Curvature regulation of the ciliary beat through axonemal twist

Predicting the locations of force-generating dyneins in beating cilia and flagella

Cilia and flagella are slender cylindrical organelles whose bending waves propel cells through fluids and drive fluids across epithelia. The bending waves are generated by dynein motor proteins, ATPases whose force-generating activity changes over time and with position along the axoneme, the motile structure within the cilium. A key question is: where, in an actively beating axoneme, are the force-generating dyneins located? Answering this question is crucial for determining which of the conformational states adopted by the dynein motors generate the forces that bend the axoneme. The question is difficult to answer because the flagellum contains a large number of dyneins in a complex three-dimensional architecture. To circumvent this complexity, we used a molecular-mechanics approach to show how the bending moments produced by single pairs of dynein motors work against elastic and hydrodynamic forces. By integrating the individual motor activities over the length of the axoneme, we predict the locations of the force-generating dyneins in a beating axoneme. The predicted location depends on the beat frequency, the wavelength, and the elastic and hydrodynamic properties of the axoneme. To test these predictions using cryogenic electron microscopy, cilia with shorter wavelengths, such as found in Chlamydomonas, are more suitable than sperm flagella with longer wavelengths because, in the former, the lag between force and curvature is less dependent on the specific mechanical properties and experimental preparation.

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.

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.

Rapid multi-plane phase-contrast microscopy reveals torsional dynamics in flagellar motion

High speed volumetric optical microscopy is an important tool for observing rapid processes in living cells or for real-time tracking of sub-cellular components. However, the 3D imaging capability often comes at the price of a high technical complexity of the imaging system and/or the requirement of demanding image analysis. Here, we propose a combination of conventional phase-contrast imaging with a customized multi-plane beam-splitter for enabling simultaneous acquisition of images in eight different focal planes. Our method is technically straightforward and does not require complex post-processing image analysis. We apply our multi-plane phase-contrast microscope to the real-time observation of the fast motion of reactivated Chlamydomonas axonemes with sub-µm spatial and 4 ms temporal resolution. Our system allows us to observe not only bending but also the three-dimensional torsional dynamics of these micro-swimmers.

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

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

How Euglena gracilis swims: Flow field reconstruction and analysis

Euglena gracilis is a unicellular organism that swims by beating a single anterior flagellum. We study the nonplanar waveforms spanned by the flagellum during a swimming stroke and the three-dimensional flows that they generate in the surrounding fluid. Starting from a small set of time-indexed images obtained by optical microscopy on a swimming Euglena cell, we construct a numerical interpolation of the stroke. We define an optimal interpolation (which we call synthetic stroke) by minimizing the discrepancy between experimentally measured velocities (of the swimmer) and those computed by solving numerically the equations of motion of the swimmer driven by the trial interpolated stroke. The good match we obtain between experimentally measured and numerically computed trajectories provides a first validation of our synthetic stroke. We further validate the procedure by studying the flow velocities induced in the surrounding fluid. We compare the experimentally measured flow fields with the corresponding quantities computed by solving numerically the Stokes equations for the fluid flow, in which the forcing is provided by the synthetic stroke, and find good matching. Finally, we use the synthetic stroke to derive a coarse-grained model of the flow field resolved in terms of a few dominant singularities. The far field is well approximated by a time-varying Stresslet, and we show that the average behavior of Euglena during one stroke is that of an off-axis puller. The reconstruction of the flow field closer to the swimmer body requires a more complex system of singularities. A system of two Stokeslets and one Rotlet, that can be loosely associated with the force exerted by the flagellum, the drag of the body, and a torque to guarantee rotational equilibrium, provides a good approximation.

Tubulin glycylation controls axonemal dynein activity, flagellar beat, and male fertility

Posttranslational modifications of the microtubule cytoskeleton have emerged as key regulators of cellular functions, and their perturbations have been linked to a growing number of human pathologies. Tubulin glycylation modifies microtubules specifically in cilia and flagella, but its functional and mechanistic roles remain unclear. In this study, we generated a mouse model entirely lacking tubulin glycylation. Male mice were subfertile owing to aberrant beat patterns of their sperm flagella, which impeded the straight swimming of sperm cells. Using cryo-electron tomography, we showed that lack of glycylation caused abnormal conformations of the dynein arms within sperm axonemes, providing the structural basis for the observed dysfunction. Our findings reveal the importance of microtubule glycylation for controlled flagellar beating, directional sperm swimming, and male fertility.

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.

Modulatory mechanisms of sliding of nine outer doublet microtubules for generating planar and half-helical flagellar waves

It has been shown that sperm flagellar motility is generated and modulated by metachronal sliding and two types of synchronous sliding of the outer doublet microtubules. Metachronal sliding propagates around the axoneme circumferentially from one doublet to another along the flagellum, whereas the two types of synchronous sliding occur synchronously throughout an extended region along the doublet microtubules. Oscillatory synchronous sliding occurs between most pairs of the nine doublet microtubules, whereas non-oscillatory synchronous sliding occurs between a specific pair of the nine doublet microtubules. These types of sliding coexist in the flagellum and create beat cycles of flagellar movement. The circumferential propagation of active sliding around the nine doublet microtubules in the metachronal sliding suggests that it is easier for a flagellum to produce helical waves than planar waves. Most sperm flagellar movements are planar to a certain extent. Therefore, mechanisms that modulate the helical waves into planar waves may be present. Structures such as the central pair microtubules in 9 + 2 sperm flagella and the fusion of fibrous-sheath and 3-,8-doublet microtubules in mammalian sperm flagella partition the nine outer doublet microtubules into two groups. Accordingly, the sliding between these two groups generates planar flagellar waves. A similar effect is caused by the sliding between a specific pair of the nine doublet microtubules of the non-oscillatory synchronous sliding, occurring in a Ca2+ concentration-dependent manner. These hard- and soft-wired systems produce the nearly planar flagellar waves required for the efficient propulsion of spermatozoa.

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.

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.

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.

The mitotic spindle is chiral due to torques within microtubule bundles

Mitosis relies on forces generated in the spindle, a micro-machine composed of microtubules and associated proteins. Forces are required for the congression of chromosomes to the metaphase plate and their separation in anaphase. However, besides forces, torques may exist in the spindle, yet they have not been investigated. Here we show that the spindle is chiral. Chirality is evident from the finding that microtubule bundles in human spindles follow a left-handed helical path, which cannot be explained by forces but rather by torques. Kinesin-5 (Kif11/Eg5) inactivation abolishes spindle chirality. Our theoretical model predicts that bending and twisting moments may generate curved shapes of bundles. We found that bundles turn by about -2 deg µm-1 around the spindle axis, which we explain by a twisting moment of roughly -10 pNµm. We conclude that torques, in addition to forces, exist in the spindle and determine its chiral architecture.

Four-dimensional analysis by high-speed holographic imaging reveals a chiral memory of sperm flagella

Here high-speed Digital Holographic Microscopy (DHM) records sperm flagellar waveforms and swimming paths in 4 dimensions (X, Z, and t). We find flagellar excursions into the Z-plane nearly as large as the envelope of the flagellar waveform projected onto the XY-plane. These Z-plane excursions travel as waves down the flagellum each beat cycle. DHM also tracks the heads of free-swimming sperm and the dynamics and chirality of rolling of sperm around their long axis. We find that mouse sperm roll CW at the maximum positive Z-plane excursion of the head, then roll CCW at the subsequent maximum negative Z-plane excursion. This alternating chirality of rolling indicates sperm have a chiral memory. Procrustes alignments of path trajectories for sequences of roll-counterroll cycles show that path chirality is always CW for the cells analyzed in this study. Human and bull sperm lack distinguishable left and right surfaces, but DHM still indicates coordination of Z-plane excursions and rolling events. We propose that sperm have a chiral memory that resides in a hypothetical elastic linkage within the flagellar machinery, which stores some of the torque required for a CW or CCW roll to reuse in the following counter-roll. Separate mechanisms control path chirality.

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.