6. MSA using DCA and branch-and-bound - Algorithms in ...

32 Comp. Sequence Analysis WS’04 ZBIT, D. Huson, (script by K. Reinert), December 17, 2004S 1S 1S 1divideS 1S 1S 1S 1S 1S 1dividedividealign optimalyconcatenateTo help determine good cut sites, we will make use of so-called additional cost matrices for each pairof sequences and each possible cut position.The additional cost is the score difference between the optimal global alignment and the best alignmentgoing through position i and j.isjtBased on this, we will determine C-optimal families of cut positions, for which the (weighted) sum ofpairwise additional costs is minimized.To define this more formally, assume we are given two sequences s p = s p1 . . . s pn and s q = s q1 . . . s qm .Let c denote a cost function and let A ∗ denote an optimal alignment of s p and s q .Let (i, j) be a pair of possible cut positions in s p and s q with 0 ≤ i ≤ n and 0 ≤ j ≤ m. We use α i s orσs i to denote the i-th prefix, or suffix, of a string s, respectively:s ps qα i s pσ i s p———–i———–——–j————–α j s qσs j qIf A is an alignment of a prefix of s p to a prefix of s q , and if B is an alignment of the two remainingsuffixes, then let A++B denote the concatenation of the two alignments.Definition For each pair (i, j) of possible cut positions of s p and s q we define the pairwise additional