Fractional Brownian Bridges for Aligned Data

Abstract

Modeling stochastic processes with fractional diffusion instead of purely Brownian-driven dynamics may better account for real-world memory effects, long-range dependencies, and anomalous diffusion phenomena that standard Brownian motion fails to capture. We incorporate fractional Brownian motion (fBM) into aligned diffusion bridges for conformational changes in proteins, utilizing a Markov approximation of fractional Brownian motion (MA-fBM) to study the effect of this generalized prior reference process on predicting future states of the protein conformations from aligned data. We observe that our generalized dynamics yield a lower root mean-squared deviation (RMSD) of $C_{\alpha}$ atomic positions in the predicted future state from the ground truth. The best performance for this task is achieved with a scaled Ornstein-Uhlenbeck (OU) reference process, which predicts $32$% of examples with an $\text{RMSD}< \overset{\circ}{A}$ on the D3PM test split, whereas purely Brownian driven dynamics achieve $0$% for this threshold.