  Global coevolutionary models of homologous protein families, as constructed
by direct coupling analysis (DCA), have recently gained popularity in
particular due to their capacity to accurately predict residue-residue contacts
from sequence information alone, and thereby to facilitate tertiary and
quaternary protein structure prediction. More recently, they have also been
used to predict fitness effects of amino-acid substitutions in proteins, and to
predict evolutionary conserved protein-protein interactions. These models are
based on two currently unjustified hypotheses: (a) correlations in the
amino-acid usage of different positions are resulting collectively from
networks of direct couplings; and (b) pairwise couplings are sufficient to
capture the amino-acid variability. Here we propose a highly precise inference
scheme based on Boltzmann-machine learning, which allows us to systematically
address these hypotheses. We show how correlations are built up in a highly
collective way by a large number of coupling paths, which are based on the
protein's three-dimensional structure. We further find that pairwise
coevolutionary models capture the collective residue variability across
homologous proteins even for quantities which are not imposed by the inference
procedure, like three-residue correlations, the clustered structure of protein
families in sequence space or the sequence distances between homologs. These
findings strongly suggest that pairwise coevolutionary models are actually
sufficient to accurately capture the residue variability in homologous protein
families.
