Fig. 3: Detecting non-native entanglements in a monomeric protein structure.

a Schematic of how self-entanglements can be detected by examining the change in the Gauss linking number g(i,j) (Eq. (5)) between a closed loop (pink) formed by the backbone segment between residues i and j that form a native contact (gold dashed line) and another pseudo-closed loop formed by the C-terminal backbone segment (blue) and a pseudo-vector (dashed green line) connecting the C-terminal residue and the start of the C-terminal segment, which begins at residue j+1. Threading of the N-terminal segment (composed of residues 1 through i−1) is determined in a similar manner. Examples of different Gauss linking numbers and their corresponding structures are shown in this hypothetical illustration. The magnitude of g(i,j) is proportional to the number of threading events of the blue segment through the pink loop, while its sign is a function of the relative positioning of primary structure vectors at crossing points between the pink and blue segments. The structure with g(i,j) = 0 exhibits no entanglement. b An example of a gain in entanglement of the protein YjgH (PDB: 1PF5), where the C-termini (cyan) threads a loop (pink) formed by the native contact between residues D72 & Y104 (gold). The black arrow indicates the location of the crossing point of the two entangled loops. c Contingency table indicating the number of trajectories that are misfolded/folded across our 122 proteins based on \({Q}_{{{{{{\rm{mode}}}}}}}\) analysis and entangled/not entangled. Indicated \(p\)-values and odds ratios were computed in SciPy using the Fisher Exact Test. d Same as (c) except contingency table displays the number of proteins that are entangled/not entangled and predicted to be slow folding/fast folding. For the purposes of this analysis, a protein is considered slow- or fast-folding if its computed folding time is above or below the median folding time from the set of 73 proteins with reliable estimates, respectively. A protein is considered entangled if ≥50% of its misfolded trajectories are entangled. All error bars are 95% confidence intervals computed from bootstrapping 10⁶ times.