as well as an X (denoting any amino acid). Note that the matrix is a mirror image of itself with respect to the diagonal. See text for details.2 Frequency. In the same way that amino acid residues cannot freely substitute for one another, the matrices also need to reflect how often particular residues occur among the entire constellation of proteins. Residues that are rare are given more weight than residues that are more common.
3 Evolution. By design, scoring matrices implicitly represent evolutionary patterns, and matrices can be adjusted to favor the detection of closely related or more distantly related proteins. The choice of matrices for different evolutionary distances is discussed below.
There are also subtle nuances that go into constructing a scoring matrix, and these are described in an excellent review by Henikoff and Henikoff (2000).
How these various factors are actually represented within a scoring matrix can be best demonstrated by deconstructing the most commonly used scoring matrix, called BLOSUM62 (Figure 3.1). Each of the 20 amino acids (as well as the standard ambiguity codes) is shown along the top and down the side of a matrix. The scores in the matrix actually represent the logarithm of an odds ratio (Box 3.1) that considers how often a particular residue is observed, in nature, to replace another residue. The odds ratio also considers how often a particular residue would be replaced by another if replacements occurred in a random fashion (purely by chance). Given this, a positive score indicates two residues that are seen to replace each other more often than by chance, and a negative score indicates two residues that are seen to replace each other less frequently than would be expected by chance. Put more simply, frequently observed substitutions have positive scores and infrequently observed substitutions have negative scores.
Box 3.1 Scoring Matrices and the Log Odds Ratio
Protein scoring matrices are derived from the observed replacement frequencies of amino acids for one another. Based on these probabilities, the scoring matrices are generated by applying the following equation:
where pi is the probability with which residue i occurs among all proteins and pj is the probability with which residue j occurs among all proteins. The quantity qi,j represents how often the two amino acids i and j are seen to align with one another in multiple sequence alignments of protein families or in sequences that are known to have a biological relationship. Therefore, the log odds ratio Si,j (or “lod score”) represents the ratio of observed vs. random frequency for the substitution of residue i by residue j. For commonly observed substitutions, Si,j will be greater than zero. For substitutions that occur less frequently than would be expected by chance, Si,j will be less than zero. If the observed frequency and the random frequency are the same, Si,j will be zero.
To explain the meaning of the numbers in the matrix more fully, imagine that two sequences have been aligned with one another, and it is now necessary to assess how well a residue in sequence A matches to a residue in sequence B at any given position of the alignment. Using the scoring matrix in Figure 3.1 as our starting point,
The values on the diagonal represent the score that would be conferred for an exact match at a given position, and these numbers are always positive. So, if a tryptophan residue (W) in sequence A is aligned with a tryptophan residue in sequence B, this match would be conferred 11 points, the value where the row marked W intersects the column marked W. Also notice that 11 is the highest value on the diagonal, so the high number of points assigned to a W:W alignment reflects not only the exact match but also the fact that tryptophan is the rarest of amino acids found in proteins. Put otherwise, the W:W alignment is much less likely to occur in general and, in turn, is more likely to be correct.
Moving off the diagonal, consider the case of a conservative substitution: a tyrosine (Y) for a tryptophan. The intersection of the row marked Y with the column marked W yields a value of 2. The positive value implies that the substitution is observed to occur more often in an alignment than it would by chance, but the replacement is not as good as if the tryptophan residue had been preserved (2 < 11) or if the tyrosine residue had been preserved (2 < 7).
Finally, consider the case of a non-conservative substitution: a valine (V) for a tryptophan. The intersection of the row marked V with the column marked W yields a value of −3. The negative value implies that the substitution is not observed to occur frequently and may arise more often than not by chance.
Although the meaning of the numbers and relationships within the scoring matrices seems straightforward enough, some value judgments need to be made as to what actually constitutes a conservative or non-conservative substitution and how to assess the frequency of either of those events in nature. This is the major factor that differentiates scoring matrices from one another. To help the reader make an intelligent choice, a discussion of the approach, advantages, and disadvantages of the various available matrices is in order.
PAM Matrices
The first useful matrices for protein sequence analysis were developed by Dayhoff et al. (1978). The basis for these matrices was the examination of substitution patterns in a group of proteins that shared more than 85% sequence identity. The analysis yielded 1572 changes in the 71 groups of closely related proteins that were examined. Using these results, tables were constructed that indicated the frequency of a given amino acid substituting for another amino acid at a given position.
As the sequences examined shared such a high degree of similarity, the resulting frequencies represent what would be expected over short evolutionary distances. Further, given the close evolutionary relationship between these proteins, one would expect that the observed mutations would not significantly change the function of the protein. This is termed acceptance: changes that can be accommodated through natural selection and result in a protein with the same or similar function as the original. As individual point mutations were considered, the unit of measure resulting from this analysis is the point accepted mutation or PAM unit. One PAM unit corresponds to one amino acid change per 100 residues, or roughly 1% divergence.
Several assumptions went into the construction of the PAM matrices. One of the most important assumptions was that the replacement of an amino acid is independent of previous mutations at the same position. Based on this assumption, the original matrix was extrapolated to come up with predicted substitution frequencies at longer evolutionary distances. For example, the PAM1 matrix could be multiplied by itself 100 times to yield the PAM100 matrix, which would represent what one would expect if there were 100 amino acid changes per 100 residues. (This does not imply that each of the 100 residues has changed, only that there were 100 total changes; some positions could conceivably change and then change back to the original residue.) As the matrices representing longer evolutionary distances are an extrapolation of the original matrix derived from the 1572 observed changes described above, it is important to remember that these matrices are, indeed, predictions and are not based on direct observation. Any errors in the original matrix would be exaggerated in the extrapolated matrices, as the mere act of multiplication would magnify these errors significantly.
There are additional assumptions that the reader should be aware of regarding the construction of these PAM matrices. All sites have been assumed to be equally mutable, replacement has been assumed to be independent of surrounding residues, and there is no consideration of
