AA1 – Life Science

Project

AA1-9

Polyhedral Geometry of Virus Capsids

Project Heads

Michael Joswig, Marta Panizzut, Bernd Sturmfels

Project Members

Oguzhan Yürük

Project Duration

01.01.2021 − 31.12.2022

Located at

TU Berlin

Description

Polytopes are fundamental objects in mathematics which show up in a big variety of scientific applications. In this project we study their role in biology and chemistry from numerous perspectives.

 

Classifications of Virus Capsids and Realization Spaces of Polytopes:

The 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. The 3-dimensional polytopes are also quite relevant in the architecture of microbiological structures such as virus capsids. Capsids are the protein shells that enclose and protect the genetic material of viruses.  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. This classification, however, is outdated due to modern measurement techniques.

Figure: A visualization of Caspar and Klug’s classification from their original work.

Nonetheless, a significantly large portion of virus capsids are spherical, exhibit icosahedral symmetry, and contain stable clusters of proteins consisting of regular pentagons. Therefore, one way to understand the possible capsid shapes is to examine the constrained realization spaces of 3-polytopes. In one direction of our research, we initiate a research on constrained realization spaces of polytopes by studying of the realizations of 3-polytopes with edges tangent to the unit sphere. In particular, we study the possible algebraic degrees that the coordinates of such constrainted realizations can have, and investigate whether these algebraic complexity measures also reflects a measure of some combinatorial complexity, see e.g. [1].

 

Software for Visualizing and Analyzing the Virus Capsids:

Protein Data Bank (PDB) is a database that contains the structural data of virus capsids as well as various other micro and macro protein structures. ViperDB is another database which solely focuses on virus capsids, whose entries can also be found in PDB. Both in PDB and in ViperDB, the spatial information about the individual atoms in a protein are stored as a special file format with .pdb extension. A .pdb file essentially stores a point configuration in 3-space where each point corresponds to a spatial information of a single atom. There are powerful visualization tools such as PyMol or ChimeraX to analyze the atomic structures stored in .pdb files, but they are not very well suited for symbolic and combinatorial computations. As icosahedral virus capsids are highly symmetric, it is desirable to study the point configurations arising from icosahedral capsids with more sophisticated combinatorial tools such as polymake. With this in mind, we have extended polymake in order to interact with the aforementioned databases, and enabled polymake to parse .pdb files as point configurations in 3-space.

Figure: Brome Mosaic Virus capsid with an icosahedral cage visualized in UCSF Chimera .

 

Nonnegativity, Supports and Newton Polytopes of Polynomials:

Another biochemistry oriented research that we pursue is about chemical reaction networks theory. For a large class of chemical reaction networks, various biochemically relevant properties of the network are accessible by studying the signs that a polynomial with parameterized coefficients can attain in the positive orthant. Due to the parameterized coefficients, usually it is not viable to certify nonnegativity numerically. However, one can use certain symbolic nonnegativity certification tools in order to describe a region in the parameter space whose elements guarantees the nonnegativity of the parameterized polynomial. In our research, we approach to symbolic nonnegativity certification by making use of the circuit polynomials. In this way, we consider the combinatorial structure behind the support and the Newton polytope of the polynomial, and use this information explicitly while constructing a symbolic certificate.

The crucial part of utilizing circuit polynomials is to understand the location of the exponents with possibly negative coefficients within the Newton polytope. If such an exponent is a vertex of the Newton polytope, then the polynomial a priori can take negative values in the positive orthant. Otherwise, we consider a collection simplices contained in the Newton polytope such that

  • the vertices of each simplex are exponents with positive coefficients, and
  • each exponent with negative coefficient is in the interior of exactly one simplex.

Then, we construct one circuit polynomial for each one of these simplices, and the nonnegativity of this polynomial is characterized by a symbolic inequality. Therefore, one gets a sufficient condition of nonnegativity by considering all of these inequalities simulatenously. Finding a good collection of simplices to cover the exponents with negative coefficients is tedious work which requires a careful analysis of the support and the Newton polytope. In one direction of our research, we consider explicit examples of reaction network models, and investigate the nonnegativity of polynomials arising from these networks by analyzing their supports and the Newton polytopes, see e.g., [2].

Left: An animation of Newton polytope of a parameterized polynomial that arise from 2-site phosphorylation. Right: The restriction of the same polynomial to the face of its Newton polytope that contains a negative (red) exponent.
 
 

 

Polytopes for Everyone:

Along with the academic part of our research, we also focus on raising awareness about our research area in the public. In this direction, we have published a new version of the educational game MatchTheNet, which was developed earlier under the Discrete Mathematics and Geometry group of TU Berlin. MatchTheNet is a web browser game in which player has to match a number of 3D polytopes with their unfoldings in 2D. In order to increase the educational value of the game in the new version, we have added short information texts about the polytopes that show up on each level.  Furthermore, we have also added various new levels to the game, some of which were inspired by the icosahedral symmetry of virus capsids.

 

Figure: A level from the educational game MatchTheNet

External Website

https://matchthenet.de/

Related Publications

  1. M. Belotti, M. Joswig, and M. Panizzut: Algebraic Degrees of 3-Dimensional Polytopes. Preprint arXiv 2103.06835 (accepted for publication by Vietnam Journal of Mathematics).
  2. P. Breiding, T. Çelik, T. Duff, A. Heaton, A. Sattelberger, L. Venturello, and O. Yürük: Nonlinear Algebra and Applications. Numerical Algebra, Control & Optimization, 2021.