mathematical models. The explicit set of nonlinear equations are derived and solved numerically. Our analysis reveals the spatio-temporal dynamics of dynein populations and flagellum shape for different regimes of motor activity, medium viscosity and flagellum elasticity. Unstable modes saturate via the coupling of dynein kinetics and flagellum shape without the need of invoking a nonlinear axonemal response. Hence, our work reveals a novel mechanism for the saturation of unstable modes in axonemal beating.

Nonlinear amplitude dynamics in flagellar beating

The physical basis of flagellar and ciliary beating is a major problem in biology which is still far from completely understood. The fundamental cytoskeleton structure of cilia and flagella is the axoneme, a cylindrical array of microtubule doublets connected by passive cross-linkers and dynein motor proteins. The complex interplay of these elements leads to the generation of self-organized bending waves. Although many mathematical models have been proposed to understand this process, few attempts have been made to assess the role of dyneins on the nonlinear nature of the axoneme. Here, we investigate the nonlinear dynamics of flagella by considering an axonemal sliding control mechanism for dynein activity. This approach unveils the nonlinear selection of the oscillation amplitudes, which are typically either missed or prescribed in mathematical models. The explicit set of nonlinear equations are derived and solved numerically. Our analysis reveals the spatio-temporal dynamics of dynein populations and flagellum shape for different regimes of motor activity, medium viscosity and flagellum elasticity. Unstable modes saturate via the coupling of dynein kinetics and flagellum shape without the need of invoking a nonlinear axonemal response. Hence, our work reveals a novel mechanism for the saturation of unstable modes in axonemal beating.

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.

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