Theory of density waves and organization of proteins in icosahedral virus capsids

Abstract

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 Y_lm 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.

---

Introduction

Almost 70 years ago, Frenkel-Konrath and Williams showed that fully infectious rod-like viral particles of the Tobacco Mosaic Virus (TMV) can be spontaneously formed in the water solution of two molecular components: purified genomic RNA and viral coat protein.¹ A dozen years later, in a similar experiment, “spherical” Cowpea chlorotic mottle virus (CCMV) was recreated in vitro for the first time.² Unlike in TMV, 180 copies of CCMV coat protein self-assemble into a shell with icosahedral symmetry (capsid) enclosing viral genome. Later it was found that in some cases, before the capsid self-assembly, identical proteins are combined into symmetrical capsomeres while still in the solution.^3–12 Thus, some small capsids (containing N = 240 or less proteins) self-assemble directly from individual proteins, while other small and larger capsids are formed from pre-assembled capsomeres. In both cases, no local energy consumption like ATP hydrolysis is needed and by changing the acidity or salinity of the solution, one can reverse initial stages of the capsid formation.^1,13–16 Therefore, as Caspar and Klug (CK) concluded long ago,¹⁷ the self-assembly of proteinaceous viral shells is akin to ordinary crystallization, the theory of which was proposed back in 1937 by L.D. Landau.^18,19

In Landau theory of crystallization and structural phase transitions, these ordering phenomena are considered as condensation of static waves of microscopic density ρ: as a result of a usual structural phase transition from a high-temperature phase, an additional component δρ in the density ρ appears, and the symmetry of the system decreases. L.D. Landau proposed to consider nonequilibrium free energy that is a potential or a functional of the density variation δρ. Within this approach, many interesting structural phase transitions have been studied. For example, Landau theory has allowed to explain the existence and stability of such remarkable objects as quasicrystals,^20–23 phase diagrams and properties of various ferroelectric materials,^24–26 as well as ordering features in different liquid crystal phases.²⁷

Within the framework of Landau theory, the self-assembly of icosahedral protein shells of small spherical viruses was also considered.^28,29 In this modified approach, the crystallization is controlled by the acidity or salinity of the protein solution instead of the temperature, density waves are usual spherical harmonics Y_lm^28,29 and the positions of individual proteins are located at maxima of the interference patterns formed by harmonics that yield a critical contribution to δρ function.

The authors^28,29 assumed that the self-assembly of icosahedral protein shells, like most known structural transitions, is driven by only one irreducible representation (IR). The IRs spanned by spherical harmonics Y_lm are indexed by a wave number l that plays a role analogous to the wave vector in the planar case. Examining icosahedral interference patterns produced by spherical harmonics with the same l, the authors^28,29 deduced structures of small shells assembled from individual proteins and demonstrated that these assemblies correspond to strictly defined odd values of l. Moreover, if l < 45, then for each permitted l there is only one icosahedral shell. In this context, it is interesting to note that the approach^28,29 allowed to rationalize the structures of several shells violating the paradigmatic Caspar and Klug model¹⁷ that describes the structures of most icosahedral capsids.

In the subsequent work,³⁰ the geometric meaning of the interference patterns^28,29 with the first odd l was revealed. It was found that these patterns correspond to chiral icosahedral spherical lattices (SLs) (also called geodesic icosahedral polyhedra) with excluded nodes located at the icosahedron symmetry axes. The first five of these SLs describe protein positions in small (N ≤ 240) shells formed from individual proteins.³⁰

In ref. 31 and 32 the modified Landau theory analogous to Brazovskii approach³³ was proposed. To explain the selectivity in the formation of left- and right-handed structures, the authors proposed a nonequilibrium thermodynamic functional, which lifts the corresponding energy degeneration. It disappeared due to specific interactions between the density waves with close even and odd values of the wave number l. In addition, in contrast to the earlier version of the crystallization theory,^28,29 where the maxima of δρ were associated with individual proteins, in ref. 31 and 32, the maxima of this function were considered as positions of capsomer centers.

Let us note, however, that a theory that rationalizes the capsids assembled from individual proteins must predict their coordinates. In addition, the handedness of any real capsid is determined precisely by the chiral shape of its asymmetric proteins, just as the modulus of the chirality vector and, accordingly, the radius of the shell are determined by the intrinsic curvature of structural units (SUs).³⁴ Thus, for such a theory to be considered as a complete one, it must also account for the fact that proteins are asymmetric and extended SUs. Moreover, we argue that if it were possible to synthesize proteins with a handedness, which is opposite to that selected by nature, then all the thermodynamic characteristics of the mirror-symmetric shell would be the same.

In the following, we start our consideration from small icosahedral capsids assembled from individual proteins. Within the framework of the theory of crystallization on a spherical surface, for the first time we consider proteins as non-point SUs and take into account their elongated chiral shape. The protein positions, as in earlier works,^28,29 are obtained as maxima of the statistical density distribution function δρ, which is an irreducible critical density wave with icosahedral symmetry and an appropriate wave number. The protein orientation is described by another irreducible critical icosahedral function ξ, whose spatial derivatives determine the basis vectors of the local coordinate system, with respect to which the local orientation of individual proteins remains constant. Within the framework of the proposed thermodynamic theory, we discuss the features of the positional and orientational order in small capsids formed from individual proteins. Then, again applying the approach of critical density waves, we develop a theory that describes the structures of icosahedral capsids assembled from capsomeres. Analyzing all structural types of these middle-size viral shells from the Protein Data Bank archive (PDB),³⁵ we show that the organization of capsomeres in them is determined by no more than two critical icosahedral density waves with close wave numbers.

Results and discussion

Positional and orientational ordering of proteins in the shells of small viruses

Following ref. 28–30, we represent the density distribution of protein molecules on the sphere as

ρ = ρ₀ + δρ, (1)

where ρ₀ is the microscopic density of the initial phase, δρ is the density variation describing the ordering of SUs. A phase transition induces spontaneous symmetry breaking, and in the ordered low-symmetry phase, there is a non-zero δρ determining the distribution of protein positions (e.g., the positions of their mass centers). Function δρ is expressed as a linear combination of spherical harmonics Y_lm.

Spherical viral shells have mostly icosahedral symmetry, so, we limit our study to this case only. Below, when constructing a phenomenological theory, we a priori assume that δρ has either I or I_h symmetry. Note that unlike ref. 28 and 29, we additionally consider functions δρ with I_h symmetry, since the location of chiral SUs at the maxima of such a function can result in a structure with I symmetry. For the small shells analyzed in this section, we suppose that only one critical IR is substantial, and the contribution of the remaining non-critical degrees of freedom to the value of δρ and crystallization is negligible. For each such IR indexed by the wave number l, the critical variation δρ_l, which represents an irreducible critical density wave (ICDW), can be written as

where A_lm are the amplitudes of the spherical harmonics Y_lm, θ and φ are the azimuthal and polar angles in the spherical coordinate system. Earlier, in ref. 28 and 29, the theory of irreducible icosahedral functions was developed, and it was shown that the wave number l of such function must satisfy the relation:

l = 15i + 10j + 6k, (3)

where i = 0, 1; j and k are positive integers. Solutions of eqn (3) with i = 1 and i = 0 correspond to IRs driving the crystallization of structures with I and I_h symmetry, respectively.

Note that the integrity basis of the group I consists of invariants of the 2nd, 6th, 10th and 15th degrees, while the basis of the group I_h does not include the 15th degree invariant. At the same time, the 2nd degree invariant (which is the radius vector squared) is simply a constant on the sphere. Thus, eqn (3) follows from the properties of the integrity basis considered above.^28,29 We emphasize that eqn (3) is equivalent to the well-known Birman criterion, according to which a phase transition from a high-symmetry phase to a low-symmetry phase is possible if and only if the IR of the order parameter (OP), which is restricted to the symmetry group of the low-symmetry phase, contains a totally symmetric representation.³⁶

The number of different solutions (j, k) of eqn (3) for a given wave number l is equal to the parametricity of δρ_l in the low-symmetry phase. For odd l < 45 and even l < 30, all functions δρ_l are single-parametric, and the irreducible normalized icosahedral functions f_l can be found by simply averaging the spherical harmonics over the group I (see Methods).^28,29 In other words, δρ_l = α_lf_l, where α_l = const and when integrated over the surface of the sphere . The first single-parametric irreducible density waves are shown in Fig. 1. For large l, when more than one solution of eqn (3) exists, the function δρ_l is expressed as , and the amplitudes α⁽ⁱ⁾_l can be determined by minimizing the appropriate free energy. The functions f⁽ⁱ⁾_l do not depend on the form of the free energy and can be found by averaging Y_lm with different values of m.

Fig. 1 First irreducible single-parametric density functions. Only the areas where the functions are positive are shown. The colour change from blue to red corresponds to a growth of the function. Panel (a) shows density waves δρ_l(θ,φ) with odd l permitted by eqn (4) and wave numbers l = 15, 21, 25, 27, 31, 33. These functions have chiral symmetry I. Panel (b) shows density waves δρ_l(θ,φ) with I_h symmetry and even numbers l = 6, 10, 12.

To analyze the structures of middle-size viral shells considered in the next section, along with single-parametric solutions, we use two-parametric solutions as well. In this case, the functions f⁽¹⁾_l and f⁽²⁾_l can be calculated by averaging over the group I the harmonics Y_ll and Y_ll−2 (if l is even) or harmonics Y_ll−1 and Y_ll−3 (if l is odd) (see Methods). After that, the obtained functions should be normalized, and, if necessary, they can always be orthogonalized (e.g., using Gram–Schmidt process).³⁷ Note that the functions f⁽¹⁾_l and f⁽²⁾_l used in the next section are normalized, but the orthogonalization procedure for these functions was not performed. We also note that if for a given l, the basis function is unique, then we do not mark f_l or α_l with a superscript.

The maxima/minima of the functions δρ_l(θ,φ) with odd l always have trivial symmetry and their positions can be filled with asymmetric SUs, on the contrary, this is not so for δρ_l with even l.³⁸ In the latter case it is possible to reduce the I_h symmetry of δρ_l to I symmetry by filling the positions of maxima with suitable chiral n-fold capsomeres. In particular, the 12 maxima of the function f₆ (see Fig. 1b, top row) can be filled with pentamers, and one can place 12 pentamers and 20 hexamers³² at the appropriate maxima of the function f₁₀ (see Fig. 1b, top row). For more detail, see the next section.

To account for the orientation of individual proteins, we introduce another icosahedral function ξ(θ,φ) that sets the basis vectors {e₁, e₂} of the local coordinate system, with respect to which all individual proteins have the same orientation. Suppose that e₁ and e₂ are parallel to the vectors ∇_‖ξ and e_r × ∇_‖ξ, where ∇_‖ = e_θ∂_θ + e_φ∂_φ/sin θ and {e_r, e_θ, e_φ} is the local basis of the spherical coordinate system.

The function ξ(θ,φ) as well as δρ(θ,φ) can be expanded in series of spherical harmonics spanning irreducible representations of O(3). The basis irreducible icosahedral functions f_l for both ξ_l and δρ_l simply coincide: ξ_l = β_lf_l. Note also that the functions ∇_‖f_l and e_r × ∇_‖f_l can be linearly expressed in terms of well-known vector spherical harmonics Y_lm = e_rY_lm, Ψ_lm = ∇Y_lm and Φ_lm = e_r × ∇Y_lm³⁹ with the same value of l. The functions f_l and ∇_‖f_l have the same parity, which is (−1)^l, while the parity of the function e_r × ∇_‖f_l is (−1)^l+1.

Following the standard ideology of Landau theory, we suppose that for a small viral shell, the orientational ordering as well as positional one is driven by only one IR. Then, as we demonstrate below, within this approximation it is possible to rationalize the orientational order of proteins forming the simplest shells of small viruses with N ≤ 240. The top row of Fig. 2 shows five viral shells, in which the mass centers of proteins are located very close to the maxima of the function δρ_l(θ,φ) with l = 15, 21, 25, 27, 31. The bottom row of Fig. 2 shows icosahedral fields ∇_‖f_m with m = 6, 6, 15, 10, 12, respectively. At the points coinciding with the maxima of δρ_l(θ,φ), the local basis vectors e₁ and e₂ (proportional to the values of ∇_‖f_m and e_r × ∇_‖f_m, respectively) are drawn. One can see that individual proteins of the viral shells presented in the top row of the figure have approximately the same orientation with respect to the local coordinate systems drawn in the bottom row.

Fig. 2 Protein orientation in real virus capsids: (a) satellite tobacco mosaic virus (4OQ8); (b) human picobirnavirus (6Z8D); (c) dengue virus (1K4R); (d) melon necrotic spot virus (2ZAH); (e) Sindbis virus (3J0F). (f)–(j) The simplest irreducible fields ∇_‖f_m and e_r × ∇_‖f_m (m = 6, 6, 15, 10, 12) determining the directions of the basis vectors of the local coordinate systems for individual proteins. The lengths of the unnormalized basis vectors (shown by black arrows) are proportional to the values of the icosahedral fields ∇_‖f_m and e_r × ∇_‖f_m Multi-coloured circles (representing the origins of the local coordinate systems) correspond to the maxima of spherical harmonics with l = 15, 21, 25, 27, 31. The coloured line segments illustrate the vector field ∇_‖f_m. The length of the segment is proportional to the field magnitude. The colour intensity gradient from blue to green shows the direction of the field ∇_‖f_m.

The structural analysis presented above was based solely on the symmetry principles. Now, let us discuss a possible construction of thermodynamics within the framework of Landau theory. First of all, we emphasize that proteins are chiral structural units and can be either left-handed or right-handed. In addition, protein synthesis is not a self-assembly process since it is carried out by cellular machinery after infection with a virus. Therefore, the capsid proteins synthesized by the cell have a strictly defined chiral shape which is dictated by the left handedness of amino acids used in all living organisms.

We would like to point out that the origin of chirality in biological nanostructures is still a matter of discussion.^40,41 An interesting experiment, providing a possible explanation for biological homochirality, was carried out quite recently.⁴⁰ The authors⁴⁰ synthesized enantiomorphically pure (either right-handed or left-handed) crystals on a ferromagnetic substrate from a racemic solution of ribo-aminooxazoline (RAO), which is an important precursor of RNA and directly determines the chirality of the assembled RNA. In the experiments,⁴⁰ depending on the spin polarization of the substrate, the chirality of the obtained crystals was also different. Similarly, if it were possible to synthesize capsid proteins with the handedness opposing that observed in vivo, then the assembled shells would also have the opposite handedness but the same thermodynamic properties including energy. Consideration of enantiomorphic shells that model viral capsids yields analogous results.⁴² Thus, although the symmetry group of the homochiral solution of viral proteins is SO(3), the symmetry group of the nonequilibrium Landau free energy must be O(3), which accounts for the virtual possibility of the existence of mirror-symmetric proteins in the solution and naturally results in the same energy (and area at the phase diagram) of enantiomorphic protein shells, one of which is real and the other is virtual.

Suppose that in a small icosahedral shell (see Fig. 2), the positions of proteins are determined by a density wave with the wave number l, and their orientation by a density wave with the wave number m. Then the corresponding nonequilibrium Landau potential can be written in the standard form, typical of the case of one critical and one secondary (improper) order parameters:

F = a₂(T, pH)I₂ + a₄I₄ + a₄I₄ + ⋯ + b₂J₂ + cM, (4)

where a_i, b₂ and c are the coefficients in the Landau expansion, the first of which a₂ depends on external parameters such as temperature, pH of the solution, pressure, concentration of proteins etc.; I_n are the n-th degree invariants of the group O(3) constructed from the coefficients (amplitudes A_lq) of δρ_l(θ,φ) expansion in terms of spherical harmonics Y_lq(θ,φ) (see eqn (2)). To consider the orientational ordering of proteins, in eqn (4), we have also added the invariant terms J₂ and M. The invariant J₂ is quadratic in the amplitudes of ξ_m(θ,φ) expansion in spherical harmonics Y_mp(θ,φ); the invariant M is the simplest “mixed” invariant, which is linear in the amplitudes of ξ_m(θ,φ) expansion.

Note that for the considered odd values of l, the term I₃ (like all other invariants of odd degrees) vanishes identically and the first part of the potential (4), containing terms with coefficients a_i, coincides with the nonequilibrium potential from ref. 28 and 29, even though the latter potential^28,29 was built as an invariant function with respect to SO(3) group.

Let us stress that self-assembly of even a small viral shell is always a first-order transition,^32,43 thus, the potential (4) must contain terms at least up to 6th degree or even higher to ensure the stability of the icosahedral phase. Let us also point out the case when a Landau potential (or a functional) depends not only on δρ but also on the specific spatial derivatives of δρ. Since the term e_r × ∇_‖f_l and the function f_l have different parities, the degeneracy of the left- and right-handed solutions^31,32 can be lifted, and one of the shells becomes more energetically favorable. However, in the light of recent experimental⁴⁰ and theoretical results,⁴² we suppose this as unnecessary complication of the theory, although earlier, following the works,^31,32 we ourselves discussed³⁸ possible terms in the Landau potential leading to an energy asymmetry in left- and right-handed octahedral shells.

After restricting the symmetry of the eqn (4) to icosahedral one, the simplest effective nonequilibrium potential (describing self-assembly of the shells shown in Fig. 2) can be written as:

F = a₂(T, pH)α_l² + a₃α_l⁴ + a₄α_l⁶ + ⋯ + b₂β_m² + c_q1(α_l)^qβ_m, (5)

where α_l and β_m are the amplitudes of δρ_l(θ,φ) and ξ_m(θ,φ) expansions in terms of the corresponding irreducible icosahedral density waves. Note that for odd m, q = 3, while for even m, q = 2.

A detailed study of such potentials can be found in ref. 44. Here we only point out that for odd l, minimization of eqn (5) with respect to α_l and β_m leads to a pair of solutions in the form of (α_l, β_m) and (−α_l, −β_m) (if m is odd) or a pair of solutions (α_l, β_m) and (−α_l, β_m) (if m is even). It occurs since amplitudes with an odd wave number change their sign under spatial inversion, and the amplitudes with an even wave number do not. These two solutions correspond to a virtual and real enantiomorphic shells, which interconverted into each other by spatial inversion. Namely, in the case of odd l (examples of the corresponding shells are shown in Fig. 2), the transformation α_l → −α_l leads to inversion of positional order. Considering the secondary OP, we note that the unit vectors e₁ and e₂ are constructed using icosahedral fields with different parity and regardless of the parity of m, the local coordinate system always changes its handedness under spatial inversion; therefore, the proposed OP correctly describes the orientation of chiral proteins in both enantiomorphic shells.

Above, we discussed only the most general thermodynamic properties of the proposed model simply to ensure that there are no contradictions within its framework and, in particular, the energies of enantiomorphic shells are equal. In the future, the model can be developed in the direction of Brazovsky approach,^31,32 while maintaining its core ideas that the chirality of a viral shell is determined by the specific shape (and handedness) of proteins. In turn, the handedness of proteins stems from the fact they are produced by cellular machinery using exclusively left-handed amino acids.

In the next section, we will consider the relationship between the concept of ICDWs and the structure of viral shells assembled from capsomeres, which are initially formed in a solution. As we show, using one or at most two icosahedral ICDWs with close wave numbers, one can rationalize arrangement of pentamers and hexamers in all known middle-size virus capsids assembled from these units.

Viral shells assembled from capsomeres and the theory of critical density waves

Individual proteins in an icosahedral shell can occupy general crystallographic positions (60-fold orbits) only, so the total number of proteins N must be equal to 60T, where T is the total number of protein orbits.¹⁷ Small capsids with T ≤ 4 can be assembled not only from monomers (individual proteins) but also from capsomeres of the single type, such as dimers, trimers, or pentamers. These symmetrical SUs are initially self-assembled from identical proteins or symmetrically arranged protein domains.

For example, T = 2 structures can be self-assembled not only from 120 monomers like Cystoviridae,⁴⁵ but also from 60 dimers (Partitiviridae) or 12 pentamers composed of dimers (decamers) (Reoviridae and Totiviridae).^4,5 The positional order in the shell comprised of 60 identical dimers can be well described by the irreducible icosahedral wave f₁₅ (see Fig. 1a, top row), since all the maxima of f₁₅ belong to the same orbit, which consists of 60 positions. Similarly, SU positions in the shell composed of 12 identical decamers can be well described by the density wave f₆ (see Fig. 1b, top row).

During the formation of some T = 1 structures, individual proteins can self-assemble into pentamers at the first stage of the virus assembly. The AaLS structure of the protein complex¹² (described by the ICDW f₆) is an example. There are many interesting pseudo-T = 3 structures (Picornaviridae, Dicistroviridae, Flaviridae, Marnaviridae, Secoviridae)⁴⁶ that are self-assembled from 60 trimers (and described by the ICDW f₁₅). An even more intriguing example of self-assembly is observed in structures composed of dimers, which in turn are organized into trimers.^9,10 Such capsomers are also called pseudohexamers. This structural organization is typical of Hepatitis B. Both T = 3 and T = 4 shells of this virus (see Fig. 3a and b) are formed in the cytoplasm during infection, with the majority (approximately 95%) being of T = 4 form.^47,48 The structure of both capsids can be rationalized in terms of a honeycomb lattice, at the vertices of which individual proteins are located. When such a lattice is mapped onto a spherical surface, in addition to hexagonal honeycombs, 12 pentagons will appear around 5-fold icosahedron axes. Furthermore, the hexagon centres (as well as the centres of pentagons) will coincide with the maxima of f₁₈ and f₂₀ harmonics describing T = 3 and T = 4 viral shells, respectively (see Fig. 3c and d).

Fig. 3 Examples of viral shells composed of pseudohexamers. (a) Hepatitis B virus T = 3 (6BVN). In this shell, each pseudohexamer consists of two types of dimers: one with proteins colored in blue and light green, and the other with proteins colored in cyan. The centers of the pseudohexamers coincide with the maxima of the icosahedral density wave f₁₈ shown in panel (c). The maxima of f₁₈ also form the spherical lattice (3,0). (b) Hepatitis B virus T = 4 (6BVF). Here, the pseudohexamers are also comprised of two types of dimers: one with proteins colored in blue and light green, and the other with proteins colored in cyan and green. Their centers coincide with the maxima of the density wave f₂₀ shown in panel (d). The maxima of f₂₀ also form the spherical lattice (2,2).

The T = 3 and T = 4 structures that are self-assembled from monomers, can be described by f₂₇ and f₃₁ harmonics,^28,29 but the same capsids may well be viewed as shells composed of pentamers and hexamers. If the difference between the capsomeres is neglected, then their location in T = 3 and T = 4 shells can be described by f₁₀ and f₁₂ harmonics.³² Nevertheless, although it is possible to distinguish motifs corresponding to pentamers and hexamers in such structures (see Fig. 2d and e), we are not aware of experimental data confirming the formation of pentamers and/or hexamers at intermediate stages of self-assembly of such small viral capsids.

Now, let us consider the larger viral shells with T > 4. First of all, we note that single-shelled capsids with T = 5 are not observed at all. Viral shells with T = 6 belonging to Polyomaviridae and Papillomaviridae families⁴⁶ always have a similar structure and are assembled from 72 identical pentamers, whose centers form a SL typical of capsids with T = 7 (see Fig. 4a). Due to this fact, a shell composed from 72 identical pentamers is also called a pseudo-T = 7 structure. An ordinary T = 7 structure (see Fig. 4b) is self-assembled from 72 capsomeres, 12 of which are pentamers, and another 60 are hexamers (Siphoviridae, Podoviridae). Fig. 4c and d show that capsomeres in T = 7 and pseudo-T = 7 shells are located at the nodes of two enantiomorphic SLs (2,1) and (1,2). To obtain these SLs, it is necessary to consider a combination of two ICDWs with wave numbers l = 15 and l = 16.³² By changing the sign of the harmonic f₁₅, one can obtain either left-handed or right-handed SL (see Fig. 4c and d).

Fig. 4 Viral shells, in which the positions of capsomere centers are characterized by the spherical lattices (2,1) and (1,2): (a) Bovine papillomavirus (3IYJ); (b) HK 97 (3QPR). (c) Two interfering icosahedral density waves with amplitudes (α₁₅, α₁₆) ≈ (1, −1) and superimposed icosahedral spherical lattice (2,1). (d) Two density waves with amplitudes (α₁₅, α₁₆) ≈ (−1, −1) and superimposed icosahedral spherical lattice (1,2). (e) Two density waves with amplitudes (α₄₀, α₄₁) ≈ (1, −1). (f) Density wave corresponding to f₄₁ harmonic.

It is interesting to note that in this and many cases considered below, the coefficients in a linear combination of ICDWs with different but close wave numbers turn out to be close by the absolute value. The latter fact is not surprising, since the icosahedral SLs under consideration are locally similar to an ordinary planar triangular lattice. In the latter case, as is well known,⁴⁹ if the density function ρ(r) of this lattice is expanded in terms of plane waves as , where k is a reciprocal lattice vector, then the amplitudes ρ_k of all waves in this expansion turn out to be equal.

Spherical structures with T = 6 and T = 7 are transient within the ICDW concept (see Fig. 4e and f). In appropriate interference patterns it is still possible to distinguish motifs related to individual proteins. Note in this context that the P22 capsid is self-assembled from 420 individual proteins, while an ordinary viral shell with T = 7 is formed from capsomers (like HK97 capsid shown in Fig. 4b).⁵⁰

Fig. 4e shows an interference pattern generated by two harmonics f₄₀ and f₄₁. As can be seen in this figure, the positions of individual proteins in each pentamer match well the maxima of the interference pattern. However, the latter, along with the maxima describing the protein locations, additionally has 12 pronounced maxima at the icosahedron vertices and 60 weak maxima near the 2-fold axes of icosahedron.

Fig. 4f shows the icosahedral function f₄₁. The maxima of this ICDW are arranged similarly to the proteins in the bacteriophage HK97 shell. Their number also coincides with the number of individual proteins (N = 420) in the capsid. However, these maxima form very distorted hexamers. For structures with T ≥ 7, ICDW combinations describing the positions of individual proteins become essentially multicomponent, and, as the result, their utilization loses its significance. However, the simplest combinations of ICDWs still describe the positions of individual capsomers (pentamers, hexamers or pseudohexamers) in the considered capsids, which, in fact, reflects the actual processes of self-assembly of T ≥ 7 structures.

All known middle-size icosahedral capsids, as well as capsids with T = 7, can be described by the Caspar and Klug model.¹⁷ Such viral shells are composed of pentamers and hexamers, the centers of which are located at the nodes of appropriate SLs. An SL with indices (h,k) has 10T + 2 nodes, where T = h² + k² + hk, and h and k are integers. Twelve of these nodes (which lie on the 5-fold rotational axes) correspond to pentamers, and the entire capsid contains 60T proteins. However, not all middle-size capsids predicted by the CK model have been found.⁴⁵ In fact, only CK structures corresponding to SLs with (h,k) = (2,1), (3,0), (3,1), (3,3), (4,0), (4,1), (4,2), (5,0), (5,1) and (6,1) are known. The SL (6,1) is characterized by T = 43, whereas the next in size experimentally observed shell has T = 169 and like other large and giant virus capsids^46,51 are self-assembled not from pentamers and hexamers, but from large, preassembled clusters, the so-called symmetrons.^52,53 Therefore, this and larger shells are out of the scope of this paper.

Below, within the approach of icosahedral ICDWs, we consider all known middle-size structures with 7 < T ≤ 43 and show that considering the maxima of two or even one critical density wave, one can obtain the location of capsomeres in all such shells. For the sake of clarity, the capsids described by achiral and chiral SLs are considered separately.

The top row of Fig. 5 shows all known³⁵ middle-size CK capsids, which correspond to achiral SLs. Let us note that achiral SLs are produced by icosahedral density waves with even wave numbers only. Furthermore, the first SLs with indices (h,0) correspond to ICDWs with the wave numbers l = 6h. The SLs with indices (h,h), where h = 1,2, correspond to the density waves f₁₀ and f₂₀. The next analogous SLs with h > 2 resemble the interference patterns produced by a pair of waves with l = 10h and l = 10h + 2. However, only in the case h = 3 (presented in Fig. 5h), the maxima of the interference pattern match relatively well the nodes of the SL (3,3).

Fig. 5 Middle-size viral shells corresponding to achiral SLs. Critical density waves for these shells are characterized by even wave numbers l. (a) Helicobacter phage KHP40 (7F2P) with T = 9. (b) Human alphaherpesvirus 3 (6LGL) with T = 16. (c) Chimpanzee adenovirus Y25 (7RD1) with T = 25; (d) Sputnik virophage (3J26) with T = 27. (e) Icosahedral density wave f₁₈ and superimposed SL (3,0). (f) Density wave f₂₄ and superimposed SL (4,0). (g) Density wave f⁽¹⁾₃₀ with superimposed SL (5,0). (h) Two interfering icosahedral density waves with amplitudes (α⁽²⁾₃₀, α₃₂) ≈ (−1, 1) and superimposed SL (3,3).

Fig. 6 shows all the structural types of middle-size capsids with chiral SLs (with T > 7) that can be found in PDB. Let us point out an interesting trend that we have identified while selecting the structures for Fig. 6. With an increase in the wave number l and a corresponding increase in T, the number of isostructural viral shells diminishes drastically. For example, there are about two dozen different T = 13 capsids, while T = 43 shell is unique.^46,51

Fig. 6 Middle-size viral shells described by chiral SLs. (a) Microcystis phage Mic1 (6J3Q); T = 13. (b) Bacteriophage phiCjT23 (7ZZZ); T = 21. (c) Haloarcula hispanica virus SH1 (6QT9); T = 28. (d) Sulfolobus turreted icosahedral virus 1 (3J31); T = 31. (e) Sulfolobus polyhedral virus 1 (6OJ0); T = 43. (f) Two interfering icosahedral density waves with amplitudes (α₂₁, α₂₂) ≈ (1, 1) and superimposed SL (1,3). (g) Two interfering icosahedral density waves with amplitudes (α₂₇, α₂₈) ≈ (−1, −1) and superimposed SL (4,1). (h) Two interfering icosahedral density waves with amplitudes (α₃₁, α₃₂) ≈ (1, 1) and superimposed SL (4,2). (i) Two interfering icosahedral density waves with amplitudes (α₃₃, α₃₄) ≈ (−1, 1) and superimposed SL (5,1). (j) Two interfering icosahedral density waves with amplitudes (α₃₉, α⁽¹⁾₄₀) ≈ (1, −1) and superimposed icosahedral SL (1,6).

The limiting factor T = 43 for the existing middle-size icosahedral viral shells can be easily understood within the developed theory. The SLs of both chiral and achiral shells with T > 43 can no longer be generated by one or at most two icosahedral density waves. As T increases, the value of l increases as well. As the result, the intrinsic parametricity (the number of solutions of eqn (3)) of the corresponding icosahedral waves must also increase. Thus, the viral shells observed in nature have the simplest SLs that can be obtained within the density wave approach.

Additionally, besides predicting the positions of capsomer centres, the developed theory can explain the orientationally consistent arrangement of pentamers and hexamers in the SL nodes. Note that SLs obtained have local symmetry elements. In the middle of each SL edge there is a local 2-fold axis, in the center of each SL triangle there is a 3-fold one, while the node itself (depending on its location) lies either on a global 5-fold axis or on a local 6-fold one. If the lattice is formed from asymmetric proteins, then an additional improper OP is required: it breaks these symmetries and, accordingly, describes the orientation of the asymmetric units. In contrast, when the SL is filled with symmetric capsomeres, there is no need to break the local symmetries, and the improper OP is simply redundant. Therefore, the orientations of neighbouring capsomeres in the capsid turn out to be interconnected by the local 2-fold and 3-fold axes of SLs obtained within our theory. In exactly the same way, orientational coordination of neighbouring capsomeres occurs in the CK geometric model.¹⁷ Surprisingly, our analysis of the Bovine papillomavirus capsid (with pseudo-T = 7 structure) shows that the orientations of three symmetry inequivalent neighbouring pentamers (one of which is located at the icosahedron vertex) are also coordinated with good precision by the corresponding local 3-fold axis.

Within the framework of the proposed density wave approach, it is also interesting to discuss a perfect match (commensurability) between protein shells in double-layered capsids from the families Cystoviridae and Reoviridae. From the structural point of view, this commensurability manifests itself in the fact that each protein of the T = 2 inner shell is located approximately under the center of the corresponding hexamer of the T = 13 outer shell.^30,54 The upper layer of the considered capsids is arranged similarly to the shell shown Fig. 6a and is described by the same combination of f₂₁ and f₂₂ harmonics (see Fig. 6f). The inner layer contains 120 proteins and corresponds to the ICDW f₂₁. Thus, the harmonic f₂₁ turns out to be common for both layers, which demonstrates the commensurability between the layers in terms of density waves.

To conclude this section, let us briefly discuss the construction of effective Landau potentials for the single-layered shells considered in this section. The capsids described by SLs (h,0), where h = 3, 4, 5 and l = 6h, represent the simplest case, which we will consider first. Here, the nonequilibrium effective potential is constructed as a power series expansion in only one amplitude α_l. Since this potential contains a cubic invariant, different signs of α_l correspond to different solutions, only one of which describes the real viral shell. Note that under spatial inversion, α_l does not change its sign and the arrangement of capsomer centers remains the same, however, the capsomeres themselves are transformed into their mirror-symmetric counterparts.

Now, let us proceed to the shells that we have rationalized using two ICDWs. Corresponding effective potential depends on two critical OPs which are the amplitudes of the first and second density wave α⁽ⁱ⁾_l and α^(j)_n, denoted as A and B, respectively. In all the above considered cases when two ICDWs condense on a spherical surface, their wave numbers l and n turn out to be adjacent in the series of values allowed by icosahedral symmetry. This fact can be easily understood within the ideas originally expressed by Lifshitz. Following ref. 55, we can assume that in a solution of proteins, the effective thermodynamic force, that prevents δρ fluctuations, has a quasi-continuous dependence on the wave number l. Under this rather obvious assumption, it becomes clear that in the Landau potential the quadratic terms corresponding to ICDWs with close values of l should have similar dependences on external thermodynamic parameters and such ICDWs tend to condense concurrently.

In the discussed potential, a term A^pB^q is symmetrically allowed if and only if the integral

over the surface S of a unit sphere does not vanish. Therefore, in the effective potential, the terms containing the ICDW amplitudes with an odd wave number should be of even powers only.

Then, in the case when both wave numbers of the first and second ICDWs are even, the potential can be constructed as follows:

F = a₂A² + a₃A³ + a₄A⁴ + ⋯ + b₂B² + b₃B³ + b₄B⁴ + ⋯ + c₁₂AB² + c₂₁A²B + c₂₂A²B² + ⋯, (6)

where a_i, b_i and c_ij are phenomenological coefficients. Usually in potentials of type (6), only a₂ and b₂ depend on external thermodynamic parameters. Note that eqn (6) contains terms, which are cubic in the first and second OPs. Therefore, the operations A → −A and B → −B change the energy (6) and, similarly to the case of shells (h,0), only one of the solutions with correct signs of A and B describes the real viral shell. Under spatial inversion, the arrangement of capsomeres as well as the amplitudes of density waves A and B do not change, but the capsomeres are transformed into their mirror-symmetric counterparts.

In the case when one of the wave numbers is odd (e.g., suppose it's l), the terms containing odd powers of A vanish in the potential (6). Minimization of this potential with respect to A and B will result in a pair of solutions (A, B) and (−A, B). These solutions have the same energy and describe virtual and real enantiomorphic shells interconverted into each other by spatial inversion (which, of course, also changes the capsomer handedness).

Thus, in all the cases considered, which include shells composed of capsomers as well as shells assembled from individual proteins, the effective nonequilibrium potential is invariant with respect to spatial inversion. This fact corresponds to the equality of energies of the real and virtual shells, the latter of which could be constructed using proteins of opposite handedness.

Conclusions

This Article continues the series of studies^28–32 devoted to the theory of protein crystallization on a spherical surface. As in our previous works,^28–30 to describe the positions of proteins and/or capsomeres in viral shells, we introduce the statistical function δρ(θ,φ), which characterizes the critical variation of density distribution and appears only in the ordered phase. This critical variation can be expanded in a series of irreducible critical density waves (ICDWs), which produce an interference pattern. The maxima of the resulting pattern determine the positions of proteins and/or capsomeres. In the case of small capsids, crystallization is driven by a single irreducible representation, so only one ICDW determines the protein positions in such shells. Since the proteins are extended structural units (SUs), in order to give a more complete description of the capsid structure, one needs to know not only the coordinates of proteins, but also their local orientation. To take into account orientational ordering of SUs, we introduce an additional improper order parameter (OP) associated with the function ξ(θ,φ), whose spatial derivatives set the basis of a local coordinate system with respect to which the orientations of proteins remain the same. As we show, in the case of small virus capsids with T ≤ 4, the orientational ordering is also described by a single irreducible representation. Therefore, the explicit form of the function ξ(θ,φ) coincides with that of an appropriate ICDW. Concluding the symmetry analysis of small capsids, we propose a nonequilibrium thermodynamic potential that depends on two order parameters (OPs): one critical and one secondary (improper) OP, which describe the positional and orientational ordering, respectively.

In the article, along with small capsids assembled from individual proteins, we also consider structures of middle-size icosahedral virus shells that are self-assembled from pentamers and hexamers. All known structural types of such shells are characterized by T ≤ 43. The next in size experimentally observed icosahedral capsid has T = 169^46,51 and belongs to the group of large and giant viruses, whose shells are formed not from individual capsomeres, but from large preassembled clusters, the so-called symmetrons.^52,53 In addition, giant shells are characterized by different principles of the structure assembly and protein packing,⁵⁶ and, therefore, are not considered in the present study.

In this context, it is also interesting to note that small viruses are always round and giant viruses have a pronounced icosahedral faceting, while the maturation process of middle-size icosahedral viruses is very often accompanied by a change in the shape from practically spherical to faceted icosahedral, although other scenarios are still possible. The appearance of icosahedral faceting in the middle-size shells can be well described within the framework of Landau theory.^56,57 The OP of such a transformation is an amplitude of a radial displacement field determined by the function f₆. The change in the sign of the OP turns the icosahedral faceting into a dodecahedral one. As have been shown in ref. 58, the OP associated with a radial displacement field proportional to the function f₁₀ can also contribute to the capsid faceting. However, to maintain clarity and conciseness of our paper, here, we do not consider possible shape changes of the middle-size shells but focus on describing the structural organization of capsomers in these capsids.

As we demonstrate, the positional order of capsomere centers in these shells is described by those SLs, which emerge as superpositions of no more than two ICDWs. Furthermore, the orientational improper OP is unnecessary since the local symmetry axes of the SLs are preserved and the 2-fold and 3-fold local axes located between the nearest SL nodes dictate the mutual orientation of capsomers as it occurs in the CK model¹⁷ (see also ref. 59).

Thus, in order to describe the structures of all known middle-size icosahedral shells with T ≤ 43, it is sufficient to consider the crystallization with only one or two ICDWs with close symmetry-allowed wave numbers. However, not all icosahedral shells with T ≤ 43 described by Caspar and Klug model are experimentally observed. Among these not observed structures, we have found only one shell (characterized by spherical lattice (3,2) and T = 19) that can be rationalized in terms of two ICDWs [with amplitudes (α₂₅, α₂₆) ≈ (1, −1)]. Therefore, the condensation of no more than two ICDWs seems to be an important general principle governing the structures of middle-size icosahedral shells.

We argue that any nonequilibrium Landau potential describing self-assembly of individual proteins or capsomeres should be invariant with respect to O(3) group rather than SO(3) group. Viral proteins, like the proteins of all other living organisms, are assembled by cellular machinery exclusively from left-handed amino acids. The latter fact explains the absolute selectivity in the formation of naturally occurring left-handed or right-handed biological structures, including viral shells. However, this selectivity bears no relation to the nonequilibrium Landau potential, which describes self-assembly from pre-made SUs. Assembly of a small viral shell from proteins that are already present in a solution is a purely physical process, which does not require the presence of a spherical substrate in the form of a genome.⁶⁰ Correspondingly, if proteins of opposite handedness were to appear in the solution, we would observe an analogous self-assembly process of a mirror-symmetric shell. Therefore, the binding energy, phase diagram, and other thermodynamic characteristics of the left- and right-handed shells must be exactly the same. The theory developed here explicitly takes this into account and all the nonequilibrium potentials proposed in the current study have two enantiomorphic solutions with equal energy, which correspond to the observed and virtual (mirror-symmetric) viral shells. In this context, the mechanism first proposed in ref. 31 and 32, which explains the prevalence of capsids with a specific chirality by the fact that one of the structures is more energetically favorable, seems unlikely to us.

In conclusion, we have developed a phenomenological thermodynamic model that describes the structures of small and middle-size viral shells. The model develops Landau's theory of crystallization and uses his idea that phase transitions and, in particular, crystallization are driven by a minimum number of critical degrees of freedom. Landau theory is a reliable symmetry-based approach, which allows one to obtain general information about the structural organization of the considered system, without making any assumptions about real interactions between its structural units. This is both an advantage of the theory and its shortcoming. If it is necessary to model details of the internal structure of the capsid, for example the actual chemical bonds between individual proteins or capsomeres, our symmetry approach would be interesting to combine with a patchy particle model, which has had success in modeling many soft matter systems.^61–64 Our results may be of interest for a wide range of scientists developing the theory of phase transitions and for specialists involved in rationalizing the structure of protein viral shells for various biotechnological applications and the development of new antiviral strategies.

Methods

Parametricity of a solution

The number η of different icosahedral irreducible density waves with a given l (in other words, parametricity) can be directly found by using the Birman criterion,³⁶i.e. by calculating the number of times the considered representation of O(3) group, restricted to I or I_h group, subduces the identity representation:where for both I and I_h groups, the summation can be carried out over the elements ĝ of the group I, |G| is the group order equal to 60, χ(ĝ) are the characters of O(3) symmetry elements.

If for a given l, the parametricity η is not zero, this implies that there are η functions with the wave number l that are invariant with respect to I or I_h group. Note that for the considered representations of O(3) with parity (−1)^l, the characters χ(ĝ) of both rotational and mirror elements coincide and are found as:

where ω is the rotation angle defined by the element ĝ. Then the parametricity η for both I and I_h symmetries readsLet us note that eqn (8) gives the same series of allowed values of l as eqn (3).

Averaging of spherical harmonics over the group I

After the symmetry-allowed values of the wave number l are established, the spherical harmonics Y_lm can be averaged using the following expression:where the summation is carried out over the elements ĝ of the group I.

If the parametricity η equals 1 and the wave number l is fixed, then the averaging procedure (9) performed for different m ∈ [−l,l] gives either 0 or, up to an amplitude A, the same function f_l(θ,φ). The result of the averaging procedure (9) can be always made real and normalized by choosing an appropriate value of A.

Let us consider first the even values of l. if η = 1, then it is convenient to average a spherical harmonic of the highest order Y_ll. If η = 2, then in order to obtain f⁽¹⁾_l and f⁽²⁾_l, we used the following harmonics: Y_ll and Y_ll−2. In the case of odd l and η = 1, it is convenient to use the spherical harmonic Y_ll−1. If η = 2, then to obtain f⁽¹⁾_l and f⁽²⁾_l, we used the harmonics Y_ll−1 and Y_ll−3.

Note that spherical function Y_lm can be calculated by repeatedly applying the lowering operator l̂₋ = −z(∂_x − i∂_y) + (x − iy)∂_z to the harmonic Y_ll = (x + iy)^l, where i is the imaginary unit.⁶⁵

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors acknowledge financial support from the Russian Science Foundation, grant no. 22-12-00105.

Notes and references

1. H. Fraenkel-Conrat and R. C. Williams, Proc. Natl. Acad. Sci. U. S. A., 1955, 41, 690–698.
2. J. B. Bancroft, G. J. Hills and R. Markham, Virology, 1967, 31, 354–379.
3. B. V. V. Prasad and M. F. Schmid, Advances in Experimental Medicine and Biology, in Viral Molecular Machines, ed. M. G. Rossmann and V. B. Rao, Springer, 2012, vol. 726, pp. 7–47.
4. D. Luque, J. M. González, D. Garriga, S. A. Ghabrial, W. M. Havens, B. Trus, N. Verdaguer, J. L. Carrascosa and J. R. Castón, J. Virol., 2010, 84, 7256–7266.
5. M. M. Poranen and D. H. Bamford, Advances in Experimental Medicine and Biology, in Viral Molecular Machines, ed. M. G. Rossmann and V. B. Rao, Springer, 2012, vol. 726, pp. 379–402.
6. H. S. Savithri and M. R. N. Murthy, Curr. Sci., 2010, 98, 346–351.
7. D. Sirohi and R. J. Kuhn, J. Infect. Dis., 2017, 216, S935–S944.
8. Z. Gao, H. Pu, J. Liu, X. Wang, C. Zhong, N. Yue, Z. Zhang, X. B. Wang, C. Han, J. Yu, D. Li and Y. Zhang, Mol. Plant-Microbe Interact., 2021, 34, 49–61.
9. K. Holmes, D. A. Shepherd, A. E. Ashcroft, M. Whelan, D. J. Rowlands and N. J. Stonehouse, J. Biol. Chem., 2015, 290, 16238–16245.
10. G. K. Shoemaker, E. Van Duijn, S. E. Crawford, C. Uetrecht, M. Baclayon, W. H. Roos, G. J. L. Wuite, M. K. Estes, B. V. V. Prasad and A. J. R. Heck, Mol. Cell. Proteomics, 2010, 9, 1742–1751.
11. R. C. Liddington, Y. Yan, J. Moulai, R. Sahli, T. L. Benjamin and S. C. Harrison, Nature, 1991, 354, 278–284.
12. E. Sasaki, D. Böhringer, M. Van De Waterbeemd, M. Leibundgut, R. Zschoche, A. J. R. Heck, N. Ban and D. Hilvert, Nat. Commun., 2017, 8, 14663.
13. J. B. Bancroft, Adv. Virus Res., 1970, 16, 99–134.
14. L. O. Liepold, J. Revis, M. Allen, L. Oltrogge, M. Young and T. Douglas, Phys. Biol., 2005, 2, S166.
15. R. Lata, J. F. Conway, N. Cheng, R. L. Duda, R. W. Hendrix, W. R. Wikoff, J. E. Johnson, H. Tsuruta and A. C. Steven, Cell, 2000, 100, 253–263.
16. I. M. Yu, W. Zhang, H. A. Holdaway, L. Li, V. A. Kostyuchenko, P. R. Chipman, R. J. Kuhn, M. G. Rossmann and J. Chen, Science, 2008, 319, 1834–1837.
17. D. L. D. Caspar and A. Klug, Cold Spring Harbor Symp. Quant. Biol., 1962, 27, 1–24.
18. L. D. Landau, Zh. Eksp. Teor. Fiz., 1937, 11, 19.
19. L. D. Landau, Zh. Eksp. Teor. Fiz., 1937, 11, 627.
20. P. Bak, Phys. Rev. B: Condens. Matter Mater. Phys., 1985, 32, 5764–5772.
21. L. Gronlund and N. D. Mermin, Phys. Rev. B: Condens. Matter Mater. Phys., 1988, 38, 3699–3710.
22. M. V. Jarić, Phys. Rev. Lett., 1985, 55, 607–610.
23. P. A. Kalugin, A. I. Kitaev and L. S. Levitov, ZhETF Pisma Redaktsiiu, 1985, 41, 119–121.
24. J. Mangeri, Y. Espinal, A. Jokisaari, S. Pamir Alpay, S. Nakhmanson and O. Heinonen, Nanoscale, 2017, 9, 1616–1624.
25. M. A. Pavlenko, F. Di Rino, L. Boron, S. Kondovych, A. Sené, Y. A. Tikhonov, A. G. Razumnaya, V. M. Vinokur, M. Sepliarsky and I. A. Lukyanchuk, Crystals, 2022, 12, 453.
26. S. Kondovych, M. Pavlenko, Y. Tikhonov, A. Razumnaya and I. Lukyanchuk, SciPost Phys., 2023, 14, 56.
27. S. Singh, Phys. Rep., 2000, 324, 107–269.
28. V. L. Lorman and S. B. Rochal, Phys. Rev. Lett., 2007, 98, 4–7.
29. V. L. Lorman and S. B. Rochal, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 77, 224109.
30. S. B. Rochal, O. V. Konevtsova, A. E. Myasnikova and V. L. Lorman, Nanoscale, 2016, 8, 16976–16988.
31. S. Dharmavaram, F. Xie, W. Klug, J. Rudnick and R. Bruinsma, Europhys. Lett., 2016, 116, 26002.
32. S. Dharmavaram, F. Xie, W. Klug, J. Rudnick and R. Bruinsma, Phys. Rev. E, 2017, 95, 62402.
33. S. Brazovskii, Zh. Eksp. Teor. Fiz., 1975, 68, 175–185.
34. J. Wagner and R. Zandi, Biophys. J., 2015, 109, 956–965.
35. H. M. Berman, T. N. Bhat, P. E. Bourne, Z. Feng, G. Gilliland, H. Weissig and J. Westbrook, Nat. Struct. Biol., 2000, 7, 957–959.
36. J. L. Birman, Phys. Rev. Lett., 1966, 17, 1216–1219.
37. S. J. Leon, Å. Björck and W. Gander, Numer. Linear Algebra Appl., 2013, 20, 492–532.
38. D. V. Chalin and S. B. Rochal, Phys. Rev. B, 2023, 107, 24102.
39. R. G. Barrera, G. A. Estevez and J. Giraldo, Eur. J. Phys., 1985, 6, 287–294.
40. S. F. Ozturk, Z. Liu, J. D. Sutherland and D. D. Sasselov, Sci. Adv., 2023, 9, eadg8274.
41. N. Globus and R. D. Blandford, Astrophys. J., 2020, 895, L11.
42. S. B. Rochal, O. V. Konevtsova, I. Y. Golushko and R. Podgornik, Soft Matter, 2023, 19, 8649–8658.
43. J. M. Johnson, J. Tang, Y. Nyame, D. Willits, M. J. Young and A. Zlotnick, Nano Lett., 2005, 5, 765–770.
44. Y. A. Izyumov and V. N. Syromyatnikov, Phase transitions and crystal symmetry, Springer Science & Business Media, 1990, vol. 38.
45. D. Nemecek, E. Boura, W. Wu, N. Cheng, P. Plevka, J. Qiao, L. Mindich, J. B. Heymann, J. H. Hurley and A. C. Steven, Structure, 2013, 21, 1374–1383.
46. C. Hulo, E. De Castro, P. Masson, L. Bougueleret, A. Bairoch, I. Xenarios and P. Le Mercier, Nucleic Acids Res., 2011, 39, D576–D582.
47. L. M. Stannard and M. Hodgkiss, J. Gen. Virol., 1979, 45, 509–514.
48. K. A. Dryden, S. F. Wieland, C. Whitten-Bauer, J. L. Gerin, F. V. Chisari and M. Yeager, Mol. Cell, 2006, 22, 843–850.
49. N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt-Saunders, 1976.
50. D. Veesler and J. E. Johnson, Annu. Rev. Biophys., 2012, 41, 473–496.
51. C. M. Shepherd, I. A. Borelli, G. Lander, P. Natarajan, V. Siddavanahalli, C. Bajaj, J. E. Johnson, C. L. Brooks and V. S. Reddy, Nucleic Acids Res., 2006, 34, D386–D389.
52. N. G. Wrigley, J. Gen. Virol., 1969, 5, 123–134.
53. V. F. Manyakov, J. Gen. Virol., 1977, 36, 73–79.
54. O. V. Konevtsova, D. S. Roshal, A. Lošdorfer Božič, R. Podgornik and S. Rochal, Soft Matter, 2019, 15, 7663–7671.
55. E. M. Lifshitz, Zh. Eksp. Teor. Fiz., 1941, 11, 269.
56. S. B. Rochal, O. V. Konevtsova and V. L. Lorman, Nanoscale, 2017, 9, 12449–12460.
57. O. V. Konevtsova, V. L. Lorman and S. B. Rochal, Phys. Rev. E, 2016, 93, 52412.
58. O. V. Konevtsova, D. S. Roshal, R. Podgornik and S. B. Rochal, Soft Matter, 2020, 16, 9383–9392.
59. V. V. Pimonov, O. V. Konevtsova and S. B. Rochal, Acta Crystallogr., Sect. A: Found. Adv., 2019, 75, 135–141.
60. S. J. Flint, L. W. Enquist, R. M. Krug, V. R. Racaniello and A. M. Skalka, Principles of virology: molecular biology, pathogenesis, and control, ASM Press, Washington, D.C., 2000.
61. Z. Zhang and S. C. Glotzer, Nano Lett., 2004, 4, 1407–1413.
62. J. P. Doye, A. A. Louis, I. C. Lin, L. R. Allen, E. G. Noya, A. W. Wilber, H. C. Kok and R. Lyus, Phys. Chem. Chem. Phys., 2007, 9, 2197–2205.
63. H. Liu, S. K. Kumar and F. Sciortino, J. Chem. Phys., 2007, 127, 084902.
64. D. Fusco and P. Charbonneau, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2013, 88, 012721.
65. J. P. Elliott and P. G. Dawber, Symmetry in Physics Vol 1: Principles and Simple Applications, University of Sussex, Brighton, 1987.

---