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.

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.

Mechanical properties of the passive sea urchin sperm flagellum.

In this study we used Triton X-100 extracted sea urchin spermatozoa to investigate the mechanical behavior of the basic 9+2 axoneme. The dynein motors were disabled by vanadate so that the flagellum is rendered a passive structure. We find that when a proximal portion of the flagellum is bent with a glass microprobe, the remainder of the flagellum distal to the probe exhibits a bend in the opposite direction (a counterbend). The counterbend can be understood from the prevailing sliding doublet model of axoneme mechanics, but does require the existence of elastic linkages between the outer doublets. Analysis of the shapes of counterbends provides a consensus value of 0.03-0.08/microm(2) for the ratio of the interdoublet shear resistance (E(S)) to the bending resistance (E(B)) and we find that the ratio E(S)/E(B) is relatively conserved for both passive flagella and transiently quiescent live flagella. This ratio expresses a fundamental mechanical property of the eukaryotic axoneme. It defines the contributions to total bending resistance derived from bending the microtubules and from stretching the interdoublet linkages, respectively. Using this ratio, and computer simulations of earlier experiments that measured the total stiffness of the flagellum, we obtain estimates of approximately 1 x 10(8) pN nm(2)/rad for E(B) and 6 pN/rad for E(S), assuming that both elasticities are linear. Our results indicate that the behavior of the flagellum is close to that predicted by a linear model for shear elasticity.

Thinking about flagellar oscillation.

Bending of cilia and flagella results from sliding between the microtubular outer doublets, driven by dynein motor enzymes. This review reminds us that many questions remain to be answered before we can understand how dynein-driven sliding causes the oscillatory bending of cilia and flagella. Does oscillation require switching between two distinct, persistent modes of dynein activity? Only one mode, an active forward mode, has been characterized, but an alternative mode, either inactive or reverse, appears to be required. Does switching between modes use information from curvature, sliding direction, or both? Is there a mechanism for reciprocal inhibition? Can a localized capability for oscillatory sliding become self-organized to produce the metachronal phase differences required for bend propagation? Are interactions between adjacent dyneins important for regulation of oscillation and bend propagation? Cell Motil. Cytoskeleton 2008. (c) 2008 Wiley-Liss, Inc.

Computer simulation of flagellar movement IX. Oscillation and symmetry breaking in a model for short flagella and nodal cilia.

A computer model of flagella in which oscillation results from regulation of active sliding force by sliding velocity can simulate the movements of very short flagella and cilia. Of particular interest are the movements of the short (2-3 microm) nodal cilia of the mammalian embryo, which determine the development of the asymmetry of the internal organs. These cilia must generate a counterclockwise (viewed from base to tip) circling motion. A three-dimensional computer model, with active force generated by a simple mathematical formulation and regulated by sliding velocity, can generate this circling motion if a time delay process is included in the control specification. Without the introduction of a symmetry-breaking mechanism, the computer models start randomly in either direction, and maintain either clockwise or counterclockwise circling. Symmetry can be broken by at least two mechanisms: (1) control of dynein activity on one outer doublet by sliding velocity can be influenced by the sliding velocity experienced on an adjacent outer doublet, or (2) a constant twist of the axoneme caused by an off-axis component of dynein force. This second mechanism appears more reasonable, but its effectiveness is highly dependent upon specifications for the elastic resistances of the model. These symmetry-breaking mechanisms need to be present only at the beginning of circling. With these models, once a circling direction is established, it remains stable even if the symmetry-breaking mechanism is removed.

Computer simulation of flagellar movement VIII: coordination of dynein by local curvature control can generate helical bending waves.

Computer simulations have been carried out with a model flagellum that can bend in three dimensions. A pattern of dynein activation in which regions of dynein activity propagate along each doublet, with a phase shift of approximately 1/9 wavelength between adjacent doublets, will produce a helical bending wave. This pattern can be termed "doublet metachronism." The simulations show that doublet metachronism can arise spontaneously in a model axoneme in which activation of dyneins is controlled locally by the curvature of each outer doublet microtubule. In this model, dyneins operate both as sensors of curvature and as motors. Doublet metachronism and the chirality of the resulting helical bending pattern are regulated by the angular difference between the direction of the moment and sliding produced by dyneins on a doublet and the direction of the controlling curvature for that doublet. A flagellum that is generating a helical bending wave experiences twisting moments when it moves against external viscous resistance. At high viscosities, helical bending will be significantly modified by twist unless the twist resistance is greater than previously estimated. Spontaneous doublet metachronism must be modified or overridden in order for a flagellum to generate the planar bending waves that are required for efficient propulsion of spermatozoa. Planar bending can be achieved with the three-dimensional flagellar model by appropriate specification of the direction of the controlling curvature for each doublet. However, experimental observations indicate that this "hard-wired" solution is not appropriate for real flagella.