Direct measurement of inter-doublet elasticity in flagellar axonemes

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.

Basal bodies bend in response to ciliary forces

Motile cilia beat with an asymmetric waveform consisting of a power stroke that generates a propulsive force and a recovery stroke that returns the cilium back to the start. Cilia are anchored to the cell cortex by basal bodies (BBs) that are directly coupled to the ciliary doublet microtubules (MTs). We find that, consistent with ciliary forces imposing on BBs, bending patterns in BB triplet MTs are responsive to ciliary beating. BB bending varies as environmental conditions change the ciliary waveform. Bending occurs where striated fibers (SFs) attach to BBs and mutants with short SFs that fail to connect to adjacent BBs exhibit abnormal BB bending, supporting a model in which SFs couple ciliary forces between BBs. Finally, loss of the BB stability protein Poc1, which helps interconnect BB triplet MTs, prevents the normal distributed BB and ciliary bending patterns. Collectively, BBs experience ciliary forces and manage mechanical coupling of these forces to their surrounding cellular architecture for normal ciliary beating.

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.

Modelling Motility: The Mathematics of Spermatozoa

In one of the first examples of how mechanics can inform axonemal mechanism, Machin's study in the 1950s highlighted that observations of sperm motility cannot be explained by molecular motors in the cell membrane, but would instead require motors distributed along the flagellum. Ever since, mechanics and hydrodynamics have been recognised as important in explaining the dynamics, regulation, and guidance of sperm. More recently, the digitisation of sperm videomicroscopy, coupled with numerous modelling and methodological advances, has been bringing forth a new era of scientific discovery in this field. In this review, we survey these advances before highlighting the opportunities that have been generated for both recent research and the development of further open questions, in terms of the detailed characterisation of the sperm flagellum beat and its mechanics, together with the associated impact on cell behaviour. In particular, diverse examples are explored within this theme, ranging from how collective behaviours emerge from individual cell responses, including how these responses are impacted by the local microenvironment, to the integration of separate advances in the fields of flagellar analysis and flagellar mechanics.

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.

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.

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

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

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.

Construction of artificial cilia from microtubules and kinesins through a well-designed bottom-up approach

Self-organized structures of biomolecular motor systems, such as cilia and flagella, play key roles in the dynamic processes of living organisms, like locomotion or the transportation of materials. Although fabrication of such self-organized structures from reconstructed biomolecular motor systems has attracted much attention in recent years, a systematic construction methodology is still lacking. In this work, through a bottom-up approach, we fabricated artificial cilia from a reconstructed biomolecular motor system, microtubule/kinesin. The artificial cilia exhibited a beating motion upon the consumption, by the kinesins, of the chemical energy obtained from the hydrolysis of adenosine triphosphate (ATP). Several design parameters, such as the length of the microtubules, the density of the kinesins along the microtubules, the depletion force among the microtubules, etc., have been identified, which permit tuning of the beating frequency of the artificial cilia. The beating frequency of the artificial cilia increases upon increasing the length of the microtubules, but declines for the much longer microtubules. A high density of the kinesins along the microtubules is favorable for the beating motion of the cilia. The depletion force induced bundling of the microtubules accelerated the beating motion of the artificial cilia and increased the beating frequency. This work helps understand the role of self-assembled structures of the biomolecular motor systems in the dynamics of living organisms and is expected to expedite the development of artificial nanomachines, in which the biomolecular motors may serve as actuators.

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.

Three-dimensional structural labeling microscopy of cilia and flagella

Locating a molecule within a cell using protein-tagging and immunofluorescence is a fundamental technique in cell biology, whereas in three-dimensional electron microscopy, locating a subunit within a macromolecular complex remains challenging. Recently, we developed a new structural labeling method for cryo-electron tomography by taking advantage of the biotin-streptavidin system, and have intensively used this method to locate a number of proteins and protein domains in cilia and flagella. In this review, we summarize our findings on the three-dimensional architecture of the axoneme, especially the importance of coiled-coil proteins. In addition, we provide an overview of the technical aspects of our structural labeling method.

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.

Flexural Rigidity and Shear Stiffness of Flagella Estimated from Induced Bends and Counterbends

Motile cilia and flagella are whiplike cellular organelles that bend actively to propel cells or move fluid in passages such as airways, brain ventricles, and the oviduct. Efficient motile function of cilia and flagella depends on coordinated interactions between active forces from an array of motor proteins and passive mechanical resistance from the complex cytoskeletal structure (the axoneme). However, details of this coordination, including axonemal mechanics, remain unclear. We investigated two major mechanical parameters, flexural rigidity and interdoublet shear stiffness, of the flagellar axoneme in the unicellular alga Chlamydomonas reinhardtii. Combining experiment, theory, and finite element models, we demonstrate that the apparent flexural rigidity of the axoneme depends on both the intrinsic flexural rigidity (EI) and the elastic resistance to interdoublet sliding (shear stiffness, ks). We estimated the average intrinsic flexural rigidity and interdoublet shear stiffness of wild-type Chlamydomonas flagella in vivo, rendered immotile by vanadate, to be EI = 840 ± 280 pN⋅μm(2) and ks = 79.6 ± 10.5 pN/rad, respectively. The corresponding values for the pf3; cnk11-6 double mutant, which lacks the nexin-dynein regulatory complex (N-DRC), were EI = 1011 ± 183 pN·μm(2) and ks = 39.3 ± 6.0 pN/rad under the same conditions. Finally, in the pf13A mutant, which lacks outer dynein arms and inner dynein arm c, the estimates were EI = 777 ± 184 pN·μm(2) and ks = 43.3 ± 7.7 pN/rad. In the two mutant strains, the flexural rigidity is not significantly different from wild-type (p > 0.05), but the lack of N-DRC (in pf3; cnk11-6) or dynein arms (in pf13A) significantly reduces interdoublet shear stiffness. These differences may represent the contributions of the N-DRCs (∼40 pN/rad) and residual dynein interactions (∼35 pN/rad) to interdoublet sliding resistance in these immobilized Chlamydomonas flagella.

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.

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

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

The axoneme, a biological template to design a swell energy recovery system

The axonemal micro-machinery, the axial skeleton and actuator of cilia and flagella of eukaryotic cells, is able to bend and twist and generates wave trains. We already demonstrated that it can be the template to construct an active trunk robot (Cibert 2013 Bioinspir. Biomim. 8 026006). The work presented in this paper describes how the axonemal model can also be useful to design worm shaped devices to convert swell energy into usable energy such as electricity. Using dynamic simulation we have designed three models either very close to the biological machinery or being a reasonable interpretation of its functioning principles. Their main advantage as compared with another existing device, Pelamis, is the additional twist degree-of-freedom that gives interesting properties to the system.

The nexin link and B-tubule glutamylation maintain the alignment of outer doublets in the ciliary axoneme

We developed quantitative assays to test the hypothesis that the N-DRC is required for integrity of the ciliary axoneme. We examined reactivated motility of demembranated drc cells, commonly termed "reactivated cell models." ATP-induced reactivation of wild-type cells resulted in the forward swimming of ∼90% of cell models. ATP-induced reactivation failed in a subset of drc cell models, despite forward motility in live drc cells. Dark-field light microscopic observations of drc cell models revealed various degrees of axonemal splaying. In contrast, >98% of axonemes from wild-type reactivated cell models remained intact. The sup-pf4 and drc3 mutants, unlike other drc mutants, retain most of the N-DRC linker that interconnects outer doublet microtubules. Reactivated sup-pf4 and drc3 cell models displayed nearly wild-type levels of forward motility. Thus, the N-DRC linker is required for axonemal integrity. We also examined reactivated motility and axoneme integrity in mutants defective in tubulin polyglutamylation. ATP-induced reactivation resulted in forward swimming of >75% of tpg cell models. Analysis of double mutants defective in tubulin polyglutamylation and different regions of the N-DRC indicate B-tubule polyglutamylation and the distal lobe of the linker region are both important for axonemal integrity and normal N-DRC function. © 2016 Wiley Periodicals, Inc.

A computational model of dynein activation patterns that can explain nodal cilia rotation

Normal left-right patterning in vertebrates depends on the rotational movement of nodal cilia. In order to produce this ciliary motion, the activity of axonemal dyneins must be tightly regulated in a temporal and spatial manner; the specific activation pattern of the dynein motors in the nodal cilia has not been reported. Contemporary imaging techniques cannot directly assess dynein activity in a living cilium. In this study, we establish a three-dimensional model to mimic the ciliary ultrastructure and assume that the activation of dynein proteins is related to the interdoublet distance. By employing finite-element analysis and grid deformation techniques, we simulate the mechanical function of dyneins by pairs of point loads, investigate the time-variant interdoublet distance, and simulate the dynein-triggered ciliary motion. The computational results indicate that, to produce the rotational movement of nodal cilia, the dynein activity is transferred clockwise (looking from the tip) between the nine doublet microtubules, and along each microtubule, the dynein activation should occur faster at the basal region and slower when it is close to the ciliary tip. Moreover, the time cost by all the dyneins along one microtubule to be activated can be used to deduce the dynein activation pattern; it implies that, as an alternative method, measuring this time can indirectly reveal the dynein activity. The proposed protein-structure model can simulate the ciliary motion triggered by various dynein activation patterns explicitly and may contribute to furthering the studies on axonemal dynein activity.

Detailed structural and biochemical characterization of the nexin-dynein regulatory complex

The nexin-dynein regulatory complex (N-DRC) is a microtubule-cross-bridging structure in cilia/flagella. The precise 3D positions of N-DRC subunits are identified using cryo–electron tomography and structural labeling. The N-DRC is purified and its composition and microtubule-binding properties were characterized.

The nexin-dynein regulatory complex (N-DRC) forms a cross-bridge between the outer doublet microtubules of the axoneme and regulates dynein motor activity in cilia/flagella. Although the molecular composition and the three-dimensional structure of N-DRC have been studied using mutant strains lacking N-DRC subunits, more accurate approaches are necessary to characterize the structure and function of N-DRC. In this study, we precisely localized DRC1, DRC2, and DRC4 using cryo–electron tomography and structural labeling. All three N-DRC subunits had elongated conformations and spanned the length of N-DRC. Furthermore, we purified N-DRC and characterized its microtubule-binding properties. Purified N-DRC bound to the microtubule and partially inhibited microtubule sliding driven by the outer dynein arms (ODAs). Of interest, microtubule sliding was observed even in the presence of fourfold molar excess of N-DRC relative to ODA. These results provide insights into the role of N-DRC in generating the beating motions of cilia/flagella.

The counterbend phenomenon in flagellar axonemes and cross-linked filament bundles

Recent observations of flagellar counterbend in sea urchin sperm show that the mechanical induction of curvature in one part of a passive flagellum induces a compensatory countercurvature elsewhere. This apparent paradoxical effect cannot be explained using the standard elastic rod theory of Euler and Bernoulli, or even the more general Cosserat theory of rods. Here, we develop a geometrically exact mechanical model to describe the statics of microtubule bundles that is capable of predicting the curvature reversal events observed in eukaryotic flagella. This is achieved by allowing the interaction of deformations in different material directions, by accounting not only for structural bending, but also for the elastic forces originating from the internal cross-linking mechanics. Large-amplitude static configurations can be described analytically, and an excellent match between the model and the observed counterbend deformation was found. This allowed a simultaneous estimation of multiple sperm flagellum material parameters, namely the cross-linking sliding resistance, the bending stiffness, and the sperm head junction compliance ratio. We further show that small variations on the empirical conditions may induce discrepancies for the evaluation of the flagellar material quantities, so that caution is required when interpreting experiments. Finally, our analysis demonstrates that the counterbend emerges as a fundamental property of sliding resistance in cross-linked filamentous polymer bundles, which also suggests that cross-linking proteins may contribute to the regulation of the flagellar waveform in swimming sperm via counterbend mechanics.

The N-DRC forms a conserved biochemical complex that maintains outer doublet alignment and limits microtubule sliding in motile axonemes

The nexin–dynein regulatory complex (N-DRC) is implicated in the control of dynein activity as a structural component of the nexin link. This study identifies several new subunits of the N-DRC and demonstrates for the first time that it forms a discrete biochemical complex that maintains outer doublet integrity and regulates microtubule sliding.

The nexin–dynein regulatory complex (N-DRC) is proposed to coordinate dynein arm activity and interconnect doublet microtubules. Here we identify a conserved region in DRC4 critical for assembly of the N-DRC into the axoneme. At least 10 subunits associate with DRC4 to form a discrete complex distinct from other axonemal substructures. Transformation of drc4 mutants with epitope-tagged DRC4 rescues the motility defects and restores assembly of missing DRC subunits and associated inner-arm dyneins. Four new DRC subunits contain calcium-signaling motifs and/or AAA domains and are nearly ubiquitous in species with motile cilia. However, drc mutants are motile and maintain the 9 + 2 organization of the axoneme. To evaluate the function of the N-DRC, we analyzed ATP-induced reactivation of isolated axonemes. Rather than the reactivated bending observed with wild-type axonemes, ATP addition to drc-mutant axonemes resulted in splaying of doublets in the distal region, followed by oscillatory bending between pairs of doublets. Thus the N-DRC provides some but not all of the resistance to microtubule sliding and helps to maintain optimal alignment of doublets for productive flagellar motility. These findings provide new insights into the mechanisms that regulate motility and further highlight the importance of the proximal region of the axoneme in generating flagellar bending.

Engineering oscillating microtubule bundles

From motility of simple protists to determining the handedness of complex vertebrates, highly conserved eukaryotic cilia and flagella are essential for the reproduction and survival of many biological organisms. Despite extensive studies, the exact mechanism by which individual components coordinate their activity to produce ciliary beating patterns remains unknown. We describe a novel approach toward studying ciliary beating. Instead of deconstructing a fully functional organelle from the top-down, we describe a process by which synthetic cilia-like structures are assembled from the bottom-up and we present methods for engineering such structures. We demonstrate how simple mixtures of microtubules, kinesin clusters, and a bundling agent assemble into structures that produce spontaneous oscillations, suggesting that self-organized beating may be a generic feature of internally driven bundles. Synthetic cilia-like structures can be assembled at high density, leading to synchronization and metachronal traveling waves, reminiscent of the waves seen in biological ciliary fields.

One of the nine doublet microtubules of eukaryotic flagella exhibits unique and partially conserved structures

The axonemal core of motile cilia and flagella consists of nine doublet microtubules surrounding two central single microtubules. Attached to the doublets are thousands of dynein motors that produce sliding between neighboring doublets, which in turn causes flagellar bending. Although many structural features of the axoneme have been described, structures that are unique to specific doublets remain largely uncharacterized. These doublet-specific structures introduce asymmetry into the axoneme and are likely important for the spatial control of local microtubule sliding. Here, we used cryo-electron tomography and doublet-specific averaging to determine the 3D structures of individual doublets in the flagella of two evolutionarily distant organisms, the protist Chlamydomonas and the sea urchin Strongylocentrotus. We demonstrate that, in both organisms, one of the nine doublets exhibits unique structural features. Some of these features are highly conserved, such as the inter-doublet link i-SUB5-6, which connects this doublet to its neighbor with a periodicity of 96 nm. We also show that the previously described inter-doublet links attached to this doublet, the o-SUB5-6 in Strongylocentrotus and the proximal 1–2 bridge in Chlamydomonas, are likely not homologous features. The presence of inter-doublet links and reduction of dynein arms indicate that inter-doublet sliding of this unique doublet against its neighbor is limited, providing a rigid plane perpendicular to the flagellar bending plane. These doublet-specific features and the non-sliding nature of these connected doublets suggest a structural basis for the asymmetric distribution of dynein activity and inter-doublet sliding, resulting in quasi-planar waveforms typical of 9+2 cilia and flagella.

Tubulin polyglutamylation regulates flagellar motility by controlling a specific inner-arm dynein that interacts with the dynein regulatory complex

The tpg1 mutant of Chlamydomonas lacks the tubulin polyglutamylase TTLL9 and is deficient in flagellar tubulin polyglutamylation. It exhibits slow swimming, whereas the double mutant with oda2 (a slow-swimming mutant that lacks outer-arm dynein) is completely nonmotile. Thus, tubulin polyglutamylation must be important for the functioning of inner-arm dynein(s). In this study, we show that the tpg1 mutation only slightly affects the motility of mutants that lack dynein "e," one of the seven species of major inner-arm dyneins, whereas it greatly reduces the motility of mutants lacking other inner-arm dynein species. This suggests that dynein e is the main target of motility regulation by tubulin polyglutamylation. Furthermore, the motility of various mutants in the background of the tpg1 mutation raises the possibility that tubulin polyglutamylation also affects the dynein regulatory complex, a dynein e-associated key regulator of flagellar motility, which possibly constitutes the interdoublet (nexin) link. Tubulin polyglutamylation thus may play a central role in the regulation of ciliary and flagellar motility. © 2012 Wiley Periodicals, Inc.

Influence of semiflexible structural features of actin cytoskeleton on cell stiffness based on actin microstructural modeling

A new actin cytoskeleton microstructural model based on the semiflexible polymer nature of the actin filament is proposed. The relationship between the stretching force and the mechanical properties of cells was examined. Experiments on deforming hematopoietic cells with distinct primitiveness from normal and leukemic sources were conducted via optical tweezer manipulation at single-cell level. The modeling results were demonstrated to be in good agreement with the experimental data. We characterized how the structural properties of the actin cytoskeleton, such as prestress, density of cross-links, and actin concentration, affect the mechanical behavior of cells based on the proposed model. Increasing prestress, actin concentration, and density of cross-links reduced cell deformation, and the cell also exhibited strain stiffening behavior with an increase in the stretching force. Compared with existing models, the proposed model exhibits a distinct feature in probing the influence of semiflexible polymer nature of the actin filament on cell mechanical behavior.

Cilia-like beating of active microtubule bundles

The mechanism that drives the regular beating of individual cilia and flagella, as well as dense ciliary fields, remains unclear. We describe a minimal model system, composed of microtubules and molecular motors, which self-assemble into active bundles exhibiting beating patterns reminiscent of those found in eukaryotic cilia and flagella. These observations suggest that hundreds of molecular motors, acting within an elastic microtubule bundle, spontaneously synchronize their activity to generate large-scale oscillations. Furthermore, we also demonstrate that densely packed, actively bending bundles spontaneously synchronize their beating patterns to produce collective behavior similar to metachronal waves observed in ciliary fields. The simple in vitro system described here could provide insights into beating of isolated eukaryotic cilia and flagella, as well as their synchronization in dense ciliary fields.

The dynein regulatory complex is the nexin link and a major regulatory node in cilia and flagella

Elegant cryoelectron tomography reveals that the nexin link between microtubule doublets in 9 + 2 axonemal structures, critical for their ability to bend, is the dynein regulatory complex.

Cilia and flagella are highly conserved microtubule (MT)-based organelles with motile and sensory functions, and ciliary defects have been linked to several human diseases. The 9 + 2 structure of motile axonemes contains nine MT doublets interconnected by nexin links, which surround a central pair of singlet MTs. Motility is generated by the orchestrated activity of thousands of dynein motors, which drive interdoublet sliding. A key regulator of motor activity is the dynein regulatory complex (DRC), but detailed structural information is lacking. Using cryoelectron tomography of wild-type and mutant axonemes from Chlamydomonas reinhardtii, we visualized the DRC in situ at molecular resolution. We present the three-dimensional structure of the DRC, including a model for its subunit organization and intermolecular connections that establish the DRC as a major regulatory node. We further demonstrate that the DRC is the nexin link, which is thought to be critical for the generation of axonemal bending.

Flagellar oscillation: a commentary on proposed mechanisms

Eukaryotic flagella and cilia have a remarkably uniform internal 'engine' known as the '9+2' axoneme. With few exceptions, the function of cilia and flagella is to beat rhythmically and set up relative motion between themselves and the liquid that surrounds them. The molecular basis of axonemal movement is understood in considerable detail, with the exception of the mechanism that provides its rhythmical or oscillatory quality. Some kind of repetitive 'switching' event is assumed to occur; there are several proposals regarding the nature of the 'switch' and how it might operate. Herein I first summarise all the factors known to influence the rate of the oscillation (the beating frequency). Many of these factors exert their effect through modulating the mean sliding velocity between the nine doublet microtubules of the axoneme, this velocity being the determinant of bend growth rate and bend propagation rate. Then I explain six proposed mechanisms for flagellar oscillation and review the evidence on which they are based. Finally, I attempt to derive an economical synthesis, drawing for preference on experimental research that has been minimally disruptive of the intricate structure of the axoneme. The 'provisional synthesis' is that flagellar oscillation emerges from an effect of passive sliding direction on the dynein arms. Sliding in one direction facilitates force-generating cycles and dynein-to-dynein synchronisation along a doublet; sliding in the other direction is inhibitory. The direction of the initial passive sliding normally oscillates because it is controlled hydrodynamically through the alternating direction of the propulsive thrust. However, in the absence of such regulation, there can be a perpetual, mechanical self-triggering through a reversal of sliding direction due to the recoil of elastic structures that deform as a response to the prior active sliding. This provisional synthesis may be a useful basis for further examination of the problem.

Asymmetry of inner dynein arms and inter-doublet links in Chlamydomonas flagella

Although the widely shared “9 + 2” structure of axonemes is thought to be highly symmetrical, axonemes show asymmetrical bending during planar and conical motion. In this study, using electron cryotomography and single particle averaging, we demonstrate an asymmetrical molecular arrangement of proteins binding to the nine microtubule doublets in Chlamydomonasreinhardtii flagella. The eight inner arm dynein heavy chains regulate and determine flagellar waveform. Among these, one heavy chain (dynein c) is missing on one microtubule doublet (this doublet also lacks the outer dynein arm), and another dynein heavy chain (dynein b or g) is missing on the adjacent doublet. Some dynein heavy chains either show an abnormal conformation or were replaced by other proteins, possibly minor dyneins. In addition to nexin, there are two additional linkages between specific pairs of doublets. Interestingly, all these exceptional arrangements take place on doublets on opposite sides of the axoneme, suggesting that the transverse functional asymmetry of the axoneme causes an in-plane bending motion.

Metachronal wave formation in a model of pulmonary cilia

A three-dimensional simulation of the formation of metachronal waves in rows of pulmonary cilia is presented. The cilia move in a two-layer fluid model. The fluid layer adjacent to the cilia bases is purely viscous while the tips of the cilia move through a viscoelastic fluid. An overlapping fixed-moving grid formulation is employed to capture the effect of the cilia on the surrounding fluid. In contrast with immersed boundary methods, this technique allows a natural enforcement of boundary conditions without the need for smoothing of singular force distributions. The fluid domains are discretized using a finite volume method. The 9 + 2 internal microtubule structure of an individual cilium is modeled using large-deflection, curved, finite-element beams. The microtubule skeleton is cross-linked to itself and to the cilium membrane through spring elements which model nexin links. The cilium membrane itself is considered to be elastic and subject to fluid stresses computed from the moving grid formulation as well as internal forces transmitted from the microtubule skeleton. A cilium is set into motion by the action of dynein molecules exerting forces between adjacent microtubules. Realistic models of the forces exerted by dynein molecules are extracted from measurements of observed cilia shapes.

The Parkin co-regulated gene product, PACRG, is an evolutionarily conserved axonemal protein that functions in outer-doublet microtubule morphogenesis

Eukaryotic cilia and flagella are highly conserved structures composed of a canonical 9+2 microtubule axoneme. Comparative genomics of flagellated and non-flagellated eukaryotes provides one way to identify new putative flagellar proteins. We identified the Parkin co-regulated gene, or PACRG, from such a screen. Male mice deficient in PACRG are sterile, but its function has been little explored. The flagellated protozoan parasite Trypanosoma brucei possesses two homologues of PACRG. We performed RNA interference knockdown experiments of the two genes independently and both together. Simultaneous ablation of both proteins produced slow growth and paralysis of the flagellum with consequent effects on organelle segregation. Moreover, using transmission electron microscopy, structural defects were seen in the axoneme, with microtubule doublets missing from the canonical 9+2 formation. The occurrence of missing doublets increased toward the distal end of the flagellum and sequential loss of doublets was observed along individual axonemes. GFP fusion proteins of both PACRG homologues localised along the full length of the axoneme. Our results provide the first evidence for PACRG function within the axoneme, where we suggest that PACRG acts to maintain functional stability of the axonemal outer doublets of both motile and sensory cilia and flagella.

The counterbend phenomenon in dynein-disabled rat sperm flagella and what it reveals about the interdoublet elasticity

Rat sperm that have been rendered passive by disabling the dynein motors with 50 muM sodium metavanadate and 0.1 mM ATP exhibit an interesting response to imposed bending. When the proximal flagellum is bent with a microprobe, the portion of the flagellum distal to the probe contact point develops a bend in the direction opposite the imposed bend. This "counterbend" is not compatible with a simple elastic beam. It can be satisfactorily explained by the sliding tubule model of flagellar structure but only if there are permanent elastic connections between the outer doublets of the axoneme. The elastic component that contributes the bending torque for the counterbend does not reset to a new equilibrium position after an imposed bend but returns the flagellum to a nearly straight or slightly curved final position after release from the probe. This suggests it is based on fixed, rather than mobile, attachments. It is also disrupted by elastase or trypsin digestion, confirming that it is dependent on a protein linkage. Adopting the assumption that the elasticity is attributed to the nexin links that repeat at 96 nm intervals, we find an apparent elasticity for each link that ranges from 1.6 to 10 x 10(-5) N/m. However, the elasticity is nonlinear and does not follow Hooke's law but appears to decrease with increased stretch. In addition, the responsible elastic elements must be able to stretch to more than 10 times their resting length without breakage to account for the observed counterbend formation. Elasticity created by some type of protein unfolding may be the only viable explanation consistent with both the extreme capacity for extension and the nonlinear character of the restoring force that is observed.

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.

Rib72, a conserved protein associated with the ribbon compartment of flagellar A-microtubules and potentially involved in the linkage between outer doublet microtubules

Ciliary and flagellar axonemes are basically composed of nine outer doublet microtubules and several functional components, e.g. dynein arms, radial spokes, and interdoublet links. Each A-tubule of the doublet contains a specialized "ribbon" of three protofilaments composed of tubulin and other proteins postulated to specify the three-dimensional arrangement of the various axonemal components. The interdoublet links hold the doublet microtubules together and limit their sliding during the flagellar beat. In this study on Chlamydomonas reinhardtii, we cloned a cDNA encoding a 71,985-Da polypeptide with three DM10 repeats, two C-terminal EF-hand motifs, and homologs extending to humans. This polypeptide, designated as Rib72, is a novel component of the ribbon compartment of flagellar microtubules. It remained associated with 9-fold arrays of doublet tubules following extraction under high and low ionic conditions, and anti-Rib72 antibodies revealed an approximately 96-nm periodicity along axonemes, consistent with Rib72 associating with interdoublet links. Following proteolysis- and ATP-dependent disintegration of axonemes, the rate of cleavage of Rib72 correlated closely with the rate of sliding disintegration. These observations identify a ribbon-associated protein that may function in the structural assembly of the axoneme and in the mechanism and regulation of ciliary and flagellar motility.

Mechanotransduction through the cytoskeleton

We constructed a model cytoskeleton to investigate the proposal that this interconnected filamentous structure can act as a mechano- and signal transducer. The model cytoskeleton is composed of rigid rods representing actin filaments, which are connected with springs representing cross-linker molecules. The entire mesh is placed in viscous cytoplasm. The model eukaryotic cell is submitted to either shock wave-like or periodic mechanical perturbations at its membrane. We calculated the efficiency of this network to transmit energy to the nuclear wall as a function of cross-linker stiffness, cytoplasmic viscosity, and external stimulation frequency. We found that the cytoskeleton behaves as a tunable band filter: for given linker molecules, energy transmission peaks in a narrow range of stimulation frequencies. Most of the normal modes of the network are spread over the same frequency range. Outside this range, signals are practically unable to reach their destination. Changing the cellular ratios of linker molecules with different elastic characteristics can control the allowable frequency range and, with it, the efficiency of mechanotransduction.

Elastic extension and jump of the flagellar nexin links: a theoretical mechanical cycle

The functions of the nexin links of a flagellar axoneme have not been clearly demonstrated. Taking into account both the elastic (Hookean) characteristics and the possible jump of the nexin links, we calculated the sliding to bending conversion of a theoretical model in a tip-ward direction step by step, according to the essential principles proposed by the geometric clutch hypothesis [Lindemann, 1994: J Theoret Biol 168:175-189]: the activity of the dynein arms depends on the transverse forces induced by the axonemal curvature. In our calculations, however, the transverse forces that are involved in the regulation of the activities of the dynein arms were due to the extension of the nexin links located upstream of a given abscissa. This allowed us to define a bent segment as the axonemal portion at whose proximal and distal ends the nexin links were relaxed, and as fully extended as possible, respectively. The model creates an apparent disorder in the orientation of the nexin links as already observed [Bozkurt and Wooley, 1993: Cell Motil Cytoskeleton 24:109-118; Wooley, 1997: J Cell Sci 110:85-94]. We propose that the nexin links are involved in a mechanical cycle, whose 3 stages are (1) rapid extension, (2) slow relaxation, and (3) stand-by. The rapid extension is compatible with the mechanical interactions between the nexin links and the inner dynein arms with which they form the dynein regulatory complex. This was correlated with the oscillating properties of the nexin links along the axoneme that allow them to be one of the regulatory elements of the local ATPase activity of the dynein arms.

A study of helical and planar waves on sea urchin sperm flagella, with a theory of how they are generated

When the spermatozoon of Echinus esculentus swims in sea water containing methyl cellulose (viscosity 1.5-4 Pa s), its flagellum may generate either a helical or a planar waveform, each type being stable. The helical wave, which is dextral, is complicated by the concurrent passage of miniature waves along it. These miniature waves have a pulsatile origin in the neck region of the spermatozoon. Our videotape analysis indicates that there are two pulses of mechanical activity for each true cycle of the helical wave. (The true helical frequency was obtained from the apparent wave frequency and the roll frequency of the sperm head, the latter being detectable in some sperm when lit stroboscopically.) The planar wave has a meander shape. During the propagation of planar waves, the sliding displacements are adjustable in either direction; moribund flagella can undergo unrestricted sliding. The planar waves are, in fact, exactly planar only at interfaces. Otherwise, there tend to be torsions in the interbend segments between planar bends. Mechanical stimulation of the flagellum can cause a sudden transition from the helical to the planar waveform. To account for the two modes of beating, we advance the hypothesis that circumferential linkages yield beyond a threshold strain. Whether this yield point is exceeded, we suggest, depends upon the balance between the active shear force and the external viscosity (among other factors). We propose that a subthreshold force originates in one array and then triggers the other dynein arrays circumferentially, but unidirectionally, around the base of the flagellum; whereas a suprathreshold force provokes bi-directional circumferential triggering. These may be the two patterns of activation that result in helical and planar waveforms, respectively. The transition from helical to planar bending may result from an increment in the force produced by the dynein motors. The pulsatile origin of the helical wave resembles behaviour described previously for spermatozoa of Ciona intestinalis and of the quail Coturnix coturnix.