Characterizing structural and kinetic ensembles of intrinsically disordered proteins using writhe

The biological functions of intrinsically disordered proteins (IDPs) are governed by the conformational states they adopt in solution and the kinetics of transitions between these states. We apply writhe, a knot-theoretic measure that quantifies the crossings of curves in three-dimensional space, to analyze the conformational ensembles and dynamics of IDPs. We develop multiscale descriptors of protein backbones from writhe to identify slow motions of IDPs and demonstrate that these descriptors provide a superior basis for constructing Markov state models of IDP conformational dynamics compared to traditional distance-based descriptors. Additionally, we leverage the symmetry properties of writhe to design an equivariant neural network architecture to sample conformational ensembles of IDPs with a denoising diffusion probabilistic model. The writhe-based frameworks presented here provide a powerful and versatile approach for understanding how the structural ensembles and conformational dynamics of IDPs influence their biological functions.