From viral capsids to self-assembling nanoparticles
Viruses are examples of biological nanoparticles. They are made of a highly symmetric protein container, the capsid, that contains the nucleic acid and self-assembles from several copies of a single protein. The relative position of the proteins in the capsids is conventionally described by the Caspar-Klug scheme, that was inspired by techniques developed by Fuller to construct geodetic domes.
Much interest has recently arisen in the engineering of self-assembling protein nanoparticles. In general, these nanoparticles do not fit into the Caspar-Klug framework, and conventional experimental methods only yield partial information on their structure: therefore, an original approach is needed to provide blueprints for the determination of their structure.
In this talk I will discuss a mathematical framework that enables the investigation of a specific class of self-assembling protein nanoparticles (SAPNs) which have been designed to act as antigen display systems for vaccines.
I will show that graph-theoretical tools, combined with symmetry methods based on the Goldberg construction for symmetric polyhedra, allow to study the topology of the protein networks. This approach unveils a hidden relation with fullerene geometries and enables the classification of the high and low symmetry particles seen in experiments.
Figure: A self-assembling protein nanoparticle formed from 180 building blocks, together with its nanoparticle graph (adapted from a figure in https://doi.org/10.1098/rsos.161092)