Eukaryotic cilia and flagella are some of the best conserved biological structures across species. Giventhe pressure natural selection imposes onto living beings to optimize their body plans, eukaryotic flagella have demonstrated to be an optimal organelle for propulsion at low Reynolds numbers. Therefore, understanding how cilia beat is of central importance to understand any biological process involving the motion of a fluid. The well conserved internal structure of cilia, the axoneme, has been thoroughly reported but there is still no scientific agreement on how oscillations can emerge from the combina-
tion of the myriad different components forming the axoneme. Current theories base their approaches on complex control mechanism to control the activity of the axonemal molecular motors, the dyneins. In the present thesis we propose a much simpler mechanism, demonstrating that no dyneins activity regulation mechanism is required since oscillations naturally emerge from the axoneme undergoing a dynamic buckling instability triggered by a system of follower forces exerted by the molecular motors.

Here, we first derive a numerical spectral methods solver to resolve the dynamics of a slender filament immersed in a low Reynolds fluid. We then use the solver to explain the emergence of oscillations in simplified synthetic versions of cilia. The structural simplicity of these experiments helps to identify the presence of follower forces and to model the structure, observing that our predictions based on the mechanical instability are in agreement with the experimental results. Given that a follower force can explain the dynamics of synthetic cilia, we then apply the model to study possible configurations of forces in the axoneme which can trigger the instability even if it is an isolated system, finding that the axoneme can indeed undergo a dynamic buckling instability leading to self-sustained oscillations reminiscent of cilia.

We finally build a minimal version of the model to capture the dynamics of cilia, simplifying the axoneme as a filament subject to a tip concentrated, tangential and compressive follower force. We use it to study the synchronization of cilia with external hydrodynamic and mechanical forcings and with another cilium, being able to capture non-linear features of flagellar dynamics which has been experimentally observed but previous simpler flagellar models fail to capture.