Buckling transition in icosahedral shells subjected to volume conservation constraint and pressure: relations to virus maturation.

Minimal energy shapes of closed, elastic shells with 12 pentagonal disclinations introduced in otherwise hexagonally coordinated crystalline lattice are studied. The geometry and the total energy of shells are studied as a function of the elastic properties of the material they are made of. Particular emphasis is put on the buckling transition of the shells, that is, a strong preference of the shell shapes to "buckle out" in spatial regions close to the pentagonal disclinations for a certain range of the elastic parameters of the problem. The transition effectively increases the mean square aspherity of shapes, making them look more like an icosahedron rather than a sphere, which is a preferred shape prior to the onset of the transition. The properties of the buckling transition are studied in cases when (i) the total volume enclosed by the elastic shell has to be fixed and when (ii) there is an internal pressure acting on the shell. This may be related to the maturation process in nonenveloped dsDNA viruses, where the insertion of the genetic material in a preformed protein shell (viral coating) may effectively impose the fixed volume and/or pressure constraint. Several scenarios that may explain the experimentally observed feature of mature viruses being more aspherical (facetted) from their immature precursors are discussed, and predictions for the elastic properties of viral coatings are obtained on the basis of the presented studies.