Novel measures of topological complexity of protein structure and function

Understanding the interplay between sequence, structure and function of biopolymers is a major problem in biology and bioengineering, at the core of protein folding, protein aggregation and cell nucleus organization and function. The single, pairwise, or multi-chain characterization of entanglement complexity becomes rigorous in the context of mathematical topology. Yet, despite the billion of structures of proteins currently available or predicted, the methods of structure characterization are rather limited. The challenge is that the majority of these polymers are not mathematical knots, since they have open ends. In this talk we will introduce a rigorous and general topological approach to analyze the structures of macromolecules. Using this new framework we can rigorously define the Jones polynomial and Vassiliev measures of open curves in 3-space, so that they are continuous functions of the curve coordinates and so that they can capture topological and geometrical complexity, with no closure scheme or diagrammatic approximation. We will apply our methods to proteins and show that these can be used to characterize all protein structures and that they correlate with experimental folding rates. By analyzing structures in the Protein Data Bank we can obtain a representation of the native topological landscape of proteins at multiple lengthscales that provides a new way of understanding their structures. When applied to the SARS-CoV-2 spike protein, we see that the local native topological landscape can predict sites where mutations can have an important impact on protein structure and possibly in viral transmissibility. These methods can thus help us understand biopolymer function and biological material properties in many contexts with the goal of their prediction and design.