Figure 1
Top panel: the blue curve is the plot of the substitution matrix elements (210 elements of the lower triangular BLOSUM62, non-rounded, expressed in bit units) sorted by increasing value; the red curve is their approximations, , obtained as the dot products of the raw, non-centered , vectors. Bottom panel: the blue curve is the same as above but with centered matrix elements (i.e., the mean of the shifted BLOSUM62 matrix is zero), the red curve is the approximation computed with the centered vectors, as described in the text. The x-axis corresponds to the sorted 210 lower triangular matrix elements, e.g., the 210th element is the diagonal element corresponding to the tryptophan, s WW - the largest element in the BLOSUM62 matrix. The y-axis corresponds to the values of the matrix elements. Notice that correlation coefficients are very similar in both cases (0.989 for the curves of the top panel vs 0.998 for the curves of the bottom panel).