Modeling Protein Sequence Evolution as an Ornstein-Uhlenbeck Process in a Latent Space

High-throughput directed evolution produces longitudinal sequence libraries that are ideal for probing local fitness neighborhoods but often underpowered for global inference tasks such as contact prediction. We present an unsupervised inference model that integrates directed-evolution sequencing time series with natural homologs. We project sequences into a low-dimensional latent space learned from the natural multiple sequence alignment and model the experimental process as an Ornstein-Uhlenbeck dynamics in that space. Maximum-likelihood estimation of the latent drift and noise parameters determines a stationary Gaussian distribution, which induces an effective Potts model in sequence space. The inferred couplings improve structural contact prediction by combining global evolutionary constraints from nature with local, experiment-specific signals. Experiments on PSE1 {beta}-lactamase and dihydrofolate reductase demonstrate the ability to identify correct complementary contacts not recovered by methods using either natural or experimental data alone, with gains concentrated in intermediate- and long-range contacts.