Sequence Comparison.pdf

Chapter 7
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.
Section 7.2 presents the Karlin and Altschul statistics of local alignment scores.
We first prove that maximal segment scores are accurately described by a geometriclike
distribution in asymptotic limit in Sections 7.2.1 and 7.2.3; we introduce the
Karlin-Altschul sum statistic in Section 7.2.4. Section 7.2.5 summarizes the corresponding
results for optimal ungapped local alignment scores. Finally, we discuss
the edge effect issue in Section 7.2.6.
The explicit theory is unknown for the distribution of local similarity scores
in the case that gaps are allowed. Hence, most studies in this case are empirical.
These studies suggest that the optimal local alignment scores also fit an extreme
value type-I distribution for most cases of interest. Section 7.3.1 describes a phase
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