Statistics for Sequence Alignment

Oct 20, 2021

Karlin-Altschul statistics

Evaluate statistical significance of hits in blast search
1990, PNAS, S Karlin & S F Altschul, Methods for assessing the statistical significance of molecular sequence features by using general scoring schemes
The expected number of ungapped alignments with score \(S\) found with random sequences \(E\) was demostrated to be

\[E(S) = Kmne^{-\lambda S}\]

\(m\) is length of sequence 1, \(n\) is length of sequence 2, \(K\) is a constant
Given the substitution matrix \(S_{i,j}\), and frequency of the character \(a_{i}\) and \(a_{j}\)

\[\sum_{i,j=1}^{r}p_{i}p_{j}e^{s_{ij} \lambda}=1\]

Alternatively, the alignment score \(S\) follows extreme value distribution (EVD), where \(u=\frac{lnKmn}{\lambda}\)

\[P(S<x)=e^{-e^{-\lambda(x-u)}}=e^{-Kmne^{-{\lambda} x}}\] \[P(S{\geq}x)=1-P(S<x)=1-e^{-Kmne^{-{\lambda} x}}=1-e^{-E(x)}\]

\(E(S)\) can be interpreted as expected value in poisson distribution
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