The intrinsic dimension of protein sequence evolution

It is well known that, in order to preserve its structure and function, a protein cannot change its sequence at random, but only by mutations occurring preferentially at specific locations. We here investigate quantitatively the amount of variability that is allowed in protein sequence evolution, by computing the intrinsic dimension (ID) of the sequences belonging to a selection of protein families. The ID is a measure of the number of independent directions that evolution can take starting from a given sequence. We find that the ID is practically constant for sequences belonging to the same family, and moreover it is very similar in different families, with values ranging between 6 and 12. These values are significantly smaller than the raw number of amino acids, confirming the importance of correlations between mutations in different sites. However, we demonstrate that correlations are not sufficient to explain the small value of the ID we observe in protein families. Indeed, we show that the ID of a set of protein sequences generated by maximum entropy models, an approach in which correlations are accounted for, is typically significantly larger than the value observed in natural protein families. We further prove that a critical factor to reproduce the natural ID is to take into consideration the phylogeny of sequences.Author summary: Protein sequence evolution is an extremely complex process, whose roles are ultimately determined by the necessity of living organisms to adapt to changes in the environment. We here address a fundamental question related with this process: in how many independent directions can a sequence evolve, without compromising the protein capability of folding and of performing its function? We find that the number of these directions is surprisingly small, of 10 or less in most of the families we considered. This property is not correctly accounted for by most of the theoretical model we considered, which predict that sequence evolution can take place in 30-40 independent directions. The only way to accomplish the task of generating low-dimensional sequences is to take into consideration sequence phylogeny.