Un modello di tipo Ginzburg-Landau per l'espansione di un capside virale icosaedrale

In this work we formulate a model for the expansion of an icosahedral viral capsid, in the framework of the Ginzburg-Landau theory of phase transitions. Most viral capsids have icosahedral symmetry. It has been observed that, before the infection process, many capsids undergo a swelling phenomenon, which creates slots through which the genetic material (RNA and DNA) comes out. This process corresponds in some respect to phase transitions in solids. Following Ginzburg and Landau, we associate to the system a free-energy which retains symmetry properties whose minima correspond to the stable phases of the system. Thanks to the property of duality between the icosahedron and the dodecahedron, we restrict to the simplest form of capsid, which is assumed to be a regular dodecahedron. We then associate to each face (pentamer) a parameter of translation \lambda_i which controls the expansion. In this way the symmetry group of the system is the icosahedral group I, which acts on the 12 faces of the dodecahedron permuting them. This action provides a representation \rho of I. Thus, we require the energy E to be a smooth function in the 12 variables \lambda_i which is invariant under the action of I, i.e. E(\rho(g)x) = E(x) for all g in I, with x = (\lambda_1, ¿,\lambda_12). Moreover, we look for polynomial forms of E. We introduce two explicit forms of the energy, using two algorithms of computational Invariant Theory. Both forms of the energy contain scalar quantities which play the role of bifurcation parameters. In this way, we can study the bifurcation patterns associated to the system. When (0,¿0) is a minimum of E, then the capsid has not yet expanded; when it loses its stability and new minima appear the capsid expands and a phase transition occurs.