J. Math. Biol. (2012) 64:745–773 DOI 10.1007/s00285-011-0425-5

Mathematical Biology

A crystallographic approach to structural transitions in icosahedral viruses Giuliana Indelicato · Paolo Cermelli · David G. Salthouse · Simone Racca · Giovanni Zanzotto · Reidun Twarock

Received: 21 December 2010 / Revised: 4 April 2011 / Published online: 25 May 2011 © Springer-Verlag 2011

Abstract Viruses with icosahedral capsids, which form the largest class of all viruses and contain a number of important human pathogens, can be modelled via suitable icosahedrally invariant finite subsets of icosahedral 3D quasicrystals. We combine concepts from the theory of 3D quasicrystals, and from the theory of structural phase transformations in crystalline solids, to give a framework for the study of the structural transitions occurring in icosahedral viral capsids during maturation or infection.

G. Indelicato (B) · D. G. Salthouse · S. Racca · R. Twarock Department of Mathematics, York Centre for Complex Systems Analysis, University of York, Heslington, York, UK e-mail: [email protected] D. G. Salthouse e-mail: [email protected] S. Racca e-mail: [email protected] R. Twarock Department of Biology, York Centre for Complex Systems Analysis, University of York, Heslington, York, UK e-mail: [email protected] P. Cermelli Dipartimento di Matematica, Università di Torino, Torino, Italy e-mail: [email protected] G. Zanzotto Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Padova, Padova, Italy e-mail: [email protected]

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As 3D quasicrystals are in a one-to-one correspondence with suitable subsets of 6D icosahedral Bravais lattices, we study systematically the 6D-analogs of the classical Bain deformations in 3D, characterized by minimal symmetry loss at intermediate configurations, and use this information to infer putative viral-capsid transition paths in 3D via the cut-and-project method used for the construction of quasicrystals. We apply our approach to the Cowpea Chlorotic Mottle virus (CCMV) and show that the putative transition path between the experimentally observed initial and final CCMV structures is most likely to preserve one threefold axis. Our procedure suggests a general method for the investigation and prediction of symmetry constraints on the capsids of icosahedral viruses during structural transitions, and thus provides insights into the mechanisms underlying structural transitions of these pathogens. Keywords Virus structure · Structural transitions in viral capsids · Bain strain · Phase transitions in quasicrystals Mathematics Subject Classification (2000)

92B05 · 74N05 · 52C25

1 Introduction Simple viruses are composed of a protein container, called the viral capsid, and genomic RNA or DNA. Structural transitions of these viral capsids, such as maturation events, are important for the infectivity of virus particles. They are therefore important targets of anti-viral therapy. Structural transitions have been intensively studied from both an experimental and a theoretical point of view (Conway et al. 2001; Rim et al. 2010; Robinson and Harrison 1982; Sherman et al. 2006; Tama and Brooks 2002, 2005). However, information on the configurations of a capsid during its transition between the (stable) start and end states of a dynamic event in the viral life-cycle are notoriously difficult to derive, both experimentally and via current computational biophysical modelling techniques. This is, among others, due to the many degrees of freedom available to the capsomers during the transition process. In this paper we explore how concepts from crystallography, which are useful in the kinematics of reconstructive structural phase transformations in crystalline solids (Bhattacharya et al. 2004; Christian 2002; Toledano and Dmitriev 1996; Wayman 1964), can be combined with concepts derived from the higher-dimensional crystallographic theory of 3D quasicrystals, to provide insights into structural properties of viral capsid during such dynamic events. The majority of viral capsids exhibit non-crystallographic icosahedral symmetry before as well as after their structural transition. Due to this, their geometries can be modelled via icosahedrally invariant point arrays that are obtained as suitable finite subsets of the vertex sets of icosahedral quasicrystals in three dimensions (Keef and Twarock 2009). Such approximant point arrays, which can be thought of as geometric boundary conditions on the layout of viral particles, can be associated with any given capsid structure (Keef et al. 2011). An example of such an approximation is shown for Cowpea Chlorotic Mottle virus (CCMV) in Fig. 1.

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Fig. 1 A 3D point array modelling the layout of a viral capsid. The approximant point array is shown superimposed on a the viral capsid seen from the outside, b a top view of a hexamer, and c a side view of a hexamer

The structural transitions of a capsid can be studied via an analysis of how the approximant icosahedral point arrays transform into each other during a structural transition. We know from the classical theory of quasicrystals (see for instance Senechal 1995) that icosahedral point arrays can be obtained as projections into 3D space of suitable subsets of the icosahedral [i.e. the simple (SC), body-centered (BCC), and face-centered cubic (FCC)] Bravais lattices in 6D (Levitov and Rhyner 1988). We can therefore, in principle, study the transitions between two viral configurations (represented by the two corresponding icosahedral point arrays) in terms of transitions between the two 6D lattices that generate them via projection. In order to obtain information on the possible capsid transitions we propose to consider, as candidate paths for the 6D lattice distortions, suitable 6D analogs of the ‘Bain deformations’. These, and related concepts, provide an important class of transition paths for the investigation of structural transformations in 3D crystalline materials (Alippi et al. 1997; Bain 1924; Bhattacharya et al. 2004; Boyer et al. 1991; Boyer 1989; Capillas et al. 2007; Kaxiras and Boyer 1992; Mehl et al. 2004; Pitteri and Zanzotto 1998, 2002; Sowa and Koch 2002; Toledano and Dmitriev 1996). They are characterized by the property of having minimal symmetry reduction at intermediate states, i.e. the configurations along a Bain path admit as symmetry group a maximal common subgroup of the symmetry groups of the initial and final states of a transition. In this paper, we develop a general procedure to determine the maximal symmetry that intermediate configurations of a structural transition may have based on these hypotheses. The strategy followed by our procedure is illustrated in Fig. 2. Based on the approximant icosahedral point sets we determine all possible transition paths between each one of the point arrays corresponding to the pre- and post-transition configurations of the viral capsid under consideration. Note that the approximant point array is not necessarily unique as different members of an ensemble of point arrays may provide equally good approximations of capsid geometry according to Keef et al. (2011). We derive the full list of admissible 6D Bain transformations between the 6D lattices associated with all possible initial and final icosahedral point arrays. By construction, these paths entail minimal intermediate symmetry loss, described, respectively, by the three maximal subgroups of the icosahedral group, i.e. the tetrahedral group A4 , and the dihedral groups D10 and D6 . The corresponding set of 6D Bain deformations is obtained by computing the centralizer in G L(6, R)

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Fig. 2 Sketch of the procedure: Bain-like transitions in 6D can be used to study conformational changes of models of 3D viral capsids

of each such maximal subgroup. In this way, we can determine for any virus the putative maximal symmetry conserved during the transition process. This information provides important insights into the mechanism underlying the structural transition. In order to illustrate our approach, we perform a systematic study of the Bain deformations for the 6D lattices related the CCMV. As 3D icosahedral point sets approximating the CCMV initial and final capsid configurations, we use the 3D point arrays obtained through affine extensions of the icosahedral group, as proposed and listed in Keef and Twarock (2009). An investigation of the CCMV capsid structures following the procedure in Keef et al. (2011) showed that the experimental data in http://viperdb.scripps.edu/ are compatible with two icosahedral point arrays from the list in Keef and Twarock (2009) for the pre-transition configuration, and 10 point arrays for the post-transition configuration (Wardman, private communication, 2010). We show here that only three of the point arrays for capsid geometry after expansion can be connected to one of the initial configurations by a Bain-type transition path. Moreover, we show that the maximal symmetry of this transition is D6 , which implies that a threefold axis plays a pivotal role in CCMV transitions. This paper is organised as follows: Sect. 2 briefly discusses how the configurations of an icosahedral capsid can be approximated by suitable quasicrystalline icosahedral point arrays which are in turn derived from suitable 6D icosahedral Bravais lattices. Section 3 relates the transitions between such 3D icosahedral point arrays to 6D lattice transition paths and gives a method for the determination of admissible Bain paths with highest intermediate lattice symmetry. Section 4 illustrates the procedure based on the example of CCMV. Section 5 gives some final comments and places our results in the wider context of models for structural transitions in viruses available in the literature. The Appendices summarize various technical information needed throughout the text.

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2 Quasicrystalline approximations of icosahedral viral capsids and their relation to 6D lattices 2.1 Discrete approximations of capsid geometry The structures of icosahedral viral capsids abide to a set of geometric boundary conditions that are encoded by a library of 3D point arrays and stems from a classification of affine extensions of the icosahedral group (Keef and Twarock 2009). The points of these arrays map around structurally or functionally important geometric features of the capsid (Keef et al. 2011). Janner pioneered the idea of representing the layouts of proteins and in particular of viruses via lattices (Janner 2005, 2006). Our approach is a natural generalisation of this, as it uses, following Keef et al. (2011), 3D icosahedrally invariant point arrays as discrete models for capsid geometry, which we prove here to be finite subsets of the vertex sets of icosahedral quasi-lattices. The library of point arrays in Keef and Twarock (2009), which we use here as discrete models for virus architecture, has a simple description in terms of superpositions of multiple translated copies of the vertex sets of the polyhedra in Fig. 3. Indeed, the descriptor of any such point array is a ‘skeletal’ double-shell structure, that is given as the union of two nested polyhedra (either an icosahedron, a dodecahedron or an icosidodecahedron) with specific relative radii corresponding to the classification in Keef and Twarock (2009). The full point array is obtained via translation of the double-shell structure by one of a finite set of values, listed in Keef and Twarock (2009), along the directions of a common symmetry axis of the polyhedra forming the double-shell. Details of the construction are summarised in Appendix A. We use here the following notation for the point arrays in Keef and Twarock (2009): The icosahedral group, denoted by I, is the 60-element group given by   I := a, b | a 2 = b3 = (ab)5 = 1 . Let I3 denote a 3D representation of I. If u, r, s are vectors pointing along a two, three, or fivefold axis of I3 , then the icosahedral orbits I3 u, I3 r and I3 s correspond to the standard polyhedra in Fig. 3 (cf. also Definition 7 in Appendix A). For every element S in the classification of icosahedral point arrays in Keef and Twarock (2009), there exists u, r and s as above such that S ≡ S(u, r, s) = I3 u ∪ I3 r ∪ (I3 u + I3 s) ∪ (I3 r + I3 s).

(1)

In the following, we call S a viral configuration, or in short, a point array. Experimental data concerning the structures of viral capsids are available as pdbfiles from either the Protein data Bank or the ViPER website (http://viperdb.scripps. edu/). Via the algorithm in Keef et al. (2011) it is possible to associate with the pdb-file of any given virus the appropriate viral configuration that best encodes the features of its geometry. For CCMV, the point arrays have been determined based on the pdb-files with ID 1cwp for the pre-, and ccmv_swln_1 for the post-transition configuration

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(Wardman, private communication, 2010). Point arrays corresponding to these configurations are shown in Fig. 4. The algorithm proposed in Keef et al. (2011) determines the icosahedral point set that best encodes the geometrical features of the viral capsid, choosing the set from the library of the affine extensions with icosahedral symmetry (Keef and Twarock 2009). Starting from the pdb data, the algorithm first locates the outermost features of the virus, and rescales and aligns the icosahedral point sets so that the external points of the set are as close as possible to them. Then, a root mean square deviation analysis (RMSD) is performed to score how well the point sets fit around the proteins of the capsid. A second score (topology score) is calculated, measuring how well the sets of points encode the features of the external surface of the capsid. The final ranking of the points set results from a combination of the RMSD and the topology scores (see also Keef and Twarock 2010 for examples).

Fig. 3 The standard polyhedra with icosahedral symmetry: the icosahedron, the dodecahedron, the icosidodecahedron

Fig. 4 First row The pre-transition state of CCMV. a Double shell skeletal structure; b the corresponding point array; c the point array superimposed to the pre-transition state of CCMV. Second row The post-transition state of CCMV. d Double shell skeletal structure; e the corresponding point array; f the point array superimposed to the post-transition state of CCMV

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After an introduction to 6D icosahedral lattices in the next subsection, we will show that these point arrays are finite subsets of the vertex sets of suitable 3D icosahedral quasicrystals and that they can be uniquely associated with icosahedral Bravais lattices in 6D. 2.2 6D icosahedral lattices Given a basis {bα }α=1,...,6 in R6 , we indicate by B ∈ G L(6, R) the matrix with columns given by the components of the vectors bα in the canonical basis {eα }α=1,...,6 of R6 . We write   6  α α L(B) = x = m bα : m ∈ Z α=1

for the Bravais lattice with basis B. Note that other lattice bases have the form B M, with M ∈ G L(6, Z). Throughout this paper we assume that all bases fulfill det B > 0, i.e. they have the same orientation in R6 . Different aspects of the symmetry of a lattice are encoded by the following groups (Pitteri and Zanzotto 2002). Definition 1 The lattice group of a lattice L(B) is given by (B) = {M ∈ G L(6, Z) : ∃Q ∈ S O(6) such that Q B = B M}, and its point group is P(B) = {Q ∈ S O(6) : ∃M ∈ G L(6, Z) such that Q B = B M}. The point group and the lattice group are related via the identity (B) = B −1 P(B)B.

(2)

The lattice group is therefore the representation of the point group in the lattice basis. The point group hence only depends on the lattice and is invariant under changes of the lattice basis, while lattice groups associated with different bases of the same lattice are conjugated in G L(6, Z). In what follows we will use the simple cubic, body-centered cubic and face-centered cubic 6D lattices, given by see, e.g. Levitov and Rhyner (1988): L SC = {x = (x1 , . . . , x6 ) : xi ∈ Z, i = 1, . . . , 6},   L BCC = x = 21 (x1 , . . . , x6 ) : xi ∈ Z, xi = x j mod 2, i, j = 1, . . . , 6 ,

 L FCC = x = 21 (x1 , . . . , x6 ) : xi ∈ Z, x j = 0 mod 2 .

(3)

Note that L SC ⊂ L FCC and L BCC ⊂ L FCC . In particular, since the canonical basis B = I is a basis of the SC lattice, all its bases are given by integer matrices

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in G L(6, Z). Moreover, the point group PC of the SC lattice [of (6!25 )-elements] coincides with the lattice group in the canonical basis, so that PC = S O(6) ∩ G L(6, Z). Note that the point groups of the three cubic lattices in (4) coincide, whilst their lattice groups do not. Indeed, they are not conjugate in G L(6, Z) (this is the defining property for lattices with distinct Bravais types, Pitteri and Zanzotto 2002). A 6D representation of the generators of I in (2.1) in the canonical basis of R6 , which is also a basis of the simple cubic lattice, is given by cf. e.g., Katz (1989) ⎞ ⎛ ⎞ 0 −1 0 0 0 0 −1 0 0 0 0 0 ⎜0 ⎜ 0 −1 0 0 0 0 ⎟ 0 −1 0 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ 0 0 0 0 0 0⎟ 0 0 0 1 0⎟ ⎟. ⎜1 b = a=⎜ ⎜0 ⎜ 0 0 0 0 0 1⎟ 0 0 0 0 1⎟ ⎟ ⎜ ⎟ ⎜ ⎝0 ⎝ 0 0 0 1 0 0⎠ 0 1 0 0 0⎠ 0 0 0 0 1 0 0 0 0 1 0 0 ⎛

(4)

The matrix group I leaves all three 6D cubic lattices (SC, FCC, BCC) invariant, and is a subgroup of their common cubic point group in the standard basis. Therefore, these lattices are all invariant under the icosahedral group. As the three 6D cubic lattices are the only 6D lattices with this property (Levitov and Rhyner 1988), they are also often referred to as the ‘icosahedral’ 6D lattices. The action of I on R6 decomposes into two non-equivalent irreducible representations of dimension 3 (Katz 1989). Therefore, the 6D space splits into the direct sum of two 3D subspaces which are invariant under icosahedral symmetry and are orthogonal to each other. In analogy to the notation used in the study of quasicrystals, we refer to these spaces as the physical (or parallel) space, and the orthogonal space, denoted by E  and E ⊥ , respectively, and we write π : R6 → E  and π ⊥ : R6 → E ⊥ for the projections onto the parallel and orthogonal spaces. In particular, following Katz (1989), we choose as physical space E  the subspace in which the orthogonal projection of to the vertices of the standard basis (eα )α=1,...,6 corresponds to the vectors pointing √ an icosahedron (see Table 1). With the notation τ = 21 (1 + 5), and τ = −1/τ , the projection π has the following matrix representation in the standard basis: ⎛

⎞ τ 0 −1 0 τ 1 ⎝1 τ 0 −τ −1 0 ⎠ . 0 1 τ 1 0 τ

(5)

We denote by I3 the 3D representation of I on E  . Q ∈ I3 corresponds to Q ∈ I in this representation if Q π v = π Qv,

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Table 1 6D vectors and their 3D projections corresponding to the vertices of the standard polyhedra Orbits in 6D modulo inversion

3D representation

Icosahedron (1, 0, 0, 0, 0, 0)

(τ, 1, 0)

(0, 1, 0, 0, 0, 0)

(0, τ, 1)

(0, 0, 1, 0, 0, 0)

(−1, 0, τ )

(0, 0, 0, 1, 0, 0)

(0, −τ, 1)

(0, 0, 0, 0, 1, 0)

(τ, −1, 0)

(0, 0, 0, 0, 0, 1)

(1, 0, τ )

Dodecahedron 1 2 (1, −1, 1, 1, −1, 1) 1 2 (−1, 1, −1, 1, 1, 1) 1 2 (1, −1, −1, −1, −1, 1) 1 2 (1, 1, −1, 1, −1, 1) 1 2 (1, 1, 1, 1, −1, −1) 1 2 (1, 1, 1, −1, 1, −1) 1 2 (−1, 1, −1, 1, −1, −1) 1 2 (−1, −1, 1, −1, −1, 1) 1 2 (1, −1, 1, 1, 1, −1) 1 2 (1, −1, −1, 1, −1, −1)

Icosidodecahedron 1 2 (1, 0, 0, −1, 0, 0) 1 2 (0, 1, 0, 0, 0, 1) 1 2 (0, 1, 1, 0, 0, 0) 1 2 (0, 0, 0, 1, 0, 1) 1 2 (1, 0, 0, 0, 0, 1) 1 2 (1, 0, 0, 0, 1, 0) 1 2 (0, 0, 0, 1, 1, 0) 1 2 (0, 0, 0, 0, 1, 1) 1 2 (0, 0, 1, 1, 0, 0) 1 2 (0, 0, 1, 0, −1, 0) 1 2 (1, 0, −1, 0, 0, 0) 1 2 (1, 1, 0, 0, 0, 0) 1 2 (0, 0, 1, 0, 0, 1) 1 2 (0, 1, 0, −1, 0, 0) 1 2 (0, 1, 0, 0, −1, 0)

(0, 1 − τ, τ ) (1, −1, 1) (1, 1, −1) (1, 1, 1) (−1, 1, 1) (τ − 1, τ, 0) (−τ, 0, 1 − τ ) (−τ, 0, −1 + τ ) (−1 + τ, −τ, 0) (0, 1 − τ, −τ ) 1 2 2 (τ, τ , −1) 1 2 2 (1, τ, τ ) 1 (−1, τ, τ 2 ) 2 1 2 2 (1, −τ, τ ) 1 2 2 (τ , 1, τ )

(τ, 0, 0) 1 2 2 (τ, −τ , 1) 1 (τ 2 , −1, τ ) 2 1 2 2 (−1, −τ, τ ) 1 2 2 (−τ , 1, τ ) 1 (τ 2 , 1, −τ ) 2 1 2 2 (τ, τ , 1)

(0, 0, τ ) (0, τ, 0) 1 2 2 (−τ, τ , 1)

Vectors are listed modulo inversion

for all v ∈ R6 . Hence, icosahedral orbits in R6 project to icosahedral orbits in R3 = E  . The same property holds for any subgroup G ⊂ I: G-orbits in R6 project to G3 -orbits in R3 , where G3 is the 3D representation of G in E  .

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In addition to the point group operations, an important symmetry operation of the FCC and BCC lattices in 6D is the quasi-dilatation, with the following matrix representation in the standard basis: ⎞ ⎛ 1 1 −1 −1 1 1 ⎜ 1 1 1 −1 −1 1 ⎟ ⎟ ⎜ 1⎜ −1 1 1 1 −1 1 ⎟ ⎟. ⎜ D= ⎜ 1 1 1 1⎟ 2 ⎜ −1 −1 ⎟ ⎝ 1 −1 −1 1 1 1⎠ 1 1 1 1 1 1 The BCC and FCC lattices are invariant under the action of D, while the SC lattice is invariant under D 3 . The transformation D belongs to the centralizer of I in G L(6, R), i.e., for every M ∈ I, D −1 M D = M, and D acts as a dilatation by a factor of τ in E  and by −τ in E ⊥ , i.e., π Dv = τ π v and π ⊥ Dv = −τ π ⊥ v for every vector v ∈ R6 . In general, we have π D k v = τ k π v and π ⊥ D k v = (−τ )k π ⊥ v for every vector v ∈ R6 and k ∈ Z. 2.3 Embedding of capsid geometry into a 6D icosahedral lattice via the cut-and-project method In order to relate a point array of the form (1) with a 6D lattice, we use the following facts: (i) The projection π is one-to-one onto its image when restricted to L, since E  is totally irrational (i.e., E  ∩ L = {0}) with respect to L SC , L FCC and L BCC . (ii) The icosahedral group commutes with the projection, so that the 6D preimages of the standard icosahedral polyhedra are, in turn, icosahedral orbits. (iii) A dilatation by a factor of τ in 3D corresponds to a symmetry operation of the cubic lattices in 6D. We use the following three-step procedure: • Step 1: We embed the standard polyhedra in Fig. 3 (see also Appendix A) into 6D according to Table 1: the standard icosahedron can be obtained via projection of the icosahedral orbit of the canonical basis vector e1 of the SC lattice, the standard icosidedecahedron via projection of the icosahedral orbits of the FCC lattice vector 1 2 (e1 + e2 ), and the standard dodecahedron via projection of the icosahedral orbit of the BCC lattice vector 21 (e1 + e2 − e3 + e4 − e5 + e6 ). • Step 2: Since the skeletal double shell structures of the point arrays are combinations of two standard polyhedra at different scalings of τ k with k ∈ Z (see Keef and Twarock 2009; Tables 2, 3), we create the 6D counterparts of standard polyhedra rescaled by τ via the action of the√quasidilatation D. For example, the 6D embedding of a icosahedron of length τ k 2 + τ , k ∈ Z is given by the icosahedral orbit of the rescaled vector D k e1 . • Step 3: The translation vectors s ∈ R3 , along which the double-shell structure is translated to generate the point arrays, also belongs to a rescaled standard poly-

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hedron. Therefore t = π −1 (s) and all its orbit belong to one of the icosahedral lattices. Hence, we associate with each point array S in (1) a unique set  of points in either L SC , L FCC or L BCC . It is called the lifted viral configuration or lifted point array, and fulfills π() = S. It follows from (1) that  is the union of icosahedral orbits in

Table 2 List of the standard polyhedra and the corresponding translations, relevant to the description of CCMV

Shell #

St. polyhedron

Translation

Scaling factor

10

ICO

DOD

τ2

11

ICO

ICO

−τ

12

ICO

ICO

1

13

ICO

ICO

τ

19

DOD

ICO

−2τ

26

DOD

DOD

τ2

27

DOD

ICO

τ 2

29

DOD

ICO

1

30

DOD

ICO

τ

44

IDD

DOD

51

IDD

ICO

52

IDD

ICO

1 2τ 1 2 1 2τ

53

IDD

ICO

1

54

IDD

ICO

1 2 2τ

55

IDD

ICO

τ

Table 3 List of generating vectors and minimal lattices for each viral configuration relevant to CCMV Viral conf.

Lattice

v

w

t

10-44

SC

26-44

BCC

(1, 0, 0, 0, 0, 0)

(1, 1, 0, 0, 1, 1)

(1, 1, 0, 0, 0, 1)

(1, 0, 0, 1, 0, −1)

(1, 0, 0, 0, 1, 0)

BCC

1 2 (1, −1, 1, 1, −1, −1)

1 2 (1, 1, −1, 1, 1, −1)

1 2 (1, 1, 1, 1, −1, −1) 1 2 (3, −1, 1, 1, −1, −1)

11-27 12-27

SC

(1, 0, 0, 0, 0, 0)

(1, 1, 0, 0, 0, 1)

(1, 0, 0, 0, 0, 0)

13-27

BCC

(1, 1, 1, 1, 1, −2)

1 2 (1, 1, 1, 1, −1, −1)

1 2 (3, −1, 1, 1 − 1, −1)

27-51

SC

(0, 1, 1, 0, −1, 0)

(0, 1, 0, 0, −1, 0)

(1, 0, 0, 0, 0, 0)

27-52

SC

(0, 1, 0, −1, −1, 0)

(0, 0, 1, 0, 1, 0)

(1, 0, 0, 0, 0, 0)

27-53

FCC

1 2 (1, −1, −1, −1, −1, 1)

1 2 (1, 0, 1, 1, 0, −1)

1 2 (3, −1, 1, 1, −1, −1)

27-54

SC

(0, 1, 0, −1, −1, 0)

(1, 0, 1, 0, 1, −1)

(1, 0, 0, 0, 0, 0)

27-55

FCC

1 2 (1, 1, 0, 2, 0, −2) 1 2 (1, 3, −1, 3, 1, −3)

1 2 (3, −1, 1, 1, −1, −1)

(1, 0, 0, 0, 0, 0)

(1, −1, 1, 0, 0, 0)

(1, 0, 0, 0, 0, 0)

27-29

BCC

1 2 (1, −1, 1, 1, 1, −1) 1 2 (1, 1, −1, 1, −1, 1)

27-30

SC

(1, 1, 0, 0, 0, 1)

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R6 and their suitable translates:  = (v, w, t) = Iv ∪ Iw ∪ (Iv + I t) ∪ (Iw + I t),

(6)

where π v = u, π w = r, π t = s. By construction, all points of a given lifted viral configuration  are points of some icosahedral lattice, and there exists a unique minimal such lattice that contains a given lifted viral configuration. We say that the lifted point array is embedded into such a lattice, and the following characterization holds: Proposition 1 The minimal lattice containing a given lifted viral configuration  is icosahedral. Proof Let  = {v 1 , . . . , v N } ∪ {w1 , . . . , w M } ∪ {t 1 , . . . , t K },where {v 1 , . . . , v N } and {w1 , . . . , w M } are distinct I-orbits, and let {t 1 , . . . , t K } be the icosahedral orbit of the translation vector. Consider the Z-module ⎧ ⎫ N M K ⎨ ⎬    L = x ∈ R6 : x = ni vi + m jwj + pk t k , ni , m j , pk ∈ Z . ⎩ ⎭ i=1

j=1

k=1

Since {v 1 , . . . , v N } , {w1 , . . . , w M }, and {t 1 , . . . , t K } are icosahedral orbits, L is invariant under I. Furthermore, L is a lattice, because {v 1 , . . . , v N } and {w1 , . . . , w M }, and {t 1 , . . . , t K } are all lattice vectors of the maximal lattice FCC. It is always possible to extract a basis of R6 from , since otherwise the subspace generated by , being invariant under I, would coincide either with E  or E ⊥ , both irrational with respect to the cubic lattices. But the elements of  are lattice vector, which leads to a contradiction. Further, L contains , and it follows that L is a 6D icosahedral lattice (cf. Senechal 1995). Finally, L is also minimal, since every lattice that contains  must also contain all integer linear combinations of vectors of , that 

is, L . A given viral configuration can trivially be embedded in infinitely many other lattices but the minimal lattice constructed in the above proposition is unique. Also, from the above proposition it follows that, since the point arrays (1) can be obtained by projection of a 6D lattice onto a completely irrational icosahedrally invariant subspace in 3D, they are subsets of an (aperiodic) icosahedral 3D quasicrystal (see also Senechal 1995). The following definition is central in our approach to the study of structural transformations of viral configurations. Definition 2 Given a lifted viral configuration  = (v, w, t), any basis {bα }α=1,...,6 of R6 such that (i) {bα } is a basis for the minimal icosahedral lattice containing ; (ii) every basis vector bα belongs to either Iv, Iw, or I t; (iii) each orbit contains at least one basis vector; is called an admissible basis for the lifted viral configuration.

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The second condition requires that the vectors of an admissible basis are representatives of icosahedral orbits of the skeletal vectors and of the translation. Since the icosahedral orbits of these three vectors are used to construct  (cf. (6)), there exists a unique  for each basis B. We write this unique viral configuration as  = (bα ) = (B).

(7)

Note that all admissible bases for a given lifted viral configuration are bases of the same 6D lattice. Furthermore, applying a rotation R ∈ I to the vectors of an admissible basis yields another admissible basis for the same lifted viral configuration, and the same is true for permutations and change of signs of the basis vectors. 3 Capsid transitions as 6D lattice transitions In the previous section we have associated a unique 6D icosahedral lattice to any given viral configuration. In this section we investigate transitions between lattices as a means of understanding transitions between the viral configurations that are embedded into them. 3.1 6D lattice transitions For a matrix group G ⊂ G L(6, Z) we have the following standard definitions (Pitteri and Zanzotto 2002): Definition 3 The normalizer N (G, K ) of G in G L(6, K ), with K = Z, Q or R, is

N (G, K ) = N ∈ G L(6, K ) : N −1 G N = G ; the centralizer Z(G, K ) of G in G L(6, K ) is

Z(G, K ) = N ∈ G L(6, K ) : N −1 G N = G, ∀G ∈ G . We also introduce the set Z + (G, K ), defined to be the connected component of the identity of the set {N ∈ Z(G, K ) : det N > 0} ⊂ G L(6, R). We now define a lattice phase transition as a continuous transformation between two lattices L(B0 ) and L(B1 ) along which some symmetry is preserved, described by a common maximal subgroup G ⊂ G L(6, Z) of the lattice groups (B0 ) and (B1 ). We restrict our attention to the icosahedral lattices L SC , L FCC or L BCC (so that, in what follows, every lattice is understood to be one of those defined in (4)), and to the maximal subgroups G ⊂ I. The latter condition characterizes the ‘Bain-type’ lattice transformations, as mentioned in Sect. 1.

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Definition 4 Let L0 and L1 be two icosahedral lattices, and let G0 be a maximal subgroup of I. We say that there exists a transition between L0 and L1 with intermediate symmetry G0 if there exists a continuous path B : [0, 1] → G L(6, R) such that (i) B(0) = B0 is a basis of L0 and B(1) = B1 is a basis of L1 ; (ii) Letting G = B0−1 G0 B0 ⊂ (B0 ), we have (B(σ )) ⊇ G, σ ∈ [0, 1].

(8)

We call the linear mapping T := B1 B0−1 : R6 → R6 ,

(9)

the transition, while the curve T (σ ) = B(σ )B0−1 is referred to as the transition path. Notice that, in the above definition, we only require that all the intermediate lattices have at least symmetry G0 : along the transition path the symmetry may increase for selected values of σ . Further, Definition 4 could be given in the more general case where I is replaced by any symmetry group of the lattice. Here we use I since we are interested in the minimal symmetry group that guarantees icosahedral symmetry after projection, because we are considering applications to icosahedral viruses. The following statements characterise lattice transitions: Proposition 2 Let G0 be a maximal subgroup of I. The following statements are equivalent: (i) There exists a transition between L0 and L1 with intermediate symmetry G0 . (ii) There exist bases B0 and B1 of L0 and L1 , such that B1 = R N B0 , where R ∈ S O(6) and N ∈ Z + (G0 , R). (iii) There exist a basis B0 of L0 , and continuous paths R : [0, 1] → S O(6), N : [0, 1] → Z + (G0 , R), such that R(0) = N (0) = I and R(1)N (1)B0 = B1 , is a basis for L1 . Proof (i) ⇒(iii). For σ ∈ [0, 1], let Gσ = B(σ )G B(σ )−1 ; recalling that G0 = B0 G B0−1 , Gσ = B(σ )B0−1 G0 B0 B(σ )−1 .

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By hypothesis, G ⊆ (B(σ )), so that Gσ ⊆ P(B(σ )): hence, G0 and Gσ are finite subgroups of S O(6) that are conjugated in G L(6, R), and it follows that they are also conjugated in S O(6). In fact, defining P(σ ) ∈ S O(6) as the rotation in the polar decomposition of B(σ )B0−1 (Pitteri and Zanzotto 2002; notice that, by the polar decomposition theorem, σ → P(σ ) is continuous), then Gσ = P(σ )G0 P(σ )−1 , which implies G0 = P(σ )−1 B(σ )B0−1 G0 B0 B(σ )−1 P(σ ). Letting K (σ ) = P(σ )−1 B(σ )B0−1 , this identity can be rewritten in the form G0 = K (σ )−1 G0 K (σ ), i.e., K (σ ) belongs to the normalizer of G0 in G L(6, R). On the other hand when G0 is a maximal subgroup of I, every element of the normalizer of G0 ⊂ PC can be written in the form K = Q N , with N in the centralizer of G0 and Q ∈ PC (cf. Appendix C). We conclude that P(σ )−1 B(σ )B0−1 = Q N (σ ), i.e., B(σ ) = R(σ )N (σ )B0 , with R(σ ) = P(σ )Q. (iii) ⇒ (i). Letting B(σ ) = R(σ )N (σ )B0 , then for every σ ∈ [0, 1] B(σ )G B(σ )−1 = R(σ )N (σ )B0 G B0−1 N (σ )−1 R(σ )−1 = R(σ )N (σ )G0 N (σ )−1 R(σ )−1 = R(σ )G0 R(σ )−1 ⊂ S O(6), so that B(σ )G B(σ )−1 ⊆ P(B(σ )) and G ⊆ (B(σ )). (ii)⇔ (iii). By definition, Z + (G0 , R) is a connected submanifold of G L(6, R), and since S O(6) is also connected, there exist continuous paths joining the identity I to R and N , respectively. We may choose for instance R(σ ) = exp(W σ ), with W such that exp(W ) = R (such W may be not unique). The converse assertion is trivial.  In this paper we focus on Bain transitions with G0 a maximal subgroup of I, i.e., one of the following three groups: (i) the tetrahedral group A4 , of order 12 and index 5; (ii) the dihedral group D10 , of order 10 and index 6; (iii) the dihedral group D6 , of order 6 and index 10.

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3.2 6D transitions compatible with projection According to Definition 4, the intermediate lattices in a transition path all have the same symmetry, since G is constant. However, the lattice may undergo a rotation in 6D space. Therefore Gσ is not fixed and its symmetry axes may rotate during the transition. Hence, the invariant subspaces E  and E ⊥ of the icosahedral group I are not in general invariant under the subgroup Gσ of the point group of the intermediate lattices. As a result, orbits of 6D vectors under Gσ do not in general project onto orbits or point sets with any symmetry in E  . To avoid this problem, and to make a general lattice transition compatible with the projection π (in such a way that the 6D symmetry be appropriately related with the symmetry of the projected 3D point sets) it is necessary to introduce the requirement that, during the transition, the parallel space E  be invariant under the point symmetry group Gσ of the intermediate states. Definition 5 With the same notation as in Definition 4, we say that a transition with intermediate symmetry G0 is compatible with projection if, in addition to (8): P(B(σ )) ⊇ Gσ ≡ G0 , σ ∈ [0, 1].

(10)

Since G0 is a subgroup of I, (10) implies that E  and E ⊥ are invariant with respect to G0 , and that G0 -orbits in 6D project onto 3D-orbits of the 3D representation of G0 in E  . The following result characterizes transitions that are compatible with projection in terms of the sole centralizers of the point group G0 in G L(6, R). Proposition 3 For a transition compatible with projection, the following statements are equivalent (cf. Proposition 2): (i) There exists a transition between L0 and L1 with intermediate symmetry G0 . (ii) There exist bases B0 and B1 of L0 and L1 such that B1 = T B0 , with T ∈ Z + (G0 , Q). (iii) There exist a basis B0 of L0 , and continuous paths T : [0, 1] → Z + (G0 , R), such that T (0) = I and where T (1)B0 = B1 is a basis of L1 . Proof We only prove that (i) implies (iii). If (10) holds, then by (2) and (8), G0 = B(σ )G B −1 (σ ),

σ ∈ [0, 1],

yielding G0 = B(σ )B0−1 G0 B0 B −1 (σ ),

123

σ ∈ [0, 1].

(11)

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Hence, T (σ ) = B(σ )B0−1 belongs to the normalizer of G0 in G L(6, R). But T (σ ) is continuous and T (0) = I , and, as shown in Appendix C, the connected component of the identity in the normalizer of G0 in G L(6, R) is contained in the centralizer, which proves the claim. 

3.3 Viral transitions Consider two viral configurations S0 and S1 , with corresponding lifted viral configurations 0 and 1 in 6D. Let 0 be embedded into L0 , and 1 into L1 . Definition 6 A viral transition between two viral configurations S0 and S1 in 3D, with intermediate symmetry G0 , is a transition obtained via the projection π from a transition between the lattices L0 and L1 in 6D, such that (i) B0 and B1 (the bases associated with the transition) are admissible for 0 and 1 , i.e., (cf. (7)) 0 = (B0 ),

1 = (B1 );

(ii) the transition is compatible with projection π . By Proposition 3, given a transition T , the possible viral transition paths are curves in Z + (G0 , R) connecting the identity with T . To derive from these paths the information about the actual intermediate structure of a viral capsid, let T (σ ) ∈ Z + (G0 , R), for σ ∈ [0, 1], be a viral transition path with intermediate symmetry G0 . Let v 0 , w 0 and t 0 be three vectors of the basis B0 such that 0 = (v 0 , w 0 , t 0 ), i.e., v 0 and w0 belong to the skeletal shells, and t 0 is in the orbit of the translation vector. For σ ∈ (0, 1), define v(σ ) = T (σ )v 0 , w(σ ) = T (σ )w0 , t(σ ) = T (σ )t 0 . By definition, v(σ ), w(σ ) and t(σ ) are vectors of the basis B(σ ) = T (σ )B0 of the intermediate lattice. We associate with the transition path T (σ ), for any σ ∈ (0, 1), a lifted viral configuration (σ ) defined as (σ ) = G0 v(σ ) ∪ G0 w(σ ) ∪ (G0 v(σ ) + G0 t(σ )) ∪ (G0 w(σ )+G0 t(σ )) ⊂ L(B(σ )). (12) The resulting point array, when projected to R3 via π , yields a family of non-icosahedral point sets S(σ ) parametrized by σ , with constant G0 -symmetry, that represents

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the intermediate viral configurations, i.e. S(σ ) = G3 u(σ ) ∪ G3 r(σ ) ∪ (G3 u(σ ) + G3 s(σ )) ∪ (G3 r(σ ) + G3 s(σ )),

(13)

where G3 denotes a representation of G0 on E  , and u(σ ) = π v(σ ), r(σ ) = π w(σ ) and s(σ ) = π t(σ ). 3.4 A procedure to determine Bain transitions The above discussion implies that the Bain transitions are special elements of the rational centralizer of the intermediate symmetry group. As shown in Appendix C, the centralizers of the three maximal subgroups of the icosahedral group in G L(6, R) are 6 × 6 matrices whose coefficients depend linearly on a finite number n of parameters, with n = 4, 6, 8 for tetrahedral, D10 and D6 transitions, respectively. Hence, in order to find transitions between two given lifted viral configurations, we use the following strategy to determine their parameter values: we solve a linear system of equations that formulates the requirement that the vectors playing the roles of descriptors of the pre-transitional configuration, be mapped into descriptors of the final configuration. For this, we write T = T ( p), with p = ( p1 , . . . , pn ) ∈ Rn , and 0 = (v 0 , w 0 , t 0 ), 1 = (v 1 , w 1 , t 1 ). Choosing v 0 ∈ Iv 0 , w 0 ∈ Iw0 , t 0 ∈ I t 0 , v 1 ∈ Iv 1 , w1 ∈ Iw1 , and t 1 ∈ I t 1 , we require that v 1 = T ( p)v 0 , w 1 = T ( p)w0 , t 1 = T ( p)t 0 .

(14)

This yields a system of equations for the unknown parameter p, whose solution (if it exists) we denote by p. Notice now that, by construction, T ( p) is in the centralizer of G0 , so that T ( p)Gv = GT ( p)v for every G ∈ G0 and v ∈ R6 . Hence, G0 -orbits in 0 are mapped into G0 -orbits in 1 . If a basis B 0 for the minimal lattice containing 0 can be extracted from the G0 -orbits of v 0 , w 0 , and t 0 , then this basis is automatically admissible for 0 . Moreover, its image B 1 = T ( p)B 0 is also admissible for 1 , and T ( p) is a viral transition between 0 and 1 . Via repeated application of the above procedure for all possible representatives of the orbits Iv 0 , Iw0 , I t 0 and Iv 1 , Iw 1 , I t 1 , we hence obtain all possible viral transitions between 0 and 1 . 4 Results 4.1 The parameter spaces for the 6D Bain transition paths According to Proposition 3, every transition with intermediate symmetry G0 , that is compatible with the projection, must belong to the centralizer of G0 . This means T ∈ Z + (G0 , Q), and the transition paths associated to T are curves T : [0, 1] → Z + (G0 , R),

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such that T (0) = I and T (1) = T . In this sense, the set Z + (G0 , R) is the parameter space for transitions with intermediate symmetry G0 . We have computed in Appendix C the centralizers of the maximal subgroups of I. From this, we have obtained the following results: (i) The parameter space for tetrahedral transition paths is four-dimensional. All tetrahedral transition paths have the form ⎛

z −x −y

−t

⎜ z t ⎜ t ⎜ ⎜ z ⎜ −y −x T (x, y, z, t) = ⎜ ⎜ ⎜ x −t −x ⎜ ⎜ −x −t x ⎝ t

x t z y

t −x

⎟ y⎟ ⎟ ⎟ −t −x ⎟ ⎟, ⎟ y t⎟ ⎟ z t⎟ ⎠ x

t −x −x

y

⎞

(15)

z

where x = x(σ ), y = y(σ ), z = z(σ ), t = t (σ ) with σ ∈ [0, 1] are curves in R4 such that T (x(0), y(0), z(0), t (0)) = I and T (x(1), y(1), z(1), t (1)) is a tetrahedral transition determined as in Sect. 3.4. (ii) The parameter space for D10 -transitions is six-dimensional. All such transition paths have the form ⎛

z ⎜x ⎜ ⎜y T = T (x, y, z, t, u, w) = ⎜ ⎜y ⎜ ⎝x u

x z x y y u

y x z x y u

y y x z x u

⎞ x t y t⎟ ⎟ y t⎟ ⎟, x t⎟ ⎟ z t⎠ u w

(16)

where x = x(σ ), y = y(σ ), z = z(σ ), t = t (σ ), u = u(σ ), w = w(σ ) with σ ∈ [0, 1] are curves in R6 such that T (x(0), y(0), z(0), t (0), u(0), w(0)) = I . T (x(1), y(1), z(1), t (1), u(1), w(1)) is a D10 -transition determined as in Sect. 3.4. (iii) The parameter space for D6 -transition paths is eight-dimensional. All such transitions have the form ⎞ ⎛ u w −w x s s ⎜ −t y v −v z −t ⎟ ⎟ ⎜ ⎜ t v y v t −z ⎟ ⎟, (17) ⎜ T = T (x, y, z, t, u, v, w, s) = ⎜ v y −t −t ⎟ ⎟ ⎜ z −v ⎝ s x −w w u s⎠ s w −x w s u where x = x(σ ), y = y(σ ), z = z(σ ), t = t (σ ), u = u(σ ), v = v(σ ), w = w(σ ), s = s(σ ) with σ ∈ [0, 1], are curves in R8 such that T (x(0),

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y(0), z(0), t (0), u(0), v(0), w(0), s(0)) = I and T (x(1), y(1), z(1), t (1), u(1), v(1), w(1), s(1)) is a D6 -transition determined as in Sect. 3.4. 4.2 Application to CCMV We apply the procedure developed in the previous sections to CCMV capsid transitions during maturation. An analysis of the CCMV capsid structure via the algorithm in Keef et al. (2011) (Wardman, private communication, 2010, based on Proposition 4.2 in Keef and Twarock 2009) shows that the pre-transition geometry of the CCMV capsid is given by one of the two following viral configurations (see Appendix A and Tables 2, 3 for the notations used to label the point-arrays): S0 ∈ {10-44, 26-44},

(18)

while the swollen form of CCMV is best approximated by one of the ten following viral configurations: S1 ∈ {11-27, 12-27, 13-27, 27-29, 27-30, 27-51, 27-52, 27-53, 27-54, 27-55}. (19) Examples for such point sets for the pre- and post-transition structures are shown in Fig. 4. We have determined all possible Bain transitions with capsid configurations given by a point array S0 in (18), and post-transition configuration given by an array S1 in (19). The results are as follows: (i) There exist no Bain transitions with either A4 or D10 symmetry between any of the initial and final configurations. (ii) There exist four Bain transitions with D6 symmetry, mapping the initial configuration 10-44 into one of the final configurations 27-52, 11-27 and 12-27. These transitions are listed in Appendix B, together with the bases of the initial configuration involved in each transition. Without further assumptions on the energetic contributions to these transitions, it is not possible to fully determine the exact transition path. However, our analysis provides fundamental insights into the likely symmetry of the transition path. Our results for CCMV imply that the configurations of the CCMV capsid during the transition will not be icosahedral, unlike the start and end structure, and will at most have D6 symmetry. D6 , a subgroup of the icosahedral group, has a representation as the dihedral group of a triangular prism, with one distinguished threefold axis. Assuming that the particle will maximise symmetry throughout the transition, this implies that a threefold axis will play a crucial role during the structural transition. This is a very interesting result because it relates to a phenomenon observed for transition events of other viruses. Indeed, structural transitions seem to start at a given symmetry axis of a spherical particle, e.g. a fivefold axis, and then propagate over the surface of the capsid like a circular wave until the entire particle has undergone the transition (Steven, private communication, 2010). This implies that intermediate configurations are presumably preserving one of the symmetry axis, and our method can be used to determine it a priori which symmetry axis this is most likely to be.

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5 Conclusive remarks The approach outlined in this paper gives a general method for the investigation and prediction of the symmetry properties of icosahedral viral capsids during structural transitions important for infection. We have developed a method that can be used to predict the symmetry of intermediate capsid configurations based on concepts from the theory of lattice transitions using discrete approximations of capsid geometry in terms of the 3D point arrays in Keef et al. (2011). As an example, we have considered in detail the case of CCMV, and have determined the likely symmetry of the intermediate capsid configurations. The proposed methods can be applied to the study of any icosahedral virus, and may give guidelines for future experimental or numerical work on the structural transitions of these viruses. For instance, in combination with considerations regarding the energetics governing the capsid transition, our results may contribute to the analysis and prediction of the actual transition path. Our procedure may therefore complement the two main existing approaches to conformational changes of viral capsids. The first approach (Tama and Brooks 2002, 2005) is based on the description of viral capsids via elastic networks, with nodes corresponding to biologically relevant positions in the capsids, such as the locations of Cα atoms, and interactions given by suitable empirical potentials. A normal mode analysis of such networks shows that many conformational changes, such as the swelling of CCMV, can be accounted for by a combination of a few collective modes of the network. It is a powerful tool that shows, for instance, that conformational changes are related to soft-mode instabilities of the elastic structure. However, it is perhaps difficult to use to actually predict details of the complex nature of the actual transition path. The second approach (Guérin and Bruinsma 2007; Klug et al. 2006; Rim et al. 2010) is an adaptation of the Ginzburg–Landau theory of phase transformations, in which the capsid is viewed as a spherical elastic shell. An order parameter related to the conformation of the capsid proteins is introduced, whose role in the energetics is determined by a double-well potential. Our model complements the above approaches, providing information about the restrictions that symmetry imposes on the parameter space for transition paths, thus reducing the computational complexity of the normal-mode analysis, and providing complementary information on the capsid structure to that encoded by the scalar order parameters used in Ginzburg–Landau models. In combination with the 3D point arrays in Keef et al. (2011), our method can be used to determine geometric constraints on the outcome of a structural transition. Given the point array that encodes the geometry of a viral capsid before the transition, it can be used to derive the point array that captures the geometry of the capsid after transition, assuming that the transition path preserves as much symmetry as possible. This information may be used in combination with experimental data, e.g. from cryo-EM, to reconstruct the geometry of the capsid after transition. Acknowledgments We thank Jess Wardman for providing us with the point arrays for CCMV using the algorithm in Keef et al. (2011). RT would like to thank the Leverhulme Trust for funding of her research team via a Research Leadership Award. PC and GI acknowledge partial funding by the research project ‘Modelli

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Matematici per la Scienza dei Materiali’ of the Università di Torino, Italy. PC, GI and AZ acknowledge partial funding by the MATHMAT and MATHXPRE projects of the Università di Padova, Italy. GI also acknowledges funding by the Marie Curie Project MATVIR.

Appendix A: Approximation of viral capsids by icosahedral point arrays We briefly recall here the representation of virus geometry in terms of the icosahedral point arrays proposed in Keef and Twarock (2009). The 3D structure of a virus with an icosahedral capsid is encoded by a configuration that consists of copies of two nested polyhedra translated along suitable vectors, providing a blueprint for the layout of the capsid protein arrangement and features. The polyhedra in the viral configuration are icosahedra, icosidodecahedra, dodecahedra (see Fig. 3), or a combination thereof, with relative scalings. The classification in Keef and Twarock (2009) shows that there are in total 342 such combinations. The following three definitions are central to this approach: √ Definition 7 The standard icosahedron of length 2 + τ , denoted by ICO, is the icosahedral orbit of the vector (τ, 1, 0) in R3 ; the standard icosidodecahedron of length τ , 2 denoted by √IDD, is the icosahedral orbit of (τ, τ , 1), and the standard dodecahedron of length 3, denoted by DOD, is the icosahedral orbit of (1, 1, 1). Definition 8 A (viral) skeletal configuration is the union of two rescaled standard polyhedra, each of the form 2h τ k ICO, 2h τ k IDD or 2h τ k DOD, with k ∈ Z and h ∈ {−1, 0, 1}. Each skeletal configuration is completely defined, modulo a global rescaling, by two numbers, that indicate its two standard polyhedra (cf. Table 3 for the list of skeletal configurations relevant to the description of CCMV). The labeling is as follows: Each label uniquely identifies a pair composed of a standard polyhedron and a translation vector along a symmetry axis of the icosahedral group, together with a scaling factor that indicates its length relative to the size of the standard polyhedron (cf. Table 2). A skeletal configuration, identified by the pair x − y, is the union of the polyhedra labeled by x and y, one of which is rescaled so that the translation vectors of both shells coincide. For instance, the skeletal configuration labeled by 10-44 is the union of an icosahedron (structure 10, translation vector τ 2 DOD), and an icosidodecahedron rescaled by 2τ (structure 44, translation vector 21 τ DOD). Note that rescaling by the factor 2τ transforms the translation vector of the structure 44 into the translation vector of the structure 10. Therefore, a skeletal configuration with label x − y uniquely identifies both the relative sizes of the two standard polyhedra and a common translation vector s x−y . Definition 9 A point array, or viral configuration, S is the point set obtained by adding to a skeletal configuration with label x − y, its translates by the icosahedral orbit of the vector s x−y , see (1).

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Appendix B: Transitions and associated lattice bases Here we list the transitions obtained with our approach as “Symmetry group, start configuration → end configuration”. Notice that all transitions have D6 symmetry. • D6 , 10-44→27-52 ⎞ ⎞ ⎛ ⎛ 0 −1 1 −3 1 1 0 1 0 0 1 0 ⎜0 ⎜ 0 0 0 0 1 −1 1 0⎟ 1 0 0⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜0 ⎜ 1 0 1 0 −1 ⎟ 0 0 0 1 −1 0 ⎟ ⎟, ⎜ ⎜ , B1044 = ⎜ T1044−2752 = ⎜ 0 0 0⎟ 0 −1 1 ⎟ ⎟ ⎟ ⎜ 1 −1 1 ⎜ 0 0 0 ⎝ 1 −3 1 −1 0 ⎝ 0 0 1 −1 1⎠ 0 1⎠ 1 −1 3 −1 1 0 −1 0 0 1 −1 1 • D6 , 10-44→27-52 ⎞ −1 1 −1 1 0 0 ⎜ 0 0 −1 1 −1 0⎟ ⎟ ⎜ ⎜ 0 −1 0 −1 0 1⎟ ⎟, ⎜ T1044−2752 = ⎜ 1 −1 0 0 0⎟ ⎟ ⎜ −1 ⎝ 0 1 −1 1 −1 0⎠ 0 1 −1 1 0 −1 ⎛

⎛

0 ⎜0 ⎜ ⎜0 B1044 = ⎜ ⎜0 ⎜ ⎝1 0

⎞ 0 0 1 0 1 0 1 1 0 0⎟ ⎟ 0 1 −1 0 −1 ⎟ ⎟, 0 −1 −1 1 0⎟ ⎟ 0 −1 0 1 1⎠ 1 0 0 1 0

• D6 , 10-44→11-27 ⎛

2 ⎜ 0 ⎜ ⎜ 0 T1044−1127 = ⎜ ⎜ 1 ⎜ ⎝ −1 −1

⎞ −1/2 1/2 1/2 −1 −1 −1/2 1/2 −1/2 1 0⎟ ⎟ 1/2 −1/2 1/2 0 −1 ⎟ ⎟, −1/2 1/2 −1/2 0 0⎟ ⎟ 1/2 1/2 −1/2 2 −1 ⎠ −1/2 −1/2 −1/2 −1 2

• D6 , 10-44→12-27 ⎞ ⎛ 1 −1 1 0 0 0 ⎜ 1 −1 0 0 −1 1 ⎟ ⎟ ⎜ ⎜ −1 0 −1 0 −1 1 ⎟ ⎟, T1044−1227 = ⎜ ⎜ −1 0 0 −1 1 1⎟ ⎟ ⎜ ⎝ 0 0 1 −1 1 0⎠ 0 −1 0 −1 0 1

⎛

0 ⎜0 ⎜ ⎜0 B1044 = ⎜ ⎜1 ⎜ ⎝0 0

⎛

B1044

0 ⎜0 ⎜ ⎜0 =⎜ ⎜1 ⎜ ⎝0 0

0 0 1 0 0 0

0 0 1 0 0 0

0 1 1 1 0 1

1 0 0 1 1 1

⎞ 1 −1 1 −1 ⎟ ⎟ 0 0⎟ ⎟, 0 0⎟ ⎟ 1 0⎠ 1 −1

⎞ 0 1 0 1 1 1 0 1⎟ ⎟ 1 −1 0 0 ⎟ ⎟. 1 −1 1 0 ⎟ ⎟ 0 0 1 0⎠ 1 0 1 1

Appendix C: Normalizers in G L(6, R) of the maximal subgroups of I We sketch here the procedure in Wijnands (1991) to compute the normalizer of a matrix group. Recall first that, for G ⊂ G L(n, R), the group of inner automorphisms I (G) is the group of automorphisms of G of the form G → M −1 G M,

∀G ∈ G and some M ∈ G.

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Let K = Z, Q, or R. In order to compute the normalizer in G L(n, K ) of a given crystallographic group G, notice first that Z(G, K ) is a normal subgroup of N (G, K ). p Hence, writing N (G, K ) = ∪i=1 Ni Z(G, K ), with Ni representatives of the cosets of Z(G, K ) in N (G, K ), the set of automorphisms of G defined by A(G) = {ϕ1 , . . . , ϕ p }, with ϕi (G) = Ni−1 G Ni , is a group containing the group of inner automorphisms, so that we may write A(G) = ∪nk=1 ϕk I (G) for some ϕ1 , . . . , ϕn ∈ A(G). Since G is finite, p and n are also finite. Hence, to compute the normalizer it is sufficient to compute the centralizer and n special elements of the normalizer. Now, notice that the elements of A(G) correspond to all possible ways of associating sets of generators of G with other generators of G with the requirement that ϕ(G) has the same eigenvalues as G. In the special cases that we treat below, this implies that ϕ(G) and G correspond to symmetry axes of the same order (i.e., for instance, both twofold axes).1 Below we implement this calculation for the maximal subgroups of I. C.1 The tetrahedral group A4 The tetrahedral group contains four threefold axes, corresponding to eight threefold rotations, and three twofold axes, in addition to the identity. Any choice of a three and a twofold rotation yields two generators of the group. The subgroup of inner automorphisms is the tetrahedral group acting on itself by conjugation, and has 12 elements. Choosing as generators ⎛

0 0 0 0 0 ⎜0 0 0 1 0 ⎜ ⎜ 0 −1 0 0 0 G3 = ⎜ ⎜0 0 −1 0 0 ⎜ ⎝1 0 0 0 0 0 0 0 0 1

⎞ 1 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0

⎛

0 ⎜0 ⎜ ⎜1 G2 = ⎜ ⎜0 ⎜ ⎝0 0

0 0 0 0 0 1

⎞ 1 0 0 0 0 0 0 1⎟ ⎟ 0 0 0 0⎟ ⎟, 0 −1 0 0⎟ ⎟ 0 0 −1 0 ⎠ 0 0 0 0

and using the notation in the following table, T1 = G 3 G 2 , T4 = G 2 G 3 , T7 = G −1 3 G2 G3, T10 = G 2 G −1 3 ,

T2 = G −1 3 , T5 = G 2 , T8 = I, T11 = G 3 G 2 G −1 3 ,

T3 = G 3 , T6 = G 3 G 2 G 3 , T9 = G −1 3 G2, −1 T12 = G −1 3 G2G3 ,

the inner automorphisms are given by (G 3 , G 2 ) → (T12 , T7 ), (G 3 , G 2 ) → (T1 , T7 ), (G 3 , G 2 ) → (T1 , T5 ), (G 3 , G 2 ) → (T4 , T11 ),

(G 3 , G 2 ) → (T3 , T11 ), (G 3 , G 2 ) → (T12 , T5 ), (G 3 , G 2 ) → (T3 , T5 ), (G 3 , G 2 ) → (T4 , T5 ),

1 We say that G ∈ I is an axis of order k = 2, 3, 5 if G k = I .

123

(G 3 , G 2 ) → (T3 , T7 ), (G 3 , G 2 ) → (T1 , T11 ), (G 3 , G 2 ) → (T12 , T11 ), (G 3 , G 2 ) → (T4 , T7 ).

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In order to compute the normalizer N (A4 , K ) of the tetrahedral group, notice that A(A4 ) is the group of automorphisms of A4 whose action on the generators is given by all possible ways of combining a three and a twofold generator. Since there are 24 possible ways of choosing a three and a twofold axis in the tetrahedral group, it follows that A(A4 ) has order 24. Furthermore, the group of inner automorphisms I (A4 ) has 12 elements, so that A(A4 )/I (A4 ) has only two elements. Therefore, every element of N (A4 , K ) can be written in one of the two forms N = QC, or N = R QC, with R ∈ G L(6, K ), Q ∈ A4 , and C ∈ Z(A4 , K ). Here, R is a fixed element of the normalizer, e.g. corresponding to the action (G 3 , G 2 ) → (T2 , T5 ). A straightforward calculation shows that we may choose R ∈ PC as ⎞ 0 −1 0 0 0 0 ⎜1 0 0 0 0 0⎟ ⎟ ⎜ ⎜0 0 0 0 0 −1 ⎟ ⎟. R=⎜ ⎜0 0 0 0 1 0⎟ ⎟ ⎜ ⎝0 0 0 −1 0 0⎠ 0 0 1 0 0 0 ⎛

The centralizers of A4 can be obtained by solving the linear equations G 3 C = C G 3 and G 2 C = C G 2 for the unknown C. The solutions have the form ⎛

z −x −y t z t

⎜ ⎜ ⎜ ⎜ −y −x z C =⎜ ⎜ x −t −x ⎜ ⎜ −x −t x ⎝ t

y

−t x t z y

t −x

⎞ t −x x y⎟ ⎟ ⎟ −t −x ⎟ ⎟ y t ⎟, ⎟ z t⎟ ⎠ −x z

(20)

with x, y, z, t ∈ K . C.2 The dihedral group D10 . The dihedral group has one fivefold axis, corresponding to four fivefold rotations, and five twofold axes, in addition to the identity. Any choice of a five and a twofold rotation yields two generators of the group. The subgroup of inner automorphisms is the dihedral group acting on itself by conjugation, and has 10 elements. Choosing as

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generators ⎛

0 ⎜1 ⎜ ⎜0 G5 = ⎜ ⎜0 ⎜ ⎝0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

1 0 0 0 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ 1

⎞ 0 −1 0 0 0 0 ⎜ −1 0 0 0 0 0⎟ ⎟ ⎜ ⎜ 0 0 0 0 −1 0⎟ ⎟, G2 = ⎜ ⎜ 0 0 0 −1 0 0⎟ ⎟ ⎜ ⎝ 0 0 −1 0 0 0⎠ 0 0 0 0 0 −1 ⎛

and using the notation D1 = G 2 , D2 = G 5 G 2 , D3 = G 5 , D4 = G 45 G 2 , 3 2 2 D5 = G 5 , D6 = G 5 , D7 = G 5 G 2 , D8 = I, D9 = G 45 , D10 = G 25 G 2 , the inner automorphisms are given by (G 5 , G 2 ) → (D9 , D1 ), (G 5 , G 2 ) → (D9 , D7 ), (G 5 , G 2 ) → (D3 , D10 ), (G 5 , G 2 ) → (D9 , D10 ), (G 5 , G 2 ) → (D3 , D4 ), (G 5 , G 2 ) → (D3 , D2 ), (G 5 , G 2 ) → (D9 , D4 ), (G 5 , G 2 ) → (D3 , D1 ), (G 5 , G 2 ) → (D3 , D7 ), (G 5 , G 2 ) → (D9 , D2 ). In order to compute the normalizer N (D10 , K ) of the dihedral group, notice that there are 20 possible ways of associating a five with a twofold generator of the D10 group, so that A(D10 ) has 20 elements. Furthermore, the group of inner automorphisms I (D10 ) has ten elements, so that A(D10 )/I (D10 ) has only two elements. Therefore, every element of N (D10 , K ) can be written in one of the two forms N = QC, or N = R QC, with R ∈ G L(6, K ), Q ∈ D10 , C ∈ Z(D10 , K ). Here R is a fixed element of the normalizer which is not in the centralizer, e.g. corresponding to the action (G 5 , G 2 ) → (D5 , D1 ). A direct calculation shows that we may choose R ∈ PC = S O(6) ∩ G L(6, Z) as ⎛

0 ⎜0 ⎜ ⎜1 R=⎜ ⎜0 ⎜ ⎝0 0

123

0 0 0 0 1 0

0 1 0 0 0 0

0 0 0 1 0 0

⎞ 1 0 0 0⎟ ⎟ 0 0⎟ ⎟. 0 0⎟ ⎟ 0 0⎠ 0 −1

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The centralizers of D10 can be obtained by solving the linear equations G 5 C = C G 5 and G 2 C = C G 2 for the unknown C. The solutions have the form ⎛

z ⎜x ⎜ ⎜y C =⎜ ⎜y ⎜ ⎝x u

x z x y y u

y x z x y u

y y x z x u

⎞ x t y t⎟ ⎟ y t⎟ ⎟, x t⎟ ⎟ z t⎠ u w

(21)

with x, y, z, t, u, w ∈ K . C.3 The dihedral group D6 . The dihedral group has one threefold axis, corresponding to two threefold rotations, and three twofold axes, in addition to the identity. Any choice of a three and a twofold rotation yields two generators of the group. The subgroup of inner automorphisms is the dihedral group acting on itself by conjugation, and has six elements. Choosing the generators ⎛

0 0 0 0 0 ⎜0 0 0 1 0 ⎜ ⎜ 0 −1 0 0 0 G3 = ⎜ ⎜0 0 −1 0 0 ⎜ ⎝1 0 0 0 0 0 0 0 0 1

⎞ 1 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0

⎞ 0 0 0 0 −1 0 ⎜ 0 0 0 −1 0 0⎟ ⎟ ⎜ ⎜ 0 0 −1 0 0 0⎟ ⎟ G2 = ⎜ ⎜ 0 −1 0 0 0 0⎟ ⎟ ⎜ ⎝ −1 0 0 0 0 0⎠ 0 0 0 0 0 −1 ⎛

(22)

and using the following notation D1 = I, D2 = G 2 , D3 = G −1 3 G2 G3, −1 D4 = G 3 , D5 = G 3 G 2 G 3 , D6 = G 23 , the inner automorphisms are given by (G 3 , G 2 ) → (D4 , D2 ) = (G 3 , G 2 ), (G 3 , G 2 ) → (D6 , D2 ), (G 3 , G 2 ) → (D6 , D5 ), (G 3 , G 2 ) → (D6 , D3 ), (G 3 , G 2 ) → (D4 , D5 ). (G 3 , G 2 ) → (D4 , D3 ), Since there are six possible ways of combining a three with a twofold generator of D6 , it follows that A(D6 ) has six elements. Furthermore, the group of inner automorphisms I (D6 ) has order 6, so that A(D6 )/I (D6 ) has only one element. Therefore, every element of N (D6 , K ) can be written in the form N = QC, with Q ∈ D6 and C ∈ Z(D6 , K ). Hence, to compute the normalizer of D6 it is sufficient to compute the centralizer by solving the linear equations G 3 C = C G 3 and G 2 C = C G 2 for the

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unknown C. The solutions have the form ⎞ u w −w x s s ⎜ −t y v −v z −t ⎟ ⎟ ⎜ ⎜ t v y v t −z ⎟ ⎟, C =⎜ ⎜ z −v v y −t −t ⎟ ⎟ ⎜ ⎝ s x −w w u s⎠ s w −x w s u ⎛

(23)

with x, y, z, t, u, w, v, s ∈ K .

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