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Models of viral capsid symmetry

as a driver of discovery in

virology andnanotechnology

Viruses are prominent examples of symmetry in biology. A better understanding of symmetry and

symmetry breaking in virus structure via mathematical modelling opens up novel perspectives on

how viruses form, evolve and infect their hosts. In particular, mathematical models of viral symmetry

pave the way to novel forms of antiviral therapy and the exploitation of viral protein containers in

bio- nanotechnology.

Pierre- Philippe Dechant

(York St John University,

U.K.)

Reidun Twarock

(University of York, U.K.)

Breaking Symmetry

Symmetry is ubiquitous in virology

Viruses are evolutionarily highly optimized molecular

machines. Understanding their inner workings sheds

light on fundamental questions in molecular biology,

biomedicine and nanotechnology. Viruses store their

genetic material inside protective protein containers

called viral capsids. Viral genomes consist of either

DNA or RNA, which can both be single or double-

stranded, and some viruses reverse transcribe between

the two. Single- stranded viruses tend to have shorter

genomes due to the relative exibility of the nucleic acid

molecule, and they oen package their genomes into the

capsid during its assembly in a co- assembly process. By

contrast, genetically more complex viruses tend to store

their genetic message in the form of the more stable

dsDNA. Whilst some of these viruses have much more

complex life cycles and less symmetric structures (e.g.

poxviruses), a surprisingly large fraction of them still

exhibit icosahedral and helical design principles (e.g.

the tailed phages or many of the recently discovered

giant viruses). In these viruses, the nucleic acid is oen

packaged into a preformed capsid using an energy-

driven molecular motor.

In the vast majority of viruses, these capsid containers

exhibit icosahedral symmetry, meaning that they look

like tiny footballs at the nanoscale. From a mathematical

point of view, this implies that the structural organization

of the capsid building blocks, called capsomers, and their

constituent protein subunits, displays a characteristic

set of rotational symmetry axes with two-, three- and

ve- fold symmetry (Figure1a). Denoting the locations

of the protein subunits in the corners of the triangles

gives rise to characteristic protein clusters (cf. clusters of

grey spheres in Figure1b). Crick and Watson provided

a biological explanation for this surprising degree of

symmetry in virology. ey argued that viruses encode

only a small number of distinct protein building blocks,

which are then repeatedly synthesized from the same

gene, as this minimizes the part of the genome required

to code for the capsid. For instance, hepatitis B virus

and phage MS2 only encode one structural protein each

and only have four genes altogether. At the same time,

building a capsid from the largest known rotational

symmetry, the icosahedral symmetry, ensures that the

maximal possible number of subunits is used to form

the capsid, thus optimizing its volume. is is known

as the principle of genetic economy. It is a consequence

of the selective pressure in viral evolution to generate

capsid structures that make genome packaging as easy

as possible, thus optimizing an essential step in any viral

replication cycle.

Mathematical models of viral symmetry

Symmetry alone is not sucient to explain all aspects of

virus architecture as can be seen from the plethora of

distinct capsid structures in nature that all obey icosahedral

symmetry. Mathematics can play a key role in formulating

the rules according to which viral capsids are organized.

e rst mathematical models of capsid architecture were

introduced by Caspar and Klug in 1962. ey are based

on the principle of quasi- equivalence, which stipulates

that protein subunits organize locally into equivalent

environments. From a mathematical point of view,

this implies that virus capsids should be describable by

surface lattices. Caspar and Klug used triangulations of

the capsid surface, in which protein organization in the

triangular facets mimics that of the icosahedral faces in the

simplest viruses. Such models can be built by drawing an

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Breaking Symmetry

icosahedral net on a hexagonal lattice and then folding this

net up into an icosahedron. In their seminal theory, they

provide a classication of virus architecture in terms of such

triangulations, deriving polyhedral models that indicate

the positions of individual capsid proteins in the surface

lattice. Indeed, it is possible to break a triangle down into

smaller triangles. An example of a virus structure that can

be modelled in this way is the hepatitis B virus, where an

icosahedral face is broken down into four triangular facets

(Figure1c). is idea leads to the triangulation number ,

which counts how many smaller triangles each icosahedral

face consists of (e.g. four in the example above). e dierent

triangles are then not necessarily symmetry- equivalent in a

mathematical sense, but are in approximately equivalent

local environments, which is why capsid architectures,

according to Caspar and Klug, are called quasi- equivalent.

A large fraction of the only recently discovered giant

viruses also exhibit this design principle (Figure1d). e

largest precisely known structure is

that occurs in Cafeteria

roenbergensis virus. It is so large that it even has a virophage

(the Mavirus virophage) associated with it. Mimiviruses

are even larger, with an estimated - number around 1000.

Similarly, they have a virophage called Sputnik, which

itself has a substantial triangulation number of 27. Caspar–

Klug- type cage architectures also occur in other areas

of science, where repeated building blocks are used to

construct higher- order structures. Examples are carbon

fullerenes, cellular compartments (such as carboxysomes),

and they have even been engineered as geodesic domes

in architecture. A major conclusion from Caspar–Klug

theory is that only certain triangulation numbers, and thus

numbers of capsid proteins, are possible.

With the advent of more rened imaging

techniques, in particular, the recent revolution in cryo-

electron microscopy (cryo- EM), it has become clear

that these Caspar–Klug- type models are too restrictive

in order to explain all known capsid architectures.

Prominent examples are the cancer- causing papilloma

viruses (e.g. human papillomavir us (HPV) in Figure2a)

that have capsids in which every protein subunit takes

on one of two distinct types of local conguration.

Viral tiling theory, a rst generalization of Caspar–

Klug theory, had been introduced to describe such

non–quasi- equivalent capsid architectures, in which

the proteins are not in approximately equivalent local

environments, as the bonds they are forming with the

surrounding proteins are not identical. is is done via

tessellations akin to the famous Penrose tiling, in which

distinct types of tiles represent the dierent types of

biological interactions. In this case, kites represent

three proteins forming a trimer interaction, and

rhombs two proteins involved in a dimer interaction.

Moreover, a generalized principle of quasi- equivalence

has recently been introduced that also encompasses

the architectures of viral capsids formed from more

than one type of capsid protein such as herpes simplex

virus (Figure2b), or the dsDNA tailed phage Basilisk.

is principle stipulates that local interactions between

Figure 1. Historic mathematical models of virus

architecture. (a) Viruses exhibit icosahedral symmetry, as

exemplied for Satellite Tobacco Necrosis Virus (pdb- id

4bcu). Examples of particle 2-, 3-, and 5- fold symmetry axes

are indicated on an icosahedral reference frame. (b) Protein

positions (grey) in an icosahedral virus model. (c) Example of

a model (here a T=4 triangulation for Hepatitis B virus based

on pdb- id 3j2v) in Caspar and Klug’s quasi- equivalence

theory; an icosahedral face subdivided into 4 triangular

facets is indicated by a black triangle. (d) Virus capsids

with very large triangulation numbers have recently been

discovered in giant viruses (here based on pdb- id 1m4x of

Paramecium Bursaria Chlorella Virus, with a T- number of 169,

shown together with an icosahedral reference frame).

Figure 2. Generalized tiling models of virus architecture.

(a) Viral Tiling theory model of Human Papillomavirus (HPV)

(based on pdb- id 3j6r); interactions between three proteins

(trimer interactions) are represented by kite- shaped tiles,

and interactions between two proteins (dimer interactions)

by rhombic tiles. (b) Herpes Simplex virus exhibits the

architecture of an Archimedean surface lattice based on the

generalised quasi- equivalence principle (based on pdb- id

6cgr).

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Breaking Symmetry

identical proteins, as well as interactions between the

same types of distinct proteins, must be the same across

the entire capsid surface. In this framework, capsid

architectures are modelled based on more general types

of lattices called Archimedean lattices. is theory

contains the hexagonal surface lattices from Caspar–

Klug theory as a special case, and has a number of

interesting implications for the geometric constraints

on viral evolution. For example, it suggests that the

size gaps between capsid architectures in Caspar–Klug

theory may be bridged by capsid structures abiding to

these more generalized lattice types. It even suggests

a way in which larger capsid architectures may have

evolved from smaller ones: the gyration of the surface

lattice, whereby the relative sizes of pentagonal and

triangular faces vary, resulting in a rotation of the

protein subunits.

Fighting viruses with mathematics

e importance of symmetry is not only conned to

the capsid surface itself, but also can manifest itself at

dierent radial levels of a virus particle. In particular, if

genome and capsid co- assemble, mediated by specic

points of contact, the symmetry from the capsid

impacts on the organization of the packaged genome.

In order to formulate these arising mathematical

constraints on genome organization, deeper

mathematical concepts called root systems are required.

By extending this concept to the specic example of

icosahedral symmetry, the symmetry of virus capsids,

it has been possible to derive a classication of nested

shell arrangements that capture virus architecture at

dierent radial levels (Figure 3a). ese structures

are not only relevant in virology, but also occur in the

context of multi- shell fullerene structures in carbon

chemistry, such as the nested carbon cages known as

carbon onions (Figure3d).

An important feature of these nested shell models is

that, when applied to viruses, they pinpoint the positions

between the genetic material of a virus and its capsid

shell (Figure 3a, inset). Such information is important

because it formulates constraints on where such contacts

can be located in the genome as a travelling salesman

problem: that is, as the combinatorial problem of how

the nodes in a network can be visited precisely once

along its edges. Indeed, by connecting all vertices

corresponding to neighbouring binding sites into a

polyhedral shell (Figure3b), the order in which contacts

are formed between secondary structure elements in

the genome and the capsid shell can be represented as

a path on a polyhedron (Figure3c). Seen through this

lens of geometry, and combined with bioinformatics

and in collaboration with the experimental team led

by Peter Stockley at the University of Leeds, it has been

possible to identify the molecular characteristics of these

contacts between the genome and the capsid shell via an

approach called Hamiltonian path analysis. is revealed

an unsuspected phenomenon: the presence of multiple

dispersed, sequence- specic contacts between capsid

and genome (secondary structure features), which

were termed packaging signals. ese act collectively

and cooperatively to orchestrate ecient co- assembly

of the capsid around its genome, akin to clothing pegs

on a washing line. Packaging signals constitute a second

code, overlaid on top of the genetic code of the virus,

that functions like a virus capsid assembly manual. eir

discovery has opened up novel avenues for antiviral

therapy that are based on both geometric and biophysical

insights into capsid assembly.

The role of symmetry breaking

Viral capsids must perform dierent functions: package

the viral genome eciently, protect their cargoes whilst

acting as a delivery vehicle, and nally, release it in

response to cues from the host environment. is is in

many cases facilitated by additional capsid components

that break the capsid’s overall icosahedral symmetry.

Prominent examples are dsDNA phages with helical

Figure 3. Radial models of virus architecture and packaging

signals. (a) Virus organisation at dierent radial levels reveals

a molecular scaling principle relating the positions and

dimensions of dierent viral components; a 3D multi- shell

model for bacteriophage MS2 is shown superimposed on

a cryo- EM map from the Ranson lab (University of Leeds).

Specic vertices of the model are positioned at the contact

sites of genomic RNA and the inner capsid surface (inset).

(b) These vertices form the corners of a polyhedron; (c)

paths connecting vertices along its edges have been used

as constraints in a bioinformatics approach (Hamiltonian

Path Analysis) to identify secondary structure elements

(packaging signals) in the viral genome in contact with the

inner capsid shell. (d) Similar multi- shell models also occur

in carbon chemistry, where several nested fullerene cages

form a carbon onion whose structures are orchestrated

collectively by an overarching symmetry principle.

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Breaking Symmetry

tails, packaging motors that enable energy- driven

internalization of the genomic DNA, or portals, such

as the stargate in Mimivirus. At the other end of the

size spectrum, one of the capsid protein dimers in the

bacteriophage MS2 capsid is replaced by maturation

protein, which enables attachment of the particle to the

bacterial pilus at this distinguished site, and thereby its

internalization into the bacterial host. Mathematical

modelling can help better understand the consequences

of such asymmetric capsid features and how they drive

vital dynamic processes in the viral life cycle. e larger

the viruses are, the more complex their structural

organization becomes. Coronaviruses have one of the

longest RNA genomes, presenting a challenge for its

packaging into the connes of the particle volume; this

is overcome by additional protein components, the

nucleocapsid (N) protein, that aid compaction of the

genome.

Turning tables on viruses –

nanotechnology mining from nature

In addition to pointing the way to new types of antiviral

strategies, a better understanding of viral geometry also

opens up novel routes for drug delivery and vaccination:

either by repurposing and optimizing viral protein

containers or by de novo engineering containers based

on similar geometric design principles. An example

of the former would be exploiting and optimizing the

virus assembly instructions as we are currently doing in

collaboration with the Stockley lab, tuning it for optimal

assembly eciency as demonstrated for satellite tobacco

necrosis virus (STNV; Figure 4a). An example of de

novo design are nanoparticles that form from a protein

building block with two dierent oligomerization

domains that spontaneously form cages with local three-

and ve- fold axes (Figure 4b). ese self- assembling

protein nanoparticles (SAPNs) share structural

similarities with papillomaviruses and can be modelled

in terms of surface tessellations analogously to viral

tiling theory. Such tiling models, indicating the positions

of individual protein chains, have been used to analyse

the particle morphologies of assembly products that

arise experimentally and to mathematically reconstruct

their surface architectures and properties. ese SAPNs

are currently used for the design of malaria vaccines.

A better understanding of virus capsids and their

symmetries has, therefore, paved the way to new antiviral

strategies and to the repurposing of the genome-

encoded virus assembly instructions for engineering

articial virus- like particles. Such particles have a host

of applications in nanotechnology, ranging from cargo

storage, over drug delivery to diagnostics. Viruses are

highly sophisticated molecular machines. We are just at

the beginning of the tantalizing journey of unravelling

how they work in detail, potentially with profound

impacts in biomedicine and nanotechnology.■

Figure 4. Articial nanocages for applications in

biomedicine and nanotechnology. (a) Knowledge of the

packaging signal- encoded virus assembly instructions

(top) enables the recoding of the nucleic acids (bottom)

to enhance their packaging properties and the assembly

eciency of the surrounding capsid, as shown here for STNV.

(b) Viral Tiling theory applied to the de novo engineering

of self- assembling protein nanoparticles used in malaria

vaccine design.

Further reading

• Crick, F.H.C. and Watson, J.D. (1956) Structure of small viruses. Nature 177, 473–475. DOI: 10.1038/177473a0

• Caspar, D.L. and Klug, A. (1962) Physical principles in the construction of regular viruses. Cold Spring Harb. Symp.

Quant. Biol. 27, 1–24. DOI: 10.1101/sqb.1962.027.001.005

• Twarock, R. (2004) A tiling approach to virus capsid assembly explaining a structural puzzle in virology. J. Theor. Biol.

226, 477–482. DOI: 10.1016/j.jtbi.2003.10.006

• Twarock, R. and Luque, A. (2019) Structural puzzles in virology solved with an overarching icosahedral design

principle. Nat. Commun. 10, 1–9. DOI: 10.1038/s41467-019-12367-3

• Dechant, P- P. , Boehm, C. and Twarock, R. (2012) Novel Kac- Moody- type ane extensions of non- crystallographic

Coxeter groups. J. Phys. A. 45,285202. DOI: 10.1088/1751-8113/45/28/285202

24 February 2021 © The Authors. Published by Portland Press Limited under the Creative Commons Attribution License 4.0 (CC BY- NC- ND)

Breaking Symmetry

Continued

• Keef, T. , Wardman, J.P. , Ranson, N.A., et al. (2013) Structural constraints on the three- dimensional geometry of simple

viruses: case studies of a new predictive tool. Acta Crystallogr. A. 69, 140–150. DOI: 10.1107/S0108767312047150

• Dechant, P.P., Wardman, J., Keef, T. and Twarock, R. (2014) Viruses and fullerenes - symmetry as a common thread? Acta

Crystallogr. A. 70, 162–167. DOI: 10.1107/S2053273313034220

• Twarock, R. , Leonov, G. and Stockley, P.G. (2018) Hamiltonian path analysis of viral genomes. Nat. Commun. 9, 2021.

DOI: 10.1038/s41467-018-03713-y

• Twarock, R. and Stockley, P.G. (2019) RNA- mediated virus assembly: mechanisms and consequences for viral evolution

and therapy. Annu. Rev. Biophys. 48, 495–514. DOI: 10.1146/annurev-biophys-052118-115611

• Indelicato, G., Wahome, N., Ringler, P. et al. (2016) Principles governing the self- assembly of coiled- coil protein

nanoparticles. Biophys. J. 110, 646–660. DOI: 10.1016/j.bpj.2015.10.057

Pierre- Philippe Dechant is a Senior Lecturer in Mathematical Sciences and the Programme Director for the

Data Science Degree Apprenticeship at York St John University. Pierre received his PhD from Cambridge,

where he worked on symmetry principles in gravitational and particle physics, before moving to York to

start work in Mathematical Virology. His research combines computational and mathematical modelling,

often involving symmetry applications that span biology, physics and algebra. Email: p.dechant@yorksj.

ac.uk

Reidun Twarock is Professor of Mathematical Virology at the University of York. She is an EPSRC Established

Career Fellow in Mathematics, a Royal Society Wolfson Fellow, and together with experimentalist Peter

Stockley from the University of Leeds, a Wellcome Trust Investigator. Reidun’s research in Mathematical

Virology, an area pioneered by her, focuses on the development of mathematical and computational

techniques to elucidate how viruses form, evolve and infect their hosts. She has won the Gold Medal of the

Institute of Mathematics and Its Applications in 2018. Email: reidun.twarock@york.ac.uk