Structure and behaviour of the sperm terminal filament

Light- and electron-microscopic observations of Ciona and Lytechinus spermatozoa show a thin terminal filament at the distal end. The terminal filament is 5-6 microns long and contains the two central microtubules and a variable number of A-tubule extensions of the peripheral doublet microtubules. The transition from the 9 + 2 region to the terminal filament is tapered more gradually in Lytechinus than in Ciona. Photographs of the movement of beating spermatozoa do not show any obvious discontinuity in curvature at the transition region. Bends are propagated smoothly off the end of the flagellum with no decrease in curvature. However, spermatozoa in which the terminal filament has been removed show a clear 'end effect'. This end effect involves a rapid unbending of bends that have reached the distal end of the flagellum. Computer simulations of flagellar models lacking a terminal filament show a similar end effect. Addition of a terminal filament to the end of the computer model can eliminate the end effect. Realistic bending behaviour of the model is obtained by using a terminal filament with a tapered elastic bending resistance in the basal portion of the terminal filament and a value of 0.03 x 10^(9) pN nm^2 in the remainder of the terminal filament. This leads to estimates of 0.01 x 10^(9) pN nm^2 for the elastic bending resistance of an individual microtubule, and 0.2 x 10^(9) pN nm^2 for the elastic bending resistance of the 9 + 2 region of the flagellum. An improvement in propulsive effectiveness by addition of a terminal filament remains to be demonstrated

Effects of antibodies against dynein and tubulin on the stiffness of flagellar axonemes

Antidynein antibodies, previously shown to inhibit flagellar oscillation and active sliding of axonemal microtubules, increase the bending resistance of axonemes measured under relaxing conditions, but not the bending resistance of axonemes measured under rigor conditions. These observations suggest that antidynein antibodies can stabilize rigor cross-bridges between outer-doublet microtubules, by interfering with ATP-induced cross-bridge detachment. Stabilization of a small number of cross-bridge appears to be sufficient to cause substantial inhibition of the frequency of flagellar oscillation. Antitubulin antibodies, previously shown to inhibit flagellar oscillation without inhibiting active sliding of axonemal microtubules, do not increase the static bending resistance of axonemes. However, we observed a viscoelastic effect, corresponding to a large increase in the immediate bending resistance. This immediate bending resistance increase may be sufficient to explain inhibition of flagellar oscillation; but several alternative explanations cannot yet be excluded

Phylogeny and floral character evolution of Mentzelia section Bicuspidaria (Loasaceae)

Mentzelia section Bicuspidaria (Loasaceae) is a monophyletic group of desert ephemerals that inhabit the complex, heterogeneous landscapes of the southwestern United States and northwestern Mexico. To investigate species circumscriptions and evolutionary relationships in M. sect. Bicuspidaria, we employed phylogeny reconstructions based on DNA sequences from the plastid trnL-trnF, trnS-trnfM, ndhF-rpl32, and rpl32-trnL regions and the nuclear ribosomal ITS and ETS regions. Due to evidence of discordant relationships reconstructed from the plastid and nuclear partitions, we used coalescent-based methods in addition to concatenated data sets to estimate the species tree. Maximum likelihood reconstructions based on the combined plastid and nuclear data and coalescent-based reconstructions inferred congruent, fully-resolved species-level phylogenies of M. sect. Bicuspidaria. A monophyletic M. sect. Bicuspidaria was composed of two main clades, which corresponded to a clade of species endemic to the United States composed of M. reflexa, M. tricuspis, and M. tridentata that was sister to a clade of species at least partially distributed in Mexico, composed of M. hirsutissima and M. involucrata. Despite the unusual floral morphology of M. reflexa, molecular reconstructions placed M. reflexa sister to M. tridentata. All species of M. sect. Bicuspidaria were monophyletic, except for M. hirsutissima, which was composed of two distinct lineages and paraphyletic with respect to M. involucrata. The northern clade of M. hirsutissima from California and Baja California was sister to M. involucrata, and both, in turn, were sister to a geographically disjunct southern clade of M. hirsutissima from Baja California Sur and Cedros Island. These phylogeny reconstructions provide evidence for the inclusion of five species in M. sect. Bicuspidaria and have uncovered cryptic diversity that has been largely unrecognized. Character state reconstructions based on the phylogeny of M. sect. Bicuspidaria suggest innovative and, at times, homoplasious floral evolution

Self-organized Beating and Swimming of Internally Driven Filaments

We study a simple two-dimensional model for motion of an elastic filament subject to internally generated stresses and show that wave-like propagating shapes which can propel the filament can be induced by a self-organized mechanism via a dynamic instability. The resulting patterns of motion do not depend on the microscopic mechanism of the instability but only of the filament rigidity and hydrodynamic friction. Our results suggest that simplified systems, consisting only of molecular motors and filaments could be able to show beating motion and self-propulsion.Comment: 8 pages, 2 figures, REVTe

Physical Aspects of Axonemal Beating and Swimming

We discuss a two-dimensional model for the dynamics of axonemal deformations driven by internally generated forces of molecular motors. Our model consists of an elastic filament pair connected by active elements. We derive the dynamic equations for this system in presence of internal forces. In the limit of small deformations, a perturbative approach allows us to calculate filament shapes and the tension profile. We demonstrate that periodic filament motion can be generated via a self-organization of elastic filaments and molecular motors. Oscillatory motion and the propagation of bending waves can occur for an initially non-moving state via an instability termed Hopf bifurcation. Close to this instability, the behavior of the system is shown to be independent of microscopic details of the axoneme and the force-generating mechanism. The oscillation frequency however does depend on properties of the molecular motors. We calculate the oscillation frequency at the bifurcation point and show that a large frequency range is accessible by varying the axonemal length between 1 and 50$\mu$m. We calculate the velocity of swimming of a flagellum and discuss the effects of boundary conditions and externally applied forces on the axonemal oscillations.Comment: 14 pages, 8 figures, REVTE

The optimal elastic flagellum

Motile eukaryotic cells propel themselves in viscous fluids by passing waves of bending deformation down their flagella. An infinitely long flagellum achieves a hydrodynamically optimal low-Reynolds number locomotion when the angle between its local tangent and the swimming direction remains constant along its length. Optimal flagella therefore adopt the shape of a helix in three dimensions (smooth) and that of a sawtooth in two dimensions (non-smooth). Physically, biological organisms (or engineered micro-swimmers) must expend internal energy in order to produce the waves of deformation responsible for the motion. Here we propose a physically-motivated derivation of the optimal flagellum shape. We determine analytically and numerically the shape of the flagellar wave which leads to the fastest swimming while minimizing an appropriately-defined energetic expenditure. Our novel approach is to define an energy which includes not only the work against the surrounding fluid, but also (1) the energy stored elastically in the bending of the flagellum, (2) the energy stored elastically in the internal sliding of the polymeric filaments which are responsible for the generation of the bending waves (microtubules), and (3) the viscous dissipation due to the presence of an internal fluid. This approach regularizes the optimal sawtooth shape for two-dimensional deformation at the expense of a small loss in hydrodynamic efficiency. The optimal waveforms of finite-size flagella are shown to depend upon a competition between rotational motions and bending costs, and we observe a surprising bias towards half-integer wave-numbers. Their final hydrodynamic efficiencies are above 6%, significantly larger than those of swimming cells, therefore indicating available room for further biological tuning

Nonlinear instability in flagellar dynamics: a notel modulation mechanism in sperm migration

Throughout biology, cells and organisms use flagella and cilia to propel fluid and achieve motility. The beating of these organelles, and the corresponding ability to sense, respond to and modulate this beat is central to many processes in health and disease. While the mechanics of flagellum–fluid interaction has been the subject of extensive mathematical studies, these models have been restricted to being geometrically linear or weakly nonlinear, despite the high curvatures observed physiologically. We study the effect of geometrical nonlinearity, focusing on the spermatozoon flagellum. For a wide range of physiologically relevant parameters, the nonlinear model predicts that flagellar compression by the internal forces initiates an effective buckling behaviour, leading to a symmetry-breaking bifurcation that causes profound and complicated changes in the waveform and swimming trajectory, as well as the breakdown of the linear theory. The emergent waveform also induces curved swimming in an otherwise symmetric system, with the swimming trajectory being sensitive to head shape—no signalling or asymmetric forces are required. We conclude that nonlinear models are essential in understanding the flagellar waveform in migratory human sperm; these models will also be invaluable in understanding motile flagella and cilia in other systems

Beating patterns of filaments in viscoelastic fluids

Many swimming microorganisms, such as bacteria and sperm, use flexible flagella to move through viscoelastic media in their natural environments. In this paper we address the effects a viscoelastic fluid has on the motion and beating patterns of elastic filaments. We treat both a passive filament which is actuated at one end, and an active filament with bending forces arising from internal motors distributed along its length. We describe how viscoelasticity modifies the hydrodynamic forces exerted on the filaments, and how these modified forces affect the beating patterns. We show how high viscosity of purely viscous or viscoelastic solutions can lead to the experimentally observed beating patterns of sperm flagella, in which motion is concentrated at the distal end of the flagella

Properties of an antiserum against native dynein 1 from sea urchin sperm flagella

Effects of an antiserum against native dynein 1 from sperm flagella of the sea urchin Strongylocentrotus purpuratus were compared with effects of an antiserum previously obtained against an ATPase-active tryptic fragment (fragment 1A) of dynein 1 from sperm flagella of the sea urchin, Anthocidaris crassispina. Both antisera precipitate dynein 1 and do not precipitate dynein 2. Only the fragment 1A antiserum precipitates fragment 1A and produces a measurable inhibition of dynein 1 ATPase activity. Both antisera inhibit the movement and the movement- coupled ATP dephosphorylation of reactivated spermatozoa. The inhibition of movement by the antiserum against dynein 1 is much less than by the antiserum against fragment 1A, suggesting that a specific interference with the active ATPase site may be required for effective inhibition of movement. Both antisera reduce the bend angle as well as the beat frequency of reactivated S. purpuratus spermatozoa, suggesting that the bend angle may depend on the activity of the dynein arms which generate active sliding