Rapid Global Alignments

How to align genomic sequences in (more or less) linear time

Methods to CHAIN Local Alignments

Sparse Dynamic ProgrammingO(N log N)

The Problem: Find a Chain of Local Alignments

(x,y) (x’,y’)

requires

x < x’y < y’

Each local alignment has a weight

FIND the chain with highest total weight

Sparse DP for rectangle chaining

• 1,…, N: rectangles

• (hj, lj): y-coordinates of rectangle j

• w(j): weight of rectangle j

• V(j): optimal score of chain ending in j

• L: list of triplets (lj, V(j), j)

L is sorted by lj L is implemented as a balanced binary tree

y

h

l

Sparse DP for rectangle chaining

Main idea:

• Sweep through x-coordinates

• To the right of b, anything chainable to a is chainable to b

• Therefore, if V(b) > V(a), rectangle a is “useless” – remove it

• In L, keep rectangles j sorted with increasing lj-coordinates sorted with increasing V(j)

V(b)V(a)

Sparse DP for rectangle chaining

Go through rectangle x-coordinates, from left to right:

1. When on the leftmost end of rectangle i, compute V(i)

a. j: rectangle in L, with largest lj < hi

b. V(i) = w(i) + V(j)

2. When on the rightmost end of i, possibly store V(i) in L:

a. j: rectangle in L, with largest lj lib. If V(i) > V(j):

i. INSERT (li, V(i), i) in L

ii. REMOVE all (lk, V(k), k) with V(k) V(i) &

lk li

i

j

Example

x

y

1: 5

3: 3

2: 6

4: 45: 2

2

56

91011

1214

1516

Time Analysis

1. Sorting the x-coords takes O(N log N)

2. Going through x-coords: N steps

3. Each of N steps requires O(log N) time:

• Searching L takes log N• Inserting to L takes log N• All deletions are consecutive, so log N per deletion• Each element is deleted at most once: N log N for all deletions

• Recall that INSERT, DELETE, SUCCESSOR, take O(log N) time in a balanced binary search tree

Putting it All Together:Fast Global Alignment Algorithms

1. FIND local alignments

2. CHAIN local alignments

FIND CHAIN

GLASS: k-mers hierarchical DP

MumMer: Suffix Tree sparse DP

Avid: Suffix Tree hierarchical DP

LAGAN CHAOS sparse DP

LAGAN: Pairwise Alignment

1. FIND local alignments

2. CHAIN local alignments

3. DP restricted around chain

LAGAN

1. Find local alignments

2. Chain -O(NlogN) L.I.S.

3. Restricted DP

LAGAN: recursive call

• What if a box is too large? Recursive application of LAGAN,

more sensitive word search

A trick to save on memory

“necks” have tiny tracebacks

…only store tracebacks

Multiple Sequence Multiple Sequence AlignmentsAlignments

Overview

• Definition

• Scoring Schemes

• Algorithms

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

• Ideally: Find alignment that maximizes probability that sequences evolved

from common ancestor, according to some phylogenetic model

• More on phylogenetic models later

x

yz

w

v

?

Scoring Function

• A comprehensive model would have too many parameters, too inefficient to optimize

• Possible simplifications

Ignore phylogenetic tree

Statistically independent columns:

S(m) = G(m) + i S(mi)

m: alignment matrixG: function penalizing gaps

Scoring Function: Sum Of Pairs

Definition: Induced pairwise alignment

A 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

Sum Of Pairs (cont’d)

• The sum-of-pairs score of an alignment is the sum of the scores of all induced pairwise alignments

S(m) = k<l s(mk, ml)

s(mk, ml): score of induced alignment (k,l)

Sum Of Pairs (cont’d)

• Heuristic way to incorporate evolution tree:

Human

Mouse

Chicken• Weighted SOP:

S(m) = k<l wkl s(mk, ml)

wkl: weight decreasing with distance

Duck

Consensus

-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC---TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGACCAG-CTATCAC--GACCGC----TCGATTTGCTCGAC

CAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC

• Find optimal consensus string m* to maximize

S(m) = i s(m*, mi)

s(mk, ml): score of pairwise alignment (k,l)

Multiple Sequence Alignments

Algorithms

1. Multidimensional Dynamic Programming

Generalization of Needleman-Wunsh:

S(m) = i S(mi)

(sum of column scores)

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

• 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) }

1. Multidimensional Dynamic Programming

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)

1. Multidimensional Dynamic Programming

2. Progressive Alignment

• Multiple Alignment is NP-complete

• Most used heuristic: Progressive Alignment

Algorithm:1. Align two of the sequences xi, xj

2. Fix that alignment

3. Align a third sequence xk to the alignment xi,xj

4. Repeat until all sequences are aligned

Running Time: O( N L2 )

2. Progressive Alignment

• When evolutionary tree is known:

Align closest first, in the order of the tree

Example:Order of alignments: 1. (x,y)

2. (z,w)3. (xy, zw)

x

w

y

z

CLUSTALW: progressive alignment

CLUSTALW: most popular multiple protein alignment

Algorithm:

1. Find all dij: alignment dist (xi, xj)

2. Construct a tree

(Neighbor-joining hierarchical clustering)

3. Align nodes in order of decreasing similarity

+ a large number of heuristics

CLUSTALW & the CINEMA viewer

MLAGAN: progressive alignment of DNA

Given N sequences, phylogenetic tree

Align pairwise, in order of the tree (LAGAN)

Human

Baboon

Mouse

Rat

MLAGAN: main steps

Given a collection of sequences, and a phylogenetic tree

1. Find local alignments for every pair of sequences x, y

2. Find anchors between every pair of sequences, similar to LAGAN anchoring

3. Progressive alignment• Multi-Anchoring based on reconciling the pairwise anchors• LAGAN-style limited-area DP

4. Optional refinement steps

MLAGAN: multi-anchoring

XZ

YZ

X/Y

Z

To anchor the (X/Y), and (Z) alignments:

Heuristics to improve multiple alignments

• Iterative refinement schemes

• A*-based search

• Consistency

• Simulated Annealing

• …

Iterative Refinement

One problem of progressive alignment:• Initial alignments are “frozen” even when new evidence comes

Example:

x: GAAGTTy: GAC-TT

z: GAACTGw: GTACTG

Frozen!

Now clear correct y = GA-CTT

Iterative Refinement

Algorithm (Barton-Stenberg):

1. Align most similar xi, xj

2. Align xk most similar to (xixj)3. Repeat 2 until (x1…xN) are aligned

4. For j = 1 to N,Remove xj, and realign to x1…xj-1xj+1…xN

5. Repeat 4 until convergence

Note: Guaranteed to converge

Iterative Refinement

For each sequence y1. Remove y2. Realign y

(while rest fixed)x

y

z

x,z fixed projection

allow y to vary

Iterative Refinement

Example: align (x,y), (z,w), (xy, zw):

x: GAAGTTAy: GAC-TTAz: GAACTGAw: GTACTGA

After realigning y:

x: GAAGTTAy: G-ACTTA + 3 matchesz: GAACTGAw: GTACTGA

Iterative Refinement

Example not handled well:

x: GAAGTTAy1: GAC-TTAy2: GAC-TTAy3: GAC-TTA

z: GAACTGAw: GTACTGA

Realigning any single yi changes nothing

Restricted MDP

Here is another way to improve a multiple alignment:

1. Construct progressive multiple alignment m

2. Run MDP, restricted to radius R from m

Running Time: O(2N RN-1 L)

Restricted MDP

Run MDP, restricted to radius R from m

x

y

z

Running Time: O(2N RN-1 L)

Restricted MDP

x: GAAGTTAy1: GAC-TTAy2: GAC-TTAy3: GAC-TTA

z: GAACTGAw: GTACTGA

• Within radius 1 of the optimal

Restricted MDP will fix it.

Optional refinement steps in MLAGAN

• Limited-area iterative refinement

• Radius-r 3-sequence refinement on each node of the tree