Viral self-assembly pathway and mechanical stress relaxation

Exact Solution for Elastic Networks on Curved Surfaces

The problem of characterizing the structure of an elastic network constrained to lie on a frozen curved surface appears in many areas of science and has been addressed by many different approaches, most notably, extending linear elasticity or through effective defect interaction models. In this Letter, we show that the problem can be solved by considering nonlinear elasticity in an exact form without resorting to any approximation in terms of geometric quantities. In this way, we are able to consider different effects that have been unwieldy or not viable to include in the past, such as a finite line tension, explicit dependence on the Poisson ratio, or the determination of the particle positions for the entire lattice. Several geometries with rotational symmetry are solved explicitly. Comparison with linear elasticity reveals an agreement that extends beyond its strict range of applicability. Implications for the problem of the characterization of virus assembly are also discussed.

Stiffness heterogeneity of small viral capsids

Nanoindentation of viral capsids provides an efficient tool in order to probe their elastic properties. We investigate in the present work the various sources of stiffness heterogeneity as observed in atomic force microscopy experiments. By combining experimental results with both numerical and analytical modeling, we first show that for small viruses, a position-dependent stiffness is observed. This effect is strong and has not been properly taken into account previously. Moreover, we show that a geometrical model is able to reproduce this effect quantitatively. Our work suggests alternative ways of measuring stiffness heterogeneities on small viral capsids. This is illustrated on two different viral capsids: Adeno associated virus serotype 8 (AAV8) and hepatitis B virus (HBV with T=4). We discuss our results in light of continuous elasticity modeling.

Scaling relation between genome length and particle size of viruses provides insights into viral life history

In terms of genome and particle sizes, viruses exhibit great diversity. With the discovery of several nucleocytoplasmic large DNA viruses (NCLDVs) and jumbo phages, the relationship between particle and genome sizes has emerged as an important criterion for understanding virus evolution. We use allometric scaling of capsid volume with the genome length of different groups of viruses to shed light on its relationship with virus life history. The allometric exponents for icosahedral dsDNA bacteriophages and NCDLVs were found to be 1 and 2, respectively, indicating that with increasing capsid size DNA packaging density remains the same in bacteriophages but decreases for NCLDVs. We argue that the exponents are largely shaped by their entry mechanism and capsid mechanical stability. We further show that these allometric size parameters are also intricately linked to the relative energy costs of translation and replication in viruses and can have further implications on viral life history.

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Capsid and genome size allometric exponent gives insights into viral life history

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The allometric exponent of NCLDVs is almost twice that of bacteriophages

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The exponent is largely shaped by the viral entry mechanism and capsid stability

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The relaxed genome size constraint allows large viruses to evolve greater autonomy

Shape selection and mis-assembly in viral capsid formation by elastic frustration

The successful assembly of a closed protein shell (or capsid) is a key step in the replication of viruses and in the production of artificial viral cages for bio/nanotechnological applications. During self-assembly, the favorable binding energy competes with the energetic cost of the growing edge and the elastic stresses generated due to the curvature of the capsid. As a result, incomplete structures such as open caps, cylindrical or ribbon-shaped shells may emerge, preventing the successful replication of viruses. Using elasticity theory and coarse-grained simulations, we analyze the conditions required for these processes to occur and their significance for empty virus self-assembly. We find that the outcome of the assembly can be recast into a universal phase diagram showing that viruses with high mechanical resistance cannot be self-assembled directly as spherical structures. The results of our study justify the need of a maturation step and suggest promising routes to hinder viral infections by inducing mis-assembly.

Ground States of Crystalline Caps: Generalized Jellium on Curved Space

We study the ground states of crystals on spherical surfaces. These ground states consist of positive disclination defects in structures spanning from flat and weakly curved caps to closed shells. Comparing two continuum theories and one discrete-lattice simulation, we first investigate the transition between defect-free caps to single-disclination ground states and show it to be continuous and symmetry breaking. Further, we show that ground states adopt icosahedral subgroup symmetries across the full range of curvatures, even far from the closure of complete shells. While superficially similar to other models of 2D "jellium" (e.g., superconducting disks and 2D Wigner crystals), the interplay between the free edge of caps and the non-Euclidean geometry of its embedding leads to nontrivial ground state behavior that is without counterpart in planar jellium models.

Mechanical stress relaxation in molecular self-assembly

Molecular self-assembly on a curved substrate leads to the spontaneous inclusion of topological defects in the growing bidimensional crystal, unlike assembly on a flat substrate. We propose in this work a quantitative mechanism for this phenomenon by using standard thin shell elasticity. The Gaussian curvature of the substrate induces large in-plane compressive stress as the surface grows, in particular at the rim of the assembly, and the addition of a single defect relaxes this mechanical stress. We found out that the value of azimuthal stress at the rim of the assembly determines the preferred directions for defect nucleation. These results are also discussed as a function of different defect combinations, like dislocations and grain boundaries or scars. In particular, the elastic model permits us to compare quantitatively the ability of various defects to relax mechanical stress. Moreover, these findings allow us to understand the progressive building-up of the typical disclination and grain boundary pattern observed for ground states of large 2D spherical crystals.

Elasticity in curved topographies: Exact theories and linear approximations

Almost all available results in elasticity on curved topographies are obtained within either a small curvature expansion or an empirical covariant generalization that accounts for screening between Gaussian curvature and disclinations. In this paper, we present a formulation of elasticity theory in curved geometries that unifies its underlying geometric and topological content with the theory of defects. The two different linear approximations widely used in the literature are shown to arise as systematic expansions in reference and actual space. Taking the concrete example of a two-dimensional crystal, with and without a central disclination, constrained on a spherical cap, we compare the exact results with different approximations and evaluate their range of validity. We conclude with some general discussion about the universality of nonlinear elasticity.

Why large icosahedral viruses need scaffolding proteins

While small single-stranded viral shells encapsidate their genome spontaneously, many large viruses, such as the herpes simplex virus or infectious bursal disease virus (IBDV), typically require a template, consisting of either scaffolding proteins or an inner core. Despite the proliferation of large viruses in nature, the mechanisms by which hundreds or thousands of proteins assemble to form structures with icosahedral order (IO) is completely unknown. Using continuum elasticity theory, we study the growth of large viral shells (capsids) and show that a nonspecific template not only selects the radius of the capsid, but also leads to the error-free assembly of protein subunits into capsids with universal IO. We prove that as a spherical cap grows, there is a deep potential well at the locations of disclinations that later in the assembly process will become the vertices of an icosahedron. Furthermore, we introduce a minimal model and simulate the assembly of a viral shell around a template under nonequilibrium conditions and find a perfect match between the results of continuum elasticity theory and the numerical simulations. Besides explaining available experimental results, we provide a number of predictions. Implications for other problems in spherical crystals are also discussed.