### Matrix

A key element in evaluating the quality of a pairwise sequence alignment is the "substitution matrix", which assigns a score for aligning any possible pair of residues. The theory of amino acid substitution matrices is described in [1], and applied to DNA sequence comparison in [2]. In general, different substitution matrices are tailored to detecting similarities among sequences that are diverged by differing degrees [1-3]. A single matrix may nevertheless be reasonably efficient over a relatively broad range of evolutionary change [1-3]. Experimentation has shown that the BLOSUM-62 matrix [4] is among the best for detecting most weak protein similarities. For particularly long and weak alignments, the BLOSUM-45 matrix may prove superior. A detailed statistical theory for gapped alignments has not been developed, and the best gap costs to use with a given substitution matrix are determined empirically. Short alignments need to be relatively strong (i.e. have a higher percentage of matching residues) to rise above background noise. Such short but strong alignments are more easily detected using a matrix with a higher "relative entropy" [1] than that of BLOSUM-62. In particular, short query sequences can only produce short alignments, and therefore database searches with short queries should use an appropriately tailored matrix. The BLOSUM series does not include any matrices with relative entropies suitable for the shortest queries, so the older PAM matrices [5,6] may be used instead. For proteins, a provisional table of recommended substitution matrices and gap costs for various query lengths is:

         Query length     Substitution matrix     Gap costs
         ------------     -------------------     ---------
         <35              PAM-30                  ( 9,1)
         35-50            PAM-70                  (10,1)
         50-85            BLOSUM-80               (10,1)
         >85              BLOSUM-62               (11,1)

### Gap Costs

The raw score of an alignment is the sum of the scores for aligning pairs of residues and the scores for gaps. Gapped BLAST and PSI-BLAST use "affine gap costs" which charge the score -a for the existence of a gap, and the score -b for each residue in the gap. Thus a gap of k residues receives a total score of -(a+bk); specifically, a gap of length 1 receives the score -(a+b).

### Lambda Ratio

To convert a raw score S into a normalized score S' expressed in bits, one uses the formula S' = (lambda*S - ln K)/(ln 2), where lambda and K are parameters dependent upon the scoring system (substitution matrix and gap costs) employed [7-9]. For determining S', the more important of these parameters is lambda. The "lambda ratio" quoted here is the ratio of the lambda for the given scoring system to that for one using the same substitution scores, but with infinite gap costs [8]. This ratio indicates what proportion of information in an ungapped alignment must be sacrificed in the hope of improving its score through extension using gaps. We have found empirically that the most effective gap costs tend to be those with lambda ratios in the range 0.8 to 0.9.

    [1] Altschul, S.F. (1991) "Amino acid substitution matrices from an information
        theoretic perspective." J. Mol. Biol. 219:555-565.
    [2] States, D.J., Gish, W. & Altschul, S.F. (1991) "Improved sensitivity of
        nucleic acid database searches using application-specific scoring matrices."
        Methods 3:66-70.
    [3] Altschul, S.F. (1993) "A protein alignment scoring system sensitive at all
        evolutionary distances." J. Mol. Evol. 36:290-300.
    [4] Henikoff, S. & Henikoff, J.G. (1992) "Amino acid substitution matrices from
        protein blocks." Proc. Natl. Acad. Sci. USA 89:10915-10919.
    [5] Dayhoff, M.O., Schwartz, R.M. & Orcutt, B.C. (1978) "A model of evolutionary
        change in proteins." In "Atlas of Protein Sequence and Structure, vol. 5,
        suppl. 3," M.O. Dayhoff (ed.), pp. 345-352, Natl. Biomed. Res. Found.,
        Washington, DC.
    [6] Schwartz, R.M. & Dayhoff, M.O. (1978) "Matrices for detecting distant
        relationships." In "Atlas of Protein Sequence and Structure, vol. 5,
        suppl. 3," M.O. Dayhoff (ed.), pp. 353-358, Natl. Biomed. Res. Found.,
        Washington, DC.
    [7] Karlin, S. & Altschul, S.F. (1990) "Methods for assessing the statistical
        significance of molecular sequence features by using general scoring
        schemes." Proc. Natl. Acad. Sci. USA 87:2264-2268.
    [8] Altschul, S.F. & Gish, W. (1996) "Local alignment statistics." Meth.
        Enzymol. 266:460-480.**
    [9] Altschul, S.F., Madden, T.L., Schäffer, A.A., Zhang, J., Zhang, Z., Miller,
        W. & Lipman, D.J. (1997) "Gapped BLAST and PSI-BLAST: a new generation of
        protein database search programs." Nucleic Acids Res. 25:3389-3402.