Substitution Matrices & MSA

1. Log-Odds Score Matrices

For amino acids i, j with background frequencies p_i, p_j and observed joint frequency q_ij:

\[ s(i, j) \;=\; \frac{1}{\lambda}\,\log\!\frac{q_{ij}}{p_i\,p_j} \]

Positive scores indicate substitutions more frequent than by chance; negative scores the reverse. The λ normalisation sets a consistent half-bit or third-bit unit system. BLOSUM62 is the de-facto default for protein alignment; BLOSUM45 and BLOSUM80 tune to distant vs. close homology.

2. PAM vs BLOSUM

Dayhoff’s PAM matrices extrapolate from closely-related proteins, producing nested PAM1, PAM40, PAM120 series by matrix multiplication. BLOSUM matrices are built directly from alignments at specific percent-identity blocks — BLOSUM62 is derived from ungapped blocks of proteins ≥62% identical. In practice BLOSUM performs better at the 30–70% identity range and is the standard.

Simulation: BLOSUM62 Visualisation

3. Multiple Sequence Alignment

MSA extends DP to multiple sequences. Exact MSA is NP-hard; practical tools use progressive (ClustalW, T-Coffee), iterative (MUSCLE, MAFFT), or consistency-based (T-Coffee) heuristics. Alignment quality depends on guide-tree order and column scoring. Benchmarks (BAliBASE, SABmark) compare tools; MAFFT L-INS-i and Clustal Omega are current defaults.

4. Profile HMMs & HMMER

Once aligned, a family can be summarised as a profile HMM: per-column emission and transition probabilities that encode position-specific conservation. HMMER (Eddy 2011) builds profile HMMs and searches databases for family members, more sensitively than BLAST for distant homology. Pfam (now InterPro) is the canonical protein-family database built on HMMs.

Key References