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Dynamic programming algorithms for pairwise alignment and maximum parsimony

This guide explores dynamic programming algorithms for calculating edit distance and maximum parsimony in sequence alignment. It addresses indel and substitution costs, outlining methods to determine the minimum transformation cost from sequence X to sequence Y using a systematic approach. The matrix M[i,j] represents the minimum edit distance, initialized based on sequence lengths and computed with previously established entries. Additionally, it discusses maximum parsimony for fixed trees, detailing how to compute costs for internal nodes based on Hamming distances.

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Dynamic programming algorithms for pairwise alignment and maximum parsimony

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  1. Dynamic programming algorithms for pairwise alignment and maximum parsimony

  2. Edit distance • Given indel cost 1, and substitution cost C, and two sequences X and Y, find the minimum cost of any edit transformation of X into Y. • How to solve this using Dynamic Programming? • M[i,j] is the minimum cost of any edit transformation of X[1…i] into Y[1…j]

  3. DP solution for edit distance • M[i,j] is the minimum cost of any edit transformation of X[1…i] into Y[1…j] • How to initialize? • How to set each entry using previously computed entries? • How to order the calculations? • Where is the answer?

  4. The DP solution • Input strings: X[1…n] and Y[1…m]. • Indel cost 1, substitution cost C>0 • M[i,j] is the edit distance of the prefix x1,x2,…,xi to the prefix y1,y2,…,yj. We need to compute M[i,j] for I=0,1,…n, and j=0,1,…m. • M[0,j]=M[j,0]=j for all j (why?) • The solution is stored in M[n,m] (why?). • How do we compute M[i,j] for the other values of i and j?

  5. The DP solution • Indel cost 1, substitution cost C>0 • M[i,j] is the edit distance of the prefix x1,x2,…,xi to the prefix y1,y2,…,yj. How do we compute M[i,j] for the other values of i and j? • If xi = yj then M[i,j] = min{M[i-1,j-1], M[i-1,j]+1, M[i,j-1]+1} Else M[i,j] = min{M[i-1,j-1] +C, M[i-1,j]+1, M[i,j-1]+1}

  6. The DP solution • Indel cost 1, substitution cost C>0 • If xi = yj then M[i,j] = min{M[i-1,j-1], M[i-1,j]+1, M[i,j-1]+1} else M[i,j] = min{M[i-1,j-1] +C, M[i-1,j]+1, M[i,j-1]+1} We compute the entries of the matrix M row-by-row (or column-by-column). The edit distance is stored in M[n,m]. If we add arrows (from each box to the box(es) which gave the lowest edit distance, we can obtain the minimum cost transformation.

  7. Maximum parsimony • Fixed tree problem: given a tree T and sequences at the leaves, compute the “length” of the tree under an optimal assignment of sequences to the internal nodes. • The “length” is the sum of the Hamming distances on the edges of the tree.

  8. MP on a fixed tree (cont.) • We solve the problem for sequences of length 1! • Let Cost(v,L) be the minimum cost of the tree rooted at v given that we label v by the letter L (so L= A, C, T or G). • What should Cost(v,L) be if v is a leaf (and so already labelled)? (Infinity if L is the wrong label, and otherwise 0.)

  9. MP on a fixed tree (cont.) • Suppose v has two children, w and x. Then Cost(v,L) = minP{Cost(w,L), Cost(w,P)+1 if P =! L} + minP{Cost(x,L), Cost(x,P)+1 if P =! L}

  10. MP on a fixed tree (cont.) • Compute Cost(v,L) for every node v and every nucleotide L, as you go from the leaves to the root. • min{Cost(root,A),Cost(root,C), Cost(root,T), Cost(root,G)} is the minimum cost achievable (i.e., the “length” of the tree for that site). • Backtrack to get the actual assignment to internal nodes.

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