Viral genome is protected by a solid protein shell (capsid) self-assembled from many copies of identical subunits (one or few proteins). The positions and orientations of subunits display high level of spatial organization well suited to modern structural methods of study. The structural and biochemical data rise a whole number of questions concerning unconventional positional order of subunits in the shell, physical mechanisms of the self-assembly and its thermodynamics. In the present work we develop the theory which explains the formation and classifies the structures of viruses with spherical topology and icosahedral symmetry. We develop an explicit method based on symmetry selection rules, which predicts the positions of centers of mass for the proteins self-assembled in the viral capsid shell. The theory describes in a uniform way both the structures satisfying the well-known Caspar and Klug model for capsid construction and those violating it. The peculiarities of the assembly thermodynamics are discussed. We also show the relation between the protein density distributions obtained and the infectivity properties of several human viruses. To illustrate the notions of the theory and the results obtained we focus on the Dengue virus (DENV) capsid.

Physical modeling of viral assembly

Gene expression program in eukaryotic cells is strongly dependent on physical state of the genome carrier. Physical state of the chromatin fiber constituting chromosomes is a key element in this program. However, despite the efforts to elucidate the structure and physical principles underlying chromatin organization, they remain not clear. On the one hand, chromosomes are highly hierarchical structures with quite different interactions involved on different scales of organization. On the other hand, the interactions involved at each scale are extremely complex (e.g. at 30 nm scale these are direct electrostatic, solvent mediated structural and polyelectrolyte mediated bridging interactions, DNA tension, etc.). Usual approach consisting in taking into account all the interactions between various molecular elements on the microscopic scale remains in this case out of reach. In the present work we suggest the approach which studies mesoscopic properties of the chromatin components and their interactions. Corresponding phenomenological theory helps to analyze the thermodynamically probable chromatin organization. We focus on liquid-crystalline order in chromatin resulting from the balance of thermal disorder and electrostatic (and mechanical) interactions. Using generally accepted experimental facts we identify robust mesogenic parameters of nucleosomes (DNA-protein nanoassemblies) at the smaller scale and show how the correlations of these parameters control the ordering into a chromatin structure at the bigger scale. The model is based on correlation of polar and chiral characteristics of nucleosomes. Phenomenological theory allows us to describe the condensed phases in aqueous solutions of nucleosomes with digested linker DNA, both in physiological conditions and in a wide range of monovalent salt concentration. Using the hypothesis of similar physical mechanism acting in condensed solutions and in the fiber in the same physiological conditions, we perform detailed symmetry analysis, construct the free energy model and reveal the thermodynamically favorable helical liquid-crystalline states of the fiber. In addition to « solenoid » and « two-start-helix » models abundantly discussed previously, we show the possibility of multi-start helix arrangements of nucleosomes in the chromatin and possible biaxiality of the structures. The effects of homogeneous mechanical force field applied to the chromatin in biochemical experiments are also studied. We show that helical state unwinding is a multi-step process and we give its structural and thermodynamic details.

Gene expression program in eukaryotic cells is strongly dependent on physical state of the genome carrier. Physical state of the chromatin fiber constituting chromosomes is a key element in this program. However, despite the efforts to elucidate the structure and physical principles underlying chromatin organization, they remain not clear. On the one hand, chromosomes are highly hierarchical structures with quite different interactions involved on different scales of organization. On the other hand, the interactions involved at each scale are extremely complex (e.g. at 30 nm scale these are direct electrostatic, solvent mediated structural and polyelectrolyte mediated bridging interactions, DNA tension, etc.). Usual approach consisting in taking into account all the interactions between various molecular elements on the microscopic scale remains in this case out of reach. In the present work we suggest the approach which studies mesoscopic properties of the chromatin components and their interactions. Corresponding phenomenological theory helps to analyze the thermodynamically probable chromatin organization. We focus on liquid-crystalline order in chromatin resulting from the balance of thermal disorder and electrostatic (and mechanical) interactions. Using generally accepted experimental facts we identify robust mesogenic parameters of nucleosomes (DNA-protein nanoassemblies) at the smaller scale and show how the correlations of these parameters control the ordering into a chromatin structure at the bigger scale.

The success of condensed matter physics in the 20th century is largely related to the deep understanding of matter organization, and following elaboration of physical concepts simplifying complex problems. Similar approach applied to the living matter reveals novel unique types of order at all scales, from individual molecules to biological tissues. At the scale typical for intracellular biomolecular assemblies, the peculiarities of the organization are mainly related to their unconventional topology and geometry, and to low dimensionality. Some of these unconventional types of order and their relation to biological functions will be illustrated : screw-like liquid-crystalline order of DNA in bacteriophages, chiral anticlinic liquid-crystalline state of nucleosomes in the chromatin fiber, 2D spherical crystals and 2D dodecahedral quasicrystals in capsids of small and medium viruses.

We develop a continuous theory of low-frequency dynamics for nanotubes with walls constituted by single-atom monolayer, the topological elasticity of which is not related to its vanishing macroscopic thickness. The applicability region of the theory proposed includes all truly two-dimensional materials such as graphene and MoS2. New comprehensive interpretation and analytical expressions for low-frequency modes in single-walled carbon nanotube (SWCNT) are given. The theory unambiguously relates the radial breathing modes of SWCNT and breathinglike modes of the double-walled carbon nanotube (DWCNT). The existing Raman data on DWCNTs are fitted better than in the frame of previous models.

Crystalline and quasicrystalline order in viral capsids