A numerical approach to the geometry of ciliary dynamics



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The motion of cilia and flagella under internal stresses has been widely studied over the past years as a way of relating the biology of these structures and the functions they perform. While the mechanism which leads to the ciliary motion is fairly well understood, its method of activation and coordination is still essentially unknown. In this paper, we present a geometric approach to modeling ciliary dynamics which suggests that the motion of a free flowing cilium can be completely described by its shape. We then introduce the numerical method of least squares piecewise polynomial projections as a potential tool for solving the systems of partial differential equations in the ciliary dynamics models. It is concluded that although stability issues arise, with some adjustments, the method could potentially be employed with success: it is possible to create computer simulations of the motion predicted by this geometric model and the results are expected to provide a good match for the biological data.



Mathematics, Biomathematics, Geometry