Download Structural Description of Viral Particles Based on Affine Extensions

Bulg. J. Phys. 34 (2007) 81–91
Structural Description of Viral Particles Based
on Affine Extensions of Non-Crystallographic
Coxeter Groups
R. Twarock
Department of Mathematics and Department of Biology, University of York,
York YO10 5DD, U.K.
Received 20 May 2007
Abstract. An essential part of a virus is its viral capsid, a protein shell that
encapsulates and hence provides protection for the viral genome. Due to the
fact that the majority of the known viruses has capsids with overall icosahedral
symmetry, group theory lends itself for their structural description. In particular, icosahedral symmetry determines the arrangement of the protein subunits
in the capsid up to the fundamental domain of the icosahedral group. Thus,
for larger viruses consisting of more than 60 protein subunits, icosahedral symmetry does not specify their arrangement completely. This has already been
observed by Caspar and Klug who suggested in their seminal paper from the
1960s to model the locations of the additional subunits based on the concept of
quasi-equivalence. However, their theory is applicable only to a subset of the
known viruses. In particular, it does not apply to the cancer-causing Papovaviridae, which are of crucial importance to the public health sector. We present here
a theory – based on affine extensions of non-crystallographic Coxeter groups –
that closes this gap and provides a structural description also for the missing
cases. Recent results concerning applications of this theory to the description of
tubular malformations, the modelling of crosslinking structures and the assembly process are also discussed.
PACS number: 87.15.B-
1 Introduction
Already in 1956 Crick and Watson observed that the majority of viruses exhibit symmetry in the structural organisation of their capsids [1]. This observation was subsequently corroborated by experimental results, which showed that
icosahedral symmetry occurs predominantly (see e.g. [2] for a review), and a
biophysical explanation for the origin of icosahedral symmetry in viruses has
recently been suggested in [3].
c 2007 Heron Press Ltd.
1310–0157 81
R. Twarock
(a)
)
(b)
)
Figure 1. Example of a viral capsid (a) with capsomeres shown in magnification in (b)
(both from the Johnson Lab [4]).
The proteins in viral capsids typically appear in clusters of three, five or six
protein subunits, called capsomeres, that are organised with icosahedral symmetry in the capsid. An example of such an arrangement is shown in Figure 1(a).
Part of this viral capsid is shown in magnification in Figure 1(b) in order to
demonstrate the organisation in terms of capsomeres: each toroidal-looking unit
corresponds to a capsomere composed of six individual protein subunits.
Cryo-electron microscopy experiments that investigate viruses in the frozen hydrated state have provided evidence for the fact that viruses formed from different types of protein subunits may follow the same type of surface lattice as
a blueprint for their structural organisation. This suggests that it should be possible to pinpoint a uniform organisational principle for the locations of protein
subunits and inter-subunit bonds for viral capsids at large. The first theory of this
kind was proposed by Caspar and Klug in their seminal paper from the 1960s [5].
It has become one of the major tools in virology for the analysis and classification of experimental data. It is so powerful that the first experimental results
deviating from Caspar-Klug Theory have initially been thought to be incorrect,
until it had been established that they rather had to be seen as pointers to the
incompleteness of Caspar-Klug Theory. For example, the Papovaviridae which
contain cancer-causing viruses and are hence of prime importance for the public
health sector fall out of the scope of this theory [6]. The first theory to capture
all the missing cases while incorporating also the results of Caspar-Klug Theory
is Viral Tiling Theory [7, 8]. It provides a new series of polyhedra that complements the series derived in Caspar-Klug theory and specifies the locations of
both protein subunits and inter-subunit bonds in viral capsids [9].
A key ingredient in this new theory are the so-called local symmetries. In contrast to symmetry operations that leave the overall structure of a geometrical
shape invariant, local symmetries only preserve a subset of a structure in a neighbourhood of the local symmetry axis. The introduction of local symmetries is
necessary in order to pinpoint the locations of protein subunits within the fun-
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Group Theoretical Description of Viral Particles
damental domain of the overall symmetry group. In particular, for viruses with
more than 60 protein subunits icosahedral symmetry is not sufficient to determine the locations of all protein subunits completely, because it does not constrain the arrangement of the proteins within the fundamental domain.
This has already been recognised by Caspar and Klug in [5], where they consider a prototype of a local symmetry that they call quasi-equivalence. However, quasi-equivalence considers only restricted types of local symmetries that
can be obtained with elementary mathematics, such as superpositions of hexagonal lattices on the surface of an icosahedron. For the types of local symmetries
required for the description of general families of viruses, more advanced mathematical tools such, as affine extensions of non-crystallographic Coxeter groups
need to be exploited.
After an introduction of the concept of local symmetry based on the example
of Caspar-Klug Theory, the affine extension of the non-crystallographic Coxeter
group H3 is introduced. It is shown how it can be implemented in order to derive
the new theory for the structural description of viral capsids. Finally, predictions
and applications of this theory are discussed.
2 A Prototype of a Local Symmetry – Caspar-Klug Theory
Caspar-Klug Theory is the first theory to predict the locations of the protein subunits in viral capsids. Local symmetries are introduced in Caspar-Klug Theory
geometrically via a superposition of a hexagonal surface lattice on the surface of
an icosahedron. This is demonstrated in Figure 2, which shows an embedding
of the twenty triangular faces of the surface of an icosahedron in a hexagonal lattice. Every inequivalent embedding of this type – allowing for rotations
and scalings of the icosahedral surface, such that vertices of the triangular faces
meet centres of the hexagons in the underlying lattice – corresponds to a differ-
Figure 2. An embedding of the surface of an icosahedron in a regular hexagonal lattice.
83
R. Twarock
Figure 3. Local symmetries via triangulations.
ent blueprint for a viral capsid. In particular, via a subsequent replacement of
each hexagon by six triangles, triangulations are obtained that are compatible
with overall icosahedral symmetry. Each triangular facet in such a triangulation
encodes the locations of three protein subunits in its corners. Hence, one obtains
clusters of five protein subunits (called pentamers) at the 5-fold axes of icosahedral symmetry, and clusters of six protein subunits (hexamers) at global and
local symmetry axes. This is demonstrated in Figure 3 (adapted from [2]). On
the left, two of the 20 triangular faces of an icosahedron are shown. By introduction of further triangular facets as shown on the right, additional symmetry
axes have been created. In particular, a local symmetry axis has appeared on the
edge between the two original faces.
The different surface lattices that are obtained in this way have been classified
in terms of the triangulation number T = h 2 + hk + k 2 with h, k ∈ N ∪ {0},
which counts the number of triangular facets per face in the corresponding triangulation. These surface lattices define a series of polyhedra, that we call the
Caspar-Klug series, because Caspar-Klug Theory predicts that viruses are organised according to the structure of the polyhedra in this series. By construction, all local symmetries are of order six and local symmetries of order five are
excluded a priori. In order to obtain polyhedra with local 5-fold symmetry axes,
a construction based on a hexagonal lattice is not suitable. Since an icosahedral
lattice does not exist in two and three dimensions, a straight-forward generalisation of the Caspar-Klug construction is not possible. However, one can use
a generalised lattice that is induced by the root lattice of D 6 via projection, or,
equivalently, the roots of the affine extension of H 3 as discussed in the following
section.
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Group Theoretical Description of Viral Particles
Figure 4. Figure illustrating the projection π of the simple roots of D6 on those of H3 .
3 Local Symmetries Based on Affine Extensions of
Non-Crystallographic Coxeter Groups
In order to introduce local symmetries for viral capsids and hence pinpoint the
locations and relative orientations of the capsomeres, we make use of the correspondence between the roots of H 3 and D6 , which is illustrated in terms of the
Dynkin diagram of D 6 , respectively the Coxeter diagram of H 3 , in Figure 4.
On the left, the Dynkin diagram of D 6 is shown – in a folded and hence slightly
unconventional form – to illustrate how the simple root vectors of D 6 (denoted
as αj , j = 1, . . . , 6) project on the root vectors of H 3 (denoted as aj , j =
1, . . . , 3). In particular, the 5 above the link in the Coxeter diagram of H 3 on the
right indicates that the angle between the corresponding root vectors is π − π5 .
√
√
Moreover, τ = 12 (1 + 5) and its Galois conjugate τ = 12 (1 − 5) denote the
two solutions of the equation x 2 = x + 1. Using the identity τ + τ = 1, the
root vectors can be expressed in terms of the standard basis in 3 dimensions:
a1 = 12 (−τ , 1, −τ )
a1 →
a3 →
τ a3 = 12 (−τ, −τ 2 , 1)
a5 →
τ a1 = 12 (1, −τ, −τ 2 )
a2 →
a2 = 12 (1, −τ, −τ )
a4 →
τ a2 = 12 (τ, −τ 2 , 1)
a6 →
a3 = 12 (−1, τ, −τ ) .
(1)
We remark that this procedure is similar to the projection from E 8 on H4 considered in [10].
The root lattice of D 6 is given by Z-linear combinations of the simple roots α j ,
j = 1, . . . , 6. Correspondingly, its projection into R 3 via π is given by Z[τ ]linear combinations of the simple root vectors aj , j = 1, . . . , 3, of H3 , where
Z[τ ] denotes the extended ring of integers Z[τ ] := {a + τ b |a, b ∈ Z}. Since
Z[τ ] is dense in R, the Z[τ ]-linear combinations of the simple root vectors of H 3
are dense in R3 . It is thus necessary to select subsets thereof that are suitable to
provide the vertex sets of the desired polyhedra. Such point sets will be derived
in the following section based on the affine extension of H 3 .
3.1 Affine extension of the non-crystallographic Coxeter groups H3
The elements of H 3 are reflections rj at planes perpendicular to the root vectors
αj in the root system Δ of H 3 [11]. Hence, the root system given by the 30
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R. Twarock
a)
(a)
b)
(b)
Figure 5. The root polytope of H3 (a) and examples of the reflection planes encoded by
the root vectors (b).
vectors [12]
(±1, 0, 0)
and all permutations
1
2 (±1, ±τ , ±τ ) and all even permutations
2(x|α )
correspond to the reflections r j (x) = x − (αj |αjj ) αj in the group.
Δ=
(2)
The root vectors of H 3 point from the centre of an icosidodecahedron (see Figure 5(a)) to its vertices, and encode the locations of the planes of reflection as
shown in Figure 5(b), where two planes are shown explicitly. For all other planes
only the geodesics marking their intersection with the surface of the sphere are
indicated.
The intersections of the planes of reflection mark the locations of the axes of
rotational symmetry of the icosahedral group. For example, the intersection of
the two planes in the figure corresponds to an axis of 5-fold symmetry, which
intersects the sphere at two of the twelve 5-fold vertices of the (spherical) icosahedron. Similarly, the intersections of the geodesics mark the locations of all
symmetry axes, and one can hence reconstruct the locations of all six 5-fold,
fifteen 2-fold and ten 3-fold axes from them.
In order to extend the root system in a way compatible with the overall symmetry group and hence find in addition local symmetry axes compatible with
overall icosahedral symmetry, one needs to consider the basis of simple roots,
from which all other root vectors can be constructed by positive or negative
Z[τ ]-linear combinations. In particular, a possible choice of simple roots in the
orthonormal basis is (see also [13]):
α1 = (0, 0, 1) ,
86
α2 = 12 (−τ , −τ, −1) ,
α3 = (0, 1, 0) .
(3)
Group Theoretical Description of Viral Particles
Their relations are encoded in the Cartan matrix
⎛
⎞
2 −1 0
2(αi | αj )
C :=
= ⎝−1 2 −τ ⎠ .
(αj |αj ) ij
0 −τ 2
(4)
By means of an affine extension this matrix is extended by an additional row
and column that encode a further root corresponding to an affine reflection. It
has been shown in [13], that the (unique) affine extended Cartan matrix of H 3 is
given by
⎛
⎞
2 0 τ 0
⎜ 0 2 −1 0 ⎟
⎟
(5)
C af f = ⎜
⎝τ −1 2 −τ ⎠ ,
0 0 −τ 2
and that the affine reflection acts as a translation T by the highest root α H =
τ α1 + 2τ α2 + τ 2 α3 = −τ ω2 = (1, 0, 0). The three other reflections are cyclic
operations of order two, and the products of any two of them correspond to
rotations around the origin as follows:
⎧
=2
M = 1 if cjk
⎪
⎪
⎨
=0
M
=
2
if
c
jk
(rj rk )M = 1 where
.
(6)
M
=
3
if
c
=
−1
⎪
jk
⎪
⎩
M = 5 if cjk = −τ, τ
The affine extended group is hence generated by the three reflections r 1 , r2 , r3
as well as T .
3.2 Construction of polyhedra with local symmetry axes
Three-dimensional point sets with local symmetries can be constructed via an
iterated action of the generators of the extended group on the origin. If the
action of the translation operator T is not restricted, R 3 would be densely filled
in this way. However, if T acts only a finite number of times, N say, while
the action of all other operations is not restricted, point sets S(N ) are obtained
that are finite subsets of cut-and-project quasicrystals with centrally symmetric
acceptance windows. For increasing N , the patches become larger and more
dense, and correspond to cut-and-project quasicrystals with increasingly larger
acceptance windows. Examples have been worked out explicitly for the twodimensional subgroup H 2 in [13], and for H 3 in [14], where applications to
carbon cage structures have been considered.
Due to the fact that T acts as a translation by the highest root, a simple expression
for these point sets can be given in terms of the root system Δ [14]:
nα α | nα ∈ N ∪ {0},
nα ≤ N ,
(7)
S(N ) :=
α∈Δ
α∈Δ
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R. Twarock
i.e. S(N ) consists of all vectors that are obtained from N ∪ {0}-linear combinations of up to N roots in Δ. These sets form the starting point for the
construction of icosahedral particles with local symmetries. In particular, the
vectors in S(N ) define nested point sets that are subsets of generalised lattices
and form the vertex sets of the desired polyhedra. Identifying these vertex sets
within S(N ) is not straight-forward, because they do not need to be isometric.
One has to specify the properties of the desired polyhedra and then search for
compatible vertex sets, as discussed, for example, in the next section.
4 Applications to Viruses
A family of viruses not covered by Caspar-Klug Theory are the Papovaviridae
and we therefore focus on this family of viruses here, even though the procedure
is applicable also to more general families of viruses. The characteristic property
of these viral capsids is the fact that all capsomeres are pentamers, i.e. clusters
of five proteins. In order to determine polyhedra with vertices in S(N ) that correspond to the surface structures of such viruses one has therefore to search for
subsets of S(N ) that correspond to vertex sets of polyhedra with 5-coordinated
vertices, i.e. vertices corresponding to local 5-fold symmetry axes. This has
been done in [9], and it has been shown in this reference that there are three different polyhedra of this type (apart from the icosahedron as a trivial case), which
are shown in Figure 6.
(a)
a)
b)
(b)
((c)
c)
Figure 6. Schematic representation of the polyhedra in the triacontahedral series. Only
facets overlapping with the fundamental domain of the symmetry groups are shown for
the medium particle ((b) – with octahedral symmetry) and the large particle ((c) – with
icosahedral symmetry). All figures are adapted from [9].
This (finite) series is called the triacontahedral series because it starts with the
triacontahedron. It corresponds to the exceptional cases that complement the
Caspar-Klug series discussed in section 2.
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Group Theoretical Description of Viral Particles
4.1 Predictions
In order to visualise the structure of the viral capsids that are encoded by these
polyhedra (or tilings), the rules of Viral Tiling Theory need to be used [8]: protein subunits are located in the corners of the tiles that meet at the local symmetry
axes in the tiling. By construction tiles represent dimer, respectively trimer, interactions, i.e. interactions between two, respectively three, protein subunits.
They are visualised schematically as spiral arms and mark the locations of the
C-terminal arm extensions of the proteins. With this interpretation, the predictions for the bonding structure implied by the polyhedra of the medium and large
particles are as illustrated in Figure 7.
In the case of the medium particle (a), the local bonding environments are identical around all pentamers, while there are two different types of bonding environments for the large particle (b). The predictions for the large particle agree
well with the experimental results in [15]. It would be interesting to validate also
the predictions for the bonding structure of the other particles experimentally.
Moreover, a crucial feature of our approach is the fact that it predicts the relative
radii of the different particles, and that there is hence only one scaling parameter
that maps the overall mathematical set-up collectively on its biological counterpart. A comparison with experiments shows excellent agreement between the
predicted radii [9] and the experimentally observed radii [16, 17].
4.2 Implications of the new theory
Due to the fact that the tiles have a biological interpretation in terms of dimerand trimer-interactions as discussed above, they can be used as building blocks
for the study of the assembly of the capsids. In particular, the information on
a)
(a)
b)
(b)
Figure 7. The bonding structure of medium (a) and large (b) particles implied by Viral
Tiling Theory.
89
R. Twarock
the bonding structure as encoded by the polyhedra in Figure 7 allows to model
pentamers together with their local bonding structure. Assembly models for viral particles corresponding to the large species in Figure 7(b) have been worked
out based on the vertex stars in this tiling in [18, 19]. In these references, combinatorial structures have been derived that encode the succession of the energetically favourable assembly intermediates under the assumption that assembly
takes place by addition of a single building block per iteration step. The information in these graphs is subsequently used to derive the concentrations of the
assembly intermediates, and to pinpoint the dependence of the assembly process
on the choices of the association energies of the inter-subunit bonds. Moreover,
the dominant pathways of assembly have been determined in [19], and it has
been investigated which geometrical properties are responsible for their distinguished role. This is a first step in developing strategies for interference with the
assembly process by misdirecting or inhibiting the assembly of the capsid.
Moreover, based on the knowledge of the bonding structure, it is possible to classify the tubular malformation that may arise during assembly. It has been shown
experimentally that, in dependence on boundary conditions, such as pH-value
and salt concentration in solution, different types of particles occur. For example, studies of the self-assembly of the major capsid protein of polyomavirus
have shown that besides spherical particles also tubular particles may appear
[17]. Based on the assumption that the same dimer- and trimer-interactions are
responsible both for the formation of the spherical particles and for the formation
of the tubular particles, the classification of the possible tubular malformations
translates into the mathematical problem of classifying tubular lattices with a
particular set of properties as carried out in [20, 21].
Another spin-off of the theory is the fact that it can be used as a basis for the
description of crosslinking structures. These are covalent bonds that occur in
certain types of viruses besides dimer- and trimer-bonds and cover the whole
capsid in a chainmail organisation, hence rendering it particularly stable [22].
We have shown that such crosslinking structures can be modelled as higher-level
tilings compatible with the polyhedra in Viral Tiling Theory [23]. This provides
a theoretical method to assess whether crosslinking is possible for a given type
of virus.
5 Discussion
It has been demonstrated that affine extensions of non-crystallographic Coxeter groups can be used to determine the structure of viral particles with overall
icosahedral symmetry in terms of the locations of their protein subunits and
inter-subunit bonds. The approach has in particular led to a new series of polyhedra that closes an important gap in Caspar-Klug Theory. Due to the correspondence between the faces of the polyhedra in that series and inter-subunit
bonds between protein-subunits in the viral capsid they represent, the theory can
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Group Theoretical Description of Viral Particles
moreover be used as a basis for the construction of assembly models, the classification of tubular malformations and the description of crosslinking structures
as discussed in subsection 4.2.
Acknowledgements
Financial support via an EPSRC Advanced Research Fellowship is gratefully
acknowledged.
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