Generic aspects of axonemal beating

We study the dynamics of an elastic rod-like filament in two dimensions, driven by internally generated forces. This situation is motivated by cilia and flagella which contain an axoneme. These hair-like appendages of many cells are used for swimming and to stir surrounding fluids. Our approach characterizes the general physical mechanisms that govern the behaviour of axonemes and the properties of the bending waves generated by these structures. Starting from the dynamic equations of a filament pair in the presence of internal forces we use a perturbative approach to systematically calculate filament shapes and the tension profile. We show that periodic filament motion can be generated by a self-organization of elastic filaments and internal active elements, such as molecular motors, via a dynamic instability termed Hopf bifurcation. Close to this instability, the behaviour of the system is shown to be independent of many microscopic details of the active system and only depends on phenomenological parameters such as the bending rigidity, the external viscosity and the filament length. Using a two-state model for molecular motors as an active system, we calculate the selected oscillation frequency at the bifurcation point and show that a large frequency range is accessible by varying the axonemal length between 1 and 50 µm. We discuss the effects of the boundary conditions and externally applied forces on the axonemal wave forms and calculate the swimming velocity for the case of free boundary conditions.