Self-foldability of Dürer net models of viral capsids: Exploring nanocapsule design via 4D printing

¹ Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
² Delaware Biotechnology Institute, University of Delaware, Newark, DE 19716, USA
³ Department of Biological Sciences, University of Delaware, Newark, DE 19716, USA
⁴ Department of Applied Engineering & Sciences, University of New Hampshire, Manchester, NH 03101, USA

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 15 May 2024
Accepted: 27 August 2025

Abstract

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.