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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]

A diagonal substitution matrix score measures enrichment over a background match, rather than simply rewarding an identical letter. tryptophan residues are comparatively rare, so a W–W match has a small chance background frequency $p_W^2$. Its observed conservation produces a much larger ratio $q_{WW}/p_W^2$ than the corresponding ratio for the more common leucine. The BLOSUM62 diagonal therefore awards W–W more than L–L. Both the background frequency and the observed pair frequency matter; rarity alone does not determine a diagonal score.^[ref]

The number specifies the sequence-identity threshold used when clustering related sequences before deriving a BLOSUM substitution matrix. At 62%, sufficiently similar sequences are grouped and their within-cluster comparisons are excluded. Lowering the threshold merges more sequences, emphasizing comparisons across greater divergence.

Consequently BLOSUM80 is suited to closer relatives, BLOSUM45 to more divergent sequences, and BLOSUM62 is intermediate. The numbers do not specify a required identity of the query and target, nor a number of evolutionary time steps.^[ref]

Read the ten residue–residue columns from the BLOSUM62 substitution matrix. In their order in the supplied sequence alignment, their contributions areThere are three gap positions, each incurring the stated linear gap penalty of four, including the leading gap. Therefore the alignment score is $\boxed{24-3(4)=12}$.

Use global sequence alignment, for example the Needleman-Wunsch algorithm with an appropriate substitution matrix and gap penalty. The two full-length globin proteins are related by orthology and are from closely related species, so the expected correspondence extends across both complete sequences. Unmatched ends should normally contribute to the score rather than being discarded.

With a linear gap penalty of magnitude $g$, the dynamic programming entry $F_{ij}$ is the best score for the first $i$ and $j$ residues. It satisfieswith $F_{i0}=-ig$, $F_{0j}=-jg$, and traceback from $F_{mn}$. An affine gap penalty uses separate match and gap states.

A substitution matrix assigns a score to aligning each pair of sequence symbols. Scores based on logarithms of observed-to-background pair frequencies turn products of relative support into additive alignment scores.