Résumé
A virus entry into a host cell depends strongly on protein arrangement in a capsid, a solid shell composed of identical proteins, which protects the viral genome from external aggression. In spite of certain similarity in organization between capsid structures and classical crystals, the generalization of solid state physics concepts taking into account specific properties of proteins has started only quite recently [1]. Almost all works devoted to physics of viruses with spherical topology postulate as an initial paradigm the existence of local hexagonal order of protein positions. Among the direct consequences of the local hexagonal order hypothesis there are: i) specific arrangements of proteins and their environments in a capsid; ii) location of topological defects of the hexagonal order in the vicinity of fivefold axes of the global icosahedral symmetry. The latter property is the basis of the theory explaining faceting of large viruses by a buckling transition associated with the 12 defects of fivefold symmetry [2]. However, there exist exceptional families of viruses which show local pentagonal order of capsid proteins and not hexagonal one. For these families, the principles of organization, defect formation and mechanical properties (including faceting) remain uncovered. In the present work we give a clear-cut explanation of the unusual pentagonal protein arrangement in viral capsids from the viewpoint of solid state physics, namely theory of quasicrystals (QC). We show that the systems considered represent the first example of matter organization in 2D nanoparticles, in which the regions with a chiral pentagonal quasicrystalline order of protein positions are arranged in a structure commensurate with the spherical topology and dodecahedral geometry. We establish the relations between the elastic properties of these unconventional QC regions and the unusual structure and geometry of the resulting viral capsid. In the frame of classical elasticity theory of QC we develop the concept of non-linear phason strain. Resulting approach allows us to calculate the protein positions and thus to explain the protein organization in exceptional viruses, in spite of its extreme complexity, and to relate local chiral QC order and non-zero curvature of the dodecahedral capsid faces. 1. V.L. Lorman and S.B. Rochal, Phys. Rev. Lett. 98, 185502 (2007); J. Thomas, Nature Nanotech. 2, 388 (2007); O.V. Konevtsova, S.B. Rochal, and V.L. Lorman, Phys. Rev. Lett. 108, 038102 (2012) 2. J. Lidmar, L. Mirny and D.R. Nelson, Phys. Rev. E 68, 0519010 (2003)