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. Significance StatementIntrinsically disordered proteins (IDPs) are essential for many cellular processes and are implicated in numerous diseases. The biological functions of IDPs are dictated by the populations of the diverse conformational states they adopt in solution and the kinetics of the conformational transitions between these states. Computer simulations are a powerful tool for studying IDPs, but analyzing simulated structural ensembles of IDPs to identify functionally important conformational states and motions remains a significant challenge. In this study, we demonstrate that writhe, a geometric descriptor from the field of knot theory that describes the crossing of curves in space, provides a powerful basis for characterizing the structural ensembles and conformational dynamics of IDPs in atomic detail. Our results provide new tools for unraveling the relationships between IDP sequences, their conformational ensembles, and their biological functions.