**Project Heads**

Michael Joswig, Marta Panizzut, Bernd Sturmfels

**Project Members**

Oguzhan Yürük

**Project Duration**

01.01.2021 − 31.12.2022

**Located at**

TU Berlin

Capsids are the protein shells that enclose and protect the genetic material of viruses. In this project we investigate geometric models of virus capsids via studying realization spaces of certain 3-dimensional polytopes under the constraints given by the capsid shape.

Caspar and Klug in 1962 characterized the capsid shapes by 3-dimensional polytopes which exhibit icosahedral symmetry, using the concept of quasi-equivalence based on Crick and Watson’s principle of genetic economy. The classification of Caspar and Klug has been outdated due to more precise measurements for virus data as a result of the advancements in cryogenic electron microscopy. In a recent paper, Twarock and Luque provided a generalized classification using Archimedean lattices. More specifically, they divide the capsid shapes into four families of polytopes, where each family is anchored at one of the Archimedean solids dodecahedron, icosidodecahedron, snub dodecahedron, and rhombicosidodecahedron. These solids exhibit a common subconfiguration of 12 regular pentagons among their facets, which correspond to particularly stable clusters of proteins, called pentons, in the virus capsid.

Figure: Pictures of the 4 Archimedean solids dodecahedron, icosidodecahedron, snub dodecahedron, and rhombicosidodecahedron from left to right.

*realization space* of a polytope comprises all polytopes (given, e.g., in terms of vertex or facet coordinates) with isomorphic face lattices. The study of realization spaces is a classical topic in geometric combinatorics, and the 3-dimensional case is well understood. By Steinitz’ Theorem from 1922, 3-polytopes can be characterized in terms of planar graphs, and their realization spaces are contractible. However, identifying the realization spaces of 3-polytopes that satisfies a given set of constraints is a more intriguing task. For instance, Schramm showed that the space of all realizations of a 3-polytope such that all edges are tangent to a smooth convex body forms a 6-dimensional manifold.

The main goal of this project is to systematically study the realization spaces of 3-polytopes, in which facets of the polytope obey the metric constraints provided by the stable penton clusters on capsids. Each of the four Archimedean solids listed above serves as the germ of an infinite family of 3-polytopes which arise via subdivisions of the boundary. The original Caspar-Klug case corresponds to the dodecahedron, and generates the family of *Goldberg polytopes*. This family of polytopes contain 12 pentagonal facets, and all other facets are hexagons.

We aim to study the three new classes of *generalized Goldberg polytopes*, which arise from the new classification of Twarock and Luque. We would like to develop a virosphere model that captures the shapes of all possible capsids as the deformations of one unified realization space. This will lead to studying difficult degeneration phenomena in realization spaces of polytopes, calling for methods from tropical geometry as well as recent advanced techniques from real algebraic geometry and numerical nonlinear algebra. Coordinates on these models can be used to analyze specific types of viruses, such as viruses from the same lineage, using tools from optimization and topological data analysis. We aim to explore the biologically meaningful metrics that we can put on our realization spaces, and by carefully controlling this metric we would like to understand the boundaries of our models, where a transition of combinatorial types occurs. If successful, this could lead to a new model for evolutionary pathways between viruses.

**External Website**

**Related Publications
**

- M. Belotti, M. Joswig, and M. Panizzut:
*Algebraic Degrees of 3-Dimensional Polytopes.*Preprint arXiv 2103.06835.