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Generative models emerge as promising candidates for novel sequence-datadriven approaches to protein design, and for the extraction of structural andfunctional information about proteins deeply hidden in rapidly growing sequencedatabases. Here we propose simple autoregressive models as highly accurate butcomputationally efficient generative sequence models. We show that they performsimilarly to existing approaches based on Boltzmann machines or deep generativemodels, but at a substantially lower computational cost (by a factor between$10^2$ and $10^3$). Furthermore, the simple structure of our models hasdistinctive mathematical advantages, which translate into an improvedapplicability in sequence generation and evaluation. Within these models, wecan easily estimate both the probability of a given sequence, and, using themodel's entropy, the size of the functional sequence space related to aspecific protein family. In the example of response regulators, we find a hugenumber of ca. $10^{68}$ possible sequences, which nevertheless constitute onlythe astronomically small fraction $10^{-80}$ of all amino-acid sequences of thesame length. These findings illustrate the potential and the difficulty inexploring sequence space via generative sequence models.

Efficient generative modeling of protein sequences using simple autoregressive models