Mechanical properties of the passive sea urchin sperm flagellum

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.

The reaction-diffusion basis of animated patterns in eukaryotic flagella

The flagellar beat of bull spermatozoa and C. Reinhardtii are modelled by a minimal, geometrically exact, reaction-diffusion system. Spatio-temporal animated patterns describe flagellar waves, analogous to chemical-patterns from classical reaction-diffusion systems, with sliding-controlled molecular motor reaction-kinetics. The reaction-diffusion system is derived from first principles as a consequence of the high-internal dissipation by the flagellum relative to the external hydrodynamic dissipation. Quantitative comparison with nonlinear, large-amplitude simulations shows that animated reaction-diffusion patterns account for the experimental beating of both bull sperm and C. Reinhardtii. Our results suggest that a unified mechanism may exist for motors controlled by sliding, without requiring curvature-sensing, and uninfluenced by hydrodynamics. High-internal dissipation instigates autonomous travelling waves independently of the external fluid, enabling progressive swimming, otherwise not possible, in low viscosity environments, potentially critical for external fertilizers and aquatic microorganisms. The reaction-diffusion system may prove a powerful tool for studying pattern formation of movement on animated structures.

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.

A Bayesian Framework to Estimate Fluid and Material Parameters in Micro-swimmer Models

To advance our understanding of the movement of elastic microstructures in a viscous fluid, techniques that utilize available data to estimate model parameters are necessary. Here, we describe a Bayesian uncertainty quantification framework that is highly parallelizable, making parameter estimation tractable for complex fluid-structure interaction models. Using noisy in silico data for swimmers, we demonstrate the methodology's robustness in estimating fluid and elastic swimmer parameters, along with their uncertainties. We identify correlations between model parameters and gain insight into emergent swimming trajectories of a single swimmer or a pair of swimmers. Our proposed framework can handle data with a spatiotemporal resolution representative of experiments, showing that this framework can be used to aid in the development of artificial micro-swimmers for biomedical applications, as well as gain a fundamental understanding of the range of parameters that allow for certain motility patterns.

Emergent three-dimensional sperm motility: coupling calcium dynamics and preferred curvature in a Kirchhoff rod model

Changes in calcium concentration along the sperm flagellum regulate sperm motility and hyperactivation, characterized by an increased flagellar bend amplitude and beat asymmetry, enabling the sperm to reach and penetrate the ovum (egg). The signalling pathways by which calcium increases within the flagellum are well established. However, the exact mechanisms of how calcium regulates flagellar bending are still under investigation. We extend our previous model of planar flagellar bending by developing a fluid-structure interaction model that couples the 3D motion of the flagellum in a viscous Newtonian fluid with the evolving calcium concentration. The flagellum is modelled as a Kirchhoff rod: an elastic rod with preferred curvature and twist. The calcium dynamics are represented as a 1D reaction-diffusion model on a moving domain, the flagellum. The two models are coupled assuming that the preferred curvature and twist of the sperm flagellum depend on the local calcium concentration. To investigate the effect of calcium on sperm motility, we compare model results of flagellar bend amplitude and swimming speed for three cases: planar, helical (spiral with equal amplitude in both directions), and quasi-planar (spiral with small amplitude in one direction). We observe that for the same parameters, the planar swimmer is faster and a turning motion is more clearly observed when calcium coupling is accounted for in the model. In the case of flagellar bending coupled to the calcium concentration, we observe emergent trajectories that can be characterized as a hypotrochoid for both quasi-planar and helical bending.

Flow-induced buckling dynamics of sperm flagella

The swimming sperm of many external fertilizing marine organisms face complex fluid flows during their search for egg cells. Aided by chemotaxis, relatively weak flows are known to enhance sperm-egg fertilization rates through hydrodynamic guidance. However, strong flows have the potential to mechanically inhibit flagellar motility through elastohydrodynamic interactions-a phenomenon that remains poorly understood. Here we explore the effects of flow on the buckling dynamics of sperm flagella in an extensional flow through detailed numerical simulations, which are informed by microfluidic experiments and high-speed imaging. Compressional fluid forces lead to rich buckling dynamics of the sperm flagellum beyond a critical dimensionless sperm number, Sp, which represents the ratio of viscous force to elastic force. For nonmotile sperm, the maximum buckling curvature and the number of buckling locations, or buckling mode, increase with increasing sperm number. In contrast, motile sperm exhibit a local flagellar curvature due to the propagation of bending waves along the flagellum. In compressional flow, this preexisting curvature acts as a precursor for buckling, which enhances local curvature without creating new buckling modes and leads to asymmetric beating. However, in extensional flow, flagellar beating remains symmetric with a smaller head yawing amplitude due to tensile forces. The flagellar beating frequency also influences the maximum curvature of motile sperm by facilitating sperm reorientation relative to the compressional axis of the flow near stagnation points. These combined simulations and experiments directly illustrate the microscopic elastohydrodynamic mechanisms responsible for inhibiting flagellar motility in flow and have possible implications for our understanding of external fertilization in dynamic marine systems.

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.

The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella

The inertialess fluid-structure interactions of active and passive inextensible filaments and slender-rods are ubiquitous in nature, from the dynamics of semi-flexible polymers and cytoskeletal filaments to cellular mechanics and flagella. The coupling between the geometry of deformation and the physical interaction governing the dynamics of bio-filaments is complex. Governing equations negotiate elastohydrodynamical interactions with non-holonomic constraints arising from the filament inextensibility. Such elastohydrodynamic systems are structurally convoluted, prone to numerical errors, thus requiring penalization methods and high-order spatio-temporal propagators. The asymptotic coarse-graining formulation presented here exploits the momentum balance in the asymptotic limit of small rod-like elements which are integrated semi-analytically. This greatly simplifies the elastohydrodynamic interactions and overcomes previous numerical instability. The resulting matricial system is straightforward and intuitive to implement, and allows for a fast and efficient computation, more than a hundred times faster than previous schemes. Only basic knowledge of systems of linear equations is required, and implementation achieved with any solver of choice. Generalizations for complex interaction of multiple rods, Brownian polymer dynamics, active filaments and non-local hydrodynamics are also straightforward. We demonstrate these in four examples commonly found in biological systems, including the dynamics of filaments and flagella. Three of these systems are novel in the literature. We additionally provide a Matlab code that can be used as a basis for further generalizations.

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.

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.

Identification of internal properties of fibres and micro-swimmers

In this paper, we address the identifiability of constitutive parameters of passive or active micro-swimmers. We first present a general framework for describing fibres or micro-swimmers using a bead-model description. Using a kinematic constraint formulation to describe fibres, flagellum or cilia, we find explicit linear relationship between elastic constitutive parameters and generalized velocities from computing contact forces. This linear formulation then permits one to address explicitly identifiability conditions and solve for parameter identification. We show that both active forcing and passive parameters are both identifiable independently but not simultaneously. We also provide unbiased estimators for generalized elastic parameters in the presence of Langevin-like forcing with Gaussian noise using a Bayesian approach. These theoretical results are illustrated in various configurations showing the efficiency of the proposed approach for direct parameter identification. The convergence of the proposed estimators is successfully tested numerically.

Curvature regulation of the ciliary beat through axonemal twist

Cilia and flagella are hairlike organelles that propel cells through fluid. The active motion of the axoneme, the motile structure inside cilia and flagella, is powered by molecular motors of the axonemal dynein family. These motors generate forces and torques that slide and bend the microtubule doublets within the axoneme. To create regular waveforms, the activities of the dyneins must be coordinated. It is thought that coordination is mediated by stresses due to radial, transverse, or sliding deformations, and which build up within the moving axoneme and feed back on dynein activity. However, which particular components of the stress regulate the motors to produce the observed waveforms of the many different types of flagella remains an open question. To address this question, we describe the axoneme as a three-dimensional bundle of filaments and characterize its mechanics. We show that regulation of the motors by radial and transverse stresses can lead to a coordinated flagellar motion only in the presence of twist. We show that twist, which could arise from torque produced by the dyneins, couples curvature to transverse and radial stresses. We calculate emergent beating patterns in twisted axonemes resulting from regulation by transverse stresses. The resulting waveforms are similar to those observed in flagella of Chlamydomonas and sperm. Due to the twist, the waveform has nonplanar components, which result in swimming trajectories such as twisted ribbons and helices, which agree with observations.

Functional anatomy of the mammalian sperm flagellum

The eukaryotic flagellum is the organelle responsible for the propulsion of the male gamete in most animals. Without exception, sperm of all mammalian species use a flagellum for swimming. The mammalian sperm has a centrally located 9 + 2 arrangement of microtubule doublets and hundreds of accessory proteins that together constitute an axoneme. However, they also possess several characteristic peri-axonemal structures that make the mammalian sperm tail function differently. These modifications include nine outer dense fibers (ODFs) that are paired with the nine outer microtubule doublets of the axoneme, and are anchored in a structure called the connecting piece located at the base. The presence of the ODFs and connecting piece, and the absence of a basal body, dictate that physical forces generated by the dynein motors are transmitted to the base of the flagellum through the ODFs. Mammalian sperm flagella also possess a mitochondrial and a fibrous sheath that encircle most of the axoneme. These sheaths and the ODFs add mechanical rigidity to the flagellum creating the functional effect of increasing bend wavelength, which requires the entrainment of more dynein motors in the production of a single wave. The sheaths also act as a retinaculum and maintain the integrity of the central axoneme when large bending torques are generated by dynein. Large torque production is crucial to the process of hyperactivation and the unique motility transitions associated with effective fertilizing capacity. Consequently, these specialized anatomical features are essential for the effective interaction of sperm with the female reproductive tract and ovum. © 2016 Wiley Periodicals, Inc.

Emergence of flagellar beating from the collective behavior of individual ATP-powered dyneins

Flagella are hair-like projections from the surface of eukaryotic cells, and they play an important role in many cellular functions, such as cell-motility. The beating of flagella is enabled by their internal architecture, the axoneme, and is powered by a dense distribution of motor proteins, dyneins. The dyneins deliver the required mechanical work through the hydrolysis of ATP. Although the dynein-ATP cycle, the axoneme microstructure, and the flagellar-beating kinematics are well studied, their integration into a coherent picture of ATP-powered flagellar beating is still lacking. Here we show that a time-delayed negative-work-based switching mechanism is able to convert the individual sliding action of hundreds of dyneins into a regular overall beating pattern leading to propulsion. We developed a computational model based on a minimal representation of the axoneme consisting of two representative doublet microtubules connected by nexin links. The relative sliding of the microtubules is incorporated by modeling two groups of ATP-powered dyneins, each responsible for sliding in opposite directions. A time-delayed switching mechanism is postulated, which is key in converting the local individual sliding action of multiple dyneins into global beating. Our results demonstrate that an overall nonreciprocal beating pattern can emerge with time due to the spatial and temporal coordination of the individual dyneins. These findings provide insights in the fundamental working mechanism of axonemal dyneins and could possibly open new research directions in the field of flagellar motility.

ATP Consumption of Eukaryotic Flagella Measured at a Single-Cell Level

The motility of cilia and flagella is driven by thousands of dynein motors that hydrolyze adenosine triphosphate (ATP). Despite decades of genetic, biochemical, structural, and biophysical studies, some aspects of ciliary motility remain elusive, such as the regulation of beating patterns and the energetic efficiency of these nanomachines. In this study, we introduce an experimental method to measure ATP consumption of actively beating axonemes on a single-cell level. We encapsulated individual sea urchin sperm with demembranated flagellum inside water-in-oil emulsion droplets and measured the axoneme's ATP consumption by monitoring fluorescence intensity of a fluorophore-coupled reporter system for ATP turnover in the droplet. Concomitant phase contrast imaging allowed us to extract a linear dependence between the ATP consumption rate and the flagellar beating frequency, with ∼2.3 × 10(5) ATP molecules consumed per beat of a demembranated flagellum. Increasing the viscosity of the aqueous medium led to modified beating waveforms of the axonemes and to higher energy consumption per beat cycle. Our single-cell experimental platform provides both new insights, to our knowledge, into the beating mechanism of flagella and a powerful tool for future studies.

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

Cilia and flagella are model systems for studying how mechanical forces control morphology. The periodic bending motion of cilia and flagella is thought to arise from mechanical feedback: dynein motors generate sliding forces that bend the flagellum, and bending leads to deformations and stresses, which feed back and regulate the motors. Three alternative feedback mechanisms have been proposed: regulation by the sliding forces, regulation by the curvature of the flagellum, and regulation by the normal forces that deform the cross-section of the flagellum. In this work, we combined theoretical and experimental approaches to show that the curvature control mechanism is the one that accords best with the bending waveforms of Chlamydomonas flagella. We make the surprising prediction that the motors respond to the time derivative of curvature, rather than curvature itself, hinting at an adaptation mechanism controlling the flagellar beat.

DOI:http://dx.doi.org/10.7554/eLife.13258.001

Mechanics of the eukaryotic flagellar axoneme: Evidence for structural distortion during bending

The sliding doublet mechanism is the established explanation that allows us to understand the process of ciliary and flagellar bending. In this study, we apply the principles of the sliding doublet mechanism to analyze the mechanics of the counterbend phenomenon in sea urchin sperm flagella. When a passive, vanadate-treated, flagellum is forced into a bend with a glass microprobe, the portion of the flagellum distal to the probe exhibits a bend of opposite curvature (counterbend) to the imposed bend. This phenomenon was shown to be caused by the induction of inter-doublet shear and is dependent on the presence of an inter-doublet shear resistance. Here we report that in sea urchin flagella there is systematically less shear induced in the distal flagellum than is predicted by the sliding doublet mechanism, if we follow the assumption that the diameter of the flagellum is uniform. To account for the reduced shear that is observed, the likeliest and most direct interpretation is that the portion of the axoneme that is forced to bend undergoes substantial compression of the axoneme in the bending plane. A compression of 30-50 nm would be sufficient to account for the shear reduction from a bend of 2 radians. A compression of this magnitude would require considerable flexibility in the axoneme structure. This would necessitate that the radial spokes and/or the central pair apparatus are easily compressed by transverse stress. © 2016 Wiley Periodicals, Inc.

The physiological role of ADP and Mg2+ in maintaining a stable beat cycle in bull sperm

Sperm flagella derive their motive power from the motor protein dynein. In this study, we show that maintenance of the flagellar beat cycle in detergent-extracted bull sperm models is highly dependent on the ratio of Mg(2+) to adenosine triphosphate (ATP). An excess of either ATP un-complexed with Mg(2+) , or an excess of Mg(2+) without an equivalent concentration of ATP, results in the loss of beat amplitude and a reduced curvature development in the beat cycle. In addition, we find that adenosine diphosphate (ADP) can stabilize the beat cycle and permit rhythmic beating across a broader range of ATP and Mg(2+) concentrations. We provide evidence that suggests that when ATP is un-complexed with Mg(2+) , it disrupts the beat cycle by reducing dynein adhesion and thereby, reduces the transmission of dynein-generated force between the doublets. Excess Mg(2+) does not act by the same mechanism and induces a condition where the flagellum is more resistant to bending. This is consistent with the idea that high Mg(2+) stabilizes rigor bridges, and ATP reduces the microtubule binding affinity of dynein. Our results may explain how intact sperm are able to sustain coordinated flagellar beating under a wide range of metabolic conditions, as intact sperm produce ADP in direct proportion to their consumption of ATP.

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.

The geometric clutch at 20: stripping gears or gaining traction?

It has been 20 years since the geometric clutch (GC) hypothesis was first proposed. The core principle of the GC mechanism is fairly simple. When the axoneme of a eukaryotic flagellum is bent, mechanical stress generates forces transverse to the outer doublets (t-forces). These t-forces can push doublets closer together or pry them apart. The GC hypothesis asserts that changes in the inter-doublet spacing caused by t-forces are responsible for the activation and deactivation of the dynein motors, that creates the beat cycle. A series of computer models utilizing the clutch mechanism has shown that it can simulate ciliary and flagellar beating. The objective of the present review is to assess where things stand with the GC hypothesis in the clarifying light of new information. There is considerable new evidence to support the hypothesis. However, it is also clear that it is necessary to modify some of the original conceptions of the hypothesis so that it can be consistent with the results of recent experimental and ultrastructural studies. In particular, dynein deactivation by t-forces must be able to occur with dyneins that remain attached to the B-subtubule of the adjacent doublet.

Computer simulation of flagellar movement X: doublet pair splitting and bend propagation modeled using stochastic dynein kinetics

Experimental observations on cyclic splitting and bending by a flagellar doublet pair are modeled using forces obtained from a model for dynein mechanochemistry, based on ideas introduced by Andrew Huxley and Terrill Hill and extended previously for modeling flagellar movements. The new feature is elastic attachment of dynein to the A doublet, which allows movement perpendicular to the A doublet and provides adhesive force that can strain attached dyneins. This additional strain influences the kinetics of dynein attachment and detachment. Computations using this dynein model demonstrate that very simple and realistic ideas about dynein mechanochemistry are sufficient for explaining the separation and reattachment seen experimentally with flagellar doublet pairs. Additional simulations were performed after adding a "super-adhesion" elasticity. This elastic component is intended to mimic interdoublet connections, normally present in an intact axoneme, that would prevent visible splitting but allow sufficient separation to cause dynein detachment and cessation of shear force generation. This is the situation envisioned by Lindemann's "geometric clutch" hypothesis for control of dynein function in flagella and cilia. The simulations show abrupt disengagement of the "clutch" at one end of a bend, and abrupt reengagement of the "clutch" at the other end of a bend, ensuring that active sliding is only operating where it will cause bend propagation from base to tip.

Waveform generation is controlled by phosphorylation and swimming direction is controlled by Ca2+ in sperm from the mosquito Culex quinquefasciatus

Most animal sperm are quiescent in the male reproductive tract and become activated after mixing with accessory secretions from the male and/or female reproductive tract. Sperm from the mosquito Culex quinquefasciatus initiate flagellar motility after mixing with male accessory gland components, and the sperm flagellum displays three distinct motility patterns over time: a low amplitude, a long wavelength form (Wave A), a double waveform consisting of two superimposed waveforms over the length of the flagellum (Wave B), and finally, a single helical waveform that propels the sperm at high velocity (Wave C). This flagellar behavior is replicated by treating quiescent sperm with trypsin. When exposed to either broad spectrum or tyrosine kinase inhibitors, sperm activated by accessory gland secretions exhibited motility through Wave B but were unable to progress to Wave C. The MEK1/2 inhibitor UO126 and the ERK1/2 inhibitor FR180204 each blocked the transition from Wave B to Wave C, indicating a role for MAPK activity in the control of waveform and, accordingly, progressive movement. Furthermore, a MAPK substrate antibody stained the flagellum of activated sperm. In the absence of extracellular Ca(2+), a small fraction of sperm swam backwards, whereas most could not be activated by either accessory glands or trypsin and were immotile. However, the phosphatase inhibitor okadaic acid in the absence of extracellular Ca(2+) induced all sperm to swim backwards with a flagellar waveform similar to Wave A. These results indicate that flagellar waveform generation and direction of motility are controlled by protein phosphorylation and Ca(2+) levels, respectively.

Biomechanical measurement of kinocilium

Vestibular hair cell bundles in the inner ear each contain a single kinocilium that has the classic 9+2 axoneme microtubule structure. Kinocilia transmit movement of the overlying otoconial membrane mass and cupula to the mechanotransducing portion of the hair cell bundle. Here, we describe how force-deflection techniques can be used to measure turtle utricle kinocilium shaft and base rotational stiffness. In this approach, kinocilia are modeled as homogenous cylindrical rods and their deformation examined as both isotropic Euler-Bernoulli beams (bending only) and transversely isotropic Timoshenko beams (combined shear and bending). The measurements fit the transversely isotropic model much better with flexural rigidity EI=10,400 pN μm(2) (95% confidence interval: 7182-13,630) and shear rigidity kGA=247 pN (180-314), resulting in a shear modulus (G=1.9 kPa) that was four orders of magnitude less than Young's modulus (E=14.1 MPa), indicating that significant shear deformation occurs within deflected kinocilia. The base rotational stiffness (κ) was measured following BAPTA treatment to break the kinocilial links that bind the kinocilium to the bundle along its shaft, and κ was measured as 177±47 pN μm/rad. These parameters are important for understanding how forces arising from head movement are transduced and encoded.

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.

Biomechanics of hair cell kinocilia: experimental measurement of kinocilium shaft stiffness and base rotational stiffness with Euler-Bernoulli and Timoshenko beam analysis

Vestibular hair cell bundles in the inner ear contain a single kinocilium composed of a 9+2 microtubule structure. Kinocilia play a crucial role in transmitting movement of the overlying mass, otoconial membrane or cupula to the mechanotransducing portion of the hair cell bundle. Little is known regarding the mechanical deformation properties of the kinocilium. Using a force-deflection technique, we measured two important mechanical properties of kinocilia in the utricle of a turtle, Trachemys (Pseudemys) scripta elegans. First, we measured the stiffness of kinocilia with different heights. These kinocilia were assumed to be homogenous cylindrical rods and were modeled as both isotropic Euler-Bernoulli beams and transversely isotropic Timoshenko beams. Two mechanical properties of the kinocilia were derived from the beam analysis: flexural rigidity (EI) and shear rigidity (kGA). The Timoshenko model produced a better fit to the experimental data, predicting EI=10,400 pN μm(2) and kGA=247 pN. Assuming a homogenous rod, the shear modulus (G=1.9 kPa) was four orders of magnitude less than Young's modulus (E=14.1 MPa), indicating that significant shear deformation occurs within deflected kinocilia. When analyzed as an Euler-Bernoulli beam, which neglects translational shear, EI increased linearly with kinocilium height, giving underestimates of EI for shorter kinocilia. Second, we measured the rotational stiffness of the kinocilium insertion (κ) into the hair cell's apical surface. Following BAPTA treatment to break the kinocilial links, the kinocilia remained upright, and κ was measured as 177±47 pN μm rad(-1). The mechanical parameters we quantified are important for understanding how forces arising from head movement are transduced and encoded by hair cells.

Simulation of cyclic dynein-driven sliding, splitting, and reassociation in an outer doublet pair

A regular cycle of dynein-driven sliding, doublet separation, doublet reassociation, and resumption of sliding was previously observed by Aoyama and Kamiya in outer doublet pairs obtained after partial dissociation of Chlamydomonas flagella. In the work presented here, computer programming based on previous simulations of oscillatory bending of microtubules was extended to simulate the cycle of events observed with doublet pairs. These simulations confirm the straightforward explanation of this oscillation by inactivation of dynein when doublets separate and resumption of dynein activity after reassociation. Reassociation is augmented by a dynein-dependent "adhesive force" between the doublets. The simulations used a simple mathematical model to generate velocity-dependent shear force, and an independent elastic model for adhesive force. Realistic results were obtained with a maximum adhesive force that was 36% of the maximum shear force. Separation between a pair of doublets is the result of a buckling instability that also initiates a period of uniform sliding that enlarges the separation. A similar instability may trigger sliding initiation events in flagellar bending cycles.

Frenetic activation of fish spermatozoa flagella entails short-term motility, portending their precocious decadence

In most species, fish spermatozoa activate their motility on contact with the external medium (sea or fresh water depending of their reproductive habitat). Their flagella immediately develop waves propagated at high beat frequency (up to 70 beats s(-1)), which propel these sperm cells at high velocity (6-10 mm min(-1)), but for a quite short period of time, usually limited to minutes. Their specific inability to restore their energy content (mostly adenosine triphosphate) fast enough relatively to their high rate of energy consumption by flagellar contributes mainly to the activity arrest of motility, as the spermatozoa need to rely on early accumulated energy prior to activation. This review of the published data explains the present understanding of physico-chemical mechanisms by which flagellar motility is activated (mostly through osmotic and ionic regulation) and then propels sperm cells at speed. It aims also to describe the gradual arrest of their motility much of which occurs within a few minutes.