  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 crosslinkers 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 spatiotemporal 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.
