Martin Weigt

Roberto Netti, Emily Hinds, Francesco Calvanese et al.

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.

Lorenzo Rosset, Roberto Netti, Anna Paola Muntoni et al.

In this methods article, we provide a flexible but easy-to-use implementation of Direct Coupling Analysis (DCA) based on Boltzmann machine learning, together with a tutorial on how to use it. The package \texttt{adabmDCA 2.0} is available in different programming languages (C++, Julia, Python) usable on different architectures (single-core and multi-core CPU, GPU) using a common front-end interface. In addition to several learning protocols for dense and sparse generative DCA models, it allows to directly address common downstream tasks like residue-residue contact prediction, mutational-effect prediction, scoring of sequence libraries and generation of artificial sequences for sequence design. It is readily applicable to protein and RNA sequence data.

Pierre Barrat-Charlaix, Anna Paola Muntoni, Kai Shimagaki et al.

Boltzmann machines (BM) are widely used as generative models. For example, pairwise Potts models (PM), which are instances of the BM class, provide accurate statistical models of families of evolutionarily related protein sequences. Their parameters are the local fields, which describe site-specific patterns of amino-acid conservation, and the two-site couplings, which mirror the coevolution between pairs of sites. This coevolution reflects structural and functional constraints acting on protein sequences during evolution. The most conservative choice to describe the coevolution signal is to include all possible two-site couplings into the PM. This choice, typical of what is known as Direct Coupling Analysis, has been successful for predicting residue contacts in the three-dimensional structure, mutational effects, and in generating new functional sequences. However, the resulting PM suffers from important over-fitting effects: many couplings are small, noisy and hardly interpretable; the PM is close to a critical point, meaning that it is highly sensitive to small parameter perturbations. In this work, we introduce a general parameter-reduction procedure for BMs, via a controlled iterative decimation of the less statistically significant couplings, identified by an information-based criterion that selects either weak or statistically unsupported couplings. For several protein families, our procedure allows one to remove more than $90\%$ of the PM couplings, while preserving the predictive and generative properties of the original dense PM, and the resulting model is far away from criticality, hence more robust to noise.