Local Alignment Statistics

The understanding of the statistical significance of local sequence alignment has improved greatly since Karlin and Altschul published their seminal work [100] on the distribution of optimal ungapped local alignment scores in 1990. In this chapter, we discuss the local alignment statistics that are incorporated into BLAST and other alignment programs. Our discussion focuses on protein sequences for two reasons. First, the analysis for DNA sequences is theoretically similar to, but easier than, that for protein sequences. Second, protein sequence comparison is more sensitive than that of DNA sequences. Nucleotide bases in a DNA sequence have higher-order dependence due to codon bias and other mechanisms, and hence DNA sequences with normal complexity might encode protein sequences with extremely low complexity. Accordingly, the statistical estimations from DNA sequence comparison are often less reliable than those with proteins.

The statistics of local similarity scores are far more complicated than what we shall discuss in this chapter.Many theoretical problems arising from the general case in which gaps are allowed have yet to be well studied, even though they have been investigated for three decades. Our aim is to present the key ideas in the work of Karlin and Altschul on optimal ungapped local alignment scores and its generalizations to gapped local alignment. Basic formulas used in BLAST are also described.

This chapter is divided into five sections. In Section 7.1, we introduce the extreme value type-I distribution. Such a distribution is fundamental to the study of local similarity scores, with and without gaps.