  Residue-residue interactions that fold a protein into a unique
three-dimensional structure and make it play a specific function impose
structural and functional constraints on each residue site. Selective
constraints on residue sites are recorded in amino acid orders in homologous
sequences and also in the evolutionary trace of amino acid substitutions. A
challenge is to extract direct dependences between residue sites by removing
indirect dependences through other residues within a protein or even through
other molecules. Recent attempts of disentangling direct from indirect
dependences of amino acid types between residue positions in multiple sequence
alignments have revealed that the strength of inferred residue pair couplings
is an excellent predictor of residue-residue proximity in folded structures.
Here, we report an alternative attempt of inferring co-evolving site pairs from
concurrent and compensatory substitutions between sites in each branch of a
phylogenetic tree. First, branch lengths of a phylogenetic tree inferred by the
neighbor-joining method are optimized as well as other parameters by maximizing
a likelihood of the tree in a mechanistic codon substitution model. Mean
changes of quantities, which are characteristic of concurrent and compensatory
substitutions, accompanied by substitutions at each site in each branch of the
tree are estimated with the likelihood of each substitution. Partial
correlation coefficients of the characteristic changes along branches between
sites are calculated and used to rank co-evolving site pairs. Accuracy of
contact prediction based on the present co-evolution score is comparable to
that achieved by a maximum entropy model of protein sequences for 15 protein
families taken from the Pfam release 26.0. Besides, this excellent accuracy
indicates that compensatory substitutions are significant in protein evolution.
