Genomic Sequence Alignment

Overview

• Dynamic programming & the Needleman-Wunsch algorithm

• Local alignment—BLAST

• Fast global alignment

• Multiple sequence alignment

• Rearrangements in genomic sequences

Biology in One Slide – Twentieth Century

…ACGTGACTGAGGACCGTGCGACTGAGACTGACTGGGTCTAGCTAGACTACGTTTTATATATATATACGTCGTCGTACTGATGACTAGATTACAGACTGATTTAGATACCTGACTGATTTTAAAAAAATATT…

…and today

Complete DNA Sequences

About 300 complete genomes have been

sequenced

Evolution

Evolution at the DNA level

…ACGGTGCAGTTACCA…

…AC----CAGTCCACCA…

Mutation

SEQUENCE EDITS

REARRANGEMENTS

Deletion

InversionTranslocationDuplication

Evolutionary Rates

OK

OK

OK

X

X

Still OK?

next generation

Sequence conservation implies function

Alignment is the key to• Finding important regions• Determining function• Uncovering the evolutionary forces

Sequence Alignment

-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC---TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC

DefinitionGiven two strings x = x1x2...xM, y = y1y2…yN,

an alignment is an assignment of gaps to positions0,…, N in x, and 0,…, N in y, so as to line up each

letter in one sequence with either a letter, or a gapin the other sequence

AGGCTATCACCTGACCTCCAGGCCGATGCCCTAGCTATCACGACCGCGGTCGATTTGCCCGAC

What is a good alignment?

Alignment: The “best” way to match the letters of one sequence with those of the other

How do we define “best”?

Alignment:A hypothesis that the two sequences come from a common ancestor through sequence edits

Parsimonious explanation:Find the minimum number of edits that transform one sequence into the other

Scoring Function

• Sequence edits: AGGCCTC

Mutations AGGACTC

InsertionsAGGGCCTC

DeletionsAGG.CTC

Scoring Function:Match: +mMismatch: -sGap: -d

Score F = (# matches) m - (# mismatches) s – (#gaps) d

How do we compute the best alignment?

AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA

AGTGACCTGGGAAGACCCTGACCCTGGGTCACAAAACTC

Too many possible alignments:

O( 2N)

Dynamic Programming

• Given two sequences x = x1……xM and y = y1……yN

• Let F(i, j) = Score of best alignment of x1……xi to y1……yj

• Then, F(M, N) == Score of best alignment

Idea: Compute F(i, j) for all i and j Do this by using F(i–1 , j), F(i, j–1), F(i–1, j–1)

Dynamic Programming (cont’d)

Notice three possible cases:

1. xi aligns to yj

x1……xi-1 xi

y1……yj-1 yj

2. xi aligns to a gap

x1……xi-1 xi

y1……yj -

3. yj aligns to a gap

x1……xi -

y1……yj-1 yj

m, if xi = yj

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(xi, yj)

F(i, j) = max F(i-1, j) – dF( i, j-1) – d

Where s(xi, yj) = m, if xi = yj; -s, if not

i-1, j-1 i-1, j

i, j-1 i, j

Example

x = AGTA m = 1y = ATA s = -1

d = -1

A G T A

0 -1 -2 -3 -4

A -1 1 0 -1 -2

T -2 0 0 1 0

A -3 -1 -1 0 2

F(i,j) i = 0 1 2 3 4

j = 0

1

2

3

Optimal Alignment:

F(4,3) = 2

AGTAA - TA

The Needleman-Wunsch Algorithm

1. Initialization.a. F(0, 0) = 0b. F(0, j) = - j dc. F(i, 0) = - i d

2. Main Iteration. Filling-in partial alignmentsa. For each i = 1……M

For each j = 1……N F(i-1,j) – d [case 1]

F(i, j) = max F(i, j-1) – d [case 2]

F(i-1, j-1) + s(xi, yj) [case 3]

UP if [case 1]Ptr(i,j) = LEFT if [case 2]

DIAG if [case 3]

3. Termination. F(M, N) is the optimal score, andfrom Ptr(M, N) can trace back optimal alignment

Alignment on a Large Scale

• Given a gene that we care about, how can we compare it to all existing DNA?

• Assume we use Dynamic Programming:

The entire genomic database

gene of interest

~105

~1011

Index-based Local Alignment

Main idea:

1. Construct a dictionary of all the words in the query

2. Initiate a local alignment for each word match between query and DB

Running Time:Theoretical worst case: O(MN)Fast in practice

query

DB

Index-based Local Alignment — BLAST

Dictionary:All words of length k (~11)Alignment initiated between exact-matching words

(more generally, between words of alignment score T)

Alignment:Ungapped extensions until score

below statistical threshold

Output:All local alignments with score

> statistical threshold

……

……

query

DB

query

scan

Index-based Local Alignment — BLAST

A C G A A G T A A G G T C C A G T

C

C

C

T

T

C C

T

G

G

A T

T

G

C

G

A

Example:

k = 4,T = 4

The matching word GGTC initiates an alignment

Extension to the left and right with no gaps until alignment falls < 50%

Output:GTAAGGTCC

GTTAGGTCC

Gapped BLAST

A C G A A G T A A G G T C C A G T

C

T

G

A

T

C C

T

G

G

A

T

T

G C

G

A

Added features:

• Pairs of words can initiate alignment

• Nearby alignments are merged

• Extensions with gaps until score < T below best score so far

Output:

GTAAGGTCCAGTGTTAGGTC-AGT

Example

Query: gattacaccccgattacaccccgattaca (29 letters) [2 mins]

Database: All GenBank+EMBL+DDBJ+PDB sequences (but no EST, STS, GSS, or phase 0, 1 or 2 HTGS sequences) 1,726,556 sequences; 8,074,398,388 total letters

>gi|28570323|gb|AC108906.9| Oryza sativa chromosome 3 BAC OSJNBa0087C10 genomic sequence, complete sequence Length = 144487 Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plus

Query: 4 tacaccccgattacaccccga 24 ||||||| |||||||||||||

Sbjct: 125138 tacacccagattacaccccga 125158

Score = 34.2 bits (17),

Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plus

Query: 4 tacaccccgattacaccccga 24 ||||||| |||||||||||||

Sbjct: 125104 tacacccagattacaccccga 125124

>gi|28173089|gb|AC104321.7| Oryza sativa chromosome 3 BAC OSJNBa0052F07 genomic sequence, complete sequence Length = 139823 Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plus

Query: 4 tacaccccgattacaccccga 24 ||||||| |||||||||||||

Sbjct: 3891 tacacccagattacaccccga 3911

http://www.ncbi.nlm.nih.gov

Efficient global alignment

S1

S2

Global alignment with the chaining approach

1. Find local alignments2. Chain them into a rough global map3. Align regions in-between

LAGAN: 1. FIND Local Alignments

1. Find Local Alignments

2. Chain Local Alignments

3. Restricted DP

Mike Brudno, Chuong Do, et al.

LAGAN: 2. CHAIN Local Alignments

1. Find Local Alignments

2. Chain Local Alignments

3. Restricted DP

Mike Brudno, Chuong Do, et al.

LAGAN: 3. Restricted DP

1. Find Local Alignments

2. Chain Local Alignments

3. Restricted DP

Mike Brudno, Chuong Do, et al.

Multiple Alignment

Definition

• Given N sequences x1, x2,…, xN: Insert gaps (-) in each sequence xi, such that

All sequences have the same length L Score of the global map is maximum

• A faint similarity between two sequences becomes significant if present in many

• Multiple alignments can help improve the pairwise alignments

Scoring Function: Sum Of Pairs

Definition: Induced pairwise alignmentA pairwise alignment induced by the multiple alignment

Example:

x: AC-GCGG-C y: AC-GC-GAG z: GCCGC-GAG

Induces:

x: ACGCGG-C; x: AC-GCGG-C; y: AC-GCGAGy: ACGC-GAC; z: GCCGC-GAG; z: GCCGCGAG

Given sequences x1, …, xN, aligned in a multiple alignment m,

S(m) = k<l wkl s(xk, xl)

A Profile Representation

• Given a multiple alignment M = m1…mn

Replace each column mi with profile entry pi Frequency of each letter in # gaps

Can think of this as a “likelihood” of each letter in each position

- A G G C T A T C A C C T G T A G – C T A C C A - - - G C A G – C T A C C A - - - G C A G – C T A T C A C – G G C A G – C T A T C G C – G G

A 1 1 .8 C .6 1 .4 1 .6 .2G 1 .2 .2 .4 1T .2 1 .6 .2- .2 .8 .4 .8 .4

Multiple Sequence Alignments

Algorithms

Generalization of Needleman-Wunsh:

S(m) = i S(mi)

(sum of column scores)

F(i1,i2,…,iN): Optimal alignment up to (i1, …, iN)

F(i1,i2,…,iN) = max(all neighbors of cube)(F(nbr)+S(nbr))

Multidimensional DP

• Example: in 3D (three sequences):

• 7 neighbors/cell

F(i,j,k) = max{ F(i-1,j-1,k-1)+S(xi, xj, xk),F(i-1,j-1,k )+S(xi, xj, - ),F(i-1,j ,k-1)+S(xi, -, xk),F(i-1,j ,k )+S(xi, -, - ),F(i ,j-1,k-1)+S( -, xj, xk),F(i ,j-1,k )+S( -, xj, xk),F(i ,j ,k-1)+S( -, -, xk) }

Multidimensional DP

Running Time:

1. Size of matrix: LN;

Where L = length of each sequence N = number of sequences

2. Neighbors/cell: 2N – 1

Therefore………………………… O(2N LN)

Multidimensional DP

Running Time:

1. Size of matrix: LN;

Where L = length of each sequence N = number of sequences

2. Neighbors/cell: 2N – 1

Therefore………………………… O(2N LN)

Multidimensional DP

Progressive Alignment

• When evolutionary tree is known:

Align closest first, in the order of the tree In each step, align two sequences x, y, or profiles px, py, to generate a new

alignment with associated profile presult

Weighted version: Tree edges have weights, proportional to the divergence in that edge New profile is a weighted average of two old profiles

x

w

y

z

pxy

pzw

pxyzw

Progressive Alignment

• When evolutionary tree is known:

Align closest first, in the order of the tree In each step, align two sequences x, y, or profiles px, py, to generate a new

alignment with associated profile presult

Weighted version: Tree edges have weights, proportional to the divergence in that edge New profile is a weighted average of two old profiles

x

w

y

z

Progressive Alignment

• When evolutionary tree is unknown:

Perform all pairwise alignments Define distance matrix D, where D(x, y) is a measure of

evolutionary distance, based on pairwise alignment Construct a tree Align on the tree

x

w

y

z?

Some useful sites

• Genome browsers Ensembl: www.ensembl.org UCSC: genome.ucsc.edu/cgi-bin/hgGateway

• Genomic alignment LAGAN: lagan.stanford.edu MAVID:

baboon.math.berkeley.edu/mavid• Protein multiple alignment

MUSCLE: www.drive5.com ProbCons: probcons.stanford.edu

Evolution at the DNA level

…ACGGTGCAGTTACCA…

…AC----CAGTCACCA…

Mutation

SEQUENCE EDITS

REARRANGEMENTS

Deletion

InversionTranslocationDuplication

Local & Global Alignment

AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA

AGTGACCTGGGAAGACCCTGAACCCTGGGTCACAAAACTC

AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA

AGTGACCTGGGAAGACCCTGAACCCTGGGTCACAAAACTC

Local Global

Glocal Alignment Problem

Find least cost transformation of one sequence into another using shuffle operations

• Sequence edits

• Inversions

• Translocations

• Duplications

• Combinations of above

AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA

AGTGACCTGGGAAGACCCTGAACCCTGGGTCACAAAACTC

SLAGAN: 1. Find Local Alignments

1. Find Local Alignments

2. Build Rough Homology Map

3. Globally Align Consistent Parts

Mike Brudno, Sanket Malde, et al.

SLAGAN: 2. Build Homology Map

1. Find Local Alignments

2. Build Rough Homology Map

3. Globally Align Consistent Parts

Mike Brudno, Sanket Malde, et al.

SLAGAN: 3. Global Alignment

1. Find Local Alignments

2. Build Rough Homology Map

3. Globally Align Consistent Parts

Mike Brudno, Sanket Malde, et al.

SLAGAN Example: Chromosome 20

Human Chromosome 20 versus Mouse Chromosome 2

• 270 Segments of conserved synteny

• 70 Inversions

SLAGAN example: HOX cluster

• 10 paralogous genes• Conserved order in Human/Mouse/Rat

SLAGAN example: HOX cluster

• 10 paralogous genes• Conserved order in Human/Mouse/Rat

Examples of shuffled regions

Hum/Mus Hum/Rat

Examples of shuffled regions

Hum/Mus Hum/Rat

Examples of shuffled regions

Hum/Mus Hum/Rat

Examples of shuffled regions

Hum/Mus Hum/Rat

Examples of shuffled regions

Hum/Mus Hum/Rat

More DNA is coming…