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7.5 Bibliographic Notes and Further Reading 147<br />
7.2<br />
The theorems and their proofs in Sections 7.2.1 and 7.2.2 are from the work<br />
of Karlin and Dembo [102]. The Karlin-Altschul sum statistic in Section 7.2.4 is<br />
reported in [101]. The results summarized in Section 7.2.5 are found in the work<br />
of Dembo, Karlin, and Zeitouni [57]. The edge effect correction presented in Section<br />
7.2.6 is first used in the BLAST program [7]. The justification of the edge<br />
effect correction is demonstrated by Altschul and Gish [6]. Edge effects are even<br />
more serious for gapped alignment as shown by Park and Spouge [157] and Spang<br />
and Vingron [182].<br />
7.3<br />
Phase transition phenomena for local similarity measures is studied by Waterman<br />
and Arratia [199, 15], Dembo, Karlin, and Zeitouni [56], and Grossmann and Yakir<br />
[82]. In general, it seems hard to characterize the logarithmic region. Toward this<br />
problem, a sufficient condition is given by Chan [39].<br />
Many empirical studies strongly suggest that the optimal local alignment scores<br />
with gaps also follow an extreme value type-I distribution in the logarithmic zone<br />
[5, 6, 51, 144, 145, 160, 181, 200]. Although this observation seems a long way<br />
from being mathematically proved, it is further confirmed in the theoretical work of<br />
Siegmund and Yakir [179] and Grossman and Yakir [82].<br />
Two types of methods for parameter estimation are presented in Section 7.3.2.<br />
The direct method is from the work of Altschul and Gish [6]. The island method,<br />
rooting in the work of Waterman and Vingron [200], is developed in the paper of<br />
Altschul et al. [5]. More heuristic methods for estimating the distribution parameters<br />
are reported in the papers of Bailey and Gribskov [21], Bundschuh [35], Fayyaz et<br />
al. [66], Kschischo, Lässig, and Yu [117], Metzler [137], Mott [145], and Mott and<br />
Tribe [146].<br />
7.4<br />
The material covered in Section 7.4.1 can be found in the manuscript of Gertz<br />
[74]. The printouts given in Section 7.4.1 are prepared using NCBI BLAST and EBI<br />
WU-BLAST web server, respectively.<br />
Miscellaneous<br />
We have studied the statistical significance of local alignment scores. The distribution<br />
of global alignment scores is hardly studied. No theoretical result is known.<br />
Readers are referred to the papers by Reich et al. [169] and Webber and Barton<br />
[201] for information on global alignment statistics.
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