Density-wave theory of the capsid structure of small icosahedral viruses

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.

Minimal Design Principles for Icosahedral Virus Capsids

The geometrical structures of single- and multiple-shell icosahedral virus capsids are reproduced as the targets that minimize the cost corresponding to relatively simple design functions. Capsid subunits are first identified as building blocks at a given coarse-grained scale and then represented in these functions as point particles located on an appropriate number of concentric spherical surfaces. Minimal design cost is assigned to optimal spherical packings of the particles. The cost functions are inspired by the packings favored for the Thomson problem, which minimize the electrostatic potential energy between identical charged particles. In some cases, icosahedral symmetry constraints are incorporated as external fields acting on the particles. The simplest cost functions can be obtained by separating particles in disjoint nonequivalent sets with distinct interactions, or by introducing interacting holes (the absence of particles). These functions can be adapted to reproduce any capsid structure found in real viruses. Structures absent in Nature require significantly more complex designs. Measures of information content and complexity are assigned to both the cost functions and the capsid geometries. In terms of these measures, icosahedral structures and the corresponding cost functions are the simplest solutions.

Relationships between RNA topology and nucleocapsid structure in a model icosahedral virus

The process of genome packaging in most of viruses is poorly understood, notably the role of the genome itself in the nucleocapsid structure. For simple icosahedral single-stranded RNA viruses, the branched topology due to the RNA secondary structure is thought to lower the free energy required to complete a virion. We investigate the structure of nucleocapsids packaging RNA segments with various degrees of compactness by small-angle x-ray scattering and cryotransmission electron microscopy. The structural differences are mild even though compact RNA segments lead on average to better-ordered and more uniform particles across the sample. Numerical calculations confirm that the free energy is lowered for the RNA segments displaying the larger number of branch points. The effect is, however, opposite with synthetic polyelectrolytes, in which a star topology gives rise to more disorder in the capsids than a linear topology. If RNA compactness and size account in part for the proper assembly of the nucleocapsid and the genome selectivity, other factors most likely related to the host cell environment during viral assembly must come into play as well.

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.

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.

Kinetics of empty viral capsid assembly in a minimal model

The efficient construction of a protective protein shell or capsid is one of the most crucial steps in the replication cycle of a virus. The formation of the simplest capsid typically proceeds by the spontaneous assembly of identical building blocks. This process can also be achieved in vitro even in the absence of genetic material, thus opening the door to the production of artificial viral cages for a myriad of applications. In this work, we analyze the efficiency and the kinetic peculiarities of this self-assembly process using Brownian Dynamics simulations. We use a minimal model that considers identical assembly units and is able to reproduce successfully the correct final architecture of spherical capsids. The selection of a specific size and structure is achieved by changing a single parameter that imposes an angular anisotropy on the interaction. We analyze how the geometrical constraints of the interaction affect the efficiency of the assembly. We find that the optimal conditions for an efficient assembly from a kinetic point of view strongly depart from the lowest capsid energy corresponding to the minimum of the potential energy landscape. Our work illustrates the important differences between the equilibrium and dynamic characteristics of viral self-assembly, and provides important insights on how to design specific interactions for a successful assembly of artificial viral cages.

Invariant theory and orientational phase transitions

The Landau theory of phase transitions has been productively applied to phase transitions that involve rotational symmetry breaking, such as the transition from an isotropic fluid to a nematic liquid crystal. It even can be applied to the orientational symmetry breaking of simple atomic or molecular clusters that are not true phase transitions. In this paper, we address fundamental problems that arise with the Landau theory when it is applied to rotational symmetry breaking transitions of more complex particle clusters that involve order parameters characterized by larger values of the l index of the dominant spherical harmonic that describes the broken symmetry state. The problems are twofold. First, one may encounter a thermodynamic instability of the expected ground state with respect to states with lower symmetry. A second problem concerns the proliferation of quartic invariants that may or may not be physical. We show that the combination of a geometrical method based on the analysis of the space of invariants, developed by Kim to study symmetry breaking of the Higgs potential, with modern visualization tools provides a resolution to these problems. The approach is applied to the outcome of numerical simulations of particle ordering on a spherical surface and to the ordering of protein shells.

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.

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.

Static and dynamic hidden symmetries of icosahedral viral capsids

Viral shells self-assemble from identical proteins, which tend to form equivalent environments in the resulting assembly. However, in icosahedral capsids containing more than 60 proteins, they are enforced to occupy not only the symmetrically equivalent locations but also the quasi-equivalent ones. Due to this important fact, static and dynamic symmetries of viral shells can include additional hidden components. Here, developing the Caspar and Klug ideas concerning the quasi-equivalence of protein environments, we derive the simplest hexagonal tilings, that in principle could correspond to the local protein order in viral shells, and apply the resulting theory to nucleocytoplasmic large dsDNA viruses. In addition, analyzing the dynamic symmetry of the P22 viral shell, we demonstrate that the collective critical modes responsible for the protein reorganization during the procapsid maturation are approximately equivalent to the normal modes of the isotropic spherical membrane with O(3) symmetry. Furthermore, we establish the relationship between the dynamic symmetry of the P22 procapsid and the protein arrangement regularities that appear only in the mature capsid.

Orientational phase transitions and the assembly of viral capsids

We present a Landau theory for large-l orientational phase transitions and apply it to the assembly of icosahedral viral capsids. The theory predicts two distinct types of ordering transitions. Transitions dominated by the l=6,10,12, and 18 icosahedral spherical harmonics resemble robust first-order phase transitions that are not significantly affected by chirality. The remaining transitions depend essentially on including mixed l states denoted as l=15+16 corresponding to a mixture of l=15 and l=16 spherical harmonics. The l=15+16 transition is either continuous or weakly first-order and it is strongly influenced by chirality, which suppresses spontaneous chiral symmetry breaking. The icosahedral state is in close competition with states that have tetrahedral, D_{5}, and octahedral symmetries. We present a group-theoretic method to analyze the competition between the different symmetries. The theory is applied to a variety of viral shells.

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.

Identification of a major intermediate along the self-assembly pathway of an icosahedral viral capsid by using an analytical model of a spherical patch

Viruses are astonishing edifices in which hundreds of molecular building blocks fit into the final structure with pinpoint accuracy. We established a robust kinetic model accounting for the in vitro self-assembly of a capsid shell derived from an icosahedral plant virus by using time-resolved small-angle X-ray scattering (TR-SAXS) data at high spatiotemporal resolution. By implementing an analytical model of a spherical patch into a global fitting algorithm, we managed to identify a major intermediate species along the self-assembly pathway. With a series of data collected at different protein concentrations, we showed that free dimers self-assembled into a capsid through an intermediate resembling a half-capsid. The typical lifetime of the intermediate was a few seconds and yet the presence of so large an oligomer was not reported before. The progress in instrumental detection along with the development of powerful algorithms for data processing contribute to shedding light on nonequilibrium processes in highly complex systems such as viruses.

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.

Modeling Viral Capsid Assembly

I present a review of the theoretical and computational methodologies that have been used to model the assembly of viral capsids. I discuss the capabilities and limitations of approaches ranging from equilibrium continuum theories to molecular dynamics simulations, and I give an overview of some of the important conclusions about virus assembly that have resulted from these modeling efforts. Topics include the assembly of empty viral shells, assembly around single-stranded nucleic acids to form viral particles, and assembly around synthetic polymers or charged nanoparticles for nanotechnology or biomedical applications. I present some examples in which modeling efforts have promoted experimental breakthroughs, as well as directions in which the connection between modeling and experiment can be strengthened.

A minimal representation of the self-assembly of virus capsids

Viruses are biological nanosystems with a capsid of protein-made capsomer units that encloses and protects the genetic material responsible for their replication. Here we show how the geometrical constraints of the capsomer-capsomer interaction in icosahedral capsids and the requirement of low frustration fix the form of the shortest and universal truncated multipolar expansion of the two-body interaction between capsomers. The structures of many of the icosahedral and related virus capsids are located as single lowest energy states of a potential energy surface built from this interaction. Our minimalist representation is consistent with other models known to produce a controllable and efficient self-assembly, and unveils relevant features of the natural design of the capsids. It promises to be very useful in physical virology and may also be of interest in fields of nanoscience and nanotechnology where similar hollow convex structures are relevant.

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.

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.

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.

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.

Vibrational dynamics of icosahedrally symmetric biomolecular assemblies compared with predictions based on continuum elasticity

Coarse-grained elastic network models elucidate the fluctuation dynamics of proteins around their native conformations. Low-frequency collective motions derived by simplified normal mode analysis are usually involved in biological function, and these motions often possess noteworthy symmetries related to the overall shape of the molecule. Here, insights into these motions and their frequencies are sought by considering continuum models with appropriate symmetry and boundary conditions to approximately represent the true atomistic molecular structure. We solve the elastic wave equations analytically for the case of spherical symmetry, yielding a symmetry-based classification of molecular motions together with explicit predictions for their vibrational frequencies. We address the case of icosahedral symmetry as a perturbation to the spherical case. Applications to lumazine synthase, satellite tobacco mosaic virus, and brome mosaic virus show that the spherical elastic model efficiently provides insights on collective motions that are otherwise obtained by detailed elastic network models. A major utility of the continuum models is the possibility of estimating macroscopic material properties such as the Young's modulus or Poisson's ratio for different types of viruses.

Topological control of life and death in non-proliferative epithelia

Programmed cell death is one of the most fascinating demonstrations of the plasticity of biological systems. It is classically described to act upstream of and govern major developmental patterning processes (e.g. inter-digitations in vertebrates, ommatidia in Drosophila). We show here the first evidence that massive apoptosis can also be controlled and coordinated by a pre-established pattern of a specific ‘master cell’ population. This new concept is supported by the development and validation of an original model of cell patterning. Ciona intestinalis eggs are surrounded by a three-layered follicular organization composed of 60 elongated floating extensions made of as many outer and inner cells, and indirectly spread through an extracellular matrix over 1200 test cells. Experimental and selective ablation of outer and inner cells results in the abrogation of apoptosis in respective remaining neighbouring test cells. In addition incubation of outer/inner follicular cell-depleted eggs with a soluble extract of apoptotic outer/inner cells partially restores apoptosis to apoptotic-defective test cells. The 60 inner follicular cells were thus identified as ‘apoptotic master’ cells which collectively are induction sites for programmed cell death of the underlying test cells. The position of apoptotic master cells is controlled by topological constraints exhibiting a tetrahedral symmetry, and each cell spreads over and can control the destiny of 20 smaller test cells, which leads to optimized apoptosis signalling.

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.