Landau theory and the emergence of chirality in viral capsids

Abstract

We present a generalized Landau-Brazovskii free energy for the solidification of chiral molecules on a spherical surface in the context of the assembly of viral shells. We encounter two types of solidification transitions. The first type is a conventional first-order phase transition from a uniform to an icosahedral state, described by a single icosahedral spherical harmonic of even l. The chiral pseudo-scalar term in the free energy does not affect the transition. The second type is anomalous: icosahedral spherical harmonics with odd l are unstable. Stability is recovered when admixture with the neighboring l + 1 icosahedral spherical harmonic is included. This is in apparent conflict with the principle of Landau theory that symmetry-breaking transitions are characterized by a single irreducible representation of the symmetry group of the uniform phase. The chiral term selects one of two isomeric mixed-l icosahedral states. A direct transition is possible only over a limited range of parameters. Outside this range, a non-icosahedral state with the symmetry of an isotropy subgroup of the icosahedral group interposes between the uniform and icosahedral states.This paper is dedicated to the memory of our friends and colleagues William Klug and Vladimir Lorman.