Classification of self-assembling protein nanoparticle architectures for applications in vaccine design

We introduce here a mathematical procedure for the structural classification of a specific class of self-assembling protein nanoparticles (SAPNs) that are used as a platform for repetitive antigen display systems. These SAPNs have distinctive geometries as a consequence of the fact that their peptide building blocks are formed from two linked coiled coils that are designed to assemble into trimeric and pentameric clusters. This allows a mathematical description of particle architectures in terms of bipartite (3,5)-regular graphs. Exploiting the relation with fullerene graphs, we provide a complete atlas of SAPN morphologies. The classification enables a detailed understanding of the spectrum of possible particle geometries that can arise in the self-assembly process. Moreover, it provides a toolkit for a systematic exploitation of SAPNs in bioengineering in the context of vaccine design, predicting the density of B-cell epitopes on the SAPN surface, which is critical for a strong humoral immune response

Expandohedra: Modeling Structural Transitions of a Viral Capsid

Inspired by natural phenomena and mathematical theory this paper presents the development of a model, based on the dodecahedron, that visualizes the structural transition and expansion of a capsid (viral protein shell)

Nested Polytopes with Non-crystallographic Symmetry Induced by Projection

Inspired by the structures of viruses and fullerenes in biology and chemistry, we have recently developed a method to construct nested polyhedra and, more generally, nested polytopes in multi-dimensional geometry with non-crystallographic symmetry. In this paper we review these results, presenting them from a geometrical point of view. Examples and applications in science and design are discussed

Classification of self-assembling protein nanoparticle architectures for applications in vaccine design

We introduce here a mathematical procedure for the structural classification of a specific class of self-assembling protein nanoparticles (SAPNs) that are used as a platform for repetitive antigen display systems. These SAPNs have distinctive geometries as a consequence of the fact that their peptide building blocks are formed from two linked coiled coils that are designed to assemble into trimeric and pentameric clusters. This allows a mathematical description of particle architectures in terms of bipartite (3,5)-regular graphs. Exploiting the relation with fullerene graphs, we provide a complete atlas of SAPN morphologies. The classification enables a detailed understanding of the spectrum of possible particle geometries that can arise in the self-assembly process. Moreover, it provides a toolkit for a systematic exploitation of SAPNs in bioengineering in the context of vaccine design, predicting the density of B-cell epitopes on the SAPN surface, which is critical for a strong humoral immune response

Representations of Uh(su(N))U_h(su(N)) derived from quantum flag manifolds

A relationship between quantum flag and Grassmann manifolds is revealed. This enables a formal diagonalization of quantum positive matrices. The requirement that this diagonalization defines a homomorphism leads to a left \Uh -- module structure on the algebra generated by quantum antiholomorphic coordinate functions living on the flag manifold. The module is defined by prescribing the action on the unit and then extending it to all polynomials using a quantum version of Leibniz rule. Leibniz rule is shown to be induced by the dressing transformation. For discrete values of parameters occuring in the diagonalization one can extract finite-dimensional irreducible representations of \Uh as cyclic submodules.Comment: LaTeX file, JMP (to appear