Mathematical Virology: Virus Capsid Class+Assembly

Objectives:
1 . Classification of virus structures via tiling methods. Extension of the classification scheme in Caspar-Klug theory to not presently covered cases.
2. Construction of models for virus capsid assembly based on item 1.
3. Construction of models for virus capsids based on pattern formation techniques.
4. Connection of our tiling methods with group theory. Implemetation in the framework of image reconstruction methods and Raman spectroscopy, with special focus on non-isometric viruses.
5. Implementation of affine extensions of non-crystallographic Coxeter groups to model multi-shell and tubular structures. Discussion of phenomena such as scaffolding in this context.
6. Discussion of fullerenes as inhibitors for virus capsid assembly based on symmetry considerations.
7. Refinement of the above models in close collaboration with biologists and scientists working in drug design.
2. Investigation of the relation with the Skyrme model in quantum field theory.

Objectives at report time:
1 . Classification of virus structures via tiling methods. Extension of the classification scheme in Caspar-Klug theory to not presently covered cases.
2. Construction of models for virus capsid assembly based on item 1.
3. Construction of models for virus assembly encorporating the functional role of the viral RNA.
4. Connection of our tiling methods with group theory. Implemetation in the framework of image reconstruction methods and Raman spectroscopy.
5. Implementation of affine extensions of non-crystallographic Coxeter groups to model multi-shell and tubular structures. Discovery of a new deometric principle in virus architecture. Demonstration of its predictive power for a number of test
viruses.
6. Refinement of the above models in close collaboration with biologists and scientists working in drug design.
7. Quantum field theory approach to virus assembly.
8. Applications to the design of synthetic DNA cages for cargo storage and transport.
9. Analysis and prediction of vibrational modes of viral capsids.
10. An energy landscape approach to assembly polymorphism.
11. Modelling of structural transitions such as swelling and maturation via a lattice-transition approach.

Report summary: In this interdisciplinary project on the interface of mathematics, biophysics and structural biology we developed new mathematical methods for the description of the structure and assembly of viruses.

We introduced Viral Tiling theory to predict the surface lattices according to which the proteins in the viral protein containers are organised. This theory generalises Caspar-Klug theory to include also the cancer-causing Papilloma and
Polyomaviridae, that could not be modelled with their approach because these viruses exhibit cluster types that are not permitted in their framework. This work sparked various applications, including the constructions of models for the
assembly of protein containers from the capsid proteins, descriptions of malformations during assembly, models for special bonding configurations (crosslinking structures) and the analysis and prediction of vibrational modes of viral capsids and has been featured by Science News.

Based on novel group theoretical tools developed by us during this project we have then been able to generalise these approaches, in which viral capsids were modelled via surface lattices, and have shown that these earlier results are
actually a consequence of a deeper level of structural organisation in viruses that had not been appreciated before. In particular, we have shown that the full three-dimensional architecture of a simple virus, i.e. a virus composed of a protein
container and genomic material such as for example a common cold virus, is implied by so-called affine extended symmetries. This has led to the discovery of a new fundamental geometric principle of viral architecture: We have shown
that the dimensions and shapes of all material components of a simple virus are all constrained collectively by its symmetry. This striking result has led to profound new insights into virus architecture and viral evolution, and is currently
exploited by us in the framework of anti-viral drug design.

In collaboration with experimental collaborators we have demonstrated the predictive power of our new approach: For a number of test viruses, we have shown that a wide spectrum of features of the genome organisation as well as of the protein structures can be predicted in striking detail. We have moreover created an automated algorithm that makes applications of the new approach efficient and easy to implement, and we are currently investigating the use of these
results in the context of image reconstruction methods.

Our results have profound consequences for the understanding and modelling of dynamic events that are important for virus function and infection. In particular, they shed new light on virus assembly. We have shown that the assembly of
viruses from their protein building blocks can be better understood based on the new insights from our symmetry approach, and that in this way the role of the viral genome during virus assembly can be incorporated into earlier models. Moreover, factoring in the correlation between different radial levels of material provided a mathematical tool to reduce drastically the complexity in the modelling of virus assembly. Together with our experimental collaborators, we have been
able to model aspects of virus assembly that could not be understood with an experimental or theoretical approach alone, hence making it a truly interdisciplinary effort.

We have also used the new mathematical techniques in areas beyond virology. In a series of papers we discussed applications to the construction of bionanocontainers from synthetic DNA in bionanotechnology, and we have shown that our new symmetry principle applies also to general protein assemblies beyond virology. Applications of techniques from quantum field theory in the context of virus assembly are currently under investigation.