Context-dependent structure formation of hairpin motifs in bacteriophage MS2 genomic RNA.

Many functions of ribonucleic acid (RNA) rely on its ability to assume specific sequence-structure motifs. Packaging signals found in certain RNA viruses are one such prominent example of functional RNA motifs. These signals are short hairpin loops that interact with coat proteins and drive viral self-assembly. As they are found in different positions along the much longer genomic RNA, the formation of their correct structure occurs as a part of a larger context. Any changes to this context can consequently lead to changes in the structure of the motifs themselves. In fact, previous studies have shown that structure and function of RNA motifs can be highly context sensitive to the flanking sequence surrounding them. However, in what ways different flanking sequences influence the structure of an RNA motif they surround has yet to be studied in detail. We focus on a hairpin-rich region of the RNA genome of bacteriophage MS2-a well-studied RNA virus with a wide potential for use in biotechnology-and systematically examine context-dependent structural stability of 14 previously identified hairpin motifs, which include putative and confirmed packaging signals. Combining secondary and tertiary RNA structure prediction of the hairpin motifs placed in different contexts, ranging from the native genomic sequence to random RNA sequences and unstructured poly-U sequences, we determine different measures of motif structural stability. In this way, we show that while some motif structures can be stable in any context, others require specific context provided by the genome. Our results demonstrate the importance of context in RNA structure formation and how changes in the flanking sequence of an RNA motif sometimes lead to drastic changes in its structure. Structural stability of a motif in different contexts could provide additional insights into its functionality as well as assist in determining whether it remains functional when intentionally placed in other contexts.

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.

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.

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.

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.

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.

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.

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.

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.

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.

pH Dependence of Charge Multipole Moments in Proteins.

Electrostatic interactions play a fundamental role in the structure and function of proteins. Due to ionizable amino acid residues present on the solvent-exposed surfaces of proteins, the protein charge is not constant but varies with the changes in the environment-most notably, the pH of the surrounding solution. We study the effects of pH on the charge of four globular proteins by expanding their surface charge distributions in terms of multipoles. The detailed representation of the charges on the proteins is in this way replaced by the magnitudes and orientations of the multipole moments of varying order. Focusing on the three lowest-order multipoles-the total charge, dipole, and quadrupole moment-we show that the value of pH influences not only their magnitudes, but more notably and importantly also the spatial orientation of their principal axes. Our findings imply important consequences for the study of protein-protein interactions and the assembly of both proteinaceous shells and patchy colloids with dissociable charge groups.

Synonymous mutations reduce genome compactness in icosahedral ssRNA viruses.

Recent studies have shown that single-stranded (ss) viral RNAs fold into more compact structures than random RNA sequences with similar chemical composition and identical length. Based on this comparison, it has been suggested that wild-type viral RNA may have evolved to be atypically compact so as to aid its encapsidation and assist the viral assembly process. To further explore the compactness selection hypothesis, we systematically compare the predicted sizes of >100 wild-type viral sequences with those of their mutants, which are evolved in silico and subject to a number of known evolutionary constraints. In particular, we enforce mutation synonynimity, preserve the codon-bias, and leave untranslated regions intact. It is found that progressive accumulation of these restricted mutations still suffices to completely erase the characteristic compactness imprint of the viral RNA genomes, making them in this respect physically indistinguishable from randomly shuffled RNAs. This shows that maintaining the physical compactness of the genome is indeed a primary factor among ssRNA viruses' evolutionary constraints, contributing also to the evidence that synonymous mutations in viral ssRNA genomes are not strictly neutral.

Serum microRNAs in patients with genetic amyotrophic lateral sclerosis and pre-manifest mutation carriers.

Knowledge about the nature of pathomolecular alterations preceding onset of symptoms in amyotrophic lateral sclerosis is largely lacking. It could not only pave the way for the discovery of valuable therapeutic targets but might also govern future concepts of pre-manifest disease modifying treatments. MicroRNAs are central regulators of transcriptome plasticity and participate in pathogenic cascades and/or mirror cellular adaptation to insults. We obtained comprehensive expression profiles of microRNAs in the serum of patients with familial amyotrophic lateral sclerosis, asymptomatic mutation carriers and healthy control subjects. We observed a strikingly homogenous microRNA profile in patients with familial amyotrophic lateral sclerosis that was largely independent from the underlying disease gene. Moreover, we identified 24 significantly downregulated microRNAs in pre-manifest amyotrophic lateral sclerosis mutation carriers up to two decades or more before the estimated time window of disease onset; 91.7% of the downregulated microRNAs in mutation carriers overlapped with the patients with familial amyotrophic lateral sclerosis. Bioinformatic analysis revealed a consensus sequence motif present in the vast majority of downregulated microRNAs identified in this study. Our data thus suggest specific common denominators regarding molecular pathogenesis of different amyotrophic lateral sclerosis genes. We describe the earliest pathomolecular alterations in amyotrophic lateral sclerosis mutation carriers known to date, which provide a basis for the discovery of novel therapeutic targets and strongly argue for studies evaluating presymptomatic disease-modifying treatment in amyotrophic lateral sclerosis.

The role of solution conditions in the bacteriophage PP7 capsid charge regulation.

We investigate and quantify the effects of pH and salt concentration on the charge regulation of the bacteriophage PP7 capsid. These effects are found to be extremely important and substantial, introducing qualitative changes in the charge state of the capsid such as a transition from net-positive to net-negative charge depending on the solution pH. The overall charge of the virus capsid arises as a consequence of a complicated balance with the chemical dissociation equilibrium of the amino acids and the electrostatic interaction between them, and the translational entropy of the mobile solution ions, i.e., counterion release. We show that to properly describe and predict the charging equilibrium of viral capsids in general, one needs to include molecular details as exemplified by the acid-base equilibrium of the detailed distribution of amino acids in the proteinaceous capsid shell.

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.

Symmetry effects in electrostatic interactions between two arbitrarily charged spherical shells in the Debye-Hückel approximation.

Inhomogeneous charge distributions have important repercussions on electrostatic interactions in systems of charged particles but are often difficult to examine theoretically. We investigate how electrostatic interactions are influenced by patchy charge distributions exhibiting certain point group symmetries. We derive a general form of the electrostatic interaction energy of two permeable, arbitrarily charged spherical shells in the Debye-Hückel approximation and apply it to the case of particles with icosahedral, octahedral, and tetrahedral inhomogeneous charge distributions. We analyze in detail how charge distribution symmetry modifies the interaction energy and find that local charge inhomogeneities reduce the repulsion of two overall equally charged particles, while sufficient orientational variation in the charge distribution can turn the minimum interaction energy into an attraction. Additionally, we show that larger patches and thus lower symmetries and wave numbers result in bigger attraction given the same variation.

Multivalent ion effects on electrostatic stability of virus-like nano-shells.

Electrostatic properties and stability of charged virus-like nano-shells are examined in ionic solutions with monovalent and multivalent ions. A theoretical model based on a thin charged spherical shell and multivalent ions within the "dressed multivalent ion" approximation, yielding their distribution across the shell and the corresponding electrostatic (osmotic) pressure acting on the shell, is compared with extensive implicit Monte-Carlo simulations. It is found to be accurate for positive or low negative surface charge densities of the shell and for sufficiently high (low) monovalent (multivalent) salt concentrations. Phase diagrams involving electrostatic pressure exhibit positive and negative values, corresponding to an outward and an inward facing force on the shell, respectively. This provides an explanation for the high sensitivity of viral shell stability and self-assembly of viral capsid shells on the ionic environment.

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.

Energies and pressures in viruses: contribution of nonspecific electrostatic interactions.

We summarize some aspects of electrostatic interactions in the context of viruses. A simplified but, within well defined limitations, reliable approach is used to derive expressions for electrostatic energies and the corresponding osmotic pressures in single-stranded RNA viruses and double-stranded DNA bacteriophages. The two types of viruses differ crucially in the spatial distribution of their genome charge which leads to essential differences in their free energies, depending on the capsid size and total charge in a quite different fashion. Differences in the free energies are trailed by the corresponding characteristics and variations in the osmotic pressure between the inside of the virus and the external bathing solution.