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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 <X2n>/<X2t> 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.