QPD: Order-Aware Learning of All-Atom Protein Dynamics

Generative models of molecular dynamics learn which conformations a protein visits, yet a trajectory is also defined by the order in which it visits them. A residue can reach the same orientation through different sequences of local rotations, and losses on endpoints or on the distribution of frames cannot tell these sequences apart. We introduce Quaternion Path Dynamics (QPD), a training objective that supervises the ordered rotations of residue backbone frames across time. QPD compares rotation increments, their products, and their commutators, and contrasts each reference path with a Hurwitz negative that keeps the endpoint and the rotation angles but changes the intermediate orientation. Added to an all-atom trajectory generator, QPD improves protein ensembles on ATLAS and first-passage kinetics on held-out CATH2 domains over the same model with QPD switched off, raising the correlation of mean first-passage times from 0.763 to 0.805. It also improves on MDGen for tetrapeptide dynamics and generates ligand dissociation paths with few clashes. Ablations and frame-shuffling tests tie these gains to the order of frames. In short, supervising how residues rotate teaches a generator how proteins move, not only where they go.