The counterbend phenomenon in flagellar axonemes and cross-linked filament bundles
KAUST Grant NumberKUK-C1-013-04
Permanent link to this recordhttp://hdl.handle.net/10754/671349
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AbstractRecent 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.
CitationGadelha, H., Gaffney, E. A., & Goriely, A. (2013). The counterbend phenomenon in flagellar axonemes and cross-linked filament bundles. Proceedings of the National Academy of Sciences, 110(30), 12180–12185. doi:10.1073/pnas.1302113110
SponsorsWe 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.