Chiral quasicrystalline order and dodecahedral geometry of small virus capsids

We propose an unconventional theory which unifies the description of quasicrystal thermodynamics and quasicrystal structure formation by combining the Landau theory of crystallization and the cluster approach to quasicrystals. The theory is illustrated on the example of pentagonal Penrose quasilattice. It employs the notion of non-linear order parameter dependent on the atomic coordinates which was developed in the theory of reconstructive phase transitions. The coordinates of the quasilattice nodes are calculated by minimizing the Landau free energy with the constraint imposed by internal organization of clusters. The correspondence is shown between the theory proposed and the conventional projection method.

These points are discussed on examples of viruses of different types, chromatin fiber and its ingredients, cell patterns induced by topological constraints in oocytes (epithelium layers on a sphere), instabilities in lipid tubules and interaction of proteins with lipid tubules in the vicinity of instabilities.

Understanding of virus capsid organization and self-assembly mechanisms helps not only to get an insight into the protein interactions which render virus infectious, but also to advance new methods in nanotechnology which use capsid self-assembly to produce virus-like nanoparticles. As in abiotic nanostructures, the obstacles along this way are related not only to the nanoscopic size of capsids but also to their unconventional topology and symmetry. We have shown that proteins in the bovine papilloma virus (and other viruses of the exceptional papovavirus families) self-assemble into unprecedented chiral QC-like structures, with chiral pentagonal order in the faces and global dodecahedron geometry of the capsid. The corresponding dodecahedron net is commensurate with the chiral pentagonal quasilattice, with the asymmetric proteins put at the nodes of the quasilattice. We developed the nonlinear phason strain concept in the frame of the classical elasticity theory of QCs. The resulting approach allowed us to calculate the protein positions and thus to explain the protein organization in papovaviruses, in spite of its extreme complexity.

Chiral quasicrystalline order in exceptional family of viruses

Chromatin is a key assembly in eucariotic cell nuclei. Despite the efforts to elucidate chromatin fiber structure, the detail of its organization remains still unknown. Historically, there are two principal classes of competing models of the 30-nm fiber: the solenoid models and the two-start helix models, though a whole series of indications of complex multi-start helix organization appeared quite recently. At the present it is impossible to resolve experimentally which model describes better the fiber structure under physiological conditions. However, certain observable features in the fiber organization are generally accepted, namely: (1) the nucleosomes are peripherally situated in the fiber and at low ionic strength form a zig-zag chain, (2) DNA linker is situated in the interior of the fiber, and (3) nucleosome-nucleosome interactions play an active role in condensation-decondensation processes. Theoretical analysis helps to elucidate which geometries are most plausibles for the fiber structure. One of the new ways to approach the problem is to consider the fiber as an ordered structure resulting from the balance of thermal disorder and electrostatic (and mechanical) interactions of its constituants. To achive this, it is necessary to identify robust parameters of the DNA-protein assembly at the smaller scale and to show how they control the fiber structure ordering at the bigger scale. Here, we propose a model based on the interactions between nucleosomes. The type of interaction proposed has allowed us previously to describe with success the condensed phases in acqueous solutions of nucleosomes with digested linker DNA, both in physiological conditions and in a wide range of monovalent salt concentration [1]. Taking into account the polar and chiral nature of nucleosomes, we perform detailed group theory analysis to construct the free energy of the native fiber and to reveal the thermodynamically favorable helical fiber structures. The effects of homogeneous mechanical torsion applied over the fiber (in biochemical experiments, rather than in single molecule ones) are also studied. We show that even in the case of very low load the helix unwinding is a multistep process and we give its geometrical and thermodynamical details. Such a scenario, known in biology since many years, had no simple explanation in previous theoretical works. 1. Lorman V., Podgornik R., Žekš B., Europhys. Lett., 69, 1017 (2005)

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)