Topological balance of cell distributions in plane monolayers.

Most of normal proliferative epithelia of plants and metazoans are topologically invariant and characterized by similar cell distributions according to the number of cell neighbors (DCNs). Here we study peculiarities of these distributions and explain why the DCN obtained from the location of intercellular boundaries and that based on the Voronoi tessellation with nodes located on cell nuclei may differ from each other. As we demonstrate, special microdomains where four or more intercellular boundaries converge are topologically charged. Using this fact, we deduce a new equation describing the topological balance of the DCNs. The developed theory is applied for a series of microphotographs of non-tumoral epithelial cells of the human cervix (HCerEpiC) to improve the image processing near the edges of microphotographs and reveal the topological invariance of the examined monolayers. Special contact microdomains may be present in epithelia of various natures, however, considering the well-known vertex model of epithelium, we show that such contacts are absent in the usual solid-like state of the model and appear only in the liquid-like cancer state. Also, we discuss a possible biological role of special contacts in context of proliferative epithelium dynamics and tissue morphogenesis.

Theory of density waves and organization of proteins in icosahedral virus capsids.

Understanding the physical principles underlying the structural organization of the proteinaceous viral shells is of major importance to advance antiviral strategies. Here, we develop a phenomenological thermodynamic theory, which considers structures of small and middle-size icosahedral viral shells as a result of condensation of a minimum number of protein density waves on a spherical surface. Each of these irreducible critical waves has icosahedral symmetry and can be expressed as a specific series of the spherical harmonics Ylm with the same wave number l. As we demonstrate, in small viral shells self-assembled from individual proteins, the maxima of one critical density wave determine the positions of proteins, while the spatial derivatives of the second one control the protein orientations on the shell surface. In contrast to the small shells, the middle-size ones are always formed from pentamers and hexamers (referred to as capsomers). Considering all such structures deposited in the Protein Data Bank, we unexpectedly found that the positions of capsomeres in these shells correspond to the maxima of interference patterns produced by no more than two critical waves with close wave numbers. This fact allows us to explain the observed limit size of the icosahedral shells assembled from pentamers and hexamers. We also construct nonequilibrium thermodynamic potentials describing the protein crystallization and discuss the reasons behind the specific handedness of the viral shells.

Close packings of identical proteins in small spherical capsids and similar proteinaceous shells.

Understanding the principles governing protein arrangement in viral capsids and structurally similar protein shells can enable the development of new antiviral strategies and the design of artificial protein cages for various applications. We study these principles within the context of the close packing problem, by analyzing dozens of small spherical shells assembled from a single type of protein. First, we use icosahedral spherical close packings containing 60T identical disks, where T ≤ 4, to rationalize the protein arrangement in twenty real icosahedral shells both satisfying and violating the paradigmatic Caspar-Klug model. We uncover a striking correspondence between the protein mass centers in the considered shells and the centers of disks in the close packings. To generalize the packing model, we consider proteins with a weak shape anisotropy and propose an interaction energy, minimization of which allows us to obtain spherical dense packings of slightly anisotropic structural units. In the case of strong anisotropy, we model the proteins as sequences of overlapping discs of different sizes, with minimum energy configuration not only resulting in packings, accurately reproducing locations and orientations of individual proteins, but also revealing that icosahedral packings that display the handedness of real capsids are energetically more favorable. Finally, by introducing effective disc charges, we rationalize the formation of inter-protein bonds in protein shells.

Topological properties and shape of proliferative and nonproliferative cell monolayers.

During embryonic development, structures with complex geometry can emerge from planar epithelial monolayers; studying these shape transitions is of key importance for revealing the biophysical laws involved in the morphogenesis of biological systems. Here, using the example of normal proliferative monkey kidney (COS) cell monolayers, we investigate global and local topological characteristics of this model system in dependence on its shape. The obtained distributions of cells by their valence demonstrate a difference between the spherical and planar monolayers. In addition, in both spherical and planar monolayers, the probability of observing a pair of neighboring cells with certain valences depends on the topological charge of the pair. The zero topological charge of the cell pair can increase the probability for the cells to be the nearest neighbors. We then test and confirm that analogous relationships take place in a more ordered spherical system with a larger fraction of 6-valent cells, namely, in the nonproliferative epithelium (follicular system) of ascidian species oocytes. However, unlike spherical COS cell monolayers, ascidian monolayers are prone to nonrandom agglomeration of 6-valent cells and have linear topological defects called scars and pleats. The reasons for this difference in morphology are discussed. The morphological peculiarities found are compared with predictions of the widely used vertex model of epithelium.

Integration of Cypoviruses into polyhedrin matrix.

Unlike in other viruses, in Cypoviruses the genome is doubly protected since their icosahedral capsids are embedded into a perfect polyhedrin crystal. Current experimental methods cannot resolve the resulting interface structure and we propose a symmetry-based approach to predict it. We reveal a remarkable match between the surfaces of Cypovirus and the outer polyhedrin matrix. The match arises due to the preservation of the common tetragonal symmetry, allowing perfect contacts of polyhedrin trimers with VP1 and VP5 capsid proteins. We highlight a crucial role of the VP5 proteins in embedding the Cypovirus into the polyhedrin matrix and discuss the relationship between the nucleoside triphosphatase activity of the proteins and their role in the superstructure formation. Additionally, we propose an electrostatic mechanism that drives the viral superstructure disassembly occurring in the alkaline environment of the insect intestines. Our study may underpin novel strategies for engineering proteinaceous nanocontainers in diverse biotechnological and chemical applications.

Packing and trimer-to-dimer protein reconstruction in icosahedral viral shells with a single type of symmetrical structural unit.

Understanding the principles of protein packing and the mechanisms driving morphological transformations in virus shells (capsids) during their maturation can be pivotal for the development of new antiviral strategies. Here, we study how these principles and mechanisms manifest themselves in icosahedral viral capsids assembled from identical symmetric structural units (capsomeres). To rationalize such shells, we model capsomers as symmetrical groups of identical particles interacting with a short-range potential typical of the classic Tammes problem. The capsomere particles are assumed to retain their relative positions on the vertices of planar polygons placed on the spherical shell and to interact only with the particles from other capsomeres. Minimization of the interaction energy enforces equal distances between the nearest particles belonging to neighboring capsomeres and minimizes the number of different local environments. Thus, our model implements the Caspar and Klug quasi-equivalence principle and leads to packings strikingly similar to real capsids. We then study a reconstruction of protein trimers into dimers in a Flavivirus shell during its maturation, connecting the relevant structural changes with the modifications of the electrostatic charges of proteins, wrought by the oxidative switch in the bathing solution that is essential for the process. We highlight the key role of pr peptides in the shell reconstruction and show that the highly ordered arrangement of these subunits in the dimeric state is energetically favored at a low pH level. We also discuss the electrostatic mechanisms controlling the release of pr peptides in the last irreversible step of the maturation process.

Hidden symmetry of the flavivirus protein shell and pH-controlled reconstruction of the viral surface.

Using recent Zika virus structural data we reveal a hidden symmetry of protein order in immature and mature flavivirus shells, violating the Caspar-Klug paradigmatic model of capsid structures. We show that proteins of the outer immature shell layer exhibit trihexagonal tiling, while proteins from inner and outer layers conjointly form a double-shelled close-packed structure, based on a common triangular spherical lattice. Within the proposed structural model, we furthermore rationalize the structural organization of misassembled non-infectious subviral particles that have no inner capsid. We consider a pH-controlled structural reconstruction of the outer shell from the trimeric to the dimeric state, and demonstrate that this transition, occurring during the virus maturation, can be induced by changes in protein charges at lower pH, leading to a decrease in the electrostatic interaction free energy. This transition could also be assisted by electrostatic attraction of shell proteins to the interposed lipid membrane substrate separating the shells.

Random nature of epithelial cancer cell monolayers.

Although the polygonal shape of epithelial cells has been drawing the attention of scientists for several centuries, only a decade and a half ago it was demonstrated that distributions of polygon types (DOPTs) are similar in proliferative epithelia of many different plant and animal species. In this study, we show that hyper-proliferation of cancer cells disrupts this universal paradigm and results in randomly organized epithelial structures. Examining non-synchronized and synchronized HeLa cervix cells, we suppose that the spread of cell sizes is the main parameter controlling the DOPT in the cancer cell monolayers. To test this hypothesis, we develop a theory of morphologically similar random polygonal packings. By analysing differences between tumoural and normal epithelial cell monolayers, we conclude that the latter have more ordered structures because of their lower proliferation rates and, consequently, more effective relaxation of mechanical stress associated with cell division and growth. To explain the structural features of normal proliferative epithelium, we take into account the spread of cell sizes in the monolayer. The proposed theory also rationalizes some highly ordered unconventional post-mitotic epithelia.

Irreversible and reversible morphological changes in the φ6 capsid and similar viral shells: symmetry and micromechanics.

Understanding the physicochemical processes occurring in viruses during their maturation is of fundamental importance since only mature viruses can infect host cells. Here we consider the irreversible and reversible morphological changes that occur with the dodecahedral φ6 procapsid during the sequential packaging of 3 RNA segments forming the viral genome. It is shown that the dodecahedral shape of all the four observed capsid states is perfectly reproduced by a sphere radially deformed by only two irreducible spherical harmonics with icosahedral symmetry and wave numbers l = 6 and l = 10. The rotation of proteins around the 3-fold axes at the Procapsid → Intermediate 1 irreversible transformation is in fact also well described with the shear field containing only two irreducible harmonics with the same two wave numbers. The high stability of the Intermediate 1 state is discussed and the shapes of the Intermediate 2 state and Capsid (reversibly transforming back to the Intermediate 1 state) are shown to be mainly due to the isotropic pressure that the encapsidated RNA segments exert on the shell walls. The hidden symmetry of the capsid and the physicochemical features of the in vitro genome extraction from the viral shell are also elucidated.

Carbon nanotube sorting due to commensurate molecular wrapping.

Single-walled carbon nanotubes (SWCNTs) can be sorted by their structural parameters using organic molecules and polymers: some of which, demonstrating a profound affinity only for specific nanotubes, form dense coatings on them. Here, analyzing well-known examples of flavin group molecules and those of 2,4-dichlorophenoxyacetic acid, we show for the first time that successful formation of the considered coatings depends on the ability of molecules to wrap around the SWCNT in a commensurate way. Commensurability provides a decrease in the free energy of the resulting bilayer system and makes the coating much more stable. Concurrently, it strongly relates the nanotube chiral vector with the geometric characteristics of the adhering molecules, which leads to revealed selection rules. If they are not satisfied, the deposition of molecules does not occur or is insignificant. The proposed theory unambiguously explains known experimental results on the formation of spiral wrappings of SWCNTs by flavin group molecules and points out other organic molecules and polymers suitable for effective CNT sorting.

Crystal-like order and defects in metazoan epithelia with spherical geometry.

Since Robert Hooke studied cork cell patterns in 1665, scientists have been puzzled by why cells form such ordered structures. The laws underlying this type of organization are universal, and we study them comparing the living and non-living two-dimensional systems self-organizing at the spherical surface. Such-type physical systems often possess trigonal order with specific elongated defects, scars and pleats, where the 5-valence and 7-valence vertices alternate. In spite of the fact that the same physical and topological rules are involved in the structural organization of biological systems, such topological defects were never reported in epithelia. We have discovered them in the follicular spherical epithelium of ascidians that are emerging models in developmental biology. Surprisingly, the considered defects appear in the epithelium even when the number of cells in it is significantly less than the previously known threshold value. We explain this result by differences in the cell sizes and check our hypothesis considering the self-assembly of different random size particles on the spherical surface. Scars, pleats and other complex defects found in ascidian samples can play an unexpected and decisive role in the permanent renewal and reorganization of epithelia, which forms or lines many tissues and organs in metazoans.

Long-range protein coupling mediated by critical low-energy modes of tubular lipid membranes.

We develop a theory of a resonant effect in protein-membrane coupling taking place in the vicinity of instabilities in tubular lipid membranes (TLMs) under longitudinal force and pressure difference constraints. Two critical low-energy modes defining the stability domain boundaries are found. We show that these modes mediate long-range TLM-protein coupling and interactions between absorbed proteins. Besides, TLM mechanical instabilities strongly influence protein desorption and protein cluster nucleation on TLMs. Model predictions can be tested over a large spectrum of mechanochemical conditions.