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R Soc Open Sci. 2017 Mar 8;4(3):160698. doi: 10.1098/rsos.160698. eCollection 2017 Mar.

Nonlinear amplitude dynamics in flagellar beating.

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Departament de Física de la Matèria Condensada, Facultat de Física , Universitat de Barcelona , Avinguda Diagonal 647, 08028 Barcelona, Spain.
Department of Mathematics, University of York, York YO10 5DD, UK; Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK.


The physical basis of flagellar and ciliary beating is a major problem in biology which is still far from completely understood. The fundamental cytoskeleton structure of cilia and flagella is the axoneme, a cylindrical array of microtubule doublets connected by passive cross-linkers and dynein motor proteins. The complex interplay of these elements leads to the generation of self-organized bending waves. Although many mathematical models have been proposed to understand this process, few attempts have been made to assess the role of dyneins on the nonlinear nature of the axoneme. Here, we investigate the nonlinear dynamics of flagella by considering an axonemal sliding control mechanism for dynein activity. This approach unveils the nonlinear selection of the oscillation amplitudes, which are typically either missed or prescribed in mathematical models. The explicit set of nonlinear equations are derived and solved numerically. Our analysis reveals the spatio-temporal dynamics of dynein populations and flagellum shape for different regimes of motor activity, medium viscosity and flagellum elasticity. Unstable modes saturate via the coupling of dynein kinetics and flagellum shape without the need of invoking a nonlinear axonemal response. Hence, our work reveals a novel mechanism for the saturation of unstable modes in axonemal beating.


dynein; flagellar beating; self-organization; spermatozoa

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