Amino Acid Substitution Scores

GNPKVKAH Here we discuss standard ways of assigning a score to each amino acid pair, i.e., to each possible column of a gap-free pairwise protein alignment. Examples of such scoring matrices include the PAM30, PAM70, BLOSUM80, BLOSUM62 and BLOSUM45 matrices that are available on NCBI’s blastp server. Such scores are appropriate for comparing two sequences about which we have no other information (as opposed to position specific scores tailored for a particular protein family). Thus, we seek a 20-by-20 array of numbers for protein sequence comparisons. Observation 1. There exist independent and reliable means of deciding if a particular scoring matrix gives good results. Given an alignment that is optimal with respect to certain scores, does each non-gap column contain letters that are derived from the same ancestral letter by replacement operations? We can use 3-dimensional structural information to decide the “correct answer.” With database searches, we care more about the scores than about the actual alignment, so, basically, the question is: how many of the known homologs of the query sequence score higher than the highest-scoring unrelated sequence? A number of protein families are extremely well studied and can be used to answer such questions. In contrast, for non-coding DNA sequences, it is difficult to determine the “correct alignment.” One approach is to use a knowledge of protein biochemistry to predict which amino-acid pairs are most likely to arise by replacement operations, and thereby obtain scores. However, it works better to simply look at a “training set” of correct alignments and observe frequencies of each kind of column. (Knowledge of biochemistry is of course important for determining the training set.) To help motivate these ideas, ask yourself, “What should be the relationship between the score for aligning two As as opposed to aligning two Ws?” The point is that A occurs much more frequently than W in a typical protein sequence. Most people will say that a W-over-W column should score higher than an A-over-A column. Intuitively, the reason is that W-over-W provides stronger evidence that the alignment is correct since it will occur in a chance alignment of unrelated sequences much less frequently than A-over-A, despite the fact that W-over-W appears somewhat less frequently in correct alignments than does A-over-A. Observation 2. We use scores that assess the likelihood that the alignment’s columns are drawn from the population of correct columns, as compared to the likelihood they are generated by chance. Accordingly, we use two statistical models of an alignment: one reflects biologically correct alignments and the other reflects chance alignments of unrelated sequences. (We’ll think of the columns as independent identically distributed (i.i.d.). Markov chains are an alternative, though more complicated.) It is the ratio of the two probabilities that interests us most. For the model of aligning unrelated sequences, we assign probabilities q(x) to amino acids x, reflecting the frequency with which they appear in protein sequences. Thus, the probability that a random alignment of unrelated sequences happens to align the sequences x1x2 . . . xn and y1y2 . . . yn is ∏ n i=1 (q(xi)q(yi)). Assuming that the 20 q-values