Sequence Alignment

Before we start, administrivia Instructor: Serafim Batzoglou, CS x Office hours: Monday 2:00-3:30 TA: Omkar Deshpande x Office hours: Wednesday 4:00-6:00 Class Website: Books: they should be in by Monday Let’s schedule Weekly Discussion Session Teams: ~3, <= 4 people: please use su.class.cs262  Individual writeups Lecture notes: prose + figures, ~4 – 8 pages  Previous year’s lecture notes are available

Comparing Genomes

Evolution at the DNA level …ACGGTGCAGTCACCA… …ACGTTGCAGTCCACCA… C SEQUENCE EDITSREARRANGEMENTS

Sequence conservation implies function Interleukin region in human and mouse 100% 40%

Sequence Alignment -AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC Definition Given two strings x = x 1 x 2...x M, y = y 1 y 2 …y N, an alignment is an assignment of gaps to positions 0,…, M in x, and 0,…, N in y, so as to line up each letter in one sequence with either a letter, or a gap in the other sequence AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC

Scoring Function Sequence edits: AGGCCTC  Mutations AGGACTC  Insertions AGGGCCTC  Deletions AGG.CTC Scoring Function: Match: +m Mismatch: -s Gap:-d Score F = (# matches)  m - (# mismatches)  s – (#gaps)  d

How do we compute the best alignment? AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA AGTGACCTGGGAAGACCCTGACCCTGGGTCACAAAACTC Too many possible alignments: O( 2 M+N )

Alignment is additive Observation: The score of aligningx 1 ……x M y 1 ……y N is additive Say thatx 1 …x i x i+1 …x M aligns to y 1 …y j y j+1 …y N The two scores add up: F(x[1:M], y[1:N]) = F(x[1:i], y[1:j]) + F(x[i+1:M], y[j+1:N])

Dynamic Programming We will now describe a dynamic programming algorithm Suppose we wish to align x 1 ……x M y 1 ……y N Let F(i,j) = optimal score of aligning x 1 ……x i y 1 ……y j

Dynamic Programming (cont’d) Notice three possible cases: 1.x i aligns to y j x 1 ……x i-1 x i y 1 ……y j-1 y j 2.x i aligns to a gap x 1 ……x i-1 x i y 1 ……y j - 3.y j aligns to a gap x 1 ……x i - y 1 ……y j-1 y j m, if x i = y j F(i,j) = F(i-1, j-1) + -s, if not F(i,j) = F(i-1, j) - d F(i,j) = F(i, j-1) - d

Dynamic Programming (cont’d) How do we know which case is correct? Inductive assumption: F(i, j-1), F(i-1, j), F(i-1, j-1) are optimal Then, F(i-1, j-1) + s(x i, y j ) F(i, j) = max F(i-1, j) – d F( i, j-1) – d Where s(x i, y j ) = m, if x i = y j ;-s, if not

Example x = AGTAm = 1 y = ATAs = -1 d = -1 AGTA A10 -2 T 0010 A-3 02 F(i,j) i = j = Optimal Alignment: F(4,3) = 2 AGTA A - TA

The Needleman-Wunsch Matrix x 1 ……………………………… x M y 1 ……………………………… y N Every nondecreasing path from (0,0) to (M, N) corresponds to an alignment of the two sequences Can think of it as a divide-and-conquer algorithm

The Needleman-Wunsch Algorithm 1.Initialization. a.F(0, 0) = 0 b.F(0, j) = - j  d c.F(i, 0)= - i  d 2.Main Iteration. Filling-in partial alignments a.For each i = 1……M For eachj = 1……N F(i-1,j-1) + s(x i, y j ) [case 1] F(i, j) = max F(i-1, j) – d [case 2] F(i, j-1) – d [case 3] DIAG, if [case 1] Ptr(i,j)= LEFT,if [case 2] UP,if [case 3] 3.Termination. F(M, N) is the optimal score, and from Ptr(M, N) can trace back optimal alignment

Performance Time: O(NM) Space: O(NM) Later we will cover more efficient methods

A variant of the basic algorithm: Maybe it is OK to have an unlimited # of gaps in the beginning and end: CTATCACCTGACCTCCAGGCCGATGCCCCTTCCGGC GCGAGTTCATCTATCAC--GACCGC--GGTCG Then, we don’t want to penalize gaps in the ends

Different types of overlaps

The Overlap Detection variant Changes: 1.Initialization For all i, j, F(i, 0) = 0 F(0, j) = 0 2.Termination max i F(i, N) F OPT = max max j F(M, j) x 1 ……………………………… x M y 1 ……………………………… y N

The local alignment problem Given two strings x = x 1 ……x M, y = y 1 ……y N Find substrings x’, y’ whose similarity (optimal global alignment value) is maximum e.g.x = aaaacccccgggg y = cccgggaaccaacc

Why local alignment Genes are shuffled between genomes Portions of proteins (domains) are often conserved

Cross-species genome similarity 98% of genes are conserved between any two mammals >70% average similarity in protein sequence hum_a : 57331/ mus_a : 78560/ rat_a : / fug_a : 36008/68174 hum_a : 57381/ mus_a : 78610/ rat_a : / fug_a : 36058/68174 hum_a : 57431/ mus_a : 78659/ rat_a : / fug_a : 36084/68174 hum_a : 57481/ mus_a : 78708/ rat_a : / fug_a : 36097/68174 “atoh” enhancer in human, mouse, rat, fugu fish

The Smith-Waterman algorithm Idea: Ignore badly aligning regions Modifications to Needleman-Wunsch: Initialization:F(0, j) = F(i, 0) = 0 0 Iteration:F(i, j) = max F(i – 1, j) – d F(i, j – 1) – d F(i – 1, j – 1) + s(x i, y j )

The Smith-Waterman algorithm Termination: 1.If we want the best local alignment… F OPT = max i,j F(i, j) 2.If we want all local alignments scoring > t For all i, j find F(i, j) > t, and trace back

Scoring the gaps more accurately Current model: Gap of length n incurs penaltyn  d However, gaps usually occur in bunches Convex gap penalty function:  (n): for all n,  (n + 1) -  (n)   (n) -  (n – 1)  (n)

General gap dynamic programming Initialization:same Iteration: F(i-1, j-1) + s(x i, y j ) F(i, j) = maxmax k=0…i-1 F(k,j) –  (i-k) max k=0…j-1 F(i,k) –  (j-k) Termination: same Running Time: O(N 2 M)(assume N>M) Space:O(NM)

Compromise: affine gaps  (n) = d + (n – 1)  e | | gap gap open extend To compute optimal alignment, At position i,j, need to “remember” best score if gap is open best score if gap is not open F(i, j):score of alignment x 1 …x i to y 1 …y j if if x i aligns to y j if G(i, j):score if x i, or y j, aligns to a gap d e  (n)

Needleman-Wunsch with affine gaps Initialization:F(i, 0) = d + (i – 1)  e F(0, j) = d + (j – 1)  e Iteration: F(i – 1, j – 1) + s(x i, y j ) F(i, j) = max G(i – 1, j – 1) + s(x i, y j ) F(i – 1, j) – d F(i, j – 1) – d G(i, j) = max G(i, j – 1) – e G(i – 1, j) – e Termination: same