Researcher(s)
- Solana Tabor, Statistics, Universtiy of Loyola Chicago
Faculty Mentor(s)
- John Jungck, Delaware Biotechnology Institute, University of Delaware
- Purba Biswas, Delaware Biotechnology Institute, University of Delaware
Abstract
Mathematical research on viral architecture has aided in the medical advancements of antidotes as well as the knowledge of virus proteins. An aperiodic tessellation of a three dimensional structure is referred to as a quasicrystal. Some viral capsids are arranged in a quasicrystal array. After 50 years of searching by the mathematical community, David Smith (2022) finally discovered a unitile that could aperiodically tessellate the plane. As virus proteins self-assemble in nature, we aim to discover whether einstein tiles replicate this spherical self-assembly with 3D printed tiles and magnets. While our lab has successfully self-assembled Caspar-Klug models of protein viral capsids, so far no self-assembly of quasicrystal arrays of viral capsids has been reported. The reason that we aim to use einstein tiles is to help create an even more accurate representation of virus architecture using their relations to the shape of quasicrystals. In order to achieve our goal we have collected data from experimenting with 2D einstein tiles. This data shows us where certain vertices and edges connect to each other within most einstein tile tessellations, allowing for insight on spherical tessellations in order to model self-assembly of a polyhedron. We analyzed the tessellations of einstein tiles with several graph theoretic measures. For example, we identified which vertices and edges of each tile interacted with one another in their respective arrangements. We classified the interactions of vertices as X, Y, and T configurations. We constructed nearest neighboring networks of the corona of tiles around an individual einstein tile. The significance of our research is to aid in the research of quasicrystals and medical nanocapsules involved in drug delivery.