Placement and Symmetry of Singularities on Curved Surfaces

In various contexts, biological structures resemble regular two-dimensional lattices made from smaller units like macromolecules (on the nanoscale) or cells (on the tissue scale). This regularity carries an inherent elastic energy penalty when the 2D manifold is also required to have intrinsic curvature: Virus capsids have to close, and arthropod eyes need to reconcile the periodicity of ommatidia with eye curvature, both of which are crucial for proper function. We derive a new theoretical criterion directly from the shape properties of the manifold that predicts whether in its ground state of mechanical energy a curved structure can be regular (defect-free) or whether defects are energetically favorable. In contrast to previous approaches, the criterion does not prescribe local or globally integrated curvature, but sets a universal value for a specific weighted integral Gaussian curvature. Verified against direct numerical computation, this formalism is simple, easy to employ, and accurate for a vast variety of curved surfaces. 


An extension of this approach furthermore elucidates the continuous or discontinuous character of this transition to disorder: The shape of the energy landscape determines if a transition preserves the symmetry of defect placement on surfaces of revolution because an energy barrier must be overcome, or breaks the symmetry due to stable intermediate positions. Our analytical theory predicts these transition characteristics a priori for general surface shapes, demonstrating a universal scale of energy barrier that is modified, and sometimes eliminated, by local shape properties. 

Our results give practical insight into the bounds of perfect order in the growth of lattices on manifolds in many biological systems that also inspire technological applications such as microlens arrays. They inform material design considerations through geometric control, while also improving our fundamental understanding.

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