Mathematical Modelling of Programmable Polymorphism of Protein Cages
Protein cages, convex polyhedral protein containers self-assembled from multiple copies of identical protein subunits, are pillars of nanotechnology. They include naturally occurring virus capsids (virus-like particles, VLPs) which are spherical as well as a plethora of particles with diverse symmetries such as tubes. VLPs have an innate ability to encapsulate nucleic acids, which makes them attractive as DNA/RNA delivery vehicles and in vaccine development. In this talk I will discuss a mathematical framework for the classification of the structures of such VLPs. This has been used, in collaboration with experimentalists, to programmably control VLPs size and symmetry for a model system, achieving larger particle sizes with higher carrying capacity than wild-type virus. I will also present a model of VLP assembly, that incorporates different assembly pathways leading to distinct particle geometries. The model reveals a mechanism by which particle size can be controlled and can thus serve as a guide for the design of desired particle morphologies for applications in virus nanotechnology. At the end, I will present a mathematical analysis of tubular particles arising from the self-assembly of variants of the Lumazine Synthase (LS) enzyme and explain the mathematical principles of tubular architectures that have recently been observed experimentally.