Landau theory of crystallization and the capsid structures of small icosahedral viruses

Abstract

International audienceA new approach to the capsid structures of small viruses with spherical topology and icosahedral symmetry is proposed. It generalizes Landau theory of crystallization to describe icosahedral viral shells self-assembled from identical asymmetric proteins. An explicit method which predicts the positions of centers of mass for the proteins constituting the shell is discussed in detail. The method is based on irreducible density distribution function which generates the protein positions. The universal form of the density distribution function which contains no fitting parameter permits to classify the capsids structures of small viruses. The theory describes in a uniform way both the structures satisfying the well-known Caspar and Klug geometrical model for capsid construction and those violating it. A group theory analysis of the Caspar and Klug model and of the “quasiequivalence” principle for protein environments in viral capsids is given. The molecular basis of difference in protein environments and peculiarities in the assembly thermodynamics are also discussed

Orientational phase transitions and the assembly of viral capsids

“…As noted by in ref. [4], the density of the Dengue virus capsid matches nicely to Y h (25) (see Fig. 19b, density maxima correspond to the ends of the dimers).…”

“…A systematic comparison between viral capsids and the odd-l icosahedral spherical harmonics can be found in refs. [4,5]. Larger viruses are associated with Y h (l) with larger odd l. For example, the intensely-studied CCMV virus, discussed in Section V, is associated with Y h (27).…”

“…The literature on orientational phase transitions has been mostly restricted to l values less than or equal to l = 6, but recently an application of orientational transitions involving larger values of l has emerged in the area of viral assembly [4,5]. Simple viruses are composed of a protein shell -the capsid -that surrounds the viral RNA or DNA genome molecule(s).…”

“…However, Y h (6) is even under inversion, as are all even icosahedral spherical harmonics. For this reason, all even l icosahedral spherical harmonics were excluded from the current Landau theory of viral assembly [4,5]. The absence of inversion symmetry is also the reason that the symmetry group G 0 of the disordered phase is SO(3) rather than O(3).…”

A minimal representation of the self-assembly of virus capsids

“…The simple geometrical construction model introduced by Caspar and Klug (CK) 2 to explain the architecture of icosahedral viruses is a milestone in modern virology. Generalizations of the CK rules properly account for the geometry of some exceptional icosahedral capsids [3][4][5][6] and other elongated virus capsids that share coordination numbers with the icosahedral ones [7][8][9] . Recent work [10][11][12] provides important further development about the effect of the geometrical and topological constraints on the capsid structure.…”