Self-foldability of Dürer Net Models of Viral Capsids: Exploring Nanocapsule Design via 4D Printing

Many viral capsids are icosahedral-like polyhedra whose folding involves a RNA-tethered string of identical proteinaceous capsomeres. The initial configuration of such tethered ccapsomeres can be represented as Dürer nets which are planar graphs of cuts of polyhedra. While there are only 2 Dürer nets of a tetrahedron and 11 Dürer nets each of a cube and an octahedron, there are 43,380 Dürer nets each of a dodecahedron and an icosahedron. If Dürer nets are three-dimensionally printed in two layers in plastic sheets with magnets on each edge and then are placed in warm water, some configurations self-fold into completed polyhedra. While some configurations self-fold easily to completion, others self-intersect and are unable to close into a complete polyhedron. These four-dimensional printing experiments have previously only allowed us to explore a few configurations. By using an origami simulator where we could investigate the folding of numerous Dürer nets, two topological invariants: the diameter of spanning trees and number of vertex connections of a Dürer net, had a significant impact on the time of folding to completion. This research has relevance to biomimetic design particularly to employing nanocapsules made of viral capsids (“virosomes”) as carriers of drugs in medical applications.