The covalent oscillator: A paradigm accounting for the sliding/bending mechanism and wave propagation in cilia and flagella

Summary— In most models of wave propagation in eucaryotic flagella and cilia, a clear distinction is made between the dynein dependent microtubule sliding which represents the oscillatory motor and the bending mechanism which regulates wave propagation. Little is known about the physical elements regulating the latter: in the present model, the bending propagation is postulated to be supported by an open/close cyclic mechanism protease/ligase dependent, which involves transient covalent links between adjacent microtubular doublets; this open/close cycle propagates in register with the powering action of the dynein motor along the exoneme. The implications of the model are discussed in relation to previous data which involve protease/ligase in the axonemal function as well as other data which can be integrated by the proposed model.     

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.     

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.     

Are the local adjustments of the relative spatial frequencies of the dynein arms and the beta-tubulin monomers involved in the regulation of the "9+2" axoneme?

The “9+2” axoneme is a highly specific cylindrical machine whose periodic bending is due to the cumulative shear of its 9 outer doublets of microtubules. Because of the discrete architecture of the tubulin monomers and the active appendices that the outer doublets carry (dynein arms, nexin links and radial spokes), this movement corresponds to the relative shear of these topological verniers, whose characteristics depend on the geometry of the wave train. When an axonemal segment bends, this induces the compressed and dilated conformations of the tubulin monomers and, consequently, the modification of the spatial frequencies of the appendages that the outer doublets carry. From a dynamic point of view, the adjustments of the spatial frequencies of the elements of the two facing verniers that must interact create different longitudinal periodic patterns of distribution of the joint probability of the molecular interaction as a function of the location of the doublet pairs around the axonemal cylinder and their spatial orientation within the axonemal cylinder. During the shear, these patterns move along the outer doublet intervals at a speed that ranges from one to more than a thousand times that of sliding, in two opposite directions along the two opposite halves of the axoneme separated by the bending plane, respecting the polarity of the dynein arms within the axoneme. Consequently, these waves might be involved in the regulation of the alternating activity of the dynein arms along the flagellum, because they induce the necessary intermolecular dialog along the axoneme since they could be an element of the local dynamic stability/instability equilibrium of the axoneme. This complements the geometric clutch model [Lindemann, C., 1994. A “geometric clutch” hypothesis to explain oscillations of the axoneme of cilia and flagella. J. Theor. Biol. 168, 175-189].     

Structural insights into microtubule doublet interactions in axonemes

Coordinated sliding of microtubule doublets, driven by dynein motors, produces periodic beating of eukaryotic cilia and flagella. Recent structural studies of the axoneme, which forms the core of cilia and flagella, have used cryo-electron tomography to reveal new details of the interactions between some of the multitude of proteins that form the axoneme and regulate its movement. Connections between the several types of dyneins, in particular, suggest ways in which their action might be coordinated. Study of the molecular architecture of isolated doublets has provided a structural basis for understanding mechanical properties related to the bending of the axoneme, and has also offered insight into the potential role of doublets in the mechanism of dynein activity regulation.     

Cyclical interactions between two outer doublet microtubules in split flagellar axonemes

The beating of cilia and flagella is based on the localized sliding between adjacent outer doublet microtubules; however, the mechanism that produces oscillatory bending is unclear. To elucidate this mechanism, we examined the behavior of frayed axonemes of Chlamydomonas by using high-speed video recording. A pair of doublet microtubules frequently displayed association and dissociation cycles in the presence of ATP. In many instances, the dissociation of two microtubules was not accompanied by noticeable bending, suggesting that the dynein-microtubule interaction is not necessarily regulated by the microtubule curvature. On rare occasions, association and dissociation occurred simultaneously in the same interacting pair, resulting in a tip-directed movement of a stretch of gap between the pair. Based on these observations, we propose a model for cyclical bend propagation in the axoneme.     

Functional modulation of IFT kinesins extends the sensory repertoire of ciliated neurons in Caenorhabditis elegans

The diversity of sensory cilia on Caenorhabditis elegans neurons allows the animal to detect a variety of sensory stimuli. Sensory cilia are assembled by intraflagellar transport (IFT) kinesins, which transport ciliary precursors, bound to IFT particles, along the ciliary axoneme for incorporation into ciliary structures. Using fluorescence microscopy of living animals and serial section electron microscopy of high pressure-frozen, freeze-substituted IFT motor mutants, we found that two IFT kinesins, homodimeric OSM-3 kinesin and heterotrimeric kinesin II, function in a partially redundant manner to build full-length amphid channel cilia but are completely redundant for building full-length amphid wing (AWC) cilia. This difference reflects cilia-specific differences in OSM-3 activity, which serves to extend distal singlets in channel cilia but not in AWC cilia, which lack such singlets. Moreover, AWC-specific chemotaxis assays reveal novel sensory functions for kinesin II in these wing cilia. We propose that kinesin II is a "canonical" IFT motor, whereas OSM-3 is an "accessory" IFT motor, and that subtle changes in the deployment or actions of these IFT kinesins can contribute to differences in cilia morphology, cilia function, and sensory perception.     

Dynein heavy chain isoforms and axonemal motility

The translocation of dynein along microtubules is the basis for a wide variety of essential cellular movements. Dynein was first discovered in the ciliary axoneme, where it causes the directed sliding between outer doublet microtubules that underlies ciliary bending. The initiation and propagation of ciliary bends are produced by a precisely located array of different dyneins containing eight or more different dynein heavy chain isoforms. The detailed clarification of the structural and functional diversity of axonemal dynein heavy chains will not only provide the key to understanding how cilia function, but also give insights applicable to the study of non-axonemal microtubule motors.     

Geometry drives the "deviated-bending" of the bi-tubular structures of the 9 + 2 axoneme in the flagellum

The axoneme "9 + 2" is basically a system constituted of a cylinder of 9 microtubule doublets surrounding a central pair of microtubules. These bi-tubular structures are considered as the support system of the active molecular complexes that generate and regulate the axonemal movement. Schoutens has calculated their moments of inertia [Schoutens, 1994: Journal of Theoretical Biology 171:163-177]. The results obtained allowed us to assume that these bi-tubular systems are endowed with dynamic properties that could be involved in the regulation of the axonemal machinery. For the first time, using the finite elements methods and the resistance of material principles, we have now calculated that the curvature of the axoneme induces the deviated-bending of the bi-tubular structures of the axoneme, because of their geometry only; they behave as beams in a framework. This approach is similar to the one used to measure the deflection of a single microtubule [Kasas et al., 2004: Chem Phys Chem 5:252-257]. These behaviors induce internal movement or constraints of either couples or triplets of doublets within the axonemal cylinder that could be directly involved in a constrained or a spontaneous "convergence/divergence" equilibrium of the cylindrical generatrices that they draw along the axonemal cylinder, which could apparently regulate the activity of the axonemal motors (the dynein arms). These results are discussed here, taking into consideration the dynamic propagation of the wave train along the flagellar axoneme, and the regulated balance between the activities of the two opposite sides of the axoneme during the beat. This study raises a few questions about the architecture-activity duo of the axonemal doublets.     

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.     

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.     

Heterogeneity of dynein structure implies coordinated suppression of dynein motor activity in the axoneme

Axonemal dyneins provide the driving force for flagellar/ciliary bending. Nucleotide-induced conformational changes of flagellar dynein have been found both in vitro and in situ by electron microscopy, and in situ studies demonstrated the coexistence of at least two conformations in axonemes in the presence of nucleotides (the apo and the nucleotide-bound forms). The distribution of the two forms suggested cooperativity between adjacent dyneins on axonemal microtubule doublets. Although the mechanism of such cooperativity is unknown it might be related to the mechanism of bending. To explore the mechanism by which structural heterogeneity of axonemal dyneins is induced by nucleotides, we used cilia from Tetrahymena thermophila to examine the structure of dyneins in a) the intact axoneme and b) microtubule doublets separated from the axoneme, both with and without additional pure microtubules. We also employed an ATPase assay on these specimens to investigate dynein activity functionally. Dyneins on separated doublets show more activation by nucleotides than those in the intact axoneme, both structurally and in the ATPase assay, and this is especially pronounced when the doublets are coupled with added microtubules, as expected. Paralleling the reduced ATPase activity in the intact axonemes, a lower proportion of these dyneins are in the nucleotide-bound form. This indicates a coordinated suppression of dynein activity in the axoneme, which could be the key for understanding the bending mechanism.     

Bases moléculaires du mouvement flagellaire

Because of their small size, cells encounter fundamentally different physical constraints when they want to move in their surrounding media than when aquatic animals want to move in water. For cells, external viscosity is the main resistance while inertia plays almost no role; then in order to move, cells will have to produce continually a force against the viscous media. During the evolution, eukaryotic cells have gained specialised structures to efficiently propel them: cilia and flagella. These thread-like appendages produce repetitive beating which consists in propagation of waves from the bottom to the tip of these structures. Cilia generally show an asymmetrical beating while flagella have a more symmetrical bend propagation. Cilia and flagella contain an almost identical internal complex machinery: the axoneme. As a consequence, the mechanisms involved in the generation of the movement are identical although the precise regulation may be different. In this paper, the reader will find a review of the molecular organisation of the ciliary and flagellar axoneme and of the role played by some of the constituting elements on the generation of the movement.     

Two anterograde intraflagellar transport motors cooperate to build sensory cilia on C. elegans neurons

Cilia have diverse roles in motility and sensory reception and their dysfunction contributes to cilia-related diseases. Assembly and maintenance of cilia depends on the intraflagellar transport (IFT) of axoneme, membrane, matrix and signalling proteins to appropriate destinations within the organelle. In the current model, these diverse cargo proteins bind to multiple sites on macromolecular IFT particles, which are moved by a single anterograde IFT motor, kinesin-II, from the ciliary base to its distal tip, where cargo-unloading occurs. Here, we describe the observation of fluorescent IFT motors and IFT particles moving along distinct domains within sensory cilia of wild-type and IFT-motor-mutant Caenorhabditis elegans. We show that two anterograde IFT motor holoenzymes, kinesin-II and Osm-3-kinesin, cooperate in a surprising way to control two pathways of IFT that build distinct parts of cilia. Instead of each motor independently moving its own specific cargo to a distinct destination, the two motors function redundantly to transport IFT particles along doublet microtubules adjacent to the transition zone to form the axoneme middle segment. Next, Osm-3-kinesin alone transports IFT particles along the distal singlet microtubules to stabilize the distal segment. Thus, the subtle coordinate activity of these IFT motors creates two sequential transport pathways.     

A model of flagellar movement based on cooperative dynamics of dynein-tubulin cross-bridges

A theoretical model based on molecular mechanisms of both dynein cross-bridges and radial spokes is used to study bend propagation by eukaryotic flagella. Though nine outer doublets are arranged within an axoneme, a simplified model with four doublets is constructed on the assumption that cross-bridges between two of the four doublets are opposed to those between the other two, corresponding to the geometric array of cross-bridges on the 6-9 and the 1-4 doublets in the axoneme. We also assume that external viscosity is zero, whereas internal viscosity is non-zero in order to reduce numerical complexity. For demonstrating flagellar movement, computer simulations are available by dividing a long flagellum into many straight segments. Considering the fact that dynein cross-bridge spacing is almost equal to attachment site spacing, we may use a localized cross-bridge distribution along attachment sites in each straight segment. Dynamics of cross-bridges are determined by a three-state model, and effects of radial spokes are represented by a periodic mechanical potential whose periodicity is considered to be a stroke distance of the radial spoke. First of all, we examine the model of a short segment to know basic properties of the system. Changing parameters relating to "activation" of cross-bridges, our model demonstrates various phenomena; for example "excitable properties with threshold phenomena" and "limit cycle oscillation". Here, "activation" and "inactivation" (i.e. switching mechanisms) between a pair of oppositely-directed cross-bridges are essential for generation of excitable or oscillatory properties. Next, the model for a flagellar segment is incorporated into a flagellum with a whole length to show bending movement. When excitable properties of cross-bridges, not oscillatory properties, are provided along the length of the flagellum and elastic links between filaments are presented at the base, then our model can demonstrate self-organization of bending waves as well as wave propagation without special feedback control by the curvature of the flagellum. Here, "cooperative interaction" between adjacent short segments, based on "cooperative dynamics" of cross-bridges, is important for wave propagation.     

Ultrastructural, tomographic and confocal imaging of the chondrocyte primary cilium in situ

Hyaline cartilage chondrocytes express one primary cilium per cell, but its function remains unknown. We examined the ultrastructure of chick embryo sternal chondrocyte cilia and their interaction with extracellular matrix molecules by transmission electron microscopy (TEM) and, for the first time, double-tilt electron tomography. Ciliary bending was also examined by confocal immunohistochemistry. Tomography and TEM showed the ciliary axoneme to interdigitate amongst collagen fibres and condensed proteoglycans. TEM also revealed the presence of electron-opaque particles in the proximal axoneme which may represent intraciliary-transport (ICT) particles. We observed a wide range of ciliary bending patterns. Some conformed to a heavy elastica model associated with shear stress. Others were acutely deformed, suggesting ciliary deflection by collagen fibres and proteoglycans with which the cilia make contact. We conclude that mechanical forces transmitted through these matrix macromolecules bend the primary cilium, identifying it as a potential mechanosensor involved in skeletal patterning and growth.     

Induction of beating by imposed bending or mechanical pulse in demembranated, motionless sea urchin sperm flagella at very low ATP concentrations

A basic feature of the movement of eukaryotic flagella is oscillation. Although flagellar oscillation is thought to be regulated by a self-regulatory feedback system including the mechanical signal of bending itself, the mechanism regulating the dynein motile activity to produce oscillation is not well understood. To elucidate the mechanism, we developed a new experimental system which allowed us to analyze the conditions necessary for the induction of oscillation. When a mechanical signal of bending or a pulse was applied by micromanipulation to a demembranated motionless sea urchin sperm flagellar axoneme at very low ATP concentrations (1-3 microM), a localized pair of bends was induced. The bend formation was often followed by further responses including propagation of the distal bend of paired bends, growth and propagation of the paired bends, and cyclical beating. The beating was induced at 2.0 microM or higher concentrations of ATP, but appeared even at 1.5 microM ATP if a few muM of ADP was also present. When the proximal half of a flagellum was attached to a microneedle, beating could not be induced in the distal free region at 2 microM ATP. These results suggest that mechanical signal is involved in the mechanism regulating the motile activity of dynein to produce oscillation. Our results also showed that the presence of a small amount of ADP and the axial difference along the flagellum are factors essential for the induction of flagellar oscillation.     

Propagation of force and the induced bending displacement along eukaryotic flagellar axoneme.

The mechanical forces responsible for inducing the bending movement of eukaryotic flagellar axonemes have components that propagate at velocities different from those for the displacement of the medium. These forces are subject to the Third Law of Mechanics which states the null-conservation of acting and reacting forces. Experimental demonstration of the propagating internal tensile force along the axoneme, when not accompanied by simultaneous bending displacement, demonstrates that the active force for the bending of the flagellar axoneme is part of the process of counterbalancing the corresponding reactive force to the ATP hydrolysis underlying the force generation.