6-D Lattice Phase Transformations in Viral Maturation

Abstract

In this paper we will show how to use the fundamentals of the math of quasi-crystallography to model the maturation of viral capsids. Viruses have had a deep influence on the evolution of life, with traces of this influence still present in our genome to this day. The study of viruses has shown promising results in furthering our understanding of mechanisms such as viral assembly and maturation. One of the most important discoveries made in the field was that viruses posses icosahedral symmetry and can be classified using Triangulation number. It was found that the location of surface protrusions on viral capsids relies on the inherent symmetry these capsids possess. We set out to use this icosahedral symmetry in the form of affine extended point arrays (AEIS) to model the viral capsid at stages of it maturation. We go about doing this by embedding the arrays into a Bravais lattice and using Bain transformations to mimic lattice phase transformations. In this paper we layout the fundamentals of generating icosahedral symmetric point sets, and the base polyhedra, along with a procedure to embed the arrays into the appropriately classified lattice structures. We also discuss how our algorithm determines which point array fit a particular viral capsid. Lastly we apply our procedure to the HK97 viral capsid at its prohead II and head II states and draw conclusions from our results. Below is a conceptual image of how our embedding and classifying works with the viral capsid, and the role of the Bain transformation.