 Solution

A BLOSUM substitution matrix is estimated from conserved, ungapped blocks of related protein sequences. Closely related sequences are clustered to reduce overrepresentation, and aligned amino acid pairs from different clusters are counted. Let $q_{ij}$ be their normalized observed pair frequency, and $e_{ij}$ the frequency expected from independent draws at the background amino acid frequencies. With unordered pairs, $e_{ii}=p_i^2$ and $e_{ij}=2p_i{p}_j$ for $i\ne j$.

For BLOSUM62, scores are rounded half-bit logarithms of these odds:A positive entry means that the aligned pair occurs more often in related sequences than the independent-background model predicts; a negative entry means less often. Adding these scores accumulates evidence across aligned positions. A score is neither a percentage identity nor a direct substitution probability.^[ref]