Boltzmann Machine Learning with a Parallel, Persistent Markov chain Monte Carlo method for Estimating Evolutionary Fields and Couplings from a Protein Multiple Sequence Alignment

Boltzmann Machines trained on evolutionary sequence data have emerged as a powerful paradigm for the data-driven design of artificial proteins. However, the relationship between model architecture, specifically parameter density, and experimental performance remains poorly understood. Here, we investigate this relationship using the Chorismate Mutase enzyme family as a model system. We compare standard fully connected Boltzmann Machines for Direct Coupling Analysis (bmDCA) with sparse models generated via progressive edge activation (eaDCA) and edge decimation (edDCA). We identify a maximum-entropy model (meDCA) along the decimation trajectory that represents an optimal balance between constraint satisfaction and the flexibility of the probability distribution. We synthesized and tested artificial sequences from all models using an in vivo complementation assay, finding that all architectures, regardless of sparsity, generate functional enzymes with high success rates, even at significant divergence from natural sequences. Despite this functional equivalence, we demonstrate that the meDCA model samples a viable sequence space that is more than fifteen orders of magnitude larger than its low-entropy counterparts. Furthermore, comparative analyses reveal that high-entropy models systematically minimize overfitting and better capture the local neutral spaces surrounding natural proteins. These findings suggest that while various models satisfying coevolutionary statistics can generate functional sequences, high-entropy Boltzmann Machines provide a superior representation of the underlying evolutionary fitness landscape.

Multiple Sequence Alignments (MSAs) provide protein language models with explicit evolutionary context, but their large depth makes subsampling unavoidable under limited token budgets. Existing strategies, including random selection, identity-based filtering, and diversity-driven sampling, are effective heuristics, yet provide limited control over the evolutionary signals retained in the subset. In this work, we recast MSA subsampling as an explicit optimization problem, where key evolutionary measures, including query identity and diversity, are treated as controllable objectives. Building on this view, we introduce AP-REASONER, an Affinity-Propagation-based factor-graph approach. With evolution-aware unary factors, exemplar-consistency factors, and two control knobs, AP-REASONER performs factor-graph reasoning through message passing to infer a fixed-budget MSA subset. Experiments on long-range contact prediction and conformational ensemble prediction show that AP-REASONER outperforms baseline subsamplers on structure-sensitive downstream tasks and enables controllable recovery of alternative protein conformations. These results highlight the value of modeling MSA subsampling as a controllable optimization problem, where factor-graph reasoning offers an effective alternative to heuristic selection.

Generating novel protein sequences that respect a family's statistical constraints typically requires training deep generative models on thousands to millions of examples. Yet most protein families are small: the median Pfam seed alignment contains only 22 sequences, a regime where learned models overfit or collapse. We propose \emph{stochastic attention} (SA), a training-free sampler that treats the modern Hopfield energy over stored sequences as a Boltzmann distribution and draws samples via Langevin dynamics. The score function is the residual of a single softmax attention operation, eliminating the need for a trained score network, pretraining data, or graphics processing units (GPUs). Across eight Pfam families spanning 37 to 420 sequences and 23 to 262 residues, SA generates sequences with low composition divergence, novelty, and structural plausibility supported by ESMFold and AlphaFold2. Compared with profile hidden Markov models (HMMs), EvoDiff, and the multiple sequence alignment (MSA) Transformer, SA is the only tested method to simultaneously achieve low composition divergence, genuine novelty, and sequence identity within each family's nearest-neighbor identity range; the others drift outside this range or produce near-copies. The critical inverse temperature is predicted from principal component analysis (PCA) dimensionality alone, enabling fully automatic operation from a seed alignment. In two domains with deep mutational scanning data, SA-generated substitutions are enriched for experimentally tolerated mutations beyond a position-matched null, and an independent language model (ESM2-650M) scores them within the natural range. Stochastic attention thus opens training-free sequence generation to the long tail of protein families too small for deep learning.