On the Packing of Stiff Rods on Ellipsoids Part I -- Geometry

We suggest a geometrical mechanism for the ordering of cortical micrtobules along cell wall. We show analytically how the shape of the cell affects the ordering of non - self - avoiding stiff rods. We find that for oblate cells the preferred orientation is azimuthal, along the equator, while for prolate spheroids with aspect ratio close to 1, the orientation is along the principal (long axis). Surprisingly, at high enough aspect ratio, a configurational phase transition occurs, and the rods no longer point along the principal axis, but at an angle to it, due to high curvature at the poles. We discuss some of the possible effects of self avoidance, using general energy considerations. This model is also applicable to other packing problems of stiff rods in spheroid cells.