In Chapter 7, we justified the use of fast alignment heuristics like BLAST by our need to quickly align large numbers of sequences in a database with a given query sequence to determine which were most similar to the query. In this chapter, we will consider statistical aspects of the set of alignment scores we might encounter when performing such a database search. The distribution of scores obviously depends on the substitution matrix employed (PAM30, BLOSUM62, PAM250, etc.), and a proof of the general result requires rather extensive use of sophisticated mathematical notation. We will avoid this by concentrating on a specific, simple scoring matrix for DNA before outlining the general result.

Like all but the most recent versions of BLAST, we will focus on gapless alignments. The theory of statistical properties of scores of alignments with gaps has been elucidated only approximately and only for special cases. The theory supports the empirical observation that their behavior is similar to the behavior of statistics without gaps.

BLAST Scores for Random DNA

Suppose that Q is a query sequence of DNA and that D is a sequence from a database. As usual for DNA, we will score +1 for matched bases and -1 for mismatched bases in alignments of D and Q. Suppose a BLAST search discovers a local gap-free alignment with a score of 13. Does this suggest that D and Q are in some way related, or could this be better explained by chance? To answer this question, we must define precisely what “by chance” means to us, and then compute – or at least estimate – the probability that a score of 13 or higher occurs under that definition.