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Potts models and variational autoencoders (VAEs) have recently gained popularity as generative protein sequence models (GPSMs) to explore fitness landscapes and predict mutation effects. Despite encouraging results, current model evaluation metrics leave unclear whether GPSMs faithfully reproduce the complex multi-residue mutational patterns observed in natural sequences due to epistasis. Here, we develop a set of sequence statistics to assess the “generative capacity” of three current GPSMs: the pairwise Potts Hamiltonian, the VAE, and the site-independent model. We show that the Potts model’s generative capacity is largest, as the higher-order mutational statistics generated by the model agree with those observed for natural sequences, while the VAE’s lies between the Potts and site-independent models. Importantly, our work provides a new framework for evaluating and interpreting GPSM accuracy which emphasizes the role of higher-order covariation and epistasis, with broader implications for probabilistic sequence models in general. Generative models have become increasingly popular in protein design, yet rigorous metrics that allow the comparison of these models are lacking. Here, the authors propose a set of such metrics and use them to compare three popular models.

We characterize equilibrium properties and relaxation dynamics of a two-dimensional lattice containing, at each site, two particles connected by a double-well potential (dumbbell). Dumbbells are oriented in the orthogonal direction with respect to the lattice plane and interact with each other through a Lennard-Jones potential truncated at the nearest neighbor distance. We show that the system's equilibrium properties are accurately described by a two-dimensional Ising model with an appropriate coupling constant. Moreover, we characterize the coarsening kinetics by calculating the cluster size as a function of time and compare the results with Monte Carlo simulations based on Glauber or reactive dynamics rate constants.

Potts models and variational autoencoders (VAEs) have recently gained popularity as generative protein sequence models (GPSMs) to explore fitness landscapes and predict the effect of mutations. Despite encouraging results, quantitative characterization and comparison of GPSM-generated probability distributions is still lacking. It is currently unclear whether GPSMs can faithfully reproduce the complex multi-residue mutation patterns observed in natural sequences arising due to epistasis. We develop a set of sequence statistics to assess the \"generative capacity\" of three GPSMs of recent interest: the pairwise Potts Hamiltonian, the VAE, and the site-independent model, using natural and synthetic datasets. We show that the generative capacity of the Potts Hamiltonian model is the largest, in that the higher order mutational statistics generated by the model agree with those observed for natural sequences. In contrast, we show that the VAE's generative capacity lies between the pairwise Potts and site-independent models. Importantly, our work measures GPSM generative capacity in terms of higher-order sequence covariation statistics which we have developed, and provides a new framework for evaluating and interpreting GPSM accuracy that emphasizes the role of epistasis.