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

TL;DR

This paper presents a geometric analysis of how stiff filaments pack inside non-isotropic containers, revealing orientation preferences and phase transitions influenced by cell shape and curvature, with implications for biological and industrial packing problems.

Contribution

It introduces a novel geometric mechanism explaining filament orientation in non-isotropic shapes, including phase transitions, with analytical insights and potential applications.

Findings

Filaments prefer equatorial orientation in oblate cells.

In near-spherical cells, filaments align along the long axis.

High aspect ratio causes a phase transition to angled filament orientation.

Abstract

We suggest a geometrical mechanism for the ordering of slender filaments inside non-isotropic containers, using cortical microtubules in plant cells and packing of viral genetic material inside capsids as concrete examples. We show analytically how the shape of the cell affects the ordering of phantom, non-self-avoiding, stiff rods. We find that for oblate cells the preferred orientation is along the equator, while for prolate spheroids with an aspect ratio close to one, 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 energy considerations. These results are relevant to other packing problems as well, such as spooling of filament in the industry or spider silk inside water droplets.