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.

Adenovirus major core protein condenses DNA in clusters and bundles, modulating genome release and capsid internal pressure.

Some viruses package dsDNA together with large amounts of positively charged proteins, thought to help condense the genome inside the capsid with no evidence. Further, this role is not clear because these viruses have typically lower packing fractions than viruses encapsidating naked dsDNA. In addition, it has recently been shown that the major adenovirus condensing protein (polypeptide VII) is dispensable for genome encapsidation. Here, we study the morphology and mechanics of adenovirus particles with (Ad5-wt) and without (Ad5-VII-) protein VII. Ad5-VII- particles are stiffer than Ad5-wt, but DNA-counterions revert this difference, indicating that VII screens repulsive DNA-DNA interactions. Consequently, its absence results in increased internal pressure. The core is slightly more ordered in the absence of VII and diffuses faster out of Ad5-VII- than Ad5-wt fractured particles. In Ad5-wt unpacked cores, dsDNA associates in bundles interspersed with VII-DNA clusters. These results indicate that protein VII condenses the adenovirus genome by combining direct clustering and promotion of bridging by other core proteins. This condensation modulates the virion internal pressure and DNA release from disrupted particles, which could be crucial to keep the genome protected inside the semi-disrupted capsid while traveling to the nuclear pore.

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.

Role of Condensing Particles in Polymer Confinement: A Model for Virus-Packed "Minichromosomes".

Confined mixtures of a polymer and nonspecifically binding particles (condensers) are studied as models for viruses containing double-stranded DNA (polymer) and condensing proteins (particles). We explore a model in which all interactions between the packed content (polymer and particles) and its confinement are purely repulsive, with only a short-range attraction between the condensers and polymer to simulate binding. In the range of physical parameters applicable to viruses, the model predicts reduction of pressure in the system effected by the condensers, despite the reduction in free volume. Condensers are found to be interspersed throughout the spherical confinement and only partially wrapped in the polymer, which acts as an effective medium for the condenser interactions. Crowding of the viral interior influences the DNA and protein organization, producing a picture inconsistent with a chromatin-like, beads-on-a-string structure. The model predicts an organization of the confined interior compatible with experimental data on unperturbed adenoviruses and polyomaviruses, at the same time providing insight into the role of condensing proteins in the viral infectious cycles of related viral families.

Distribution of DNA-condensing protein complexes in the adenovirus core.

Genome packing in adenovirus has long evaded precise description, since the viral dsDNA molecule condensed by proteins (core) lacks icosahedral order characteristic of the virus protein coating (capsid). We show that useful insights regarding the organization of the core can be inferred from the analysis of spatial distributions of the DNA and condensing protein units (adenosomes). These were obtained from the inspection of cryo-electron tomography reconstructions of individual human adenovirus particles. Our analysis shows that the core lacks symmetry and strict order, yet the adenosome distribution is not entirely random. The features of the distribution can be explained by modeling the condensing proteins and the part of the genome in each adenosome as very soft spheres, interacting repulsively with each other and with the capsid, producing a minimum outward pressure of ∼0.06 atm. Although the condensing proteins are connected by DNA in disrupted virion cores, in our models a backbone of DNA linking the adenosomes is not required to explain the experimental results in the confined state. In conclusion, the interior of an adenovirus infectious particle is a strongly confined and dense phase of soft particles (adenosomes) without a strictly defined DNA backbone.

Ejecting phage DNA against cellular turgor pressure.

We examine in vivo ejection of noncondensed DNA from tailed bacteriophages into bacteria. The ejection is dominantly governed by the physical conditions in the bacteria. The confinement of the DNA in the virus capsid only slightly helps the ejection, becoming completely irrelevant during its last stages. A simple calculation based on the premise of condensed DNA in the cell enables us to estimate the maximal bacterial turgor pressure against which the ejection can still be fully realized. The calculated pressure (~5 atm) shows that the ejection of DNA into Gram-negative bacteria could proceed spontaneously, i.e., without the need to invoke active mechanisms.

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.

How simple can a model of an empty viral capsid be? Charge distributions in viral capsids.

We investigate and quantify salient features of the charge distributions on viral capsids. Our analysis combines the experimentally determined capsid geometry with simple models for ionization of amino acids, thus yielding a detailed description of spatial distribution for positive and negative charges across the capsid wall. The obtained data is processed in order to extract the mean radii of distributions, surface charge densities, as well as dipole moment densities. The results are evaluated and examined in light of previously proposed models of capsid charge distributions, which are shown to have to some extent limited value when applied to real viruses.

Protein-DNA interactions determine the shapes of DNA toroids condensed in virus capsids.

DNA toroids that form inside the bacteriophage capsid present different shapes according to whether they are formed by the addition of spermine or polyethylene glycol to the bathing solution. Spermine-DNA toroids present a convex, faceted section with no or minor distortions of the DNA interstrand spacing with respect to those observed in the bulk, whereas polyethylene glycol-induced toroids are flattened to the capsid inner surface and show a crescent-like, nonconvex shape. By modeling the energetics of the DNA toroid using a free-energy functional composed of energy contributions related to the elasticity of the wound DNA, exposed surface DNA energy, and adhesion between the DNA and the capsid, we established that the crescent shape of the toroidal DNA section comes from attractive interactions between DNA and the capsid. Such attractive interactions seem to be specific to the PEG condensation process and are not observed in the case of spermine-induced DNA condensation.

Stability of elastic icosadeltahedral shells under uniform external pressure: application to viruses under osmotic pressure.

We discuss the stability of icosadeltahedral shells subjected to a uniform external load in the form of an isotropic pressure. We demonstrate that there exists a universal critical buckling pressure scaling form that defines a locus of buckling instabilities. The parameter that uniquely determines this scaling form is shown to be the Föppl-von Karman number of nonpressurized shells. Numerical results are interpreted in terms of scaling forms for buckling instabilities of spheres and cylinders under isotropic mechanical pressure, and are applied to the case of viruses under osmotic pressure.

Spontaneous curvature as a regulator of the size of virus capsids.

We investigate the physical reasons underlying the high monodispersity of empty virus capsids assembled in thermodynamical equilibrium in conditions of favorable pH and ionic strength. We propose that the high fidelity of the assembly results from the effective spontaneous curvature of the viral protein assemblies and the corresponding bending rigidity that penalizes curvatures which are larger and smaller from the spontaneous one. On the example of hepatitis B virus, which has been thoroughly studied experimentally in the context of interest to us, we estimate the magnitude of bending rigidity that is needed to suppress the appearance of aberrant capsid structures (approximately 60k(B)T). Our approach also demonstrates that the aberrant capsids that can be classified within the Caspar-Klug framework are in most circumstances likely to be smaller from the regular ones, in agreement with the experimental findings.

Nonspecific interactions in spontaneous assembly of empty versus functional single-stranded RNA viruses.

We investigate the influence of salt concentration, charge on viral proteins and the length of single-stranded RNA (ssRNA) molecule on the spontaneous assembly of viruses. Only the nonspecific interactions are assumed to guide the assembly, i.e., we exclude any chemical specificity that may lock the viral proteins and ssRNA in preferred configurations. We demonstrate that the electrostatic interactions screened by the salt in the solution impose strong limits on viral composition that can be achieved by spontaneous assembly. In particular, we show that viruses whose ssRNA carries more than twice the amount of charge that is located on the viral proteins, cannot be assembled spontaneously. We find that the spatial distribution of protein charge is important for the energetics of the assembly. We also show that the pressures that act on the viruses as a result of attractive protein-ssRNA electrostatic interactions are at least an order of magnitude smaller than is the case with bacteriophage viruses that contain double-stranded DNA molecule.

Role of electrostatic interactions in the assembly of empty spherical viral capsids.

We examine the role of electrostatic interactions in the assembly of empty spherical viral capsids. The charges on the protein subunits that make the viral capsid mutually interact and are expected to yield electrostatic repulsion acting against the assembly of capsids. Thus, attractive protein-protein interactions of nonelectrostatic origin must act to enable the capsid formation. We investigate whether the interplay of repulsive electrostatic and attractive interactions between the protein subunits can result in the formation of spherical viral capsids of a preferred radius. For this to be the case, we find that the attractive interactions must depend on the angle between the neighboring protein subunits (i.e., on the mean curvature of the viral capsid) so that a particular angle(s) is (are) preferred energywise. Our results for the electrostatic contributions to energetics of viral capsids nicely correlate with recent experimental determinations of the energetics of protein-protein contacts in the hepatitis B virus [P. Ceres A. Zlotnick, Biochemistry 41, 11525 (2002)].

Energies of sp(2) carbon shapes with pentagonal disclinations and elasticity theory.

Energies of a certain class of fullerene molecules (elongated, contracted and regular icosahedral fullerenes) are numerically calculated using a microscopic description of carbon-carbon bonding. It is shown how these results can be interpreted and comprehended using the theory of elasticity that describes bending of a graphene plane. Detailed studies of a wide variety of structures constructed by application of the same general principle are performed, and analytical expressions for energies of such structures are derived. Comparison of numerical results with the predictions of a simple implementation of elasticity theory confirms the usefulness of the latter approach.

Buckling transition in icosahedral shells subjected to volume conservation constraint and pressure: relations to virus maturation.

Minimal energy shapes of closed, elastic shells with 12 pentagonal disclinations introduced in otherwise hexagonally coordinated crystalline lattice are studied. The geometry and the total energy of shells are studied as a function of the elastic properties of the material they are made of. Particular emphasis is put on the buckling transition of the shells, that is, a strong preference of the shell shapes to "buckle out" in spatial regions close to the pentagonal disclinations for a certain range of the elastic parameters of the problem. The transition effectively increases the mean square aspherity of shapes, making them look more like an icosahedron rather than a sphere, which is a preferred shape prior to the onset of the transition. The properties of the buckling transition are studied in cases when (i) the total volume enclosed by the elastic shell has to be fixed and when (ii) there is an internal pressure acting on the shell. This may be related to the maturation process in nonenveloped dsDNA viruses, where the insertion of the genetic material in a preformed protein shell (viral coating) may effectively impose the fixed volume and/or pressure constraint. Several scenarios that may explain the experimentally observed feature of mature viruses being more aspherical (facetted) from their immature precursors are discussed, and predictions for the elastic properties of viral coatings are obtained on the basis of the presented studies.

Shapes of minimal-energy DNA ropes condensed in confinement.

Shapes of a single, long DNA molecule condensed in a confinement of a virus capsid are described as conformations optimizing a model free energy functional accounting for the interplay between the bending energy of the DNA and the surface energy of the DNA bundled in a "rope". The rope is formed by bundled DNA brought together by (self-)attractive interactions. The conformations predicted by the model depend on the shape of the confinement, the total amount of the packed DNA but also on the relative contributions of the bending and surface energies. Some of the conformations found were not predicted previously, but many previously proposed DNA conformations, some of which are seemingly contradictory, were found as the solutions of the model. The results show that there are many possible packing conformations of the DNA and that the one which realizes in a particular virus depends on the capsid geometry and the nature of condensing agents.

Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media.

We develop a theory describing propagation of spatially and temporally incoherent light in noninstantaneous nonlinear media, and predict the existence of modulation instability of "white" light. We find that the modulation instability of white light is fundamentally a collective effect, where all the temporal frequencies participate in the formation of a pattern, and self-adjust their respective contributions.

Lattice-gas Poisson-Boltzmann approach for sterically asymmetric electrolytes.

We derive a modification of the Poisson-Boltzmann (PB) equation brought about by the finite and different sizes of mobile ions. The equation we obtain contains explicitly the size asymmetry of the electrolyte, enabling one to apply the PB approach to the cases where this effect may be of importance. We show that the modification we propose predicts the asymmetry of the double-layer capacitance. Our approach thus provides a transparent analytical framework for a previously proposed heuristic modification of the PB equation, which also predicts the same effect. In addition, a detailed exposition of the lattice-based calculation of the entropy, accounting for steric, spatial correlations, enables us to clearly pinpoint some of the drawbacks of such an approach, which are not entirely obvious in previous attempts to include the steric effects in the PB equation.

Statistical analysis of sizes and shapes of virus capsids and their resulting elastic properties.

From the analysis of sizes of approximately 130 small icosahedral viruses we find that there is a typical structural capsid protein, having a mean diameter of 5 nm and a mean thickness of 3 nm, with more than two thirds of the analyzed capsid proteins having thicknesses between 2 nm and 4 nm. To investigate whether, in addition to the fairly conserved geometry, capsid proteins show similarities in the way they interact with one another, we examined the shapes of the capsids in detail. We classified them numerically according to their similarity to sphere and icosahedron and an interpolating set of shapes in between, all of them obtained from the theory of elasticity of shells. In order to make a unique and straightforward connection between an idealized, numerically calculated shape of an elastic shell and a capsid, we devised a special shape fitting procedure, the outcome of which is the idealized elastic shape fitting the capsid best. Using such a procedure we performed statistical analysis of a series of virus shapes and we found similarities between the capsid elastic properties of even very different viruses. As we explain in the paper, there are both structural and functional reasons for the convergence of protein sizes and capsid elastic properties. Our work presents a specific quantitative scheme to estimate relatedness between different proteins based on the details of the (quaternary) shape they form (capsid). As such, it may provide an information complementary to the one obtained from the studies of other types of protein similarity, such as the overall composition of structural elements, topology of the folded protein backbone, and sequence similarity.