Generic aspects of axonemal beating

We study the dynamics of an elastic rod-like filament in two dimensions, dr iven by internally generated forces. This situation is motivated by cilia a nd flagella which contain an axoneme. These hair-like appendages of many ce lls are used for swimming and to stir surrounding fluids. Our approach char acterizes the general physical mechanisms that govern the behaviour of axon emes 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 mot ion can be generated by a self-organization of elastic filaments and intern al active elements, such as molecular motors, via a dynamic instability ter med Hopf bifurcation. Close to this instability, the behaviour of the syste m is shown to be independent of many microscopic details of the active syst em and only depends on phenomenological parameters such as the bending rigi dity, the external viscosity and the filament length. Using a two-state mod el for molecular motors as an active system, we calculate the selected osci llation frequency at the bifurcation point and show that a large frequency range is accessible by varying the axonemal length between 1 and 50 mum. We discuss the effects of the boundary conditions and externally applied forc es on the axonemal wave forms and calculate the swimming velocity for the c ase of free boundary conditions.