Changes in total charge on spike protein of SARS-CoV-2 in emerging lineages.

Motivation: Charged amino acid residues on the spike protein of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) have been shown to influence its binding to different cell surface receptors, its non-specific electrostatic interactions with the environment, and its structural stability and conformation. It is therefore important to obtain a good understanding of amino acid mutations that affect the total charge on the spike protein which have arisen across different SARS-CoV-2 lineages during the course of the virus' evolution. Results: We analyse the change in the number of ionizable amino acids and the corresponding total charge on the spike proteins of almost 2200 SARS-CoV-2 lineages that have emerged over the span of the pandemic. Our results show that the previously observed trend toward an increase in the positive charge on the spike protein of SARS-CoV-2 variants of concern has essentially stopped with the emergence of the early omicron variants. Furthermore, recently emerged lineages show a greater diversity in terms of their composition of ionizable amino acids. We also demonstrate that the patterns of change in the number of ionizable amino acids on the spike protein are characteristic of related lineages within the broader clade division of the SARS-CoV-2 phylogenetic tree. Due to the ubiquity of electrostatic interactions in the biological environment, our findings are relevant for a broad range of studies dealing with the structural stability of SARS-CoV-2 and its interactions with the environment. Availability and implementation: The data underlying the article are available in the Supplementary material.

Evolutionary changes in the number of dissociable amino acids on spike proteins and nucleoproteins of SARS-CoV-2 variants.

The spike protein of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is responsible for target recognition, cellular entry, and endosomal escape of the virus. At the same time, it is the part of the virus that exhibits the greatest sequence variation across the many variants which have emerged during its evolution. Recent studies have indicated that with progressive lineage emergence, the positive charge on the spike protein has been increasing, with certain positively charged amino acids (AAs) improving the binding of the spike protein to cell receptors. We have performed a detailed analysis of dissociable AAs of more than 1400 different SARS-CoV-2 lineages, which confirms these observations while suggesting that this progression has reached a plateau with Omicron and its subvariants and that the positive charge is not increasing further. Analysis of the nucleocapsid protein shows no similar increase in positive charge with novel variants, which further indicates that positive charge of the spike protein is being evolutionarily selected for. Furthermore, comparison with the spike proteins of known coronaviruses shows that already the wild-type SARS-CoV-2 spike protein carries an unusually large amount of positively charged AAs when compared to most other betacoronaviruses. Our study sheds light on the evolutionary changes in the number of dissociable AAs on the spike protein of SARS-CoV-2, complementing existing studies and providing a stepping stone towards a better understanding of the relationship between the spike protein charge and viral infectivity and transmissibility.

Scaling properties of RNA as a randomly branching polymer.

Formation of base pairs between the nucleotides of a ribonucleic acid (RNA) sequence gives rise to a complex and often highly branched RNA structure. While numerous studies have demonstrated the functional importance of the high degree of RNA branching-for instance, for its spatial compactness or interaction with other biological macromolecules-RNA branching topology remains largely unexplored. Here, we use the theory of randomly branching polymers to explore the scaling properties of RNAs by mapping their secondary structures onto planar tree graphs. Focusing on random RNA sequences of varying lengths, we determine the two scaling exponents related to their topology of branching. Our results indicate that ensembles of RNA secondary structures are characterized by annealed random branching and scale similarly to self-avoiding trees in three dimensions. We further show that the obtained scaling exponents are robust upon changes in nucleotide composition, tree topology, and folding energy parameters. Finally, in order to apply the theory of branching polymers to biological RNAs, whose length cannot be arbitrarily varied, we demonstrate how both scaling exponents can be obtained from distributions of the related topological quantities of individual RNA molecules with fixed length. In this way, we establish a framework to study the branching properties of RNA and compare them to other known classes of branched polymers. By understanding the scaling properties of RNA related to its branching structure, we aim to improve our understanding of the underlying principles and open up the possibility to design RNA sequences with desired topological properties.

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.

Dense packings of geodesic hard ellipses on a sphere.

Packing problems are abundant in nature and have been researched thoroughly both experimentally and in numerical models. In particular, packings of anisotropic, elliptical particles often emerge in models of liquid crystals, colloids, and granular and jammed matter. While most theoretical studies on anisotropic particles have thus far dealt with packings in Euclidean geometry, there are many experimental systems where anisotropically-shaped particles are confined to a curved surface, such as Pickering emulsions stabilized by ellipsoidal particles or protein adsorbates on lipid vesicles. Here, we study random close packing configurations in a two-dimensional model of spherical geodesic ellipses. We focus on the interplay between finite-size effects and curvature that is most prominent at smaller system sizes. We demonstrate that on a spherical surface, monodisperse ellipse packings are inherently disordered, with a non-monotonic dependence of both their packing fraction and the mean contact number on the ellipse aspect ratio, as has also been observed in packings of ellipsoids in both 2D and 3D flat space. We also point out some fundamental differences with previous Euclidean studies and discuss the effects of curvature on our results. Importantly, we show that the underlying spherical surface introduces frustration and results in disordered packing configurations even in systems of monodispersed particles, in contrast to the 2D Euclidean case of ellipse packing. This demonstrates that closed curved surfaces can be effective at introducing disorder in a system and could facilitate the study of monodispersed random packings.

Measure of overlap between two arbitrary ellipses on a sphere.

Various packing problems and simulations of hard and soft interacting particles, such as microscopic models of nematic liquid crystals, reduce to calculations of intersections and pair interactions between ellipsoids. When constrained to a spherical surface, curvature and compactness lead to non-trivial behaviour that finds uses in physics, computer science and geometry. A well-known idealized isotropic example is the Tammes problem of finding optimal non-intersecting packings of equal hard disks. The anisotropic case of elliptic particles remains, on the other hand, comparatively unexplored. We develop an algorithm to detect collisions between ellipses constrained to the two-dimensional surface of a sphere based on a solution of an eigenvalue problem. We investigate and discuss topologically distinct ways two ellipses may touch or intersect on a sphere, and define a contact function that can be used for construction of short- and long-range pair potentials.

Mechanics of inactive swelling and bursting of porate pollen grains.

The structure of pollen grains, which is typically characterized by soft apertures in an otherwise stiff exine shell, guides their response to changes in the humidity of the environment. These changes can lead to desiccation of the grain and its infolding but also to excessive swelling of the grain and even its bursting. Here we use an elastic model to explore the mechanics of pollen grain swelling and the role of soft, circular apertures (pores) in this process. Small, circular apertures typically occur in airborne and allergenic pollen grains so that the bursting of such grains is important in the context of human health. We identify and quantify a mechanical weakness of the pores, which are prone to rapid inflation when the grain swells to a critical extent. The inflation occurs as a sudden transition and may induce bursting of the grain and release of its content. This process crucially depends on the size of the pores and their softness. Our results provide insight into the inactive part of the mechanical response of pollen grains to hydration when they land on a stigma as well as bursting of airborne pollen grains during changes in air humidity.

RNA Secondary Structures Regulate Adsorption of Fragments onto Flat Substrates.

RNA is a functionally rich molecule with multilevel, hierarchical structures whose role in the adsorption to molecular substrates is only beginning to be elucidated. Here, we introduce a multiscale simulation approach that combines a tractable coarse-grained RNA structural model with an interaction potential of a structureless flat adsorbing substrate. Within this approach, we study the specific role of stem-hairpin and multibranch RNA secondary structure motifs on its adsorption phenomenology. Our findings identify a dual regime of adsorption for short RNA fragments with and without the secondary structure and underline the adsorption efficiency in both cases as a function of the surface interaction strength. The observed behavior results from an interplay between the number of contacts formed at the surface and the conformational entropy of the RNA molecule. The adsorption phenomenology of RNA seems to persist also for much longer RNAs as qualitatively observed by comparing the trends of our simulations with a theoretical approach based on an ideal semiflexible polymer chain.

Site Correlations, Capacitance, and Polarizability From Protein Protonation Fluctuations.

We generalize the Kirkwood-Shumaker theory of protonation fluctuation for an anisotropic distribution of dissociable charges on a globular protein. The fluctuations of the total charge and the total dipole moment, in contrast to their average values, depend on the same proton occupancy correlator, thus exhibiting a similar dependence also on the solution pH. This has important consequences for the Kirkwood-Shumaker interaction and its dependence on the bathing solution conditions.

Electrostatic interactions between the SARS-CoV-2 virus and a charged electret fibre.

While almost any kind of face mask offers some protection against particles and pathogens of different sizes, the most efficient ones make use of a layered structure where one or more layers are electrically charged. These electret layers are essential for the efficient filtration of difficult-to-capture small particles, yet the exact nature of electrostatic capture with respect to the charge on both the particles and the electret fibres as well as the effect of the immediate environment remain unclear. Here, we explore in detail the electrostatic interactions between the surface of a single charged electret fibre and a model of the SARS-CoV-2 virus. Using Poisson-Boltzmann electrostatics coupled to a detailed spike protein charge regulation model, we show how pH and salt concentration drastically change both the scale and the sign of the interaction. Furthermore, the configuration of the few spike proteins closest to the electret fibre turns out to be as important for the strength of the interaction as their total number on the virus envelope, a direct consequence of spike protein charge regulation. The results of our work elucidate the details of virus electrostatics and contribute to the general understanding of efficient virus filtration mechanisms.

Relative humidity in droplet and airborne transmission of disease.

A large number of infectious diseases are transmitted by respiratory droplets. How long these droplets persist in the air, how far they can travel, and how long the pathogens they might carry survive are all decisive factors for the spread of droplet-borne diseases. The subject is extremely multifaceted and its aspects range across different disciplines, yet most of them have only seldom been considered in the physics community. In this review, we discuss the physical principles that govern the fate of respiratory droplets and any viruses trapped inside them, with a focus on the role of relative humidity. Importantly, low relative humidity-as encountered, for instance, indoors during winter and inside aircraft-facilitates evaporation and keeps even initially large droplets suspended in air as aerosol for extended periods of time. What is more, relative humidity affects the stability of viruses in aerosol through several physical mechanisms such as efflorescence and inactivation at the air-water interface, whose role in virus inactivation nonetheless remains poorly understood. Elucidating the role of relative humidity in the droplet spread of disease would permit us to design preventive measures that could aid in reducing the chance of transmission, particularly in indoor environment.

Mechanical design of apertures and the infolding of pollen grain.

When pollen grains become exposed to the environment, they rapidly desiccate. To protect themselves until rehydration, the grains undergo characteristic infolding with the help of special structures in the grain wall-apertures-where the otherwise thick exine shell is absent or reduced in thickness. Recent theoretical studies have highlighted the importance of apertures for the elastic response and the folding of the grain. Experimental observations show that different pollen grains sharing the same number and type of apertures can nonetheless fold in quite diverse fashions. Using the thin-shell theory of elasticity, we show how both the absolute elastic properties of the pollen wall and the relative elastic differences between the exine wall and the apertures play an important role in determining pollen folding upon desiccation. Focusing primarily on colpate pollen, we delineate the regions of pollen elastic parameters where desiccation leads to a regular, complete closing of all apertures and thus to an infolding which protects the grain against water loss. Phase diagrams of pollen folding pathways indicate that an increase in the number of apertures leads to a reduction of the region of elastic parameters where the apertures close in a regular fashion. The infolding also depends on the details of the aperture shape and size, and our study explains how the features of the mechanical design of apertures influence the pollen folding patterns. Understanding the mechanical principles behind pollen folding pathways should also prove useful for the design of the elastic response of artificial inhomogeneous shells.

Hidden symmetry of the anomalous bluetongue virus capsid and its role in the infection process.

Clear understanding of the principles that control the arrangement of proteins and their self-assembly into viral shells is very important for the development of antiviral strategies. Here we consider the structural peculiarities and hidden symmetry of the anomalous bluetongue virus (BTV) capsid. Each of its three concentric shells violates the paradigmatic geometrical model of Caspar and Klug, which is otherwise well suited to describe most of the known icosahedral viral shells. As we show, three icosahedral spherical lattices, which are commensurate with each other and possess locally hexagonal (primitive or honeycomb) order, underlie the proteinaceous shells of the BTV capsid. This interpretation of the multishelled envelope allows us to discuss the so-called "symmetry mismatch" between its layers. We also analyze the structural stability of the considered spherical lattices on the basis of the classical theory of spherical packing and relate the proximity of the outer spherical lattice to destabilization with the fact that during infection of the cell VP2 trimers are detached from the surface of the BTV capsid. An electrostatic mechanism that can assist in this detachment is discussed in detail.

Spherical structure factor and classification of hyperuniformity on the sphere.

Understanding how particles are arranged on the surface of a sphere is not only central to numerous physical, biological, soft matter, and materials systems but also finds applications in computational problems, approximation theory, and analysis of geophysical and meteorological measurements. Objects that lie on a sphere experience constraints that are not present in Euclidean (flat) space and that influence both how the particles can be arranged as well as their statistical properties. These constraints, coupled with the curved geometry, require a careful extension of quantities used for the analysis of particle distributions in Euclidean space to distributions confined to the surface of a sphere. Here, we introduce a framework designed to analyze and classify structural order and disorder in particle distributions constrained to the sphere. The classification is based on the concept of hyperuniformity, which was first introduced 15 years ago and since then studied extensively in Euclidean space, yet has only very recently been considered also for spherical surfaces. We employ a generalization of the structure factor on the sphere, related to the power spectrum of the corresponding multipole expansion of particle density distribution. The spherical structure factor is then shown to couple with cap number variance, a measure of density variations at different scales, allowing us to analytically derive different forms of the variance pertaining to different types of distributions. Based on these forms, we construct a classification of hyperuniformity for scale-free particle distributions on the sphere and show how it can be extended to include other distribution types as well. We demonstrate that hyperuniformity on the sphere can be defined either through a vanishing spherical structure factor at low multipole numbers or through a scaling of the cap number variance-in both cases extending the Euclidean definition, while at the same time pointing out crucial differences. Our work thus provides a comprehensive tool for detecting global, long-range order on spheres and for the analysis of spherical computational meshes, biological and synthetic spherical assemblies, and ordering phase transitions in spherically distributed particles.

Role of metallic core for the stability of virus-like particles in strongly coupled electrostatics.

Electrostatic interactions play important roles in the formation and stability of viruses and virus-like particles (VLPs) through processes that often involve added, or naturally occurring, multivalent ions. Here, we investigate the electrostatic or osmotic pressure acting on the proteinaceous shell of a generic model of VLPs, comprising a charged outer shell and a metallic nanoparticle core, coated by a charged layer and bathed in an aqueous electrolyte solution. Motivated by the recent studies accentuating the role of multivalent ions for the stability of VLPs, we focus on the effects of multivalent cations and anions in an otherwise monovalent ionic solution. We perform extensive Monte-Carlo simulations based on appropriate Coulombic interactions that consistently take into account the effects of salt screening, the dielectric polarization of the metallic core, and the strong-coupling electrostatics due to multivalent ions. We specifically study the intricate roles these factors play in the electrostatic stability of the model VLPs. It is shown that while the insertion of a metallic nanoparticle by itself can produce negative, inward-directed, pressure on the outer shell, addition of only a small amount of multivalent counterions can robustly engender negative pressures, enhancing the VLP stability across a wide range of values for the system parameters.

Anomalous multipole expansion: Charge regulation of patchy inhomogeneously charged spherical particles.

Charge regulation is an important aspect of electrostatics in biological and colloidal systems, where the charges are generally not fixed but depend on the environmental variables. Here, we analyze the charge regulation mechanism in patchy inhomogeneously charged spherical particles, such as globular proteins, colloids, or viruses. Together with the multipole expansion of inhomogeneously charged spherical surfaces, the charge regulation mechanism on the level of linear approximation is shown to lead to a mixing between different multipole moments depending on their capacitance-the response function of the charge distribution with respect to the electrostatic potential. This presents an additional anomalous feature of molecular electrostatics in the presence of ionic screening. We demonstrate the influence of charge regulation on several examples of inhomogeneously charged spherical particles, showing that it leads to significant changes in their multipole moments.

Electrostatics-Driven Inflation of Elastic Icosahedral Shells as a Model for Swelling of Viruses.

We develop a clear theoretical description of radial swelling in virus-like particles that delineates the importance of electrostatic contributions to swelling in the absence of any conformational changes. The model couples the elastic parameters of the capsid-represented as a continuous elastic shell-to the electrostatic pressure acting on it. We show that different modifications of the electrostatic interactions brought about by, for instance, changes in pH or solution ionic strength are often sufficient to achieve the experimentally observed swelling (∼10% of the capsid radius). Additionally, we derive analytical expressions for the electrostatics-driven radial swelling of virus-like particles that enable one to quickly estimate the magnitudes of physical quantities involved.

Compactness of viral genomes: effect of disperse and localized random mutations.

Genomes of single-stranded RNA viruses have evolved to optimize several concurrent properties. One of them is the architecture of their genomic folds, which must not only feature precise structural elements at specific positions, but also allow for overall spatial compactness. The latter was shown to be disrupted by random synonymous mutations, a disruption which can consequently negatively affect genome encapsidation. In this study, we use three mutation schemes with different degrees of locality to mutate the genomes of phage MS2 and Brome Mosaic virus in order to understand the observed sensitivity of the global compactness of their folds. We find that mutating local stretches of their genomes' sequence or structure is less disruptive to their compactness compared to inducing randomly-distributed mutations. Our findings are indicative of a mechanism for the conservation of compactness acting on a global scale of the genomes, and have several implications for understanding the interplay between local and global architecture of viral RNA genomes.

From discrete to continuous description of spherical surface charge distributions.

The importance of electrostatic interactions in soft matter and biological systems can often be traced to non-uniform charge effects, which are commonly described using a multipole expansion of the corresponding charge distribution. The standard approach when extracting the charge distribution of a given system is to treat the constituent charges as points. This can, however, lead to an overestimation of multipole moments of high order, such as dipole, quadrupole, and higher moments. Focusing on distributions of charges located on a spherical surface - characteristic of numerous biological macromolecules, such as globular proteins and viral capsids, as well as of inverse patchy colloids - we develop a novel way of representing spherical surface charge distributions based on the von Mises-Fisher distribution. This approach takes into account the finite spatial extension of individual charges, and leads to a simple yet powerful way of describing surface charge distributions and their multipole expansions. In this manner, we analyze charge distributions and the derived multipole moments of a number of different spherical configurations of identical charges with various degrees of symmetry. We show how the number of charges, their size, and the geometry of their configuration influence the behavior and relative importance of multipole magnitudes of different order. Importantly, we clearly demonstrate how neglecting the effect of charge size leads to an overestimation of high-order multipoles. The results of our work can be applied to construct analytical models of electrostatic interactions and multipole expansion of charged particles in diverse soft matter and biological systems.

Varieties of charge distributions in coat proteins of ssRNA+ viruses.

A major part of the interactions involved in the assembly and stability of icosahedral, positive-sense single-stranded RNA (ssRNA+) viruses is electrostatic in nature, as can be inferred from the strong pH- and salt-dependence of their assembly phase diagrams. Electrostatic interactions do not act only between the capsid coat proteins (CPs), but just as often provide a significant contribution to the interactions of the CPs with the genomic RNA, mediated to a large extent by positively charged, flexible N-terminal tails of the CPs. In this work, we provide two clear and complementary definitions of an N-terminal tail of a protein, and use them to extract the tail sequences of a large number of CPs of ssRNA+ viruses. We examine the pH-dependent interplay of charge on both tails and CPs alike, and show that-in contrast to the charge on the CPs-the net positive charge on the N-tails persists even to very basic pH values. In addition, we note a limit to the length of the wild-type genomes of those viruses which utilize positively charged tails, when compared to viruses without charged tails and similar capsid size. At the same time, we observe no clear connection between the charge on the N-tails and the genome lengths of the viruses included in our study.