Bending of the “9+2” axoneme analyzed by the finite element method

Abstract

Many data demonstrate that the regulation of the bending polarity of the "9+2" axoneme is supported by the curvature itself, making the internal constraints central in this process, adjusting either the physical characteristics of the machinery or the activity of the enzymes involved in different pathways. Among them, the very integrated Geometric Clutch model founds this regulation on the convenient adjustments of the probability of interaction between the dynein arms and the beta-tubulin monomers of the outer doublet pairs on which they walk. Taking into consideration (i) the deviated bending of the outer doublets pairs (Cibert, C., Heck, J.-V., 2004. Cell Motil. Cytoskeleton 59, 153-168), (ii) the internal tensions of the radial spokes and the tangential links (nexin links, dynein arms), (iii) a theoretical 5 microm long proximal segment of the axoneme and (iv) the short proximal segment of the axoneme, we have reevaluated the adjustments of these intervals using a finite element approach. The movements we have calculated within the axonemal cylinder are consistent with the basic hypothesis that found the Geometric Clutch model, except that the axonemal side where the dynein arms are active increases the intervals between the two neighbor outer doublet pairs. This result allows us to propose a mechanism of bending reversion of the axoneme, involving the concerted ignition of the molecular engines along the two opposite sides of the axoneme delineated by the bending plane.

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

“…These biophysical parameters are essential to investigations based on modeling and simulation (12,13,45). In particular, during ciliary beating, active dynein forces are balanced by both internal elastic forces within the axoneme and external viscous fluid drag (8,33).…”

Finite element models of flagella with sliding radial spokes and interdoublet links exhibit propagating waves under steady dynein loading

“…The models in the current paper are more faithful representations of the axoneme than prior models (Bayly & Dutcher, ), including up to nine doublets and accounting for the possibility of sliding between components. Cibert, Toscano, Pensee, and Bonnet () also used the finite element method to model the axoneme, but only calculated equilibrium, not dynamic, deformations. Under various loading conditions with a range of plausible parameters, the current models (i) show propagating, oscillatory waveforms at physically reasonable frequencies, and (ii) exhibit changes in frequency, amplitude, and symmetry consistent with experimental observations.…”

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

“…For simplification purpose the longitudinal fibers are constituted by homogeneous series of cylinders whose each element is linked to its two neighbors by four torsion springs defined in two perpendicular planes and whose mechanical characteristics are equal (table 1). Actually, the bending of the outer doublets of microtubules induces the deviated bending of the axoneme [32,38,39] that could be involved in the regulation of the axonemal activity [40]. One of the ends of the polarized transversal links is embedded into one longitudinal fiber and the other one consists in a carriage that rolls freely along the other longitudinal fiber they link (figures 3 and 4).…”

“…The proximal end of the model is an important element where the three longitudinal fibers and the three longitudinal series of actuators are necessarily anchored (figures 5 and 6). This end is necessarily compliant and must involve three springs (table 3) that allow the variations of the intervals between the embedded sites of the longitudinal fibers, because of the constraints due to the bending and/or twisting [38,39,41].…”

“…In spite of our basic knowledge of the axonemal machinery and its movements [2,16,22,31,36,39,[50][51][52][53][54][55] it is really hard to construct a reasonable demonstrator of the device because of the large number of mechanical and dynamical problems that must be solved to make it an efficient swell power generator. However, we tried to answer five questions related to this necessary 'dynamic' architecture that must be maintained on the sea/ocean surface.…”