In this talk we discuss how new advances in knot theory lead to novel metrics of structure of biopolymers that are continuous functions in the space of configurations. We apply our methods to proteins and show that these enable us to create a new framework for understanding protein folding, which is validated by experimental data and is complementary to existing approaches. In particular, our methods enable the definition of the Topological Landscape of Proteins and its quantitative and qualitative analysis as a continuum. The methods are general and apply to quantify structural complexity and compare structures in different contexts, such as the SARS-CoV-2 Spike protein and tau protein. We employ the writhe and Local Topological Entropy (LTE), to investigate the evolutionary dynamics of proteins. Notably, we find the evolutionary behavior of thierodoxin to be well correlated with LTE of the native state. Furthermore, our results reveal a strong correlation between these topological parameters and established dynamical measures, such as the Dynamical Flexibility Index (DFI), indicating for the first time that the local topology/geometry of the native state reflects the dynamical landscape of a protein. These results suggest that these topological metrics could serve as valuable reaction coordinates, bridging the gap between protein structure and dynamics.
