Is there an optimal substitution matrix for contact prediction with correlated mutations?

Correlated mutations in proteins are believed to occur in order to preserve the protein functional folding through evolution. Their values can be deduced from sequence and/or structural alignments and are indicative of residue contacts in the protein three-dimensional structure. A correlation among pairs of residues is routinely evaluated with the Pearson correlation coefficient and the MCLACHLAN similarity matrix. In literature, there is no justification for the adoption of the MCLACHLAN instead of other substitution matrices. In this paper, we approach the problem of computing the optimal similarity matrix for contact prediction with correlated mutations, i.e., the similarity matrix that maximizes the accuracy of contact prediction with correlated mutations. We describe an optimization procedure, based on the gradient descent method, for computing the optimal similarity matrix and perform an extensive number of experimental tests. Our tests show that there is a large number of optimal matrices that perform similarly to MCLACHLAN. We also obtain that the upper limit to the accuracy achievable in protein contact prediction is independent of the optimized similarity matrix. This suggests that the poor scoring of the correlated mutations approach may be due to the choice of the linear correlation function in evaluating correlated mutations.