Geometry drives the ?deviated-bending? of the bi-tubular structures of the 9+2 axoneme in the flagellum

The mechanics of cilia and flagella: What we know and what we need to know

In this review, we provide a condensed overview of what is currently known about the mechanical functioning of the flagellar/ciliary axoneme. We also present a list of 10 specific areas where our current knowledge is incomplete and explain the benefits of further experimental investigation. Many of the physical parameters of the axoneme and its component parts have not been determined. This limits our ability to understand how the axoneme structure contributes to its functioning in several regards. It restricts our ability to understand how the mechanics of the structure contribute to the regulation of motor function. It also confines our ability to understand the three-dimensional workings of the axoneme and how various beating modes are accomplished. Lastly, it prevents accurate computational modeling of the axoneme in three-dimensions.

Beating to rotational transition of a clamped active ribbon-like filament

We present a detailed study of a clamped ribbon-like filament under a compressive active force using Brownian dynamics simulations. We show that a clamped ribbon-like filament is able to capture beating as well as rotational motion under the compressive force. The nature of oscillation is governed by the torsional rigidity of the filament. The frequency of oscillation is almost independent of the torsional rigidity. The beating of the filament gives a butterfly-shaped trajectory of the free-end monomer, whereas rotational motion yields a circular trajectory on a plane. The binormal correlation and the principal component analysis reveal the butterfly, elliptical, and circular trajectories of the free end monomer. We present a phase diagram for different kinds of motion in the parameter regime of compressive force and torsional rigidity.

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

The axonemal micro-machinery, the axial skeleton and actuator of cilia and flagella of eukaryotic cells, is able to bend and twist and generates wave trains. We already demonstrated that it can be the template to construct an active trunk robot (Cibert 2013 Bioinspir. Biomim. 8 026006). The work presented in this paper describes how the axonemal model can also be useful to design worm shaped devices to convert swell energy into usable energy such as electricity. Using dynamic simulation we have designed three models either very close to the biological machinery or being a reasonable interpretation of its functioning principles. Their main advantage as compared with another existing device, Pelamis, is the additional twist degree-of-freedom that gives interesting properties to the system.

Bending, twisting and beating trunk robot bioinspired from the '3 + 0' axoneme

The axoneme is the skeleton and motor axis of flagella and cilia in eukaryotic organisms. Basically it consists of a series of longitudinal fibers (outer doublets of microtubules) that design a cylinder and whose sliding, due to the coordinated activities of dedicated molecular motors (the dynein arms), is converted into a bending because outer doublets pairs are stabilized by elastic links (the nexine molecules). In spite of these interesting mechanical properties, mechanical and robotics engineers have never considered this amazing molecular machinery as a model. The aim of this paper is to propose the robotic design and the kinematic modeling of the '3 + 0' axoneme that makes motile the flagellum of Diplauxis hatti, the simplest that exists. The model that we propose bends and twists and combines the two movements. It is able to propagate wave trains that could be involved in the development of biomimetic actuators of various mechanisms such as (sub)aquatic robotic propellers as well as robotic trunks.

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

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.

Flagellar and ciliary beating: the proven and the possible

The working mechanism of the eukaryotic flagellar axoneme remains one of nature's most enduring puzzles. The basic mechanical operation of the axoneme is now a story that is fairly complete; however, the mechanism for coordinating the action of the dynein motor proteins to produce beating is still controversial. Although a full grasp of the dynein switching mechanism remains elusive, recent experimental reports provide new insights that might finally disclose the secrets of the beating mechanism: the special role of the inner dynein arms, especially dynein I1 and the dynein regulatory complex, the importance of the dynein microtubule-binding affinity at the stalk, and the role of bending in the selection of the active dynein group have all been implicated by major new evidence. This Commentary considers this new evidence in the context of various hypotheses of how axonemal dynein coordination might work.

Torsion of the central pair microtubules in eukaryotic flagella due to bending-driven lateral buckling

Inspired by recent interest in torsion of the central pair microtubules in eukaryotic flagella, a novel thin-walled elastic beam model is suggested to study critical condition under which uniform bending of a flagellum will cause lateral/torsional buckling of the central pair. The model is directed to the central pair itself and the role of all surrounding cross-linkings inside the flagellum is modeled as an equivalent surrounding elastic medium. The model predicts that bending-driven torsion of the central pair does occur when the radius of curvature of the bent flagellum reduces to a moderate critical value typically of tens of microns. In particular, this critical value is almost independent of the flagellum length, and more sensitive to the parameters defining the surrounding elastic medium than the shear modulus of microtubules. The predicted wavelengths of the torsional buckling mode are insensitive to the flagellum length and comparable to some known related experimental data. These results indicate that torsion of the central pair microtubules in flagella is inevitable as a result of bending-driven lateral buckling. This offers an entirely new insight into the ongoing research on the mechanism of the central pair torsion.