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# Radial models of virus architecture and packaging signals. (a) Virus organisation at different radial levels reveals a molecular scaling principle relating the positions and dimensions of different 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). Specific 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.

Source publication

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 ex...

## Contexts in source publication

**Context 1**

... order to formulate these arising mathematical constraints on genome organization, deeper mathematical concepts called root systems are required. By extending this concept to the specific example of icosahedral symmetry, the symmetry of virus capsids, it has been possible to derive a classification of nested shell arrangements that capture virus architecture at different radial levels (Figure 3a). These 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 (Figure 3d). ...

**Context 2**

... extending this concept to the specific example of icosahedral symmetry, the symmetry of virus capsids, it has been possible to derive a classification of nested shell arrangements that capture virus architecture at different radial levels (Figure 3a). These 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 (Figure 3d). ...

**Context 3**

... 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. ...

**Context 4**

... 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 (Figure 3b), 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 (Figure 3c). 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. ...

**Context 5**

... 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 (Figure 3b), 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 (Figure 3c). 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. ...

**Context 6**

... order to formulate these arising mathematical constraints on genome organization, deeper mathematical concepts called root systems are required. By extending this concept to the specific example of icosahedral symmetry, the symmetry of virus capsids, it has been possible to derive a classification of nested shell arrangements that capture virus architecture at different radial levels (Figure 3a). These 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 (Figure 3d). ...

**Context 7**

... extending this concept to the specific example of icosahedral symmetry, the symmetry of virus capsids, it has been possible to derive a classification of nested shell arrangements that capture virus architecture at different radial levels (Figure 3a). These 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 (Figure 3d). ...

**Context 8**

... 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. ...

**Context 9**

... 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 (Figure 3b), 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 (Figure 3c). 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. ...

**Context 10**

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## Citations

... This fact has profound implications for the natural world around us, because objects consisting of identical building blocks that are 'maximally symmetric' display icosahedral symmetry. This includes most viruses and many fullerenes, as well as artificial nanocages in bionanotechnology and geodesic domes in architecture [13]. Even before any of these examples were known, icosahedral symmetry had inspired Plato to formulate a 'unified theory of everything' in his dodecahedral 'ordering principle of the universe'. ...

Recent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of $$H_3\rightarrow H_4$$ H 3 → H 4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple construction of the $${\mathrm {Pin}}$$ Pin and $${\mathrm {Spin}}$$ Spin covers. Using this connection with $$H_3$$ H 3 via the induction theorem sheds light on geometric aspects of the $$H_4$$ H 4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of $${\mathrm {Cl}}(3)$$ Cl ( 3 ) , including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.

... More recently, it has been shown that taken together, these two facts suggest that there could be more than one packaging signal, with multiple signals in fact dispersed throughout the genome [1,2]. This is because the capsid is symmetric, and the packaging signal mechanism functions via interaction between viral RNA and the coat protein (CP). ...

Realistic evolutionary fitness landscapes are notoriously difficult to construct. A recent cutting-edge model of virus assembly consists of a dodecahedral capsid with 12 corresponding packaging signals in three affinity bands. This whole genome/phenotype space consisting of 312 genomes has been explored via computationally expensive stochastic assembly models, giving a fitness landscape in terms of the assembly efficiency. Using latest machine-learning techniques by establishing a neural network, we show that the intensive computation can be short-circuited in a matter of minutes to astounding accuracy.

... This fact has profound implications for the natural world around us, because objects Dechant consisting of identical building blocks that are 'maximally symmetric' display icosahedral symmetry. This includes most viruses and many fullerenes, as well as artificial nanocages in bionanotechnology and geodesic domes in architecture [13]. Even before any of these examples were known, icosahedral symmetry had inspired Plato to formulate a 'unified theory of everything' in his dodecahedral 'ordering principle of the universe'. ...

Recent work has shown that every 3D root system allows the construction of a correponding 4D root system via an `induction theorem'. In this paper, we look at the icosahedral case of $H_3\rightarrow H_4$ in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonn\'e theorem, giving a simple construction of the Pin and Spin covers. Using this connection with $H_3$ via the induction theorem sheds light on geometric aspects of the $H_4$ root system (the $600$-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of Cl(3), including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.