Self-assembly is usually considered a parallel process while self-folding and origami are usually considered to be serial processes. We believe that these distinctions do not hold in actual experiments. Based upon our experience with 4D printing, we have developed three additional hybrid classes: (1) templated-assisted (tethered) self-assembly: e.g., when RNA is bound to viral capsomeres, the subunits are constricted in their interactions to have aspects of self-folding as well; (2) self-folding can depend upon interactions with the environment; for example, a protein synthesized on a ribosome will fold as soon as peptides enter the intracellular environment in a serial process whereas if denatured complete proteins are put into solution, parallel folding can occur simultaneously; and, (3) in turbulent environments, chaotic conditions continuously alternate processes. We have examined the 43,380 Dürer nets of dodecahedra and 43,380 Dürer nets of icosahedra and their corresponding duals: Schlegel diagrams. In order to better understand models of self-assembly of viral capsids, we have used both geometric (radius of gyration, convex hulls, angles) and topological (vertex connections, leaves, spanning trees, cutting trees, and degree distributions) perspectives to develop design principles for 4D printing experiments. Which configurations fold most rapidly? Which configurations lead to complete polyhedra most of the time? By using Hamiltonian circuits of the vertices of Dürer nets and Eulerian paths of cutting trees of polyhedra unto Schlegel diagrams, we have been able to develop a systematic sampling procedure to explore the 86,760 configurations, models of a T1 viral capsid with 60 subunits and to test alternatives with 4D printing experiments, use of Magforms^TM, and origami models to demonstrate via movies the five processes described above.

Many viral capsids are varieties of icosahedral-like polyhedra. The production of many mature viral capsids involves folding an RNA-tethered set of a string of identical proteinaceous cap-someres. The initial configuration of such tethered collections can be represented as DUrer 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 Durer nets each of a dodecahedron and an icosahedron. If various Durer 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. Unfortunately, while some configurations self-fold easily to completion in a short interval, 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. Here we report on using an origami simulator where we could investigate the folding of numerous Duürer nets. We found that two topological invariants: the diameter of spanning trees and number of vertex connections of a Durer net, had a significant impact on the time of folding to completion. Also, more symmetrical Durer nets fold faster and to completion than more irregular configurations. This research has relevance to biomimetic design particularly to employing nanocapsules made of viral capsids ("virosomes") as carriers of drugs in medical applications.