Modeling Quasicrystal Viral Capsids: Self-Assembly of Hats and Turtles

• Papa Boateng, Biochemistry, Colby College

• John Jungck, Biological Sciences and Mathematical Sciences, University of Delaware

Our research explores how unique geometric shapes called ‘hats’ and ‘turtles’ can help model the structures of certain viruses with complex patterns. These viruses have quasicrystalline patterns, meaning their structure is ordered but not repetitive, unlike traditional viral capsids that are usually symmetrical. Quasicrystalline capsids are viral shells that follow these unique, non-repetitive, aperiodic patterns, making them challenging to model with regular patterns of interactions between individual tiles. ‘Hats’ and ‘turtles’ are examples of aperiodic or Einstein tiles, shapes that tile a plane in a non-repeating pattern. Using Caspar-Klug theory, which explains how proteins naturally form viral capsids, and Twarock-Konevtsova models, which apply these ideas to quasicrystals, we study how these shapes come together on their own—a process known as self-assembly. By focusing on the self-assembly and spherical tiling of ‘hats’ and ‘turtles,’ we aim to create tiny capsules, or nanocapsules, that mimic the geometry of quasicrystalline viral capsids. We use 3D design and laser printing to create planar and spherical tiling models of these shapes and test their assembly. Our analysis includes studying the interactions of vertices and edges in these tilings, using graph theory to understand which vertices and edges of the tiles interact in different arrangements. We observed four different types of vertex-vertex connections occurring with much higher frequency than the other 101 possible interactions. These nanocapsules can carry drugs to specific parts of the body, like tumors. Once they reach their target, the nanocapsules can be heated to release the drugs locally, making delivery of more precise treatments more effective and reducing side effects. This research bridges complex viral geometry with practical medical uses, offering new ways to design nanocapsules for efficient and precise drug delivery. By combining quasicrystal self-assembly principles with biomedical engineering, we aim to improve drug delivery systems and enhance patient outcomes.