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.

Multipole analysis of the strain-mediated coupling between proteins adsorbed at tubular lipid membrane surface.

The tubular lipid membranes (TLMs) pulled out from vesicles are often used in in vitro studies of the interactions between curvature-inducing proteins and highly curved membranes. The protein molecules adsorbed at the membrane surface deform the TLM and couple with each other due to the induced strain. Here we propose an approach which models the single curvature-inducing protein action on the lipid bilayer by the multipole, the superposition of the point forces applied to the membrane in the region of the protein adsorption. We show that to be localized in the area of the protein size at the TLM surface, the force multipoles satisfying the mechanical equilibrium conditions should be composed of three or more point forces. The protein coupling energy mediated by the membrane strain is studied in detail. In the region of the tubular membrane stability the maximal distance between two neighboring interacting protein-induced force multipoles is estimated to be of the order of the TLM cross section perimeter. In the vicinity of the TLM instability in the region of the vanishing stretching force applied to the TLM, the interaction radius increases drastically. The high affinity of the single curvature-inducing protein molecule to the regions in the vicinity of the TLM ends is explained and related to the boundary conditions in the experimental set-ups. The reasons for the aggregate formation on the membrane surface are also discussed.

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.

Complex instability of axially compressed tubular lipid membrane with controlled spontaneous curvature.

Tubular lipid membranes (TLMs) are formed by an external pulling force from artificial or biological bilayer vesicles and can be subsequently stabilized by incorporating proteins or amphiphilic polymers into the lipid bilayer. The arising spontaneous curvature of the lipid sheet allows switching off the pulling force without TLM destabilization. However, here we show that during this process two different thermal fluctuation modes drastically increase their amplitudes making fluctuations of the TLM much greater than its radius. Due to the system's proximity to the critical fluctuation point, a weak axial compressive force is sufficient to destabilize the TLM. Its absolute value is shown to be much smaller than that of the pulling force required for the initial lipid nanotube formation. Induced complex instability was studied in the frame of Landau phase transition theory. The process involves two consecutive second-order phase transitions and leads to the tube deformation combining annular corrugation with completely unconventional chiral buckling.

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.

Theory of a reconstructive structural transformation in capsids of icosahedral viruses.

A theory of a reconstructive structural transformation in icosahedral capsid shells is developed for a whole family of virulent human viruses. It is shown that the reversible rearrangement of proteins during the virus maturation transformation is driven by the variation in the wave number l associated with the protein density distribution function. The collective displacement field of protein centers from their positions in the initial (procapsid) and the final (capsid) two-dimensional icosahderal structures is derived. The amplitude of the displacement field is shown to be small and it minimizes the calculated free energy of the transformation. The theory allows us to propose a continuous thermodynamical mechanism of the reconstructive procapsid-to-capsid transformation. In the frame of the density-wave approach, we also propose to take an equivalent plane-wave vector as a common structural feature for different icosahedral capsid shells formed by the same proteins. Using these characteristics, we explain the relation between the radii of the procapsid and capsid shells and generalize it to the case of the viral capsid polymorphism.

Density-wave theory of the capsid structure of small icosahedral viruses.

We apply Landau theory of crystallization to explain and to classify the capsid structures of small viruses with spherical topology and icosahedral symmetry. We develop an explicit method which predicts the positions of centers of mass for the proteins constituting the viral capsid shell. Corresponding density distribution function which generates the positions has a universal form without any fitting parameter. The theory describes in a uniform way both the structures satisfying the well-known Caspar and Klug geometrical model for capsid construction and those violating it.

Cytoskeleton influence on normal and tangent fluctuation modes in the red blood cells.

We argue that the paradoxal softness of the red blood cells (RBC) in fluctuation experiments is apparent. We show that the effective surface shear modulus mus of the RBC obtained from fluctuation data and that measured in static deformation experiments have the same order of magnitude. In the RBC model developed for this purpose the spectrin network cytoskeleton with the bulk shear modulus estimated as mu approximately 105-165 Pa contributes to both normal and tangent fluctuations of the system and confines the membrane fluctuations. The calculated ratio of the mean-square amplitudes / is 2-3 orders of magnitude smaller than it is in the free membrane with the same bending and shear moduli.

Viscoelastic dynamics of spherical composite vesicles.

A micromechanical model for the low-frequency dynamics of spherical composite vesicles (CVs) is proposed. Solid-like viscoelastic properties of the CVs are taken into account. The equations of motion of a CV surrounded by a viscous liquid are derived. They have discrete solutions which describe linearly coupled stretching and bending relaxation modes and an independent shear mode. The qualitative difference between the bending modes excited in a spherical vesicle and that in a flat membrane is demonstrated. The shear elasticity of the CVs gives an essential contribution to the relaxation rate of the bending mode at small wave numbers. It is also shown that even in an incompressible spherical vesicle with a finite shear modulus, the bending mode involves both radial and tangent displacements. These reasons make both in-plane and out-of-plane low-frequency responses of the CV quite different with respect to those of the flat membrane. To compare our theoretical results with published experimental data, the power spectra of the actin-coated CV are calculated.

Theory of chiral periodic mesophases formed from an achiral liquid of bent-core molecules.

The theory of unconventional one-dimensional periodic mesophases forming in a liquid of achiral bent-core molecules is presented. The order parameter is a polarization wave. Three distinct phases which can be stabilized directly from the isotropic liquid phase are associated with linear, circular, and elliptic polarizations of the wave. The elliptic polarization leads to the structure of the commonly studied B2 "bent-core" mesophase whereas the recently discovered C(p) phase may be assigned to the linear polarization. We present the molecular configurations and macroscopic properties of the stable states. Their behavior under chiral doping and electric field application are investigated and the corresponding phase diagrams are calculated.