Sperm structure of Trichoptera. III. Hydropsychidae,polycentropodidae and philopotamidae (Annulipalpia)

Spermatozoa from 9 species belonging to 3 families of trichopteran suborder Annulipalpia (Philopotamidae : Philopotamus ludificatus, P. montanus, Wormaldia occipitalis, W. copiosa; Polycentropodidae: Plectrocnemia geniculata, Polycentropus mortoni, P. irroratus, Cyrnus trimaculatus; Hydropsychidae: Hydropsyche pellucidula) were studied by light, scanning, and transmission electron microscopy. The absence of axonemal dynein arms and a consequent sperm immotility are characteristic of the species studied. The presence of 7, rather than 2, central microtubules is shared by Polycentropodidae and some Philopotamidae. A greater variability in the sperm axoneme was found within Philopotamidae than in the other families. The species of the genus Wormaldia have an axial vesicle rather than microtubules and their axonemes have a 9 + 9 + 0 or 13 + 13 + 0 pattern. Hydropsychidae have spermatozoa provided with numerous finger-like appendages containing microtubular doublets. The progressive loss of sperm flagellum and motility within the group is discussed.     

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.     

Sperm axoneme of two rows of doublets reversely oriented in the gall-midge Lestremia (Diptera, Cecidomyiidae)

The spermatozoon of Lestremia lacks an acrosome and has a giant centriole that gives origin to a giant axoneme with about 150 doublets. The axonemal doublets, disposed in two opposite rows oriented antiparallel, have A doublets with two dynein arms and a B tubule filled with dense proteinaceous material. Mitochondria fuse in two derivatives and show cristae and a longitudinal crystallized axis. The probable origin of the giant axoneme is hypothesized and the more prolonged motility of Lestremia sperm in comparison with that of other gall midges is related to the presence of a more precise axonemal organization. The spermatological results agree with the systematic position of Lestremiinae at the base of the evolutionary trend of the family Cecidomyiidae.     

Ultrastructural studies on euspermatozoa and paraspermatozoa in Mantispidae (Insecta, Neuroptera)

Previous studies have demonstrated the presence of sperm dimorphism in the Mantispidae Perlamantispa perla. We extended the study on several other mantidflies. In all the examined species the occurrence of euspermatozoa (typical) and paraspermatozoa (atypical) was established. The euspermatozoa are characterized by the presence of a cylindrical nucleus surrounded by an envelope that fans out laterally into two thin wings of different length. The acrosome seems to be missing. The nucleus is surrounded by extracellular material. The flagellum is provided with a 9 + 9 + 2 axonemal pattern; the accessory tubules contain 16 protofilaments and the intertubular material has the distribution typical of the taxon. Two elongated accessory bodies flank partially the axoneme and connect this structure with the mitochondrial derivatives. The flagellar axoneme of paraspermatozoa consists of an axoneme and two giant mitochondrial derivatives filled with large globular units. The axoneme exhibits a 9 + 9 + 2 pattern, in which the central 9 + 2 units have a normal structure, in that the microtubular doublets are provided with both dynein arms and radial links. On the contrary, the nine accessory microtubules have a large diameter and their tubular wall consists of 40 protofilaments. This comparative study provided evidences about the uniformity of sperm ultrastructure in Mantispidae. The function of non-fertilizing giant sperm in mantidflies is discussed.     

Sperm structure of Trichoptera. IV. Rhyacophilidae and Glossosomatidae

The families Rhyacophilidae and Glossosomatidae (Trichoptera) are considered to be the most primitive ones within the order. We examined the spermatozoa of members of these families to see whether their ultrastructure is consistent with an ancestral position. Axonemal structures, after fixation with a tannic acid-containing fixative, have been shown to be particularly useful as taxonomical indicators. It was found that 4 members of Rhyacophilidae, representing 3 subgenera (Rhyacophila, Pararhyacophila, and Hyporhyacophila) all have motile spermatozoa, with a 9 + 9 + 2 axoneme in which inner (but no outer) dynein arms are present. The accessory tubules have a wall consisting of 17 protofilaments, decreasing to 16 near the distal end, whereas the examined member of Glossosomatidae, Catagapetus nigrans, has accessory tubules with 18 protofilaments and a 9 + 9 + 2 axoneme with inner dynein arms and with motility similar to the Rhyacophilidae. Sperm motility is consistent with the inclusion of these 2 families within the suborder Integripalpia, and the axonemal pattern 9 + 9 + 2 indicates that the families indeed occupy a primitive position within Trichoptera.     

Structural-functional relationships of the dynein,spokes, and central-pair projections predicted from an analysis of the forces acting within a flagellum

In the axoneme of eukaryotic flagella the dynein motor proteins form crossbridges between the outer doublet microtubules. These motor proteins generate force that accumulates as linear tension, or compression, on the doublets. When tension or compression is present on a curved microtubule, a force per unit length develops in the plane of bending and is transverse to the long axis of the microtubule. This transverse force (t-force) is evaluated here using available experimental evidence from sea urchin sperm and bull sperm. At or near the switch point for beat reversal, the t-force is in the range of 0.25-1.0 nN/ micro m, with 0.5 nN/ micro m the most likely value. This is the case in both beating and arrested bull sperm and in beating sea urchin sperm. The total force that can be generated (or resisted) by all the dyneins on one micron of outer doublet is also approximately 0.5 nN. The equivalence of the maximum dynein force/ micro m and t-force/ micro m at the switch point may have important consequences. Firstly, the t-force acting on the doublets near the switch point of the flagellar beat is sufficiently strong that it could terminate the action of the dyneins directly by strongly favoring the detached state and precipitating a cascade of detachment from the adjacent doublet. Secondly, after dynein release occurs, the radial spokes and central-pair apparatus are the structures that must carry the t-force. The spokes attached to the central-pair projections will bear most of the load. The central-pair projections are well-positioned for this role, and they are suitably configured to regulate the amount of axoneme distortion that occurs during switching. However, to fulfill this role without preventing flagellar bend formation, moveable attachments that behave like processive motor proteins must mediate the attachment between the spoke heads and the central-pair structure.     

The spermatogenesis and sperm structure of Acerentomon microrhinus (Protura, Hexapoda) with considerations on the phylogenetic position of the taxon

The spermatogenesis of the proturan Acerentomon microrhinus Berlese, (Redia 6:1–182, 1909) is described for the first time with the aim of comparing the ultrastructure of the flagellated sperm of members of this taxon with that of the supposedly related group, Collembola. The apical region of testes consists of a series of large cells with giant polymorphic nuclei and several centrosomes with 14 microtubule doublets, whose origin is likely a template of a conventional 9-doublet centriole. Beneath this region, there are spermatogonial cells, whose centrosome has two centrioles, both with 14 microtubule doublets; the daughter centriole of the pair has an axial cylinder. Slender parietal cells in the testes have centrioles with nine doublet microtubules. Spermatocytes produce short primary cilia with 14 microtubule doublets. Spermatids have a single basal body with 14 microtubule doublets. Anteriorly, a conical dense material is present, surrounded by a microtubular basket, which can be seen by using an α-anti-tubulin antibody. Behind this region, the basal body expresses a long axoneme of 14 microtubule doublets with only inner arms. An acrosome is lacking. The nucleus is twisted around the apical conical dense structure and the axoneme; this coiling seems to be due to the rotation of the axoneme on its longitudinal axis. The posterior part of the axoneme forms three turns within the spermatid cytoplasm. Few unchanged mitochondria are scattered in the cytoplasm. Sperm consist of encysted, globular cells that descend along the deferent duct lumen. Some of them are engulfed by the epithelial cells, which thus have a spermiophagic activity. Sperm placed in a proper medium extend their flagellar axonemes and start beating. Protura sperm structure is quite different from that of Collembola sperm; and on the basis of sperm characters, a close relationship between the two taxa is not supported.     

The spermatogenesis and sperm structure of Timema poppensis (Insecta: Phasmatodea)

The ultrastructure of spermatogenesis and spermatozoa was studied in Timema poppensis Vickery & Sandoval, 1999, a putative basal taxon of Phasmatodea. The apical portion of testis follicles consists of spermatogonial cells with polymorphic nuclei. Primary spermatocytes display very short primary cilia originating from the peripheral centrosomes. Early spermatids develop a conspicuous “nebenkern” consisting of fused mitochondria. They have a single peripheral centriole with microtubular triplets, which expresses a 3.6-μm-long cilium featuring a 9?+?2 axonemal pattern. In a later stage, the centriole and the ciliary shaft displace toward the inner part of the cytoplasm by an infolding of the plasma membrane. Mature spermatids exhibit a derived centriole with microtubule doublets devoid of dynein arms, which is equipped with a dense arc-like outer structure. Ciliary degeneration was not observed during spermiogenesis. Spermatozoa are short flagellate cells about 55–60?μm in length. They are characterized by a three-layered acrosomal complex. The distinctive bell-shaped morphology of the acrosome vesicle is likely an autapomorphic trait of Timema. The flagellum has a 9?+?9?+?2 axoneme, two accessory bodies, two flattened cisterns, and two elongated mitochondrial derivatives. Results support the hypothesis that Phasmatodea, comprising Timema?+?Euphasmatodea, form a monophyletic group. The presence of 17 protofilaments in the wall of accessory microtubules and the flattened configuration of the flagellum are potential apomorphic groundplan features of the order. Within Phasmatodea, a key evolutionary divergence was from the conventional insect spermiogenesis and sperm structure of Timema, to the unusual spermiogenetic process and peculiar sperm structure of Euphasmatodea. As a result, Timema retains more sperm character states found in the polyneopteran ground pattern, while Euphasmatodea have evolved outstanding sperm autapomorphies, like the loss of mitochondria and flattened cisterns, and the presence of strongly expanded accessory bodies.     

Ultrastructure of the sperm of Dolania americana Edmunds and Traver (Ephemeroptera : Behningiidae)

The ultrastructure of the mature sperm of the mayfly, Dolania americana Edmunds and Traver (Ephemeroptera : Behningiidae), is described from scanning and transmission electron microscopy. The head is 0.7–1 μm wide and 4.6–6.9 μm long, rodlike, and topped by a short, rounded acrosome 0.4 μm long and 0.6 μm wide. The flagellum is 5–6 times the head length and is flattened, except for a thin, tubelike terminal portion. The axoneme pattern is 9-9-1 (9 outer singlet microtubules, 9 doublet microtubules, and a central dark element) and is new for Ephemeroptera. The inner dynein arms are conspicuous and outer arms are lacking, and radial spokes and a central sheath are prominent. A densely-staining and bi-lobed accessory body lies adjacent to the axoneme. A mitochondrial derivative with regularly arranged transverse-to-oblique cristae lies adjacent to the accessory body.     

Testing the geometric clutch hypothesis

The Geometric Clutch hypothesis is based on the premise that transverse forces (t-forces) acting on the outer doublets of the eukaryotic axoneme coordinate the action of the dynein motors to produce flagellar and ciliary beating. T-forces result from tension and compression on the outer doublets when a bend is present on the flagellum or cilium. The t-force acts to pry the doublets apart in an active bend, and push the doublets together when the flagellum is passively bent and thus could engage and disengage the dynein motors. Computed simulations of this working mechanism have reproduced the beating pattern of simple cilia and flagella, and of mammalian sperm. Cilia-like beating, with a clearly defined effective and recovery stroke, can be generated using one uniformly applied switching algorithm. When the mechanical properties and dimensions appropriate to a specific flagellum are incorporated into the model the same algorithm can simulate a sea urchin or bull sperm-like beat. The computed model reproduces many of the observed behaviors of real flagella and cilia. The model can duplicate the results of outer arm extraction experiments in cilia and predicted two types of arrest behavior that were verified experimentally in bull sperm. It also successfully predicted the experimentally determined nexin elasticity. Calculations based on live and reactivated sea urchin and bull sperm yielded a value of 0.5 nN/microm for the t-force at the switch-point. This is a force sufficient to overcome the shearing force generated by all the dyneins on one micron of outer doublet. A t-force of this magnitude should produce substantial distortion of the axoneme at the switch-point, especially in spoke or spoke-head deficient motile flagella. This concrete and verifiable prediction is within the grasp of recent advances in imaging technology, specifically cryoelectron microscopy and atomic force microscopy.     

Ultrastructure of spermatids and spermatozoa in the cyclostomatous bryozoan Tubulipora (Bryozoa,Cyclostomata)

Summary Differentiation of spermatids to mature spermatozoa in the bryozoan Tubulipora liliacea was studied by transmission electron microscopy. The spermatozoon of Tubulipora is of a filiform, modified type, and has evolved from the primitive type as an adaptation to a specialized biology of fertilization. The head of the spermatozoon consists of a small, conical acrosome capping an elongated, cylindrical, anteriorly tapering nucleus. A basal invagination in the nucleus contains the proximal portion of the axoneme and a dense attachment matrix. The flagellar axoneme has the typical 9+2 structure. Four elongated rodshaped mitochondria with typical cristae surround the axoneme in the cylindrical middle piece. Granular electron-dense material is accumulated in the form of four columns alternating with four long cylindrical mitochondria. The mitochondrial middle piece is separated externally from the tail region by an involution of the plasma membrane. The tail region contains a cytoplasmic sheath with accessory fibers surrounding the axoneme. Nine outer, coarse fibers extend posteriorly paralleling the nine doublets of the axoneme. The coarse fibers develop from electron-dense plate-like structures associated with the doublets of the axoneme. A characteristic feature in spermiogenesis is that spermatozoa develop in tetrads. There seem to be significant differences in spermatozoan ultrastructure between the three bryozoan classes Stenolaemata, Gymnolaemata, and Phylactolaemata. The differences indicate different lines of evolution of fertilization biology in these groups.Abbreviations used in the figures a acrosome - av acrosomal vesicles - ax axoneme - c coarse fiber - d electron dense rod - m mitochondrion - mp middle piece - Scale bars=0.5 m - mt microtubule - n nucleus - ne nuclear envelope - p nuclear protrusion - pm plasma membrane - t tail     

Comparative morphology of spermatozoa and reproductive systems of zorapteran species from different world regions (Insecta,Zoraptera)

The male and female reproductive apparatus of Zorotypus magnicaudelli (Malaysia), Zorotypus huxleyi (Ecuador) and Zorotypus weidneri (Brazil) were examined and documented in detail. The genital apparatus and sperm of the three species show only minor differences. The testes are larger in Z. magnicaudelli. Z. huxleyi lacks the helical appendage in the accessory glands. A long cuticular flagellum is present in Z. magnicaudelli and in the previously studied Zorotypus caudelli like in several other species, whereas it is absent in Z. weidneri, Z. huxleyi, Zorotypus hubbardi, Zorotypus impolitus and Zorotypus guineensis. Characteristic features of the very similar sperm are the presence of: a) two dense arches above the axoneme; b) a 9 + 9+2 axoneme with detached subtubules A and B of doublets 1 and 6; c) the axonemal end degenerating with enlarging accessory tubules; d) accessory tubules with 17 protofilaments; e) three accessory bodies beneath the axoneme; and f) two mitochondrial derivatives of equal shape. The first characteristic (a) is unknown outside of Zoraptera and possibly autapomorphic. The sperm structure differs distinctly in Z. impolitus and Z. hubbardi, which produce giant sperm and possess a huge spermatheca. The presence of the same sperm type in species either provided with a sclerotized coiled flagellum in males or lacking this structure indicates that a different organization of the genital apparatus does not necessarily affect the sperm structure. The flagellum and its pouch has probably evolved within Zoraptera, but it cannot be excluded that it is a groundplan feature and was reduced several times. The fossil evidence and our findings suggest that distinct modifications in the genital apparatus occurred before the fragmentation of the Gondwanan landmass in the middle Cretaceous.     

Ultrastructure of the flagellar apparatus inPyramimonas octopus (Prasinophyceae)

Summary While green algae usually lack one of the outer dynein arms in the axoneme, flagella of the octoflagellated prasinophytePyramimonas octopus possess dynein arms on all peripheral doublets. The outer dynein arm on doublet no. 1 is modified, and additional structures are associated with doublets no. 2 and 6. The flagellar scales are asymmetrically arranged. Thus the two rows of thick flagellar hairscales are displaced towards doublet no. 6,i.e., in the direction of the effective stroke of each flagellum. The underlayer of small scales includes two nearly opposite double rows scales, arranged in the longitudinal direction of the flagellum. The hairscales emerge from these rows. The double rows are separated on one side by 9, on the other by 11 rows of helically arranged scales. The central pair of microtubules twists, but the axoneme itself (represented by the 9 peripheral doublets), does not seem to rotate. The flagella are arranged in two groups, showing modified 180° rotational symmetry. The effective strokes of the two central flagella are exactly opposite, while the other flagella beat in six intermediate directions.     

ULTRASTRUCTURE OF THE FLAGELLUM IN SPERM OF CKCINELLA SEPTEMPUNCTATA

Abstract  Using cell whole mount preparation and ultrathin section technique, the ultrastructure of the flagellum in the sperm of Coccinella septempunctata L. was examined with transmission electron microscope. The flagellum is made up of a classic 9+9+2 axoneme containing two similar crystallized mitochondria1 derivatives, two accessory bodies, which are divided in to two portions, an osmiophilic dense crescent and a spongy one, and a non-crystalline body. At the end of the flagellum, only the axoneme is present, it loses the two central microtubules but retains the nine doublets with dynein arms and the nine accessory microtubules.     

七星瓢虫精子鞭毛的超微结构(英文)

应用细胞整装制备和超薄切片技术,在透射电子显微镜下检查了七星瓢虫成熟精子鞭毛的超微结构。精子鞭毛是由一个典型的9＋9＋2轴丝,两个同形结晶的线粒体衍生物,两个附体（每个附体具有两部分,一个嗜锇致密月牙体和一个海绵月牙体）和一个非结晶体组成,在鞭毛终端部,仅存的轴丝失去了两个中央微管保留了9个具有动力蛋白臂的双微管和9个附微管。     

Pih1d3 is required for cytoplasmic preassembly of axonemal dynein in mouse sperm

Axonemal dynein complexes are preassembled in the cytoplasm before their transport to cilia, but the mechanism of this process remains unclear. We now show that mice lacking Pih1d3, a PIH1 domain–containing protein, develop normally but manifest male sterility. Pih1d3^−/− sperm were immotile and fragile, with the axoneme of the flagellum lacking outer dynein arms (ODAs) and inner dynein arms (IDAs) and showing a disturbed 9+2 microtubule organization. Pih1d3 was expressed specifically in spermatogenic cells, with the mRNA being most abundant in pachytene spermatocytes. Pih1d3 localized to the cytoplasm of spermatogenic cells but was not detected in spermatids or mature sperm. The levels of ODA and IDA proteins were reduced in the mutant testis and sperm, and Pih1d3 was found to interact with an intermediate chain of ODA as well as with Hsp70 and Hsp90. Our results suggest that Pih1d3 contributes to cytoplasmic preassembly of dynein complexes in spermatogenic cells by stabilizing and promoting complex formation by ODA and IDA proteins.     

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.     

Gene trap mutation of murine Outer dense fiber protein-2 gene can result in sperm tail abnormalities in mice with high percentage chimaerism

Background  

Outer dense fiber protein 2, Odf2, is a major component of the outer dense fibers, ODF, in the flagellum of spermatozoa. ODF are associated with microtubule doublets that form the axoneme. We recently demonstrated that tyrosine phosphorylation of Odf2 is important for sperm motility. In the course of a study of Odf2 using Odf2 mouse knockout lines we observed that males of a high percentage chimaerism, made using XL169 embryonic stem cells, were infertile, whereas mice of low-medium percentage chimaerism were fertile.     

Structure and ultrastructure of the spermatozoa of Bephratelloides pomorum (Fabricius) (Hymenoptera: Eurytomidae)

The spermatozoa of Bephratelloides pomorum are very long and fine. Each spermatozoon measures about 620 μm in length by 0.38 μm in diameter and, when seen under the light microscope, appears to be wavy along its entire length. The head, which is approximately 105 μm, comprises a small acrosome and a nucleus. The acrosome is made up of a cone-shaped acrosomal vesicle surrounding the perforatorium and the anterior end of the nucleus. Innumerable filaments radiate from it. The perforatorium has a diameter equal to that of the nucleus at their junction, where it fits with a concave base onto the rounded nuclear tip. The nucleus is helicoidal and completely filled with homogeneous compact chromatin. It is attached to the tail by a very long and quite electron-dense centriolar adjunct that extends anteriorly from the centriole in a spiral around the nucleus for approximately 8.5 μm. The tail consists of an axoneme with the 9+9+2 microtubule arrangement pitched in a long helix, as well as a pair of spiraling mitochondrial derivatives (with regularly arranged cristae) that coil around the axoneme, and two small accessory bodies. As well as the spiraling of the nucleus, mitochondrial derivatives and axonemal microtubules, the sperm of B. pomorum present other very different morphological features. These features include the acrosome and centriolar adjunct, both of which differentiate the spermatozoa from the majority of sperm found in other Hymenoptera. In addition these structural variations demonstrate that the sperm of chalcidoids provide characteristics that can certainly prove useful for future phylogenetic analysis at the subfamily level and, possibly, the genus too.     

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.