On the near-linear correlation of the eigenvalues across BLOSUM matrices

The BLOSUMmatrices estimate the likelihood for one amino acid to be substituted with another, and are commonly used in sequence alignments. Each BLOSUMmatrix is associated with a parameter x—the matrix elements are computed based on the diversity among sequences of no more than x% similar. In an earlier work, Song et al. observed a property in the BLOSUM matrices—eigendecompositions of the matrices produce nearly identical sets of eigenvectors. Furthermore, for each eigenvector, a nearly linear trend is observed in all its eigenvalues. This property allowed Song et al. to devise an iterative alignment and matrix selection process to produce more accurate matrices. In this paper, we investigate the reasons behind this property of the BLOSUM matrices. Using this knowledge, we analyze the situations under which the property holds, and hence clarify the extent of the earlier method’s validity.