pH-induced morphological changes of proteinaceous viral shells.

Changes in environmental pH can induce morphological changes in empty proteinaceous shells of bacteriophages in vitro that are very similar to changes occurring in viral capsids in vivo after encapsidation of DNA. These changes in capsid shape and size cannot be explained with a simple elastic model alone. We propose a new theoretical framework that combines the elasticity of thin icosahedral shells with the pH dependence of capsid charge distribution. Minimization of the sum of elastic and electrostatic free energies leads to equilibrium shapes of viral shells that depend on a single elastic parameter and the detailed configuration of the imbedded protein charges. Based on the in vitro shell reconstructions of bacteriophage HK97 we elucidate the details of how the reversible transition between Prohead II and Expansion Intermediate II states of the HK97 procapsid is induced by pH changes, as well as some other features of the bacteriophage maturation.

Quasicrystalline and crystalline types of local protein order in capsids of small viruses.

Like metal alloys and micellar systems in soft matter, the viral capsid structures can be of crystalline and quasicrystalline types. We reveal the local quasicrystalline order of proteins in small spherical viral capsids using their nets of dodecahedral type. We show that the structure of some of the viral shells is well described in terms of a chiral pentagonal tiling, whose nodes coincide with centers of mass of protein molecules. The chiral protein packing found in these capsids originates from the pentagonal Penrose tiling (PPT), due to a specific phason reconstruction needed to fit the protein order at the adjacent dodecahedron faces. Via examples of small spherical viral shells and geminate capsid of a Maize Streak virus, we discuss the benefits and shortcomings of the usage of a dodecahedral net in comparison to icosahedral one, which is commonly applied for the modeling of viral shells with a crystalline local order.

Assembly of the most topologically regular two-dimensional micro and nanocrystals with spherical, conical, and tubular shapes.

We consider how to control the extension of curvature-induced defects in the hexagonal order covering different curved surfaces. In these frames we propose a physical mechanism for improving structures of two-dimensional spherical colloidal crystals (SCCs). For any SCC comprising of about 300 or less particles the mechanism transforms all extended topological defects (ETDs) in the hexagonal order into the point disclinations. Perfecting the structure is carried out by successive cycles of the particle implantation and subsequent relaxation of the crystal. The mechanism is potentially suitable for obtaining colloidosomes with better selective permeability. Our approach enables modeling the most topologically regular tubular and conical two-dimensional nanocrystals including various possible polymorphic forms of the HIV viral capsid. Different HIV-like shells with an arbitrary number of structural units (SUs) and desired geometrical parameters are easily formed. Faceting of the obtained structures is performed by minimizing the suggested elastic energy.

Hidden symmetry of small spherical viruses and organization principles in "anomalous" and double-shelled capsid nanoassemblies.

We propose the principles of structural organization in spherical nanoassemblies with icosahedral symmetry constituted by asymmetric protein molecules. The approach modifies the paradigmatic geometrical Caspar and Klug (CK) model of icosahedral viral capsids and demonstrates the common origin of both the "anomalous" and conventional capsid structures. In contrast to all previous models of "anomalous" viral capsids the proposed modified model conserves the basic structural principles of the CK approach and reveals the common hidden symmetry underlying all small viral shells. We demonstrate the common genesis of the "anomalous" and conventional capsids and explain their structures in the same frame. The organization principles are derived from the group theory analysis of the positional order on the spherical surface. The relationship between the modified CK geometrical model and the theory of two-dimensional spherical crystallization is discussed. We also apply the proposed approach to complex double-shelled capsids and capsids with protruding knob-like proteins. The introduced notion of commensurability for the concentric nanoshells explains the peculiarities of their organization and helps to predict analogous, but yet undiscovered, double-shelled viral capsid nanostructures.

Theory of morphological transformation of viral capsid shell during the maturation process in the HK97 bacteriophage and similar viruses.

We consider the symmetry and physical origin of collective displacement modes playing a crucial role in the morphological transformation during the maturation of the HK97 bacteriophage and similar viruses. It is shown that the experimentally observed hexamer deformation and pentamer twist in the HK97 procapsid correspond to the simplest irreducible shear strain mode of a spherical shell. We also show that the icosahedral faceting of the bacteriophage capsid shell is driven by the simplest irreducible radial displacement field. The shear field has the rotational icosahedral symmetry group I while the radial field has the full icosahedral symmetry I_{h}. This difference makes their actions independent. The radial field sign discriminates between the icosahedral and the dodecahedral shapes of the faceted capsid shell, thus making the approach relevant not only for the HK97-like viruses but also for the parvovirus family. In the frame of the Landau-Ginzburg formalism we propose a simple phenomenological model valid for the first reversible step of the HK97 maturation process. The calculated phase diagram illustrates the discontinuous character of the virus shape transformation. The characteristics of the virus shell faceting and expansion obtained in the in vitro and in vivo experiments are related to the decrease in the capsid shell thickness and to the increase of the internal capsid pressure.

Soft spherical nanostructures with a dodecagonal quasicrystal-like order.

We develop a theory which predicts curvature-related structural peculiarities of soft spherical nanostructures with a dodecagonal local arrangement of subunits. Spherical templates coated with a thin film of a soft quasicrystal (QC)-forming material constitute the most promising direction to realize these nanostructures. Disordered and perfect spherical nanostructures are simulated using two approaches. The first of them models a random QC-like spherical nanostructure with extended curvature-induced topological defects similar to scars in colloidal spherical crystals. The second approach is inspired by the physics of viral capsids. It deals with the most regular spherical nanostructures with a local QC-like order derived from three well-known planar dodecagonal tilings. We explain how the additional QC-like degrees of freedom assist the nanostructure stabilization and determine the point defect number and location without extended scar formation. Unusual for nanoassemblies snub cube geometry is shown to be the most energetically favorable global organization of these spherical QC nanostructures.

Chiral quasicrystalline order and dodecahedral geometry in exceptional families of viruses.

On the example of exceptional families of viruses we (i) show the existence of a completely new type of matter organization in 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, (ii) generalize the classical theory of quasicrystals (QCs) to explain this organization, and (iii) establish the relation between local chiral QC order and nonzero curvature of the dodecahedral capsid faces.