How to Disassemble a Virus Capsid

A Computational Approach

Claudio Alexandre Piedade, Ant

onio E. N. Ferreira and Carlos Cordeiro

Laborat

orio de FTICR e Espectrometria de Massa Estrutural, Centro de Qu

ımica e Bioqu

ımica,

Departamento de Qu

ımica e Bioqu

ımica, Faculdade de Ci

encias, Universidade de Lisboa, 1749-016 Lisboa, Portugal

Keywords:

Virus, Capsid Disassembly, Combinatorial Geometry, Symmetry Groups, Structural Biology.

Abstract:

In contrast with the assembly process of virus particles, which has been the focus of many experimental and

theoretical studies, the disassembly of virus protein capsids, a key event during infection, has generally been

overlooked. Although the nature of the intracellular triggers that promote subunit disassembly may be diverse,

here we postulate that the order of subunit removal is mainly determined by each virus structural geometry

and the strength of subunit interactions. Following this assumption, we modelled the early stages of virus

disassembly of T = 1 icosahedral viruses, predicting the sequence of removal of up to ﬁve subunits in a sample

of 51 structures. We used combinatorics and geometry, to ﬁnd non-geometrically identical capsid fragments

and estimated their energy by three different heuristics based on the number of weak inter-subunit contacts.

We found a main disassembly pathway common to a large group of viruses consisting of the removal of a

triangular trimer. Densoviruses lose a square-shaped tetramer while Human Adenoviruses lose a pentagon-

shaped pentamer. Results were virtually independent of the heuristic measure used. These ﬁndings suggest

that particular subunit interactions might be an important target for novel antiviral drugs designed to interfere

with capsid disassembly.

1 INTRODUCTION

Viruses are intracellular parasites that replicate inside

living cells, by using its genetic and protein synthesis

machinery to create new copies (Poranen et al., 2002;

Mateu, 2013). Viral particles are composed by a nu-

cleic acid (DNA or RNA, single or double stranded)

and, in many cases, a capsid, a proteic structure that

protects the genetic information in between infections

(Caspar and Klug, 1962; Poranen et al., 2002; Prasad

and Schmid, 2012). Round-shaped viruses have a

ﬁxed number of proteins or asymmetric units sur-

rounding the genetic material in a icosahedral sym-

metry (Caspar and Klug, 1962; Prasad and Schmid,

2012). This arrangement was ﬁrst described by Cas-

par and Klug in 1962, who developed a method to

classify the icosahedral symmetry by triangulating the

icosahedron facets. The Triangulation number (or T -

number) represents the number of equilateral trian-

gles that compose a triangular face of the icosahedron

(Prasad and Schmid, 2012).

Many studies have addressed the virus assembly

process, both experimentally and theoretically (re-

viewed in (Perlmutter and Hagan, 2015) and (Mateu,

2013)). Viruses’ capsid subunits are held together by

non-covalent interactions (Zlotnick, 1994; Zlotnick

et al., 1999; Zlotnick, 2003) such as electrostatic salt

bridges, hydrophobic contacts and hydrogen bonds.

The effects of a myriad of these weak interactions is

a globally stable capsid (Zlotnick and Stray, 2003).

Virus assembly is spontaneous in vitro under close-

to-physiological conditions (Mateu, 2013) and the re-

sulting capsids are stable for a long period of time.

Results derived from theoretical models of virus as-

sembly consistently agree on the prediction that three

trimeric protomers, pentamers of trimers and capsids

with just a triangular icosahedron face missing are

stable intermediate forms that precede the formation

of a complete capsid (Zlotnick, 1994; Zlotnick et al.,

1999; Reddy et al., 1998; Reddy and Johnson, 2005;

Rapaport, 2008; Rapaport, 2010). If capsid disassem-

bly was just the reverse process of capsid assembly,

these ﬁnding would suggest that the same structures

would be found during disassembly. In high contrast

to the number of studies focusing on the assembly

process, there are very few experimental studies on

capsid disassembly pathways. Since it is expected

that intermediates of this process are transient and

very difﬁcult to detect experimentally, models of the

disassembly pathways are always difﬁcult to conﬁrm

Piedade C., Ferreira A. and Cordeiro C.

How to Disassemble a Virus Capsid - A Computational Approach.

DOI: 10.5220/0006249802170222

In Proceedings of the 10th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2017), pages 217-222

ISBN: 978-989-758-214-1

217

experimentally. Moreover, the assembly and disas-

sembly pathways might not be symmetric since some

viruses undergo maturation steps after assembly com-

pletion, such as proteolysis, cross-linking or confor-

mational change (Zlotnick and Stray, 2003). Castel-

lanos et al. (Castellanos et al., 2012), using Atomic

Force Microscopy (AFM), observed the removal of a

triangle of subunits (an icosahedral face) when force

was applied over the Minute Virus of Mice, followed

sometimes by the loss of an adjacent triangle. In

other cases, a removal of a pentamer of triangles (15

proteins) was observed. This follows the predictions

of the assembly models (Reddy et al., 1998; Reddy

and Johnson, 2005; Rapaport, 2008; Rapaport, 2010).

Horton and Lewis (Horton and Lewis, 1992) stated

that the energy minimums appear every multiple of

three subunits. Ortega-Esteban et al. (Ortega-Esteban

et al., 2013) have shown also by AFM that for Hu-

man Adenoviruses the disassembly, both for mature

and immature capsids, start by the loss of a pentagon

of proteins.

The work described here tackles the lack of the-

oretical studies targeting the disassembly process re-

lying on rigorous geometrical and combinatorial con-

siderations, focusing primarily on T = 1 icosahedral

capsid structures. This work in progress has the long-

term goal of providing insights into the disassembly

process, which can be used in the development of an-

tiviral drugs interfering with intermediate states of the

process.

2 METHODS

2.1 Combinatorics and Symmetry

Virus capsid structures with T = 1, although hav-

ing icosahedral symmetry, are not icosahedron-like,

as they contain 60 proteins instead of only 20. The

60-face polyhedron, in which each triangular face of

the icosahedron is divided into three faces, is the Del-

toidal Hexecontahedron. Due to the symmetry of the

Deltoidal Hexecontahedron, the combinations of re-

moving N proteins from a capsid (





) leads to re-

dundancy since some are symmetrically equivalent,

and the real number is smaller. The exact number of

combinations can be obtained by studying the Burn-

side’s Lemma (Burnside, 1909). To build up the

group of permutations associated with the rotations

of a Deltoidal Hexecontahedron, the faces were num-

bered as shown in Figure 1, followed by the analy-

sis of the transformations done on the numbering of

the faces by the symmetry operators of the symme-

try group I (Vincent, 2001). Applying the different

Figure 1: Graph representation of the Rhombicosidodeca-

hedron. Each vertex represents a subunits of the T = 1 virus

capsid. Edges represent geometrical edges.

rotations on the symmetry axes of a Deltoidal Hex-

econtahedron, results in a permutation group. These

permutations can be seen as permutations of faces of

the Deltoidal Hexecontahedron or vertices of its dual,

the Rhombicosidodecahedron. Since the graph of a

Rhombicosidodecahedron (Figure 1) allows the study

of each protein as a vertex, in what follows we mainly

used that perspective to develop the following meth-

ods of the analysis of virus disassembly.

2.2 Structures of Viruses’ Capsids

Capsids’ atomic coordinates were obtained from the

Protein Data Bank (PDB, http://www.rcsb.org/,

(Berman et al., 2000)). Only T = 1 cap-

sids, present on ViperDB were considered

(http://viperdb.scripps.edu, (Carrillo-Tripp et al.,

2009)). The capsid structures analysed in this work

were divided into groups according to the criteria of

similarity of the infection host and the genes codify-

ing capsid proteins. The list is presented in Table 2,

with 51 structures divided into 14 groups. Human

Adenovirus are mainly T = 25 and the structures

studied here are a stable capsid formed with the

pentagons of the T = 25, forming a Dodecahedron

(named from now on as Human Adenovirus Pt-Dd or

HAPD).

2.3 Energy Calculation

To calculate the energy of the full capsid and its dis-

assembly products, the number of hydrogen bonds,

salt bridges and hydrophobic contacts between cap-

sid subunits were counted (intra-protein contacts were

ignored). Hydrogen bonds (N

) were counted if ac-

ceptor and donor’s atoms indicated in Table 1 were

BIOINFORMATICS 2017 - 8th International Conference on Bioinformatics Models, Methods and Algorithms

218

Table 1: Donor and Acceptor’s Atoms used to calculate Hy-

drogen Bonds.

Amino acids

Donor

ARG(N

; N

η1

; N

η2

), ASN(N

δ2

), CYS(S

GLN(N

ε2

), HIS(N

δ1

; N

ε2

), LYS(N

SER(O

), THR(O

γ1

), TRP(N

ε1

), TYR(O

)

Acceptor

ASN(O

δ1

), ASP(O

δ1

; O

δ2

), GLN(O

ε1

GLU(O

ε1

; O

ε2

), HIS(N

δ1

; N

ε2

SER(O

), THR(O

γ1

), TYR(O

)

at a distance less than or equal to 4.0

A. A salt bridge

bound (N

) was counted if a positively charged atom

of an acidic amino acid (ASP and GLU) was found

within 4.0

A of a negatively charged atom of a basic

amino acid (ARG and LYS). A hydrophobic contact

) was counted when β-carbons of the residues

ALA, VAL, LEU, ILE, MET, PHE, TYR and TRP,

were found at the distance less than or equal to 7.0

We can calculate a heuristic measure of total energy

of a complete capsid or capsid fragment (Equations

(1) to (3)).

= N

+ N

(1)

= 20 × N

+ N

(2)

= 100 × N

+ 10 × N

+ N

(3)

In Heuristic I (Equation (1)), only the total num-

ber of inter-subunit bonds in the capsid complex is

taken into account; the three types of bounds are

equally weighted. Heuristic II (Equation (2)) gives

Salt Bridges 20 times more energy than Hydrogen

Bonds. Hydrophobic Contacts was considered to be

energetically equivalent to making Hydrogen Bonds,

since a water ”cage” is destroyed when hiding hy-

drophobic amino acids, breaking Hydrogen Bonds

made by the water itself to hold this cage (Atkins and

De Paula, 2006). On the other hand, on Heuristic III

(Equation (3)) an increasing power of 10 was given to

each type of inter-subunit bond, in the known order of

strength of this types of contacts (Salt Bridges > Hy-

drogen Bonds > Hydrophobic Contacts) (Atkins and

De Paula, 2006).

2.4 Removal of Proteins and Graph

Representation

Each possible capsid fragment can be represented by

a 60 element binary vector, depending on the presence

or absence of each protein given the subunit number-

ing of Figure 1. A list of non-redundant binary vectors

representing capsid fragments was then obtained as

follows: for every combination of N indexes of a size

60 vector, a binary vector was generated with zeros

in those positions and ones elsewhere; permutations

of the permutation group mentioned above were ap-

plied to this vector; if any of the resulting vectors was

not identical to another previously obtained it would

be appended to the list. Thus, a set of non geometri-

cally identical capsid fragments was obtained. Their

representation as a subgraph of Figure 1 allowed us

to check if disconnected fragments were obtained at

each step, so that the largest fragment was retained

and the search for non redundant forms was also ap-

plied to this fragment. Finally, the fragment’s the en-

ergy was calculated according to each heuristic.

2.5 Optimal Path for Disassembly

For all the possibilities of removing N proteins, a tree

of all the paths that the virus disassembly process

could take was built, rooted at the intact virus cap-

sid (N = 0). Edges were given a transition energy

weight, based on the average heuritic energy per pro-

tein, following Equation 4. The size of conﬁguration

i is equal to 60− N

, where N

was the number of pro-

teins removed.

transition i→ j

size

−

size

(4)

The Bellman-Ford algorithm was used on this tree

to calculate the shortest path from the complete vi-

ral capsid to every possibility of a capsid with 60 − N

proteins, based on the weights of the edges of the tree,

recording the ﬁve shortest paths for every N and the

corresponding ﬁnal conﬁguration.

2.6 Implementation

All methods were implemented in Python 2.7

using scientiﬁc computing modules (igraph

Python bindings (Csardi and Nepusz, 2006)).

All the code is available on GitHub repository:

https://github.com/CAPiedade/Virus-Disassembly.

3 RESULTS AND DISCUSSION

The disassembly paths of the different capsids are

portrayed in Table 2 considering heuristic I. It can

be observed that most Parvoviruses (marked with

a †) follow the same sequence of disassembly:

{1} → {1,10} → {1, 10, 23} → {1, 2,10,23} →

{1,2,10,22,23}. Fragment {1, 10, 23} represents

the removal of a triangle of proteins on the capsid

structure (Figure 2.A). As observed in the mechanical

removal of proteins with AFM (Castellanos et al.,

2012), the Minute Mice Virus (on our work repre-

sented by the group of the Rodent Protoparvovirus)

How to Disassemble a Virus Capsid - A Computational Approach

219

Table 2: Capsid groups and minimal energy paths taken to reach the removal of N subunits using heuristic I. Viruses marked

with † are Parvoviruses (Family Parvoviridae).

Groups 1 2 3 4 5

Adeno-Associated Virus†

(3j4p, 3ra8, 4iov, 3j1q, 3ntt, 3ra2, 3ux1,

2g8g, 1lp3, 2qa0, 3ra4, 3ra9, 3raa, 4rso, 5egc)

{1}

{1,10}

{1,10,23}

{1,2,10,23}

{1,2,10,22,23}

Bovine Parvovirus† (4qc8)

Human Parvovirus† (1s58)

Porcine Parvovirus† (1k3v)

Canine and Feline Panleukopenia Virus†

(1c8f, 1c8h, 1c8e, 1c8g, 1fpv, 1p5y, 1c8d,

1p5w, 4dpv, 1ijs, 2cas)

{1,9,24,54}

{1,2,10,23}

Rodent Protoparvovirus†

(1mvm, 1z14, 4g0r, 1z1c, 2xgk, 4gbt)

{1,2,10,23}

Avian Birnavirus (1cwd)

Porcine Circovirus (3r0r, 3jci)

{1,2,22,42}

{1,2,10,23}

Bombyx mori Densovirus† (3p0s)

{1,2} {1,2,22,23}Galleria mellonella Densovirus† (1dnv)

Penaeus stylirostris Densovirus† (3n7x)

{1,2,23} {1,2,6,10,23}

Satellite Tobacco Mosaic Virus

(2buk, 4bcu, 1a34, 4oq8)

{1,6}

{1,2,6}

{1,2,23}

{1,2,6,7}

{1,2,6,24}

{1,2,6,23}

{1,2,6,10,24}

{1,2,3,23,42}

{1,2,6,10,23}

Hepatitis E Virus (HEV) (2ztn, 2zzq, 3hag)

{1,2,23}

{1,2,10,23}

{1,2,6,23}

{1,2,6,10,23}

Human Adenovirus Pt-Dd (1x9t, 4aqq, 4ar2)

{1,2} {1,2,3} {1,2,3,4} {1,2,3,4,5}

tends to start the disassembly process by losing

a triangular block, which is then followed by the

removal of the adjacent triangle. This is supported

by our results since most disassembly pathways lead

to the removal of a triangular structure, commonly

composed of proteins {1, 10,23}. The theoretical

studies of Rapaport et al. (Rapaport, 2008; Rapaport,

2010) reveal the existence of long-lived transient

structures with just one last triangle of proteins

missing to form the complete capsid. The results

of Reddy et al. (Reddy et al., 1998) using both

individual proteins and trimers show the same

for capsid assembly (Reddy et al., 1998), sug-

gesting that the path for disassembly also follows

the triangle removal. Fragment {1,2,10,22,23}

represents the removal of a trapezium-like shape,

centred on the triangle {1,10,23} (Figure 2.C)

Supposing this trend continues, it is not hard to

see that there is a chance of removing the proteins

around the ﬁve-fold axis. These could be the fol-

low up steps of the disassembly of these capsids,

potentially resulting in the loss of 15 proteins such

as {1,2,3,4, 5, 6, 10,22,23,33,34, 41, 42,49,50}.

Castellanos et al.’s results conﬁrms the predictions of

Reddy et al. (Reddy et al., 1998), since the lowest

energy conﬁguration, just before the complete capsid,

was missing a pentamer of triangles (corresponding

to the 15-protein combinations described before),

inclusively for Parvoviruses (Reddy and Johnson,

2005). The removal of four proteins has more than

one possibility. Most parvoviruses lose subunits

{1,2,10,23} (Figure 2.B). Densovirus, on the other

hand, tend to lose proteins {1,2,22,23}, a conﬁg-

uration leading to a square hole on the two-fold

symmetry of the capsid (Figure 2.D). We might

speculate that the inter-subunit interactions on these

viruses are different from the others parvoviruses.

To verify this, we took the ratio between the number

of intersubunit contacts [22 ↔ 23] and the number

of intersubunit contacts [10 ↔ 23], the only two

interactions between capsid proteins that are different

between {1,2, 22, 23} and the common parvovirus

structure {1,2,10,23}, respectively. These results

are shown on Table 3, where it can be observed that

Parvoviruses have a lower ratio than Densovirus,

suggesting that

#[22↔23]

#[10↔23]

> 0.82 ⇒ {1,2,22,23}

#[22↔23]

#[10↔23]

< 0.80 ⇒ {1,2,10,23}

Further analysis of these and other ratios will provide

some insight into why some structures are oriented

into particular pathways. Removing of a square-

shaped group of proteins raises the idea that the higher

the symmetry of the structures removed, the higher is

their chance to be lost. Structures with PDB

1c8f

and 1c8h differ only inside its group for the removal

of the four proteins {1, 9, 24, 54}. Avian Birnavirus

follow the same path as the majority of Parvovirus,

as well as Porcine Circovirus, with the exception of

structure with PDB

3jci which loses {1,2,22,42},

BIOINFORMATICS 2017 - 8th International Conference on Bioinformatics Models, Methods and Algorithms

220

Figure 2: Disassembly stages of different viruses. Parvoviruses (PDB

1s58) path {1,10,23} (A), {1,2,10,23} (B) and

{1,2,10,22,23} (C). Densovirus (PDB

1dnv) path {1,2,22,23} (D). HEV (PDB

2zzq) path {1,2,6,10,23} (E). HAPD

(PDB

4aqq) path {1,2,3,4,5} (F). Structures are represented showing molecular surfaces. UCSF Chimera (Pettersen et al.,

2004) was used for rendering.

Table 3: Ratio of contacts between proteins #[22 ↔ 23] and #[10 ↔ 23].

Ratio of the number

of contacts

Densovirus Parvovirus

1dnv 3n7x 3p0s 1s58 3raa

#[22↔23]

#[10↔23]

≈ 0,95

≈ 1,36

≈ 0,82

≈ 0,80

≈ 0,56

a reﬂection of {1, 2, 10,23}. HEV capsids loses

proteins on the order {1} → {1, 6} → {1, 2,23} ⇒

{1,2,6,23}

{1,2,10,23}

⇒ {1,2,6,10,23}. Although distinguish-

able from the Parvovirus, HEV loses a pentamer that

forms a triangular hole on the capsid structure (Fig-

ure 2.E). HAPD follows the disassembly path {1} →

{1,2} → {1,2,3} → {1, 2, 3,4} → {1,2,3,4, 5}, an

exception to all the others. The Human Adenovirus

group is the only undergoing a ﬁve-fold removal

of the proteins, removing the pentamer {1,2,3,4,5}

(Figure 2.F). On Figure 2.F we can observe the struc-

ture of a HAPD, which is formed by very condensed

clusters of pentagons, having very few contacts with

the 2-fold and 3-fold proteins. The distance between

the pentagonal clusters on the full capsid structure

might make it easier for this set of subunits to be re-

moved, in opposition to creating a bigger gap by re-

moving, for example, proteins {1,2, 10, 22,23}. Our

results are supported by those of Ortega-Esteban et al.

(Ortega-Esteban et al., 2013) which showed, through

AFM, a loss of a pentagon-shaped pentamer of pro-

teins for Human Adenoviruses. Satellite Mosaic To-

bacco Virus disassembly was to diverse. The results

from this group were not analysed any further since

due to the small size of the set.

We also investigated the effect of the other two

heuristics (Equations 2 and 3). Results were very

similar to heuristic I, with a slightly higher degree

of branching. Nevertheless, the loss of three and ﬁve

proteins was conserved in all heuristics.

4 CONCLUSIONS

Our study began with the goal of investigating

whether there was a common disassembly pathway

among different virus families and, if not, if there

would be a conserved disassembly path for each fam-

ily. Results on this work support the idea that, for a

large cluster of T = 1 viruses, there is a common dis-

assembly pathway. This cluster is composed by some

Parvoviruses (Adeno-Associated Virus, Bovine Par-

vovirus, Human Parvovirus, Porcine Parvovirus, Ro-

dent Protoparvovirus, Canine and Feline Panleukope-

nia Virus), Avian Birnavirus, Hepatitis E Virus and

Porcine Circovirus. From our results, we can specu-

late that the results of Castellanos et al. (Castellanos

et al., 2012) might be applicable to more families of

viruses besides the Rodent Protoparvovirus. Denso-

viruses (from Bombyx mori, Galleria mellonella and

Penaeus stylirostris) are exceptional for their ex-

clusive removal of proteins {1,2,22,23}, with no

branching to alternative possibilities. On the other

hand, Human Adenoviruses have a very particular

pathway of disassembly, establishing that not all dis-

assemblies proceed through the removal of triangular

protein trimers, but also through the removal of pro-

tein pentagons. Using a combinatorial method based

on symmetry and geometry, with a 60-subunit model

(Deltoidal Hexecontahedron) and not a 20-subunit

model (Icosahedron) or 12-subunit model (Dodeca-

hedron), under a heuristic using the number of weak

How to Disassemble a Virus Capsid - A Computational Approach

221

contacts, we obtained results comparable to those

found in previous literature. Furthermore, the usage

of a 60-subunit model allows the study of cases such

as those of Human Adenoviruses, which do not lose

triangular faces of the Icosahedron, but a pentagon

of faces of the Deltoidal Hexecontahedron. Nonethe-

less, an increase of the sample size is an important

follow-up step, since it would give a better insight into

the trends observed for the different viruses studies.

Moreover, the number of subunits removed from the

capsid structures should be above 5 to offer a perspec-

tive of what would be the next steps on the disassem-

bly pathway. Improvement of the algorithms as well

as the processing power could provide more informa-

tive results beyond 5. Understanding the way viruses’

capsids disassemble can be used to interfere with the

inter-subunit interactions, specially those who hold

the triangular trimer in the complete capsid. Antiviral

drugs can thus be design to disrupt these interactions.

ACKNOWLEDGEMENTS

Work supported by project RECI/BBB-

BEP/0104/2012 from Fundac¸

ao para a Ci

encia

e Tecnologia, Portugal. The funders had no role in

study design, data collection and analysis, decision to

publish, or preparation of the manuscript.

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