Origami of Viral Capsids

• Caitlyn Zeller, Environmental Engineering, University of Delaware

• John Jungck, Department of Mathematics and Department of Biology, University of Delaware

How can a fun craft learned as a kid model viral capsid behavior? Origami is the art of paper folding, behaving similarly to the self-folding of biological molecules. Dodecahedrons contain 12 pentagons and icosahedrons contain 20 triangles. A Dürer net is a 2D configuration of a 3D polyhedra. Self-assembly is an object formed simultaneously while self-folding is an object formed one step at a time. Jungck’s ongoing research has been exploring the geometric and topological perspectives of over 86,000 dodecahedron and icosahedron Dürer nets (2023). However, observing the self-assembly of every net through 3D printing would be time consuming and economically challenging. Furthermore, Kaplan investigated their “building-game” model creation to analyze self-folding without using physical models (2014). Utilizing Ghassaei’s online origami simulator in our research to analyze the self-folding of various nets decreases the economic and time demand posed from physical modeling (2018). Gathering various Dürer nets of different vertex connections from an online database supported the use of the online origami simulator to determine the length of assembly and if intersection occurs, focusing on why certain net configurations intersect by edge, by face, or not at all. What are the similarities between nets that intersect the same way? Are the similarities consistent among all dodecahedrons and icosahedrons? Vertex connections for dodecahedrons are at 36-degree angles and icosahedrons at 60-degree angles in the Dürer net. The icosahedron and dodecahedron preliminary results conclude that the greater the vertex connections, the faster the assembly with lower intersection rates and vice versa for fewer vertex connections. Modeling viral capsid self-folding and self-assembly supports further research into self-assembling biological molecules’ behavioral properties. Applications of this research include drug delivery and combating viruses containing icosahedral viral capsid structures without using traditional antiviral medication.