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Sequence comparison: Dynamic programming Genome 559: Introduction to Statistical and Computational Genomics Prof. James H. Thomas

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Sequence comparison overview Problem: Find the “best” alignment between a query sequence and a target sequence. To solve this problem, we need –a method for scoring alignments, and –an algorithm for finding the alignment with the best score. The alignment score is calculated using –a substitution matrix –gap penalties. The algorithm for finding the best alignment is dynamic programming (DP).

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A simple alignment problem. Problem: find the best pairwise alignment of GAATC and CATAC. Use a linear gap penalty of -4. Use the following substitution matrix: ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10

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How many possibilities? How many different possible alignments of two sequences of length n exist? GAATC CATAC GAATC- CA-TAC GAAT-C C-ATAC GAAT-C CA-TAC -GAAT-C C-A-TAC GA-ATC CATA-C

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How many possibilities? How many different alignments of two sequences of length n exist? GAATC CATAC GAATC- CA-TAC GAAT-C C-ATAC GAAT-C CA-TAC -GAAT-C C-A-TAC GA-ATC CATA-C 5 2.5x10 2 10 1.8x10 5 20 1.4x10 11 30 1.2x10 17 40 1.1x10 23 FYI for two sequences of length m and n, possible alignments number:

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DP matrix GAATC C A T A C ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 i 012345012345 j 0 1 2 3 etc. 5 The value in position ( i,j ) is the score of the best alignment of the first i positions of one sequence versus the first j positions of the other sequence. GA CA init. row and column

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DP matrix GAATC C A 5 1 T A C Moving horizontally in the matrix introduces a gap in the sequence along the left edge. GAA CA- ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10

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DP matrix GAATC C A 5 T 1 A C Moving vertically in the matrix introduces a gap in the sequence along the top edge. GA- CAT ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10

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DP matrix GAATC C A 5 T 0 A C Moving diagonally in the matrix aligns two residues GAA CAT ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10

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Initialization GAATC 0 C A T A C ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 Start at top left and move progressively

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Introducing a gap GAATC 0-4 C A T A C G-G- ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10

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GAATC 0-4 C A T A C -C-C ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 Introducing a gap

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Complete first row and column GAATC 0-4-8-12-16-20 C-4 A-8 T-12 A-16 C-20 ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 ----- CATAC

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Three ways to get to i=1, j=1 GAATC 0-4 C-8 A T A C ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 G- -C j 0 1 2 3 etc. i 012345012345

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GAATC 0 C-4-8 A T A C ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 -G C- Three ways to get to i=1, j=1 j 0 1 2 3 etc. i 012345012345

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GAATC 0 C-5 A T A C GCGC ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 Three ways to get to i=1, j=1 i 012345012345 j 0 1 2 3 etc.

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Accept the highest scoring of the three GAATC 0-4-8-12-16-20 C-4-5 A-8 T-12 A-16 C-20 ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 Then simply repeat the same rule progressively across the matrix

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DP matrix GAATC 0-4-8-12-16-20 C-4-5 A-8? T-12 A-16 C-20 ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10

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DP matrix GAATC 0-4-8-12-16-20 C-4-5 A-8 ? T-12 A-16 C-20 -4 0 -G CA G- CA --G CA- -4 -9-12 ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10

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DP matrix GAATC 0-4-8-12-16-20 C-4-5 A-8 -4 T-12 A-16 C-20 -4 0 -G CA G- CA --G CA- -4 -9-12 ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10

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DP matrix GAATC 0-4-8-12-16-20 C-4-5 A-8-4 T-12? A-16? C-20? ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10

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DP matrix GAATC 0-4-8-12-16-20 C-4-5 A-8-4 T-12-8 A-16-12 C-20-16 ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10

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DP matrix GAATC 0-4-8-12-16-20 C-4-5? A-8-4? T-12-8? A-16-12? C-20-16? ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10

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DP matrix GAATC 0-4-8-12-16-20 C-4-5-9 A-8-45 T-12-81 A-16-122 C-20-16-2 What is the alignment associated with this entry? ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 (just follow the arrows back - this is called the traceback)

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DP matrix GAATC 0-4-8-12-16-20 C-4-5-9 A-8-45 T-12-81 A-16-122 C-20-16-2 ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 -G-A CATA

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GAATC 0-4-8-12-16-20 C-4-5-9 A-8-45 T-12-81 A-16-122 C-20-16-2? ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 Continue and find the optimal global alignment, and its score. DP matrix

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GAATC 0-4-8-12-16-20 C-4-5-9-13-12-6 A-8-451-3-7 T-12-810117 A-16-1221176 C-20-16-271117 ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 Best alignment starts at lower right and follows traceback arrows to top left

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GAATC 0-4-8-12-16-20 C-4-5-9-13-12-6 A-8-451-3-7 T-12-810117 A-16-1221176 C-20-16-271117 ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 GA-ATC CATA-C One best traceback

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GAATC 0-4-8-12-16-20 C-4-5-9-13-12-6 A-8-451-3-7 T-12-810117 A-16-1221176 C-20-16-271117 ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 GAAT-C -CATAC Another best traceback

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GAATC 0-4-8-12-16-20 C-4-5-9-13-12-6 A-8-451-3-7 T-12-810117 A-16-1221176 C-20-16-271117 ACGT A10-5 0 C 10-5 0 G 0 10-5 T 0 10 GA-ATC CATA-C GAAT-C -CATAC

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Multiple solutions When a program returns a single sequence alignment, it may not be the only best alignment but it is guaranteed to be one of them. In our example, all of the alignments at the left have equal scores. GA-ATC CATA-C GAAT-C CA-TAC GAAT-C C-ATAC GAAT-C -CATAC

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DP in equation form Align sequence x and y. F is the DP matrix; s is the substitution matrix; d is the linear gap penalty.

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DP equation graphically take the max of these three

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Dynamic programming Yes, it’s a weird name. DP is closely related to recursion and to mathematical induction. We can prove that the resulting score is optimal.

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Summary Scoring a pairwise alignment requires a substitution matrix and gap penalties. Dynamic programming is an efficient algorithm for finding an optimal alignment. Entry ( i,j ) in the DP matrix stores the score of the best-scoring alignment up to that position. DP iteratively fills in the matrix using a simple mathematical rule.

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ACGT A10-50 C 10-50 G0 10-5 T 0 10 GAATC 0 A A T T C Practice problem: find a best pairwise alignment of GAATC and AATTC d = -4

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