Geometry drives the “deviated‐bending” of the bi‐tubular structures of the 9+2 axoneme in the flagellum

Abstract

The axoneme "9 + 2" is basically a system constituted of a cylinder of 9 microtubule doublets surrounding a central pair of microtubules. These bi-tubular structures are considered as the support system of the active molecular complexes that generate and regulate the axonemal movement. Schoutens has calculated their moments of inertia [Schoutens, 1994: Journal of Theoretical Biology 171:163-177]. The results obtained allowed us to assume that these bi-tubular systems are endowed with dynamic properties that could be involved in the regulation of the axonemal machinery. For the first time, using the finite elements methods and the resistance of material principles, we have now calculated that the curvature of the axoneme induces the deviated-bending of the bi-tubular structures of the axoneme, because of their geometry only; they behave as beams in a framework. This approach is similar to the one used to measure the deflection of a single microtubule [Kasas et al., 2004: Chem Phys Chem 5:252-257]. These behaviors induce internal movement or constraints of either couples or triplets of doublets within the axonemal cylinder that could be directly involved in a constrained or a spontaneous "convergence/divergence" equilibrium of the cylindrical generatrices that they draw along the axonemal cylinder, which could apparently regulate the activity of the axonemal motors (the dynein arms). These results are discussed here, taking into consideration the dynamic propagation of the wave train along the flagellar axoneme, and the regulated balance between the activities of the two opposite sides of the axoneme during the beat. This study raises a few questions about the architecture-activity duo of the axonemal doublets.

Are the local adjustments of the relative spatial frequencies of the dynein arms and the β-tubulin monomers involved in the regulation of the “9+2” axoneme?

“…In fact, it was observed that the longitudinal constraints regulate the dynamic instability of the microtubules (Janson et al, 2003;Warner, 1976), and that the wall of the microtubule is elastic, because it accumulates energy (Hunyadi et al, 2005). The microtubules oscillate in different modes when they are subjected to an external constraint (Kasas et al, 2004a, b;Kis et al, 2002), and might be involved in a ''deviated bending'' (Cibert and Heck, 2004). The microtubules are clearly force-sensitive, being involved in tensegrity and in the mechanical transduction of a signal (Gundersen and Cook, 1999;Ingber, 1997Ingber, , 2003a, and it is impossible to exclude that they are allosteric effectors (Bray and Duke, 2004).…”

Flagellar and ciliary beating: the proven and the possible

“…Also, because the switching mechanism depends on the total stress acting across the axonemal scaffold, imposed external forces that stress the axoneme will also alter the switch-point. Cibert and Heck have suggested that, when the doublets are bent, they also experience a torsional force because they are not radially symmetric (Cibert and Heck, 2004). They suggested that the torsional force might play a role in breaking the dynein attachments and in terminating episodes of activity in a geometric-clutch-type stress-based regulatory scheme.…”

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

“…Because the orientations of the ODPs are constant during their rotation around the axonemal cylinder, the apparent lengths of the inner and of the outer dynein arms vary in such a way that those of the outer dynein arms are shortened as compared to those of the inner dynein arms (Cibert and Heck, 2004). This mechanism could be involved in the discrimination of the functions of the inner and outer rows of dynein arms (Brokaw, 1999).…”

“…The beams that represent the ODPs have a full discoid cross section whose radius equals 8 nm. Their Young modulus and Poisson's ratio are taken from (Schoutens, 1994;Tuszynski et al, 2005); their flexure inertias are given by Schoutens, 1992 andHeck, 2004. Young's modulus is that calculated by Schoutens, 1992. The shear coefficients (k y and k z ) are obtained by considering a hollow cylinder.…”

“…In this context, using a finite element approach (Cibert and Heck, 2004), we have studied the deformation of the proximal 5 m long segment of a "9+2" axoneme to analyze what might be the topological events occurring at this level, accounting for the synchronous sliding hypothesis (Brokaw, 1996;Goldstein, 1976).…”