Topological entanglement in proteins

Proteins are essential molecules for life, responsible for a wide range of biological functions due to their three-dimensional structures. In the recent decades, proteins having entangled native structures, such as knots, raised interest regarding their folding mechanisms, stability, and biological relevance. Among these entangled structures, non-covalent lasso motifs, in which a protein portion threads through another acting as a loop closed by non-bonded interactions, have been observed in a significant fraction of proteins, although their biological functions and folding mechanisms remain largely unexplored. This Thesis reviews the current knowledge of non-covalent lassos in globular proteins and extends it to membrane proteins. We show that one fifth of membrane protein domains hosts at least a non-covalent lasso, with monotopic proteins being more similar to globular proteins compared to transmembrane ones. At variance with globular protein, we find a different chirality bias for multiple winding motifs in transmembrane proteins. We speculate that the origin of this difference lies in their different biogenesis machineries. To study how proteins can fold non-covalent lasso motifs, we employ a structure-based, coarse-grained model to explore the refolding of the natively entangled RD1 and the non-entangled src SH3 domain. Using molecular dynamics, we validate the model by recapitulating the two-state behavior observed at equilibrium for both proteins. Thanks to a collective variable based on the Gaussian entanglement to keep track of entanglement changes, we observe a complex refolding landscape for RD1 characterized by two kinetic transient intermediates: a longer-lived, partially folded but disentangled intermediate that acts as a kinetic trap, and a short-lived, mostly misfolded but entangled one that ensures a fast folding. Overall, results are compatible with available experiments, highlighting the effects of the native entanglement on folding kinetics and corroborating the "lasso-closing contacts fold later" mechanism predicted on general grounds. Inspired by the effectiveness of coarse-grained and structure-based models, we discuss the Wako-Saitô-Muñoz-Eaton model, a 1D lattice model with binary variables in which coupling constants are derived from the protein native structure, extended with a solvation term. Compared to previous extensions, hydrophobicity is inferred from the protein sequence without any calorimetric data to fit model parameters. We validate the model by analytically studying the transition state ensembles polarization of four SH3 domains, finding a general agreement with experimental data. Thanks to it computational efficiency, we plan to use this WSME with solvation for a large set of proteins, including those hosting non-covalent lasso motifs. This Thesis provides new insights into the folding mechanisms of proteins hosting non-covalent lasso motifs and introduces computational tools that efficiently explore their equilibrium and refolding kinetics features.