The counterbend phenomenon in flagellar axonemes and cross-linked filament bundles

Gadêlha, Hermes; Gaffney, Eamonn A.; Goriely, Alain

doi:10.1073/pnas.1302113110

Cited by 42 publications

(83 citation statements)

References 25 publications

(97 reference statements)

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“…The development of models of dynein regulation and flagellar motion remains an active topic of research. New details of the mechanics of flagella are still being uncovered [35] as are the mechanisms of synchronization between flagella [36,37]. The stability properties and propagation directions of all modes should be considered in evaluating proposed mathematical models and their underlying hypotheses.…”

Section: Discussionmentioning

confidence: 99%

Analysis of unstable modes distinguishes mathematical models of flagellar motion

Bayly

Wilson

2015

J. R. Soc. Interface.

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The mechanisms underlying the coordinated beating of cilia and flagella remain incompletely understood despite the fundamental importance of these organelles. The axoneme (the cytoskeletal structure of cilia and flagella) consists of microtubule doublets connected by passive and active elements. The motor protein dynein is known to drive active bending, but dynein activity must be regulated to generate oscillatory, propulsive waveforms. Mathematical models of flagellar motion generate quantitative predictions that can be analysed to test hypotheses concerning dynein regulation. One approach has been to seek periodic solutions to the linearized equations of motion. However, models may simultaneously exhibit both periodic and unstable modes. Here, we investigate the emergence and coexistence of unstable and periodic modes in three mathematical models of flagellar motion, each based on a different dynein regulation hypothesis: (i) sliding control; (ii) curvature control and (iii) control by interdoublet separation (the 'geometric clutch' (GC)). The unstable modes predicted by each model are used to critically evaluate the underlying hypothesis. In particular, models of flagella with 'sliding-controlled' dynein activity admit unstable modes with non-propulsive, retrograde (tip-to-base) propagation, sometimes at the same parameter values that lead to periodic, propulsive modes. In the presence of these retrograde unstable modes, stable or periodic modes have little influence. In contrast, unstable modes of the GC model exhibit switching at the base and propulsive base-to-tip propagation.

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Section: Discussionmentioning

confidence: 99%

Analysis of unstable modes distinguishes mathematical models of flagellar motion

Bayly

Wilson

2015

J. R. Soc. Interface.

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“…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).…”

Section: Discussionmentioning

confidence: 99%

“…The elastic resistance to interdoublet sliding, as well as the flexural rigidity, is needed to characterize its bending mechanics. The interdoublet shear stiffness of the axoneme is manifested through a characteristic phenomenon: the distal counterbend response to proximal bending (12)(13)(14). During large bending deformation, induced by a glass needle in the proximal portion of a sea urchin sperm flagellum, the distal portion of the flagellum exhibited a bend with the opposite curvature; this is referred to as the ''counterbend'' (13).…”

Section: Introductionmentioning

confidence: 99%

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

Wilson

Okamoto

et al. 2016

Biophysical Journal

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Motile cilia and flagella are whiplike cellular organelles that bend actively to propel cells or move fluid in passages such as airways, brain ventricles, and the oviduct. Efficient motile function of cilia and flagella depends on coordinated interactions between active forces from an array of motor proteins and passive mechanical resistance from the complex cytoskeletal structure (the axoneme). However, details of this coordination, including axonemal mechanics, remain unclear. We investigated two major mechanical parameters, flexural rigidity and interdoublet shear stiffness, of the flagellar axoneme in the unicellular alga Chlamydomonas reinhardtii. Combining experiment, theory, and finite element models, we demonstrate that the apparent flexural rigidity of the axoneme depends on both the intrinsic flexural rigidity (EI) and the elastic resistance to interdoublet sliding (shear stiffness, ks). We estimated the average intrinsic flexural rigidity and interdoublet shear stiffness of wild-type Chlamydomonas flagella in vivo, rendered immotile by vanadate, to be EI = 840 ± 280 pN⋅μm(2) and ks = 79.6 ± 10.5 pN/rad, respectively. The corresponding values for the pf3; cnk11-6 double mutant, which lacks the nexin-dynein regulatory complex (N-DRC), were EI = 1011 ± 183 pN·μm(2) and ks = 39.3 ± 6.0 pN/rad under the same conditions. Finally, in the pf13A mutant, which lacks outer dynein arms and inner dynein arm c, the estimates were EI = 777 ± 184 pN·μm(2) and ks = 43.3 ± 7.7 pN/rad. In the two mutant strains, the flexural rigidity is not significantly different from wild-type (p > 0.05), but the lack of N-DRC (in pf3; cnk11-6) or dynein arms (in pf13A) significantly reduces interdoublet shear stiffness. These differences may represent the contributions of the N-DRCs (∼40 pN/rad) and residual dynein interactions (∼35 pN/rad) to interdoublet sliding resistance in these immobilized Chlamydomonas flagella.

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“…3D). Therefore, a more plausible explanation is dynamic buckling, which is common in thin structures such as flagellar axonemes and cross-linked filament bundles (59). For sea urchin sperm, a buckling instability was suggested as an explanation for asymmetric compressed beat patterns at high viscosities (60).…”

Section: Discussionmentioning

confidence: 99%

Bimodal rheotactic behavior reflects flagellar beat asymmetry in human sperm cells

Bukatin

Kukhtevich

Stoop

et al. 2015

Proc. Natl. Acad. Sci. U.S.A.

125

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Rheotaxis, the directed response to fluid velocity gradients, has been shown to facilitate stable upstream swimming of mammalian sperm cells along solid surfaces, suggesting a robust physical mechanism for long-distance navigation during fertilization. However, the dynamics by which a human sperm orients itself relative to an ambient flow is poorly understood. Here, we combine microfluidic experiments with mathematical modeling and 3D flagellar beat reconstruction to quantify the response of individual sperm cells in time-varying flow fields. Single-cell tracking reveals two kinematically distinct swimming states that entail opposite turning behaviors under flow reversal. We constrain an effective 2D model for the turning dynamics through systematic large-scale parameter scans, and find good quantitative agreement with experiments at different shear rates and viscosities. Using a 3D reconstruction algorithm to identify the flagellar beat patterns causing left or right turning, we present comprehensive 3D data demonstrating the rolling dynamics of freely swimming sperm cells around their longitudinal axis. Contrary to current beliefs, this 3D analysis uncovers ambidextrous flagellar waveforms and shows that the cell's turning direction is not defined by the rolling direction. Instead, the different rheotactic turning behaviors are linked to a broken mirror symmetry in the midpiece section, likely arising from a buckling instability. These results challenge current theoretical models of sperm locomotion.sperm swimming | rheotaxis | fluid dynamics | microfluidics | simulations T axis, the directed kinematic response to external signals, is a defining feature of living things that affects their reproduction, foraging, migration, and survival strategies (1-4). Higher organisms rely on sophisticated networks of finely tuned sensory mechanisms to move efficiently in the presence of chemical or physical stimuli. However, various fundamental forms of taxis are already manifest at the unicellular level, ranging from chemotaxis in bacteria (5) and phototaxis in unicellular green algae (2) to the mechanical response (durotaxis) of fibroblasts (6) and rheotaxis (7, 8) in spermatozoa (3, 9-12). Over the last few decades, much progress has been made in deciphering chemotactic, phototactic, and durotactic pathways in prokaryotic and eukaryotic model systems. In contrast, comparatively little is known about the physical mechanisms that enable flow gradient sensing in sperm cells (3, 9-13). Recent studies (3, 12) suggest that mammalian sperm use rheotaxis for long-distance navigation, but it remains unclear how shear flows alter flagellar beat patterns in the vicinity of surfaces and, in particular, how such changes in the beat dynamics affect the steering process. Answering these questions will be essential for evaluating the importance of chemical (14) and physical (4) signals during mammalian fertilization (15-17).A necessary requirement for any form of directed kinematic response is the ability to change the direction of loco...

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The counterbend phenomenon in flagellar axonemes and cross-linked filament bundles

Cited by 42 publications

References 25 publications

Analysis of unstable modes distinguishes mathematical models of flagellar motion

Analysis of unstable modes distinguishes mathematical models of flagellar motion

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

Bimodal rheotactic behavior reflects flagellar beat asymmetry in human sperm cells

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