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

34 Comp. Sequence Analysis WS’04 ZBIT, D. Huson, (script by K. Reinert), December 17, 2004AGT0013C1002T3210C_{s_1,s_2}G0 00 0012310T0 1 20 C 1 2 0 1 2 31C_{s_1,s_3}AG1 0 0 1 2G0 2 1 0 01C_{s_2,s_3}Set all pairwise weights to 1 and assume ĉ 1 = 1. It is not difficult to verify that (c 2 , c 3 ) = (1, 0) isC-optimal with respect to ĉ 1 , sinceC(1, 1, 0) = C s1 ,s 2[1, 1] + C s1 ,s 3[1, 0] + C s2 ,s 3[1, 0] = 0.Given this cut the best possible alignment has cost 7. The unique optimal alignment has cost 6 andcan be achieved with the cut (2, 1) which is also C-optimal with respect to ĉ 1 .C -T C-T -C T -CTA ++ GT = AGT AG ++ T = AGT- G- -G- -G - -Gcut(1,1,0) cut (1,2,1)cost 7 cost 6Assume we have a multiple sequence alignment program MSA that we can use to solve small instancesof the problem of aligning k sequences. The DCA algorithm can be summarized as follows:Algorithm DCA(s 1 , s 2 , . . . , s k , L)Input: sequences s 1 , . . . , s k and cut-off LOutput: alignment of s 1 , . . . , s kbeginif max{n 1 , . . . n k } ≤ L thenreturn MSA(s 1 , s 2 , . . . , s k )elseSet ĉ 1 := ⌈ n 12⌉Set (c 2 , . . . , c k ) := C-opt((s 1 , . . . , s k ), ĉ 1 )return DCA(α cˆ1s 1, α c 2s 2, . . . , α c ksk) ++DCA(σĉ 1 1, σc 2 2, . . . , σc k k)endThe function C-opt can naively be implemented as nested loops that run over all possible values0, . . . , n i , ∀i and return the cut position with the minimal multiple additional costs.The computation of C-opt requires at most O(Nk 2 n 2 ) steps, with n = max i {|s i |} and N = ∏ i |s i|.A direct improvement of this procedure is the combination of the loops with a simple branch-andboundprocedure that cuts off combinations of the c 2 , . . . , c k , if already a partial sum of the additionalcost term exceeds the minimal cost found so far.T6.2 Naïve Dynamic programmingThe straightforward extension of the Needleman-Wunsch algorithm to compute an optimal (W)SOPcostalignment A ∗ for k sequences s 1 , . . . , s k using dynamic programming takes time O(n k ), withn = min i |s i |. Obviously this is only practical for very few, short sequences (k = 3, 4)Our goal now is to develop a branch-and-bound approach.In the following we will use the edit graph representation of a (multiple) alignment. In this representationfinding an optimal alignment corresponds to finding a shortest path in a directed acyclic graph(DAG).