On the geometry of regular icosahedral capsids containing disymmetrons

TL;DR

This paper develops a geometric classification of icosahedral virus capsids incorporating disymmetrons, extending previous models that only included trisymmetrons and pentasymmetrons, and provides formulas and invariants for these structures.

Contribution

It introduces a comprehensive geometric classification of icosahedral capsids with disymmetrons, including size formulas and invariants, expanding prior models that excluded disymmetrons.

Findings

Classified all possible icosahedral structures with disymmetrons

Derived formulas for symmetron sizes and parity restrictions

Presented methods using invariants for configuration classification

Abstract

Icosahedral virus capsids are composed of symmetrons, organized arrangements of capsomers. There are three types of symmetrons: disymmetrons, trisymmetrons, and pentasymmetrons, which have different shapes and are centered on the icosahedral 2-fold, 3-fold and 5-fold axes of symmetry, respectively. In 2010 [Sinkovits & Baker] gave a classification of all possible ways of building an icosahedral structure solely from trisymmetrons and pentasymmetrons, which requires the triangulation number T to be odd. In the present paper we incorporate disymmetrons to obtain a geometric classification of icosahedral viruses formed by regular penta-, tri-, and disymmetrons. For every class of solutions, we further provide formulas for symmetron sizes and parity restrictions on h, k, and T numbers. We also present several methods in which invariants may be used to classify a given configuration.