Elasticity theory of the maturation of viral capsids

Abstract

Many viral capsids undergo a series of significant structural changes following assembly, a process known as maturation. The driving mechanisms for maturation usually are chemical reactions taking place inside the proteins that constitute the capsid (“subunits”) that produce structural changes of the subunits. The resulting alterations of the subunits may be directly visible from the capsid structures, as observed by electron microscopy, in the form of a shear shape change and/or a rotation of groups of subunits. The existing thin shell elasticity theory for viral shells does not take account of the internal structure of the subunits and hence cannot describe displacement patterns of the capsid during maturation. Recently, it was proposed for the case of a particular virus (HK97) that thin shell elasticity theory could in fact be generalized to include transformations of the constituent proteins by including such a transformations as a change of the stress-free reference state for the deformation free energy. In this study, we adopt that approach and illustrate its validity in more generality by describing shape changes occurring during maturation across different T-numbers in terms of subunit shearing. Using phase diagrams, we determine the shear directions of the subunits that are most effective to produce capsid shape changes, such as transitions from spherical to facetted capsid shape. We further propose an equivalent stretching mechanism offering a unifying view under which capsid symmetry can be analyzed. We conclude by showing that hexamer shearing not only drives the shape change of the viral capsid during maturation but also is capable of lowering the capsid elastic energy in particular for chiral capsids (e.g., T=7) and give rise to pre-shear patterns. These additional mechanisms may provide a driving force and an organizational principle for virus assembly.

Full text

Elasticity theory of the maturation of viral capsids
Luigi E. Perotti
a,1
, Ankush Aggarwal
a,1,2
, Joseph Rudnick
b
, Robijn Bruinsma
b
,
William S. Klug
a,
n
a
Mechanical and Aerospace Engineering Department, University of California, Los Angeles, CA 90095, United States
b
Physics and Astronomy Department, University of California, Los Angeles, CA 90095, United States
article info
Article history:
Received 9 May 2014
Received in revised form
20 December 2014
Accepted 3 January 2015
Available online 21 January 2015
Keywords:
Conformational changes
Buckling transition
Reference configuration
Virus maturation
Virus assembly
abstract
Many viral capsids undergo a series of significant structural changes following assembly, a
process known as maturation. The driving mechanisms for maturation usually are che-
mical reactions taking place inside the proteins that constitute the capsid (“subunits”) that
produce structural changes of the subunits. The resulting alterations of the subunits may
be directly visible from the capsid structures, as observed by electron microscopy, in the
form of a shear shape change and/or a rotation of groups of subunits. The existing thin
shell elasticity theory for viral shells does not take account of the internal structure of the
subunits and hence cannot describe displacement patterns of the capsid during matura-
tion. Recently, it was proposed for the case of a particular virus (HK97) that thin shell
elasticity theory could in fact be generalized to include transformations of the constituent
proteins by including such a transformations as a change of the stress-free reference state
for the deformation free energy. In this study, we adopt that approach and illustrate its
validity in more generality by describing shape changes occurring during maturation
across different T-numbers in terms of subunit shearing. Using phase diagrams, we de-
termine the shear directions of the subunits that are most effective to produce capsid
shape changes, such as transitions from spherical to facetted capsid shape. We further
propose an equivalent stretching mechanism offering a unifying view under which capsid
symmetry can be analyzed. We conclude by showing that hexamer shearing not only
drives the shape change of the viral capsid during maturation but also is capable of
lowering the capsid elastic energy in particular for chiral capsids (e.g.,
T
7
=
) and give rise
to pre-shear patterns. These additional mechanisms may provide a driving force and an
organizational principle for virus assembly.
&2015 Elsevier Ltd. All rights reserved.
1. Introduction
X-ray crystallography and electron microscopy (EM) have revealed that the protein shells (capsids) of most spherical
viruses are assembled into lattices of five- and six-fold tiles (pentamers and hexamers) that have the symmetries of an
icosahedron, the platonic solid formed from 20 (three-fold symmetric) equilateral triangular faces connected by 30 (two-
fold symmetric) edges and 12 (five-fold symmetric) vertices (see Fig. 1). For all but the smallest and simplest of these shells,
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Journal of the Mechanics and Physics of Solids
http://dx.doi.org/10.1016/j.jmps.2015.01.006
0022-5096/&2015 Elsevier Ltd. All rights reserved.
n
Corresponding author.
E-mail address: klug@ucla.edu (W.S. Klug).
1
The authors contributed equally to the paper.
2
Present address: Institute for Computational Engineering & Sciences, University of Texas at Austin, TX 78712, United States.
Journal of the Mechanics and Physics of Solids 77 (2015) 86–108

the icosahedral lattice tiling scheme, first discovered by Caspar and Klug (1962), places the individual proteins (often all
chemically identical) in distinct local symmetry environments (see Fig. 1). Based on this fact, Caspar and Klug anticipated
that the proteins in different symmetry positions would have similar but slightly different shapes—i.e., they are “quasi-
equivalent.”They argued that the assembly of the capsid structure would be guided not by purely geometric principles, but
rather by physics—specifically, by minimization of the deformation energy cost. X-ray and EM measured structures have
confirmed that distinct quasi-equivalent proteins often differ in atomic structure by small “conformational”deformations.
It follows from the Caspar–Klug (CK) argument that the deformation of quasi-equivalent capsid protein subunits will
generally produce a state of pre-stress within the icosahedral shell. A theory based on Kirchhoff–Love thin shell elasticity
was constructed by Lidmar et al. (2003) (LNM) for icosahedral capsids and later generalized for non-icosahedral shells,
predicts the emergence of pentamers as disinclination defects in an otherwise hexagonal lattice of capsid proteins. Com-
pressive eigen-stresses from the CK lattice construction act as a driving force for outward “buckling”of the pentamers found
at the five-fold icosahedral vertices. In this telling, the overall shape of an icosahedral capsid is determined by the di-
mensionless Föppl von Kármán (FvK) number
YR /
C
2
γ
κ=
, which defines the ratio between stretching (Y) and bending
stiffness (
C
κ
) through the capsid radius R. Below a critical value of
15
0γ
≈
, the bending energy dominates and the virus shell
is close to a sphere, while above this value the stretching energy dominates and the capsid becomes more faceted, in closer
resemblance to a perfect icosahedron. These predictions are consistent with a trend observed in X-ray and EM structures—
larger capsids tend to be more faceted than smaller capsids. Likewise this theory has been successful in explaining the
formation of conical capsid shapes (Nguyen et al., 2005), capsid polymorphism (Zandi et al., 2004), and the nonlinear
mechanical response under atomic force microscopy (AFM) loading (Klug et al., 2006,2012;Roos et al., 2010).
Shape transitions are known to occur in particles of many virus types during ‘maturation’events, in which the proteins in
the capsid assembly undergo significant conformational changes, often yielding a final mature head configuration that is
more faceted than the initial prohead configuration, as exemplified by the well-studied bacteriophage HK97 (cf. Ross et al.,
2005 and Fig. 2). Maturation can be initiated by a cleavage chemical reaction or it may be induced bychanges in the physico-
chemical environment, such as a change in pH or the insertion of the genome into the capsid.
Within LNM theory, a change of the capsid shape during maturation must be interpreted as a buckling transition driven
by changes in the effective mechanical parameters (such as Yand
C
κ
) of the protein shells, perhaps linked to changes in the
effective thickness or bonding structure due to the local conformational changes. However, experimental measurements of
elastic properties of HK97 by AFM (Roos et al., 2012) suggest such changes are likely not large enough to produce the
Fig. 1. (Left) Icosahedron two, three and five folds symmetry sites. (Right) Structure of NωV virus, containing four distinct quasi-equivalent subunits (red,
green, blue, yellow) in each symmetry-equivalent icosahedral crystallographic “asymmetric”unit. NωV image from VIPERdb (Carrillo-Tripp et al., 2009).
(For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
Fig. 2. (Left) Maturation of bacteriophage HK97 from prohead “P-II”state to head “EI-II”state involves unshearing of skewed hexamers to symmetric
conformations. (Right) Similar to the HK97 maturation process, λ-phage virus changes from prohead state to head state involve unshearing of skewed
hexamers to symmetric conformations. Coordinates obtained from ViperDB (Shepherd et al., 2006) and EMDB (Kinjo et al., 2012) and rendered in Chimera
(Pettersen et al., 2004).
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108 87

requisite change in asphericity. Moreover, the focus on elastic moduli distracts from a second feature that is frequently
found coincident with increased faceting—the hexamers commonly take on skewed shapes in the initial prohead config-
uration, and are made symmetric in the final head configuration (Fig. 2). Detailed studies of the protein structure of HK97
indicate that the change in shape of the hexamer is the result of relaxation of internal stress following the release of
chemical bonds holding together certain protein domains (
Δ
domains). This relaxation of internal stress of the proteins is the
causal agent of the shape change. Indeed skewed hexamers are found in the immature conformations of a host of different
viruses, including P22 (Jiang et al., 2003) and
λ
-phage (Dokland et al., 1993) with the same “T¼7”CK triangulation number
structure as HK97, as well as viruses with different T-numbers: marine bacteriophage SIO-2 (Lander et al., 2012)(T¼12),
bacteriophage T5 (Preux et al., 2013)(T¼13), and herpes simplex virus (HSV) (Cheng et al., 2002)(T¼16). These viruses first
assemble into a spherical procapsid with skewed hexamers, which undergo a symmetrization or “unshearing”as the viruses
mature. Shape change of the hexamers (Johnson, 2010;Steven et al., 2005) may be triggered by chemical events that change
the internal energy of each hexamer, such as changes in pH (e.g.,
λ
-phage, HK97), interaction with scaffolding proteins (e.g.,
P22) or the proteolytic cleavage of subunit domains (e.g., HK97).
In this work we develop an elasticity theory for the maturation of viral shells that connects the mechanics of global shell
morphology directly to the local state of the hexamers. Previously (Aggarwal et al., 2012;Aggarwal, 2012) we showed that
shearing of the stress-free reference configuration of capsid hexamers can produce an effect opposite to the disinclination
driven buckling of the LNM theory, i.e., a “reverse buckling”transition with no change in material properties. Our previous
work provided an explanation for the specific case of the P-II to EI-II maturation step of the
T
7
=
bacteriophage HK97, where
the kinematics of hexamer pre-shear were fitted by the observation to the specific molecular structure of the P-II state. Here
we consider the more general problem of how a pre-skewing or pre-stressing of the reference shapes of capsomers affects
the mechanics of icosahedral shells of arbitrary T-number. Further, we will show how pre-shearing of the basic hexamer
units can lead to a large range of shape changes in viral capsids, effectively changing their Föppl von Kármán number
γ
.Note
that maturation often also entails radial swelling, as is evident in Fig. 2 . While such isotropic expansion of the protein shell
may also impact capsid shape, this effect is small. We therefore neglect it and focus purely on isochoric conformational
shear.
In the following, we illustrate the generality of this shearing/unshearing mechanism, identifying general features of the
mechanics of shells with skewed capsomers over a range of T-numbers that explain common experimental observations of
the arrangement of skew orientations, and provide a driving force and an organizational principle during self-assembly. In
studying this new mechanism we also explore its possible interaction with the forward buckling transition proposed by
Lidmar et al. (2003). Lastly, we present a unifying view of the shearing transformation in terms of pre-stretch, which is
particularly useful for evaluating symmetry of skewed-capsomer shells.
2. A shell theory for prestrained viral capsids
2.1. Kinematics for icosahedral thin shells
The Caspar–Klug classification scheme (Caspar and Klug, 1962) for structure of icosahedral viruses can be formulated by
aligning a flat equilateral hexagonal sheet of capsomers with the unfolded flat template for an icosahedron such that
Fig. 3. Caspar–Klug construction: T¼4 and
T7(laevo
)
=
shells are constructed from hexamer lattices. Notice that T¼7 is a chiral structure while T¼4is
achiral.
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–10888

icosahedral vertices are connected by integral steps hand kalong the hexamer lattice basis vectors (see Fig. 3). Trimming
along the boundaries of the icosahedral template removes a wedge of angle
/3π
from each of the capsomers positioned over
an icosahedral vertex. As the template is folded along its edges (green in Fig. 3), the joining of adjacent boundary edges leads
to a Volterra construction of five-fold disclinations at each of the 12 icosahedral vertices. The triangular faces connecting
these 12 pentamers are tiled by a variable number of hexamers, with the total number of protein subunits
N
T
60=
, where
Th k hk
22
=++
is referred to as the “triangulation number”. Virus structures with h¼kor either h¼0ork¼0 have a mirror
symmetry, i.e. they are unchanged if hand kare switched. On the other hand,
h
k≠
and h,
k0
≠
result in chiral structures.
Following the work by Lidmar et al. (2003) (LNM) we construct a Kirchhoff–Love elastic shell theory for icosahedral
capsids, describing the deformation of the shell in terms of the bending and stretching of a two-dimensional surface
:
.A
fundamental assumption of the LNM theory is that the flat equilateral hexamer lattice is the stress-free reference config-
uration. Based on this, the CK construction can be understood as localizing (infinite) curvatures and bending stresses along
the icosahedral edges. Elastic relaxation will generally lead to a combined state of membrane stretching and bending of the
shell surface. To capture capsomer conformational change, we extend the LNM theory to include a second source of pre-
stress: local changes in the stress-free configuration of individual capsomers. We retain as our reference configuration the
flat lattice defining the icosahedral tiling of the CK construction scheme as depicted in Fig. 3, with the crucial distinction that
this reference state is no longer assumed stress-free.
We parametrize material points on the mid-surface by curvilinear coordinates
:
ss
(
,)
12
∈
. Positions of material points on
the mid-surface in the CK reference icosahedron are specified by reference mapping
:ssX(, ):
12 3
→
. Likewise positions of
material points on the mid-surface in the (current) deformed configuration are given by mapping
ssx(,
)
12
. The deformation
mapping from the CK reference state to the current configuration is
xX(). (1)φ=
To model local conformational changes we imagine that the shell is formed by a CK construction from capsomers that are
pre-deformed in isolation, prior to assembly (Fig. 4). Thus for a single isolated capsomer we can define a conformational
deformation mapping
c
φ
, which maps points Xin a symmetric hexamer of the CK reference configuration to position yin a
skewed stress-free configuration,
yX(). (2)
c
φ=
Because this mapping is defined piecewise over individual hexamers in isolation, we can use it to describe incompatible
conformational changes, i.e., piecewise continuous deformations of the individual hexamers that could break the CK tiling
(see Fig. 5). We next define an elastic deformation mapping from the stress-free configuration of an isolated hexamer to the
current deformed configuration,
xy(). (3)
e
φ=
Conformational strains associated with the transition between the symmetric CK reference state and skewed stress-free
state are then captured by the intra-capsomer “conformational deformation gradient”
FX(). (4)
cc
φ=∇
Likewise elastic strains between the skewed stress-free state and the current deformed state are described by the
Fig. 4. Decomposition of the deformation mapping
φ
into conformational part
c
φ
and elastic part
e
φ
. The decomposition is defined only piecewise over
each hexamer, allowing for conformational incompatibility between hexamers, while retaining global compatibility of the total deformation
φ
. In-
compatible conformational shearing creates in-plane pre-stress (see Section 2.2) in the icosahedral reference state, in addition to the bending pre-stresses
introduced during the CK construction (Fig. 3).
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108 89

intra-capsomer “elastic deformation gradient”
Fy(). (5)
ee
φ=∇
Within an isolated hexamer, it follows that these mappings form a composition
()
xX y X,() (). (6)
ec e c
φφφ φ φ φ=○ = = =
While the stress-free configurations of hexamers defined by Eq. (2) may be incompatible, integrity of the assembled shell
dictates that the deformation mapping (1) between the CK configuration and the deformed configuration must be one-to-
one. We can therefore understand the elastic mapping as restoring the compatibility broken by the conformational
mapping. The strains then decompose multiplicatively, with the total deformation gradient given as
FX FF() . (7)
ec ec
φφφ=∇ =∇ ∇ =
This multiplicative decomposition is analogous to that used in the context of finite kinematics plasticity (e.g. Lee and Liu,
1967;Lee, 1969) and growth mechanics (e.g. Himpel et al., 2005;Lubarda and Hoger, 2002), wherein decomposition of
deformation mappings is generally impossible because the growth or plastic deformations (analogous to
F
c
) may be point-
wise incompatible, and therefore non-integrable globally. Indeed we could in the present context take Eq. (7) as a starting
point for a model of capsid conformational skew, treating
F
c
as an independent primary field without deriving it from a
mapping
c
φ
. However, the conformational skewing of hexamers is most simply modeled by taking
F
c
from a piecewise
homogeneous isochoric shear deformation, which is trivially integrable. We find explicit expression of
c
φ
to provide a
convenient way to parameterize and compute
F
c
(Section 2.3) in the finite element implementation of the model (Sections
2.4 and 2.5).
2.2. Elastic energy
The elastic strain energy
Π
of the virus shell is modeled as the sum of two parts: a bending contribution
Π
b
due to the
curvature of the capsid shell and a stretching contribution
Π
s
due to in-plane deformations. We assume that the capsid
subunits do not possess an intrinsic curvature and therefore a flat configuration is the reference state from which incre-
ments in bending energy should be measured. Accordingly, we will use the full deformation mapping
φ
to evaluate the
bending energy
Π
b
as
:WdsW H Kbb[] () , () 1
2(2 ) , (8)
bbbCG
2
∫
φΠκκ==+
where
C
κ
is the bending modulus,
G
κ
is the Gaussian curvature modulus,
H
b
tr=
and
K
b
det=
are the mean and Gaussian
curvatures, and ds represents an infinitesimal element of the surface Sin the deformed configuration. Kirchhoff–Love ki-
nematics define the curvature tensor bin terms of second derivatives of the surface position mapping
φ
(see Appendix A).
This form of the bending energy is equivalent to that of a standard Kirchhoff–Love shell using the isotropic form of Hooke's
law, with
Yh /(12(1 )
)
C22
κ
ν=−
and
(1 )
GC
κ
κν=− −
in terms of thickness h, 2-D Young's modulus Y, and Poisson's ratio
ν
.It
is also important to note that for a closed shell of constant modulus
G
κ
the Gauss–Bonnet theorem ensures that the second
term in Eq. (8) is equal to
4
G
πκ
, a constant. Therefore, without loss of generality the last term in Eq. (8) can be dropped when
minimizing the virus capsid elastic energy. It also bears emphasis that the bending energy
Π
b
in Eq. (8) depends exclusively
on the geometry of the final deformed configuration. It is independent of parameterization, and could in principle be
Pre-Shear
Pre-Shear
Fig. 5. Conformational shearing of hexamers along “right-handed”shear direction (defined later in Section 3.1.1). (Left) Piecewise homogeneous shearing
of isolated hexamers would break the compatibility of the CK icosahedral construction (T¼7 shown). (Right top) On a three-fold symmetry axis adjacent
edges of hexamers are stretched differentially. (Right bottom) On a five-fold symmetry axis, the disclination angle is reduced for a ring of hexamers and a
gap is opened between the hexamers and the central pentamer.
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–10890

computed based on either
φ
or
e
φ
. We write
Π
b
in terms of
φ
because
φ
and
F
c
are the independent variables in our
formulation, and because
φ
is globally C
1
continuous, in contrast to
e
φ
which is only piecewise continuous over individual
capsomers.
The in-plane elastic energy
Π
s
is measured with respect to a sheared reference state, and is therefore a function of the
elastic part of the deformation gradient
F
e
. By frame indifference this energy must depend on the elastic deformation
through its right Cauchy-Green deformation tensor
C
FF
ee
Te
=
. Because experimentally observed deformations are rather
large (
10
%
∼
), we model the in-plane energy by invoking a large-strain split into volumetric and isochoric (shear) parts. We
use a simple energy function that maintains this decomposition, previously suggested for biomembranes by Evans and
Skalak (1979):
:
⎛
⎝
⎜⎞
⎠
⎟
WdSW J J
FC C C
[, ] ( ) , ( ) 2(1) 2
tr( ) 2. (9)
s c se se se
2
∫
φΠκμ
==−+−
where
κ
s
is area-stretching modulus,
μ
is the shear modulus, and
J
ds
dS
=
is the ratio of area after and before deformation
calculated as
⎡
⎣
⎢⎤
⎦
⎥
IJ CC
1
2tr ( ) tr( ) . (10)
ee
222
== −
Linearization of stresses computed from Eq. (9) shows that
κ
s
and
μ
are related to linearized 2D Young's modulus Yand
Poisson's ratio
ν
through the expressions
Y
2(1 ),(11a)
s
κν
=−
Y
2(1 ).(11b)
μν
=+
In all our calculations, we assume
1/3
ν
=
and therefore
Y3/4
s
κ
=
and
Y3/8
μ
=
.
2.3. Parameterizing the conformational deformation
The conformational deformation is modeled as a simple shear deformation of magnitude
η
in the plane of the flat
reference configuration. Let
uu
U
{,
}
12
=
be an orthonormal basis spanning the reference plane (Figs. 6 and 7) and defining
the orientation of the conformational shear such that
XuuXX X() ( ) (12)
cuu u
12
122
φη=+ +
where
Xu
X
u=·
αα
are the components of
X
in the frame U. Then
F
c
is
uuF1 (13)
c12
η=+ ⊗
where
uu
1
=⊗
α
α
is the 2-D identity tensor in the reference plane. The component representation is then
Fig. 6. A simple shear conformational deformation described by
F
c
may be decomposed in a pure stretch deformation (U) followed by a pure rotation (
R
)
using the right polar decomposition
F
R
U
c
=
.
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108 91

⎡
⎣
⎢⎤
⎦
⎥
uuFFF() , [()] 1
01 (14)
cc
UcUη
=⊗ =
αβ αβ αβ
If, as in a finite element computer implementation, tensorial quantities are expressed in components relative to a global ‘lab’
frame
EEEE{, ,
}
123
=
, then placing the reference configuration in the
E
1
-
E
2
plane we can define
θ
as the shear orientation
uEEcos sin (15a)
112
θθ=+
uEEsin cos (15b)
212
θθ=− +
such that (14) can be re-written as
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
EEFFF() , [()] 1 cos sin cos
sin 1 sin cos .
(16)
cc
EcE2
2
ηθθ η θ
ηθ ηθθ
=⊗ =
−
−+
αβ αβ αβ
Eq. (16) defines the skewing or conformational deformation gradient
F
c
in terms of shear magnitude
η
and direction
θ
, which
may be assigned based upon experimental observations and symmetry considerations or alternatively considered as
“conformational degrees of freedom,”with respect to which the elastic energy could be minimized.
In order to construct the simplest model consistent with experimental observations, we assign these shear magnitude
and direction variables to be piecewise constant in each hexamer. While hexon conformational deformations are most often
described as involving a shear of the hexamers, it may also be informative to describe the deformation in terms of its
principal stretches, by a standard polar decomposition. We write the right polar decomposition of
F
c
as
F
RU
c
=
where
vv vv
U
FF() 1
c
Tc1/2 11 22
λλ
==⊗+⊗
with maximum principal stretch given by
1tan
2
λ
ηβ=+
where
⎜⎟
⎛
⎝
⎞
⎠
tan 212
2
βηη
=+ +
gives the counterclockwise angle of the principal stretch directions relative to the Uframe
vuucos sin (17a)
112
ββ=+
vuusin cos . (17b)
212
ββ=− +
Relative to the global lab frame E, the principal stretch angle is
α
θβ=+
, yielding stretch tensor components
Fig. 7. Reference equilateral triangle in X
1
–X
2
plane including the curvilinear coordinates s
1
and s
2
along edges 3-1 and 3-2 used to compute the surface
basis vectors on the virus capsid.
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–10892

⎡
⎣
⎢
⎢
⎢
⎢
⎛
⎝
⎜⎞
⎠
⎟
⎛
⎝
⎜⎞
⎠
⎟
⎤
⎦
⎥
⎥
⎥
⎥
U[]
cos 1sin sin cos 1
sin cos 1sin 1cos
.
(18)
E
22
22
λα
λαααλ
λ
ααλ
λλα
λα
=
+−
−+
μν
The conformational shear state may be parameterized equivalently by
(
,)ηθ
in Eq. (16),orby
(
,)λα
in Eq. (18).
2.4. Equilibrium
Equilibrium configurations of a virus capsid are computed by minimizing its total elastic energy, which is a function of
the actual deformation
φ
and the conformational deformation
F
c
,
FF[, ] [] [, ]. (19)
cb sc
φφφΠΠΠ=+
Because we model conformational changes in the shell as piecewise homogeneous deformations in isolated hexamers, we
can be more explicit about the parameterization of these deformations in terms of pre-shear magnitudes
η
i
and angles
θ
i
(
iN1, ,
hexamers
=…
)
F[, ] [] [, , ], (20)
cb s
ii
φφφΠΠΠηθ=+
or equivalently in terms of pre-stretch magnitudes
1tan
iii
λ
ηβ=+
and angles
α
i
.
We may choose to think physically of the conformational parameters (
η
i
and
θ
i
,or
λ
i
and
α
i
) as being fixed by the
chemistry of the capsid proteins, in which case equilibrium of the shell would entail minimization of the total energy with
respect to the deformation
φ
only. In this view, the chemical driving forces that produce conformational change of hexamers
in isolation will inject a state of pre-strain and pre-stress into the assembled shell. More generally, the conformational
parameters would have an additional free energy
g
(, )ηθ
, such that its sum with the elastic energy in (19) would be
minimized with respect to both the conformational variables and the mechanical deformation. Here we consider the two
extremes: first fixing
η
and
θ
for the hexamer according to experimental observation, and later setting g¼0 and minimizing
the elastic energy to identify “elastically optimal”states of conformational strain.
Most generally, for minimization of the energy with respect to
φ
and
F
c
, vanishing of the variation of (19) produces two
conditions (see Appendix B)
:
nmddS()0, (21a)
,,
∫φδδ·+· =
αααα
:
⎡
⎣
⎢⎤
⎦
⎥
()
dSFP F:0, (21b)
Tec1
∫δ=
−
(21a) representing the weak form of equilibrium, or balance of physical forces on the deformed configuration, and (21b) the
weak form of balance of configurational forces on the reference configuration.
2.5. Finite element approximation
In our work, we set the reference configuration Xof each triangular element to be the equilateral triangle with edge
length hrepresented in Fig. 7.
The kinematics of the virus capsid undergoing maturation is modeled using Kirchhoff–Love thin shell theory (Ti-
moshenko et al., 1959;Green and Zerna, 1992), according to which the shell director remains planar and perpendicular to
the shell middle surface during deformation. As a consequence, the elastic energy of the capsid may be written in terms of
displacements only, without using additional rotations fields, which, in turn, may be expressed as the displacements first
spatial derivatives. The shell bending energy therefore contains second derivatives and requires C
1
-continuous finite ele-
ment methods in order to guarantee numerical convergence. In our calculations we use the finite element proposed in Cirak
et al. (2000), which is based on the non-local Loop approximation scheme and provides C
1
continuity of the displacement
field.
According to classic finite element formulation, we express the continuum position field
x
as a function of discrete nodal
positions
x
a
ss Nssxx(, ) (, ), (22)
h
a
N
aa12
1
12
nodes
∑
=
=
where hrepresents the chosen mesh size and
Nss(,
)
a12
are the finite element shape functions. Different shape functions are
used to compute the position field
x
in the stretching and bending energy calculation. Piecewise linear C
0
-continuous
Lagrange interpolations are used in the stretching energy calculation. C
1
-continuous Loop's subdivision shape functions are
instead used in the calculation of the bending energy term, satisfying therefore the convergence requirement previously
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108 93

highlighted. We refer to Cirak and Ortiz (2001) and Cirak et al. (2005) for further implementation details, validation and
numerical examples regarding the subdivision surface shell finite elements used herein.
Numerical energy minimization is performed after inserting Eq. (22) in Eq. (19) and integration over the virus capsid is
obtained using a one-point Gauss quadrature rule per element. We note that integration is never performed on the skewed
broken and discontinuous configuration y. A quasi-Newton algorithm Limited Memory-BFGS is used (Zhu et al., 1997) in our
calculations to overcome convergence problems due to instabilities such as buckling transitions between two configurations
of the virus capsid.
3. Modeling the maturation process
We begin the study of our model by illustrating the effect of pre-shear on shell deformation. First we observe that when
zero pre-shear is assigned, i.e.
FI
c
=
, the present theory is nearly identical to that of Lidmar et al. (2003), differing only in the
specific form of the in-plane elastic energy
Π
s
, to which Lidmar et al. (2003) assigned Hookean form. In such a case the
closed shell constructed from the Caspar–Klug lattice would correspond to the zero in-plane strain energy state, but would
have infinite bending energy along the icosahedral edges. As Lidmar et al. (2003) showed, the minimum energy (equili-
brium) configuration for finite bending and stretching moduli interpolates between a perfect icosahedron (minimal in-plane
energy) and a sphere (minimal bending energy). The result of this competition shown in Fig. 8 (right column) for a T¼7
shell with
125
0γ
=
is noticeably faceted in shape with an asphericity roughly matching that of the partly matured EI-II
conformation of bacteriophage HK97 (Fig. 2), but differing from the CK icosahedron with curvature relaxed along the ico-
sahedral edges. The folding of the sheet of hexamers along icosahedral edges can be understood as a source of pre-stress or
geometric incompatibility induced by the CK construction, which is partially relaxed by elastic stretching. The peak values of
stretch, as represented in Fig. 8 by both invariants I
1
and Jof the in-plane stretch tensor, occur at the five-fold icosahedral
vertices, which as noted above are disclination defects created by the CK construction. Values of
J
1<
at the vertices indicate
2.0125
2.025
2.0375
2
2.05
0.725
0.85
0.975
0.6
1.1
Fig. 8. Simulated transition from prohead (left) to head (right) state for a
T
7
=
virus capsid. The deformation invariants
I
CJtr( )/
e1
=
and
I
J
2=
are shown
together with the pre-shear direction in the prohead state and hexamers outline in the head state (due to lack of invariants discontinuities in the head
state, hexamers are indistinguishable without outline). Finite element models consist of
1
.1 10
5
×
elements and
1
.6 10
5
×
degrees of freedom. Material
moduli are chosen so that the FvK number is
125
0γ
=
, which gives the shell an asphericity roughly matching that of the partly matured EI-II conformation
of bacteriophage HK97 (Fig. 2). Notice that the modest increase in radius observed in Fig. 2 during the P-II to EI-II transition is not imposed during the
simulated transition. Hence, the change in macroscopic shell morphology is driven solely by the conformational change and not by an imposed change of
the FvK number.
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–10894

large compressive strains. Apart from these concentrations of strain focused on the defects, the deformation field is ev-
erywhere exceedingly smooth, with in-plane stretch ratios very close to 1.
With an additional conformational change to the reference state of the hexamers (
FI
c
≠
), the flat hexagonal sheet and
the folded CK icosahedron must store in-plane elastic energy in addition to the energy of bending along the icosahedral
edges. Conformational strain therefore adds a second source of incompatibility and pre-stress to the CK construction to be
relaxed by elastic deformation. To simulate the skewed-hexamer immature prohead conformation of bacteriophage HK97
we assign a pre-shear by Eq. (13), with shear directions
u
1,2
and magnitude
0.1
2η
=
consistent with the experimental
structure (Fig. 2)ofaT¼7 shell. To examine the mechanical effects of conformational change independent of changes in
material properties, we assign the same moduli as for the EI-II state, with
125
0γ
=
. The signature of conformational in-
compatibility is evident in the relaxed elastic strain field (Fig. 8, left column): strain concentrations at the icosahedral
vertices remain but are joined by large strains building up along the interfaces between capsomers. Both invariants of the
elastic in-plane deformation tensor
C
e
are discontinuous along these interfaces, owing to the conformational incompatibility
of
F
c
. In order to enforce compatibility of the final equilibrium configuration, the occurring elastic deformation must also be
incompatible, resulting in deformation discontinuities and consequent stress concentrations. Discontinuities in elastic de-
formation are noticeably absent along interfaces between hexamers that are pre-sheared along parallel directions; they
exist only along interfaces where
F
c
is discontinuous: hexamer–pentamer edges, and hexamer–hexamer edges where
θ
jumps by 60°.
In addition to “microscopic”changes in deformation pattern, we observe that hexamer skew also affects “macroscopic”
shell morphology. Specifically, for the values of
η
and
θ
in Fig. 8 the shell is much less faceted. This feature is also matched in
the experimental structure of the skewed-hexamer capsid (Fig. 2). This suggests that local capsomer conformational
changes can be interpreted as a driving force for changing the global morphology. The mechanisms behind this effect can be
understood by focusing on the geometry of a ring of hexamers surrounding a pentamer (cf. Fig. 5 and Aggarwal et al., 2012;
Aggarwal, 2012). For the shear orientations
θ
evident in the experimental structure (Fig. 2), shear magnitudes
0.1
η
≈
have
two important effects: reducing the disclination angle, and increasing the mean circumference of the hexamer ring. These
mechanisms both contribute to the reduction in compressive stresses in the central pentamer, and together create a “reverse
buckling”transition, which opposes the disclination-driven buckling discovered by Lidmar et al. (2003). For moderately
positive values of
η
this produces shapes with very low asphericity.
While this analysis provides some compelling explanation of the P-II to EI-II transition in HK97, it leaves open questions
about the effect of conformational strain in other orientations, and in shells with different material properties. To further
explore these mechanisms, we next study the mechanics of T¼7 shells with a range of FvK numbers, and with all three of
the pre-shear orientation patterns along the hexamers maximal diagonals allowed by icosahedral symmetry. To examine the
generality of the mechanisms for capsids both larger and smaller than T¼7 we study two other CK structures, T¼4 and
T¼16.
3.1. Effect of shear orientation and FvK number
The analysis of a T¼7 shell (Fig. 8) illustrated the mechanical effects of hexamer pre-shear on the key measure of
asphericity in the relaxed capsid shell. Here we explore how this measure depends on (i) material properties of the shell, as
reflected in the FvK number
γ
, and (ii) magnitude and (iii) orientation of the conformational pre-shear of the hexamers. We
define the asphericity quantitatively as a measure of the deviation from a perfect sphere (Lidmar et al., 2003)
AR
R,(23)
2
2
=〈Δ 〉
〈〉
where
RR RΔ= −〈
〉
and angle brackets
〈
·
〉
denote average over the shell surface. For our finite element models, these
averages are computed over the vertices of the mesh. For reference, we note that A¼0 for a perfect sphere and
A0.002
6
≈
for a perfect icosahedron.
We begin by further probing the T¼7 virus structure from Section 3, and subsequently explore commonalities and
differences among T number structures by considering also T¼4 and T¼16 shells.
3.1.1. T¼7 shells
The T¼7 shell is the largest CK structure having a single class of symmetry equivalent hexamers. Specifically, all the
hexamers are surrounded by one pentamer and five other hexamers. Therefore, to maintain icosahedral symmetry any
choice of a conformational pre-shear in one hexamer determines the pre-shear in all other hexamers as well. To narrow our
parameterization of conformational deformations, we consider only icosahedrally symmetric configurations with pre-shear
directions oriented parallel to the three maximal hexamer diagonals, (i.e., lines connecting hexamers’opposite vertices).
Accordingly there are three such shear directions to be investigated (Fig. 10):
(a) along the diagonal incident to the rightmost (or first in counterclockwise order) vertex shared with the adjacent
pentamer;
(b) along the diagonal incident to the leftmost (or last in counterclockwise order) vertex shared with the adjacent pentamer;
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108 95

(c) along the diagonal parallel to the edge shared with the adjacent pentamer.
Considering a single pentamer and its ring of adjacent hexamers, orientations (a) and (b) are chiral isomers; we therefore
refer to (a) as the “right-handed”conformation and (b) as the “left handed”conformation. Orientation (c) we refer to as the
“parallel”conformation. These labels represent a slight abuse of the language in that the deformed shapes will all be chiral
for any nonzero shear magnitude. Moreover, chirality of the shear orientations is distinct from the chirality of the C–K
structure. However, in reference to an isolated ring of hexamers and their central pentamer the labels provide a geome-
trically intuitive way of classifying the shear orientations independent of T-number.
The right-handed pre-shear orientation, corresponds to hexamer skew observed experimentally for HK97, P22 and
λ
-
phage viruses (Ross et al., 2005;Jiang et al., 2003;Dokland et al., 1993)(Figs. 2 and 8). Slightly positive values of
η
in this
orientation have the effect of opposing the buckling transition, producing close to spherical shapes. As
η
is increased further
(up to
0.3
5η
=
, here with
240
0γ
=
), the equilibrium configuration transitions toward a dodecahedral shape, uncovering a
reversed buckling transition in which pentamers are flat and hexamers are pointed outward. For negative values of
η
, a right-
handed pre-shear has the opposite effect of further driving the buckling of the pentamers, producing a stellated equilibrium
shape. Neither the dodecahedral nor the stellated configurations are obtainable only by changing the capsid FvK number,
highlighting the broader reach of the hexamer pre-shearing mechanisms with respect to the LNM model.
Prior analysis (Aggarwal et al., 2012) has identified two mechanisms driving these convex (spherical and dodecahedral)
and concave (stellated) transitions, both of which can be understood from Fig. 5. First, the disclination angle is reduced for
0η
>
, and increased for
0η
<
. Second, the ring of hexamers is stretched circumferentially and pushed outward radially for
0η
>
(or compressed inward for
0η
<
). By manipulating the disclination angle or injecting either compressive or tensile
radial stress, these mechanisms act directly to accentuate or suppress the disclination-induced buckling transition of Lidmar
et al. (2003).
Because for an isolated pentamer and ring of hexamers the left handed pre-shear orientation is the mirror image of the
right-handed orientation, it makes sense that the shapes are qualitatively the same, only with the sign of
η
reversed.
Although qualitatively the same, because the T¼7 structure itself is chiral, these two orientations are not perfect mirror
image shapes. To be quantitative in our comparison, we plot the asphericity Aversus
γ
and
η
(Fig. 10). The effect of
0η
≠
is
more generally pronounced for right-handed pre-shear, meaning that right-handed pre-shear is slightly more effective at
reversing or enhancing the buckling transition. In particular, for large FvK numbers (
100
0γ
>
) we see that the minimum
value of asphericity for the left handed orientation is slightly larger than that of the right-handed orientation. Interestingly,
we also observe that the critical value of pre-shear
0.2
η
≈
, which produces minimum capsid asphericity, is insensitive to the
FvK number and therefore the capsid material properties. Furthermore, this result remains valid across the range of T
numbers, as we will demonstrate by example in the following examination of T¼4 and T¼16. This finding is consistent with
the idea that the effects of pre-shear are isolated to the hexamer ring surrounding each pentamer (Aggarwal et al., 2012;
Aggarwal, 2012).
In contrast to the right- and left handed orientations, pre-shear along the “parallel”direction, for both positive and
negative values of
η
, has limited impact on shell shape (Figs. 9 and 10). Here the dominant factor determining the capsid
global shape is the FvK number, modulating the LNM buckling transition. In other words, pre-shear oriented parallel to the
pentamer edges is ineffective in coupling to the buckling transition. In the small strain limit as analyzed in Aggarwal et al.
(2012) and Aggarwal (2012), such a shear acts only to rotate the central pentamer, without inducing any circumferential
strain in the ring of hexamers. This logic is consistent with the relative insensitivity of asphericity to
η
for this “parallel”
direction as compared to the left- and right-handed orientations—by neither changing the disclination angle nor inducing
any radial stress, pre-shear parallel to the pentamer edges should be expected to neither enhance nor suppress the buckling
transition.
Moreover there is a second way to rationalize the
η
-insensitivity of asphericity in the parallel shear direction that is
helpful in projecting how these results should extend to other T-numbers. For an isolated hexamer-ring surrounding a
pentamer, pre-shears of opposite signs
η+
and
η
−
would be exact mirror images. We can conclude then by symmetry that for
an isolated ring of hexamers the asphericity is an even function of pre-shear, with
AA() ()ηη−= +
. It must therefore be
that
A(0)η=
is an extremum, either a minimum or a maximum. In either case
A()η
will be locally flat, and therefore
insensitive to small changes in
η
. For C–K shells like T¼7, chirality of the structure breaks this symmetry, but still we might
expect by this logic to have less sensitivity than for the pre-shear orientations that lack the
η±
symmetry. Furthermore, by
this logic we can guess that achiral C–K shells with hexamer pre-shear oriented parallel to pentamer edges will have either a
minimum or maximum in asphericity at
0η
=
, with shell shape insensitive to
η
over some range. In the following we check
this projection by analyzing achiral shells of triangulation number T¼4 and T¼16 .
3.1.2. Phase diagrams for T ¼4 virus structure
The putative mechanisms underlying the “reverse buckling transition”were predicted to act independent of the T-
number and here we examine how they extend to different T-structures. The study of the conformational changes in a T¼7
viral capsid highlighted that the chirality of the shell influences how pre-shear orientation and magnitude affect the capsid
shape. In particular, absence of chirality would render the structure insensitive to a shear along a “parallel”orientation.
Here we first consider a T¼4 virus shell to investigate the effect of removing the structural intrinsic chirality proper of a
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–10896

T¼7 capsid from the results presented in the previous section. Similarly to T¼7 virus capsid, we consider three pre-shear
directions associated to a T¼4 structure: a right and a left handed direction, and a parallel direction. A pre-shear in either
the right or left handed direction results in equilibrium configurations comparable to the ones observed for a T¼7 virus
capsid (in Figs. 11 and 9 equilibrium configurations are computed using the same FvK number to allow a straightforward
comparison). Phase diagrams obtained for T¼4 and T¼7 capsids are also qualitatively very similar. The significant difference
between them is due to the absence of intrinsic structural chirality in a T¼4 capsid. Consequently, in the case of T¼4, the
phase diagrams obtained for right and left handed directions and positive/negative pre-shear magnitudes are exactly mirror
images (Fig. 11a and b).
The absence of a structure chirality also renders completely ineffective pre-shear in the parallel direction (Fig. 11). In this
case, the capsid shape is fully determined by the FvK number. In contrast, the structural chirality proper of the T¼7 pro-
moted small changes of the capsid equilibrium configuration even in the case of pre-shear in the parallel direction (Fig. 9).
3.1.3. Phase diagram for T ¼16 virus structure
A key feature of larger capsids, with
T
7
>
is the presence of distinct classes of hexamers defined by their transformation
under operations in the icosahedral symmetry group. For instance, all T¼4 hexamers are centered on icosahedral edges.
Each of these hexamers can be transformed into any of the others by a five-, three-, or two-fold rotation about a corre-
sponding icosahedral symmetry axis. Likewise the T¼7 hexamers are found in rings around pentamers, and each can be
transformed into any other by one of the icosahedral symmetry rotations.
To investigate the effect of conformation shear in larger capsids with different hexamer classes, we examine a T¼16 virus
structure, typical, for instance, of the herpes simplex virus (HSV). Indeed, T¼16 hexamers may be categorized in three
distinct groups: (1) face centered hexamers with one vertex at the three fold sites (2) edge-centered hexamers, and
(3) hexamers in rings surrounding each pentamer. As a consequence, there exist
3
2
7
3=
icosahedrally symmetric pre-shear
configurations if only pre-shear directions along maximal diagonals are considered. Here, among all possibilities, we in-
vestigate only a specific configuration, in which the hexamers surrounding the pentamers are pre-sheared in order to pre-
stretch the pentamers edges. Simple elasticity analysis and the previous results obtained in Sections 3.1.1 and 3.1.2 predict
Fig. 9. Minimum elastic energy configurations of T¼7 shells with
240
0γ
=
and with hexon pre-shears parallel to the three possible maximal diagonal
directions: (a) “right-handed”(b) “left handed”, and (c) “parallel”. The relaxed shapes are shown colored by their local curvature. (For interpretation of the
references to color in this figure caption, the reader is referred to the web version of this article.)
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108 97

that this configuration should maximize the sensitivity of the pre-shear
η
on the capsid shape and asphericity A. In par-
ticular we chose the “right-handed”configuration to assign pre-shear to the hexamers surrounding the pentamers. The
edge-centered hexamers are then pre-sheared in the same direction as the hexamers surrounding a pentamer so that no
pre-shear discontinuity is introduced between edge hexamers. However, it remains unclear what pre-shear direction must
be assigned to the face-centered hexamers and we decided to not assign any pre-shear to these hexamers. This choice
allows us to determine if the 5-ring of hexamers are sufficient to reverse the buckling transition in larger T-number shell
and the face centered hexamers are unnecessary in this regard. The effect of pre-shear in the face centered hexamers will be
considered in the analyses presented in Sections 3.2 and 3.2.1.
A positive pre-shear leads to a spherical virus configuration first and subsequently to a dodecahedral shape for large
η
values (
0.3
5η
=
for
γ
≂2400). Negative values of pre-shear result instead in a stellated equilibrium configuration (Fig. 13).
An equivalent structural response was observed in the case of T¼7 and T¼4 viral capsids subjected to a similar right-
handed pre-shear configuration. We observe that, as predicted, the mechanisms of reverse buckling transitions carry over to
the larger T¼16 virus capsid.
The results presented in this section have shown how pre-shearing of the hexamers leads to a large range of shape
changes without modifying the capsid elastic properties. In this context, we may think at the hexamer pre-shear changing
an effective FvK number and providing a physical explanation of the geometrical changes observed experimentally during
capsid maturation. An effective FvK number
eff
γ
may be obtained by intersecting the asphericity surfaces
A(, )ηγ
in Figs. 10,
12 and 13 with a plane parallel to the
(
,
)
ηγ
plane, i.e.
AA() { (, )
}
o
eff
γ
ηγηγ=| =
. Here, we chose the value
A1.0 10
o4
=×
−
to
define the transition between the un-buckled and buckled shell configuration (Fig. 14), but qualitatively equivalent
(
)
eff
γ
η
phase diagrams may be obtained for other values of A
o
. These curves mark the boundaries in
(
,
)
ηγ
phase space between
smooth and facetted morphologies. Combinations of
η
and
γ
“below”the curves will produce smooth, nearly spherical
shapes, while those “above”the curves will be facetted, ether icosahedral, stellated, or dodecahedral. Fig. 14a confirms that
for T¼7 shells the right-handed pre-shear direction is the most effective in changing the viral capsid asphericity at a given
value of
η
. In contrast, as discussed in the foregoing, the parallel pre-shear direction is the least effective for all T-numbers
Fig. 10. Asphericity Aof a T¼7 virus structure as a function of FvK number γand pre-shear magnitude ηfor (a) “right-handed”(b) “left handed”, and (c)
“parallel”shear orientations. “right-handed”orientation corresponds to the observed structure of skewed hexons in HK97, P22 and λ-phage viruses.
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–10898

and the FvK number does not change with
η
along this direction, i.e.
eff
γγ
≈
and it is determined by the capsid elastic
properties. As expected, the
eff
γ
curves intersect at
0η
=
since
(0)
eff
γ
η
γ
==
for all T-numbers. Since the T¼4 capsid is
achiral, the
eff
γ
curves computed for the T¼4 shell using the right and left handed pre-shear directions are mirror sym-
metric, in agreement with the results presented in Section 3.1.2.
3.2. Do minimum energy capsids have icosahedral symmetry?
In Section 3.1 we analyzed the effect of several pre-shear magnitudes and directions on the shape and asphericity of T¼4,
T¼7, and T¼16 viral structures. Although this analysis extended the application of our model beyond the specific pre-shear
orientation evident in the HK97 experimental data (Fig. 2), it was nonetheless still restricted to a discrete set of config-
urations with prescribed
η
and
θ
. Moreover, in the foregoing we have chosen only pre-shear configurations that respect
icosahedral symmetry, in both magnitude
η
and direction
θ
. However, there is not a clear physical reason to impose a priori
these restrictions. To consider the more general question of what conformational pre-shears would be energetically optimal,
we allow both the pre-shear magnitude
η
and pre-shear direction
θ
for each hexamer to vary continuously and in-
dependently. We assign independent
η
and
θ
degrees of freedom to each hexamer, defining locally a piecewise uniform pre-
shear
F
c
, and we minimize the resulting capsid elastic energy numerically with respect to the full complement of degrees of
freedom: the shell nodal positions and the hexamer pre-shear magnitudes and directions. In this augmented version of the
model, the pre-shear configurations are neither restricted among a discrete number of prescribed choices, nor forced to be
icosahedral symmetric. In general, the physical driving force which determines the orientation and magnitude of pre-shear
is still unknown. Here we assume that there is no chemical potential driving the values of
η
and
θ
. Each hexamer pre-shear
magnitude and direction is then determined so as to render the capsid elastic energy as small as possible.
Considering pre-shear magnitude and direction as degrees of freedom also carries a more subtle advantage. Even if in
some cases it is possible to estimate
η
and
θ
from experimental images, these values are measured in the capsid assembled
configuration and only estimate the pre-shear modeled here through the conformational deformation gradient
F
c
but does
not correspond to it exactly. Indeed, in our model
F
c
represents the uniform pre-shear of each isolated hexamer, before they
are deformed elastically to construct the assembled capsid (Fig. 5, left). Considering
η
and
θ
as degrees of freedom avoid the
Fig. 11. Relaxed shapes of T¼4 shells with
240
0γ
=
and with hexon pre-shears parallel to the three possible maximal diagonal directions: (a) “right-
handed”(b) “left handed”, and (c) “parallel”. The minimum elastic energy configurations are colored by their local radius. (For interpretation of the
references to color in this figure caption, the reader is referred to the web version of this article.)
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108 99

necessity to set their values a priori and to identify them with the pre-shear observed in the capsid assembled configuration.
In all the analyses presented in the present section, the elastic energy of the capsid is minimized with respect to the pre-
shear conformational degrees of freedom
η
and
θ
defined on each hexamer in addition to the capsid deformation mapping
(i.e. displacements) degrees of freedom.
As a first case, we analyze a T¼7 virus capsid with FvK number equal to
125
0
∼
, which is identical to the T¼7 capsid
considered at the beginning of Section 3. The minimum elastic energy configuration is close to spherical and corresponds to
non-zero values of pre-shear (Fig. 15(b)) even when the latter is not imposed a priori by construction. Therefore, the elastic
energy cost associated with the assembly of a capsid is reduced if the hexamers are pre-sheared and, if the hexamers are
free to shear, then it is predicted that pre-shear would be spontaneous. In other words, pre-sheared hexamers assemble more
efficiently than hexamers with a unique symmetric elastic reference configuration and consequently pre-shear can be a me-
chanism to aid assembly. Our theory suggests that assembly of chiral capsids is likely to proceed from hexamer units that
have a tendency to shear along a symmetry axis.
The pre-shear direction resulting from the minimization of the capsid elastic energy coincides, within about 3°, with the
right-handed configuration chosen previously (Fig. 8) based on experimental observations. This direction, according with
the observations reported in Section 3.1.1, is more effective than the parallel shear direction in inducing capsid shape
changes for a given pre-shear magnitude. Any pre-shear magnitude corresponds to an energy cost related to the strain/
stress discontinuities across the hexamer edges. This in-plane energy cost increases with the magnitude of the pre-shear. On
the other hand, pre-shear may also lower the capsid asphericity and consequently the capsid bending energy. According to
this energy trade-off, we expected the energy minimization process to select a right-handed pre-shear direction, which, at
equal pre-shear magnitude, is the most effective pre-shear direction in changing the capsid asphericity. In other words, the
right-handed direction lowers the capsid bending energy for the same in-plane energy cost.
Interestingly, both the resulting pre-shear magnitudes and pre-shear directions respect icosahedral symmetry in the
minimum elastic energy configuration for a T¼7 structure, even though each hexamer possesses its own conformational
degrees of freedom and non-symmetric configurations are a priori allowed.
In this framework we can compare minimum energy configurations obtained with or without pre-sheared hexamers’
reference configuration. Without including any hexamer pre-shear, the capsid asphericity Aat equilibrium is decided solely
by the FvK number
γ
: larger values of
γ
correspond to larger values of A(Fig. 16). A similar trend is observed also when
Fig. 12. Asphericity Aof a T¼4 virus structure as a function of FvK number γand pre-shear magnitude ηfor (a) “right-handed”(b) “left handed”, and (c)
“parallel”shear orientations.
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108100

pre-shear is included as a degree of freedom but the asphericity versus
γ
curve is shifted toward larger values of
γ
, effectively
delaying the buckling transition.
We observe that the asphericity versus
γ
curves are closer for small (∼10
2
) and large (∼10
4
) values of
γ
. Therefore, we can
deduce that pre-shear is more effective in changing the capsid asphericity in the intermediate range, while the virus
equilibrium configuration is mostly dictated by
γ
at the extremes.
Larger values of pre-shear
η
are allowed at small
γ
values, which correspond to low in-plane Young's modulus relative to the
bending modulus. In this case, in plane strain discontinuities caused by large values of
η
are accommodated because their
energetic cost is insignificant at the global level if they contribute to lower the bending energy. On the contrary, small
η
values
are obtained at large values of
γ
, for which lowering the bending contribution to the total energy is not as important as lowering
the in-plane energy part. In essence, there is an effective elastic penalty on
η
, which depends on the relative energy cost of the
strain incompatibilities along the hexamers edges: the elastic energy penalty is small at low values of
γ
and grows with
γ
.This
trend is clearly shown in Fig. 16, in which the optimal value of
η
seems to converge to zero as
γ
→∞
and to some finite value
(
0.1
5
≈
in the current case) as
0γ
→
. The strain incompatibilities (not shown here) on the viral shells used to construct Fig. 16
grow larger as
0γ
→
confirming that the pre-shear
η
is suppressed by the increased dominance of in-plane elasticity as
γ
→∞
.
In all the analyses presented so far, the pre-shear directions are initialized according to the right-handed configuration
discussed in Section 3.1.1 and proper, for instance, of HK97. During the analyses, the pre-shear directions
θ
are then allowed
Fig. 13. Phase diagram and equilibrium configurations for T¼16 virus capsid as a function of pre-shear magnitude ηand FvK number γ. The shown capsids
at equilibrium are obtained for
240
0γ
≃
and are colored by their local radius. Similar reverse buckling transition as the ones shown in Figs. 9 and 11 is
observed. We note that the minimum asphericity in a T¼16 viral capsid corresponds to
0.2
5η
≈
while it corresponds to
0.
2η
≈
for a T¼4 and T¼7 shells.
(For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
Fig. 14. FvK number as a function of pre-shear magnitude ηat fixed asphericity
A1.0 10
o4
=×
−
. Different T-number structures (from left to right:
T
7
=
,
T4=
, and
T16=
) and pre-shear directions (right-handed, left handed and parallel) are considered.
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108 101

to vary continuously and their values is determined by minimizing the capsid elastic energy. Since the energy may be non-
convex with respect to the conformational degrees of freedom
η
and
θ
, it is possible to identify local energy minima
dependent on the chosen initial condition. Therefore, we want to investigate if the obtained results depend on the choice of
Fig. 15. Initial and final equilibrium configurations obtained for a T¼7 virus capsid with
125
0γ
∼
. The capsid elastic energy has been minimized with
respect to displacement and conformational degrees of freedom per each hexamer. (a), (d), and (g) represent three possible initial configurations with
different pre-shear directions: right-handed direction, parallel direction and left handed direction, respectively. (b), (e), and (h) illustrate the minimum
energy equilibrium configurations corresponding to (a), (d), and (g) respectively in terms of pre-shear directions (double single arrows) and magnitude
(contour plot). (c), (f), and (i) depict the minimum energy equilibrium configurations corresponding to (a), (d), and (g) respectively in terms of pre-stretch
directions (double pointed arrows) and magnitude (contour plot). (For interpretation of the references to color in this figure caption, the reader is referred
to the web version of this article.)
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108102

initialization and, to this end, we repeat the energy minimization analyses starting from two additional initial pre-shear
directions: parallel initial direction (Fig. 15d) and left handed initial direction (Fig. 15g). We find that the final equilibrium
configurations (Fig. 15 (e) and (h)) obtained starting from parallel and left handed initial directions possess exactly the same
elastic energy and the same final pre-shear directions and magnitudes. Equal elastic energy and pre-shear magnitudes are
also found at equilibrium if the pre-shear directions are initialized according to the right-handed direction (Fig. 15a).
However, a right-handed initialization of the pre-shear directions leads to pre-shear directions at equilibrium rotated by
93≈
°
with respect to the left handed and parallel initialized analyses (Fig. 15 (e) and (h)). This apparent difference is resolved
if we express the conformational part
F
c
of the deformation gradient (Eq. (7)) in terms of its polar decomposition, i.e.
F
R
U
c
=
. The polar decomposition of
F
c
highlights that all the pre-shear directions
θ
and magnitudes
η
computed starting
from different initializations correspond to the same right stretch tensor
URF
T
c
=
and therefore are exactly equivalent. In
other words,
F
c
corresponding to the three pre-shear configurations differs only by a rigid rotation and lead to the same
elastic energy.
In Section 3.1.3 we examined the effect of a specific pre-shear configuration on the equilibrium shape of a T¼16 viral
capsid. In that case, we chose the pre-shear directions to maximize the effect of pre-shear in changing the capsid asphericity
based on previous results and simply elasticity analyses. Moreover, for simplicity we neglected pre-shear in the face cen-
tered hexamers. A complete set of analyses to investigate the effect of pre-shear in a T¼16 capsid should consider different
pre-shear directions and magnitudes in the different hexamers. Even assuming that icosahedral symmetry is preserved and
limiting the pre-shear direction to be along the maximal diagonal, there are
3
2
7
3=
orientations to be considered since
there are three distinct symmetry-equivalent hexamer types in a T¼16 capsid. Furthermore, for each orientation, different
values of
η
, both positive and negative, need to be investigated. It becomes evident that the number of analyses grows
quickly with the T-number characterizing the viral capsid and the task of fully exploring the possible pre-shear config-
urations becomes more difficult. This problem is resolved if conformational degrees of freedom are introduced. In this
framework, no hypothesis related to pre-shear icosahedral symmetry is necessary and an infinite number of pre-shear
configurations may be considered at once since
η
and
θ
vary continuously on each hexamer. The minimization process is in
charge of choosing the pre-shear directions and magnitudes which are energetically most favorable and decide which pre-
shear patterns are optimal (Fig. 17b). However, allowing the pre-shear magnitudes and directions to be degrees of freedom,
it becomes difficult to analyze the final equilibrium configuration and discern if icosahedral symmetry is preserved, given
that different shear magnitude/direction combinations are equivalent. This issue will be addressed in Section 3.2.1.
3.2.1. Unifying view in terms of pre-stretch degree of freedom
In previous Section 3.2, we showed that different distributions of pre-shear directions and magnitudes in the final
equilibrium configuration are actually equivalent when viewed in terms of the right stretch tensor U. In order to achieve a
unifying view of the results we replace the pre-shear conformational degree of freedom with an equivalent pre-stretch
degree of freedom. Specifically, as presented in Section 2.3, we parametrize
vv vv
F
U(1/ )
cc 12 2
2
λλ== ⊗+ ⊗
in terms of pre-
stretch magnitude
λ
and pre-stretch direction
α
, where
α
is the angle between the lab frame
EE
(
,)
12
and local frame
vv
(
,
)
12
(Fig. 6). As a consequence of rewriting
F
c
in terms of pre-stretch, the conformational degrees of freedom defined in-
dependently per each hexamer are now the pre-stretch magnitude
λ
and pre-stretch direction
α
.
Using the pre-stretch parametrization of
F
c
and the pre-stretch degrees of freedom, we repeat the analyses presented in
Fig. 16. Pre-shear magnitude ηand asphericity Aversus FvK number γcurves. Dashed line corresponds to simulations with no pre-shear, while continuous
lines correspond to results obtained including pre-shear as degrees of freedom. Circles markers are used to plot asphericity, while square markers identify
shear magnitude η.
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108 10 3

Section 3.2 for T¼7 and T¼16 virus structures using exactly the same elastic properties and geometry. In the case of T¼7
virus structure, the same minimum energy configuration is obtained (Fig. 15(c), (f), and (i)), independently of the initial
condition chosen for the pre-stretch directions (Fig. 15(a), (d), and (g)). As expected, the equilibrium configurations com-
puted using pre-stretch and pre-shear degrees of freedom are exactly equivalent in terms of elastic energy, asphericity and
FvK number. The polar decomposition of the conformational deformation gradient
F
c
also shows that the computed pre-
stretch magnitudes and directions are equivalent to the pre-shear configurations presented in Fig. 15(b), (e), and (h).
The pre-stretch based description of conformational changes clearly shows that the pre-stretch is aligned with the
pentamers edges and pre-stretch propagates from pentamer to pentamer. More precisely, all the hexamers intersecting an
icosahedral edge between two pentamers have the same pre-stretch directions. In the case of a T¼16 structure, pentamers
are connected by a line of hexamers centered on the icosahedral edge (as in every T-number structure with h¼0ork¼0)
and pre-stretch propagates in a straight line from pentamer to pentamer. In the case of a T¼7 capsid the path connecting
the pentamers by steps through the centers of adjacent hexamers is not straight and pre-stretch cannot propagate through a
straight line of hexamers from pentamer to pentamer. However, pentamers are always joined by dimers of hexamers in-
tersecting the same icosahedral edge and sharing the same pre-stretch direction (hexamers bidomains are evident in Fig. 15
(c), (f), and (i)).
Since any shell can have multiple pre-shear configurations that correspond to the same pre-stretch state, parametrizing
F
c
in terms of pre-stretch degrees of freedom is the only way to define the conformational state uniquely. The use of pre-
stretch degrees of freedom becomes even more advantageous to interpret the results for larger T number structure, such as
T¼16, for which the number of equivalent but apparently different pre-shear states is higher due to a larger number of
hexamers. The results shown in Fig. 17(b) obscure the icosahedral symmetry of the minimum energy configuration, which
instead becomes clear if
F
c
is parametrized in terms of pre-stretch and the capsid elastic energy is minimized directly with
respect to pre-stretch degrees of freedom
λ
and
α
(Fig. 17(c)). The two equilibrium configurations computed with pre-shear
or pre-stretch degrees of freedom are otherwise exactly equivalent (equal elastic energies, asphericity, and FvK number). We
note that, as in the T¼7 capsid, icosahedral symmetry is respected but the pre-stretch corresponding to the minimum
elastic energy is not equivalent in all the hexamers of a T¼16 shell. Instead, hexamers within each of the three symmetry
classes have the same pre-stretches, but different symmetry classes select different pre-stretch magnitudes and directions
to minimize the global capsid elastic energy.
4. Conclusions
We have presented an elasticity theory to explain the in-plane displacements and shape transitions occurring during
virus shell maturation with hexamer shearing due to local changes in the protein structure as the driving force. This results
in changes in the capsid asphericity and the global shape of the capsid such as buckling transitions and reverse buckling
transitions. Symmetric, undeformed hexamer subunits lead to a faceted icosahedral shell representative, for instance, of the
Fig. 17. Initial (a) and final (b, c) equilibrium configurations obtained for a T¼16 virus capsid with
128
0γ
∼
. The capsid elastic energy has been minimized
with respect to displacement and conformational degrees of freedom per each hexamer. (a) Initial configuration in terms of pre-shear degrees of freedom,
(b) minimum energy configuration with final pre-shear directions and colored according to pre-shear magnitudes in each hexamer, and (c) minimum
energy configuration including final pre-stretch directions and pre-stretch magnitudes in each hexamer. The final minimum energy configuration depicted
in terms of pre-stretch (c) clearly shows icosahedral symmetry even though the initial conditions are non-symmetric (a). (For interpretation of the
references to color in this figure caption, the reader is referred to the web version of this article.)
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108104

“EI-II”state in the maturation process of HK97. On the contrary, pre-sheared or pre-stretched hexamers lead to a more
spherical virus capsid, characteristic of the prohead state. Beside the spherical to faceted transition, our theory uncovers a
broader range of dodecahedral and stellated capsid shapes. These capsid configurations could not be explained solely by
changes in the shell material properties and classic thin shell theory. The results presented suggest that the proper fra-
mework to model virus shell conformational changes should include the tools typical of growth mechanics and plasticity,
e.g. the multiplicative decomposition of the deformation gradient that allows modeling pre-shearing of the hexamer
subunits.
The effect of subunits conformational changes on the final capsid configuration depends on several key factors, among
which are the capsid elastic properties and the direction of pre-stretch. We showed this important interplay with the help of
phase diagrams and demonstrated that a right or left handed direction maximizes the effect of the subunit conformational
change (in the case of achiral virus structures, parallel directions are completely ineffective). As observed in experimental
images, the same effective right-handed direction is chosen, for instance, by T¼7 viral capsids such as HK97,
λ
-phage, and
P22. We furthermore notice that hexamers pre-shear delays the transition from spherical to faceted shells with respect to
the case controlled solely by the Föppl von Kármán number
γ
, i.e. at equal FvK number, the asphericity is lower in viral
capsids composed of pre-sheared hexamers. The effect of pre-shear on the shell asphericity is greater for intermediate
values of
γ
whereas the capsid shape is primarily dictated by the FvK number at small (FvK
10
0
⪅
) or large (FvK
104
⪆
) values.
In transitioning from small to large FvK numbers, the value of the pre-shear
η
corresponding to the minimum elastic energy
decreases. This observation may be explained in terms of the elastic energy cost of the discontinuities in the elastic strains
along the hexamer edges. At small values of
γ
, incompatibilities are accommodated because their elastic energy is small
since the in-plane Young's modulus is small relative to the bending modulus. Vice-versa, at large values of
γ
, which cor-
respond to large values of the in-plane Young's modulus with respect to the bending modulus, strain discontinuities are
energetically unfavorable and are lowered by decreasing the value of pre-shear
η
.
The model we have presented is not only able to describe the conformational changes occurring during the first stages of
the virus maturation process but, under the assumptions presented in the foregoing, is capable of predicting pre-stretch
patterns in the early stages of an assembled virus shell. Specifically, we notice that pre-stretches are aligned along the
pentamers edges and pentamers are connected by a path of hexamers pre-stretched in the same direction. Moreover, pre-
deformed hexamers lower the capsid elastic energy, effectively favoring assembly with respect to the case in which sym-
metric undeformed hexamer subunits form the virus shell. In other words, pre-deformed hexamers not only determine the
capsid spherical shape but may also play a fundamental role in the capsid assembly process.
The current model also predicts that the pre-stretch magnitudes and directions configurations at equilibrium for T¼7
and T¼16 structures (Section 3.2.1) respect icosahedral symmetry, even if, a priori, the presented model allows for non-
symmetric configurations since each hexamer possesses independent conformational degrees of freedom. Although ico-
sahedral symmetry is preserved, the minimum elastic energy configuration corresponds to different pre-stretch magnitudes
and directions in different hexamers. The presence of different
λ
and
α
would be consistent with the hypothesis that there
exist “soft modes”of deformation, i.e., modes of deformation that cost less or no energy and are available to the virus capsid
to lower its elastic energy during the assembly process. Although we are not aware of any experimental evidence that
conclusively demonstrates the presence of soft modes at the capsomer level, our model makes the clear prediction that such
modes would be made evident by varied amounts of conformational strain in different capsomers.
The predictions obtained using the current model are derived by minimizing the capsid elastic energy with respect to
pre-stretch (or pre-shear) degrees of freedom. However, the elastic energy may be only one part of the physical potential
driving the hexamers conformational changes and pre-stretch may not be a free independent variable in the full capsid as it
was treated in the present context. Instead, an additional potential may be added to the model to favor particular pre-stretch
(or pre-shear) magnitudes and/or directions. The addition of this type of “conformational”potential may result in different
pre-stretch configurations, in which also icosahedral symmetry may be lost. An avenue of research we have currently
undertaken is aimed at investigating the effect of limiting the conformational flexibility of the hexamers through the in-
clusion of such additional “conformational”potential. There may be non-symmetric capsid configurations that possess a
lower energy with respect to the symmetric case and this hypothesis will be assessed analyzing several T number structures
and minimization of free energy will be considered also through Monte Carlo simulations.
Acknowledgments
The authors gratefully acknowledge support from NSF Grant nos. DMR-1006128 and DMR-1309423. W.S.K. acknowledges
partial support from NSF Grant no. CMMI-0748034.
Appendix A. Thin shell geometry
For completeness, we review here the basic equations describing thin shell geometry and deformation to derive the
expressions of the mean and Gaussian curvatures used in Eq. (8).
Denoting with
ssx(, )
12
¯
and
ssx(,
)
12
the shell reference and deformed configurations as function of the curvilinear
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108 10 5

coordinates
ss
(
,
)
12
, the covariant basis vectors are defined as
ss
axxa xx,, (A.1)
,,
¯=∂¯
∂≡¯=∂
∂≡
αααα αα
in the undeformed and deformed configurations respectively. In Eq. (A.1) and hereafter, Greek indices take values 1 and 2.
Moreover, subsequent derivations are carried out explicitly only for the deformed configuration since the reference
quantities with an overbar are analogous.
The dual, or contravariant, basis vectors
aα
in the deformed configuration are constructed in order to satisfy the relation
aa .(A.2)
δ·=
αββ
α
By definition it follows that the covariant and contravariant components of the surface metric tensor are, respectively,
aaaa aa,. (A.3)
=· =·
αβ α β αβ α β
The shell director is defined as the unit normal to the surface or, in terms of the covariant basis vectors
a
α
,as
da aa
aa
.(A.4)
312
12
≡= ×
|×|
Based on the previous definitions, the symmetric curvature tensor is
bbda da .(A.5)
,,
=− · = · =
αβ α β α β βα
The curvature strains are then defined as the difference between the deformed and reference curvatures as
bb.(A.6)
ρ=−
¯
αβ αβ αβ
Bending of an isotropic shell can be described in terms of two invariants of the curvature strain tensor, the mean and
Gaussian curvature strains. The mean curvature strain is one half of the trace of the curvature strain tensor, i.e.,
HSS
2,, (A.7)
ρ==
α
α
and the Gaussian curvature strain is the determinant of the curvature strain tensor, i.e.,
Kdet( ). (A.8)
ρ=β
α
Appendix B. Energy weak form
The total potential energy of the virus shell is a function of both the deformation field
φ
and the conformational de-
formation tensor
F
c
and can be written in a compact form as
:
⎡
⎣
⎢⎤
⎦
⎥⎡
⎣
⎢⎤
⎦
⎥
()()
WWdSFF,,,,. (B.1)
cb sc
,,
∫
φφφφφΠ=+
ααβ
Since the in-plane energy
WF(,
)
sc
φ
is split into volumetric and isochoric (deviatoric) parts (Eq. (9)), and the con-
formational deformation gradient
F
c
represents a purely isochoric deformation (Eq. (13)), W
s
can be rewritten as
() () ()
WWWFFF,, (B.2)
s c sVOL sDEV e,,
φ=+
where
FFF,(B.3a)
ec1
=−
⎜⎟
⎛
⎝
⎞
⎠
WJF2(1), (B.3b)
sVOL s
,2
κ
=−
⎡
⎣
⎢⎤
⎦
⎥
JCCCFF
1
2tr ( ) tr( ) , , (B.3c)
T22
=− =
⎛
⎝
⎜⎞
⎠
⎟⎛
⎝
⎜⎞
⎠
⎟
WJ
FC
2
tr( ) 2. (B.3d)
sDEV e
e,
μ
=−
Therefore
L.E. Perotti et al. / J. Mech. Phys. Solids 77 (2015) 86–108106

:
⎡
⎣
⎢⎤
⎦
⎥⎡
⎣
⎢⎤
⎦
⎥
()()()
WWWdSFFF,,, . (B.4)
c b sVOL sDEV e
,, ,,∫
φφφφΠ=++
ααβ
Based on Eq. (B.4), the first variation of the shell elastic energy is
::
⎜⎟⎜⎟
⎡
⎣
⎢⎤
⎦
⎥⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
dS dSFnamd PFPF,::, (B.5)
cbee,∫∫
φδΠ δ δ δ δ=·+· + +
αααα
where
n
b
α
and
m
α
are membrane and bending stresses (e.g. Cirak and Ortiz, 2001),
a
α
and
d
,
α
are defined in Appendix A, and
WW
P
FPF
,.
sVOL
e
sDEV
e
,,
=∂
∂=∂
∂
Since
F
FF
ec1
=
−
,
()
FFF FF (B.6)
ecc
11
δδ δ=+
−−
Moreover, given
()
FF 1
0
cc
1
δδ==
−
,
() ()
FF FF F FFF0and (B.7)
cccccc
cc
11 1 11
δδ δ δ+= =−
−− − −−
Using Eqs. (B.6) and (B.7),
P
F:
ee
δ
can be rewritten in terms of the variations of the primary unknown fields
F
[
,]
c
φ
or
FF
[
,]
c
:
() ( )
PF P FF PFFFF:: : (B.8a)
ee e ceccc
111
δδ δ=−
−−−
() ( )
PF F F F PF F:: (B.8b)
ecTcTTecTc
δδ=−
−−−
In terms of
F
c1−
, Eq. (B.8b) simplifies to
() ()
PF PF F FP F::: (B.9)
ee e
cTTec1
δδδ=−
−−
Since
F
aA=⊗
α
α
, and consequently
FaAδδ=⊗
α
α
, Eq. (B.9) can be rewritten as
()()
PF PFA a FP F:: (B.10)
ee e
cTTec1
δδδ=··−
αα
−−
Combining Eqs. (B.10) and (B.5), we obtain
:
⎡
⎣
⎢⎤
⎦
⎥⎡
⎣
⎢⎤
⎦
⎥
()
dSFnamdFPF,:, (B.11a)
cTec
,1
∫
φδΠ δ δ δ=·+·−
αααα−
nn PPF A(). (B.11b)
becT
=++ ·
ααα
−
It follows from Eq. (B.11b) that minimization of the capsid energy (Eq. (B.1)) is achieved by imposing
:
⎡
⎣
⎢⎤
⎦
⎥dSna md 0, (B.12a)
,∫δδ·+· =
αααα
:
⎡
⎣
⎢⎤
⎦
⎥
()
dSFP F:0, (B.12b)
Tec1
∫δ=
−
where
Fc1
δ−
can be computed in terms of the pre-shear
(
,)ηθ
or pre-stretch
(
,)λα
conformational degrees of freedom.
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