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

Hermes Gadelha, Eamonn A. Gaffney, Alain Goriely

Research output: Contribution to journalArticlepeer-review

37 Scopus citations


Recent observations of flagellar counterbend in sea urchin sperm show that the mechanical induction of curvature in one part of a passive flagellum induces a compensatory countercurvature elsewhere. This apparent paradoxical effect cannot be explained using the standard elastic rod theory of Euler and Bernoulli, or even the more general Cosserat theory of rods. Here, we develop a geometrically exact mechanical model to describe the statics of microtubule bundles that is capable of predicting the curvature reversal events observed in eukaryotic flagella. This is achieved by allowing the interaction of deformations in different material directions, by accounting not only for structural bending, but also for the elastic forces originating from the internal cross-linking mechanics. Large-amplitude static configurations can be described analytically, and an excellent match between the model and the observed counterbend deformation was found. This allowed a simultaneous estimation of multiple sperm flagellum material parameters, namely the cross-linking sliding resistance, the bending stiffness, and the sperm head junction compliance ratio. We further show that small variations on the empirical conditions may induce discrepancies for the evaluation of the flagellar material quantities, so that caution is required when interpreting experiments. Finally, our analysis demonstrates that the counterbend emerges as a fundamental property of sliding resistance in cross-linked filamentous polymer bundles, which also suggests that cross-linking proteins may contribute to the regulation of the flagellar waveform in swimming sperm via counterbend mechanics.
Original languageEnglish (US)
Pages (from-to)12180-12185
Number of pages6
Issue number30
StatePublished - 2013
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2021-09-21
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: We thank C. B. Lindemann and K. A. Lesich for many helpful discussions and for providing details from sperm buckling experiments, including the micrographs analyzed in this work, and also gratefully acknowledge C. J. Brokaw for continued insight. H.G. thanks the CAPES Foundation (Grant BEX 4676/06-8); WYNG Foundation; and Trinity Hall, University of Cambridge. This publication is based on work supported, in part, by Award KUK-C1-013-04 from King Abdullah University of Science and Technology. A.G. is a Wolfson Royal Society Merit Holder and is supported by a Reintegration Grant under European Commission Framework VII.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

  • General


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