The Statistics of Semi-Probabilistic Alignment

Computer-assisted sequence comparison has become an integral part of modern molecular biology. Two types of algorithms have been used: those which search for the optimal alignment (as exemplified by the Smith-Waterman algorithm [1]), and those which identify likely alignments (as exemplified by the HMM-based “Sequence Alignment Modules” [2]). In each case, the quality of alignment is summarized by an alignment score S; the latter is typically taken to be the logarithm of the total likelihood in the probabilistic approaches. An important goal common to the study of all algorithms is to understand the score distribution P (S) for the appropriate null models. This distribution gives the probability that a high scores could have arisen by chance and is therefore more meaningful to homology detection than the alignment scores themselves. Rigorous results on such background statistics are known only for the gapless alignment [3], whose score distribution follows the so-called Gumbel form, P (S) = KMNλe−λS−KMN exp(−λS) , for long sequence lengths M and N . There exist explicit formulae relating the hundreds of alignment parameters to the two Gumbel parameters λ and K. For the gapped Smith-Waterman alignment, ample empirical evidences suggest that the null score distribution still obeys the Gumbel form. But the dependences of the two Gumbel parameters on the alignment score functions are very complicated and largely unknown. For probabilistic alignments, the log-likelihood score does not even satisfy Gumbel distribution as was recently shown [4]. Combining the optimal and the probabilistic approaches to sequence alignment, we developed a new “hybrid” algorithm for which the alignment score is still Gumbel-distributed, and the Gumbel parameters can be computed accurately and rapidly, for a large class of scoring functions (including the position-specific ones), and for a wide range of sequence lengths (down to ∼ 100 amino acids), without the need of extensive simulation as is commonly done. We have independently checked using sequences generated by simple evolution models that the fidelity of homology detected by the hybrid algorithm is comparable to or better than that of the Smith-Waterman algorithm [4].