Multiple Sequence Multiple Sequence AlignmentsAlignments

Multiple alignment

• Pairwise alignment– Infer biological relationships from string

similarity

• Multiple alignment– Infer string similarity from biological

relationships

Biological Motivations

• One of the most essential tools in molecular biology – Finding highly conserved sub-regions or embedded

patterns of a set of biological sequences– Production of consensus sequence– Estimation of evolutionary distance between sequences– Prediction of protein secondary/tertiary structure– To find conserved regions

• Local multiple alignment reveals conserved regions• Conserved regions usually are key functional regions, prime targets

for drug developments

• Practically useful methods only since D. Sankoff (1987) based on phylogenetics– Before 1987 they were constructed by hand – The basic problem: no dynamic programming approach

can be used

Alignment between globins (human beta globin, horse beta globin, human alpha globin, horse alpha globin, cyanohaemoglobin,

whale myoglobin, leghaemoglobin) produced by Clustal. Boxes mark the seven alpha helices composing each globin.

.

Definition

• Given strings x1, x2 … xk a multiple (global) alignment is a matrix of k rows and A columns where each row represents a sequence and a column contains a symbol from each sequence or gaps symbols (at least one non gap)

Multiple Sequence Alignment

Matrix 3 rows 8 colums

Family representations

• Outcome of multiple alignment• Three kinds

– Profile representation• Frequencies of symbols in each column• Weight vector• Alignment to a profile

– Consensus sequence representation• Steiner string

– Signature representation• PROSITE, BLOCKS databases• Regular expression

Scoring Function

• Ideally:– Find alignment that maximizes probability that

sequences evolved from common ancestorx

yzw

v

?

Multiple Sequence Alignment

• Mult-Seq-Align allows to detect similarities which cannot be detected with Pairwise-Seq-Align methods.

• Detection of family characteristics.

Three questions:

1. Scoring

2. Computation of Mult-Seq-Align.

3. Family representation.

A fragment of multiple alignment of 7 kinases.

ClustalW program from SRS server.

Scoring: SP (sum of pairs)

SP – the sum of pairwise scores of all pairs of symbols in the column.

SP3(-,A,A) = (-,A)+(-,A)+(A,A)

SP Total Score = sum over all columns

(-,-) = 0

Induced pairwise alignment

Induced pairwise alignment or projection of a multiple alignment.

a(S1, S2 )

a(S2, S3)

a(S1, S3)

(-,-) = 0

SP Total Score = i<j score[ a(Si, Sj ) ]

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)

Optimal solution

• Multidimensional Dynamic Programming

• Generalization of pair-wise alignment

• For simplicity, assume k sequences of length n

• The dynamic programming array is k-dimensional hyperlattice of length n+1 (including initial gaps)

• The entry F(i1, …, ik) represents score of optimal alignment for s1[1..i1], … sk[1..ik]

• Initialize values on the faces of the hyperlattice

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

Sum Of Pairs

• The sum-of-pairs (SP) score of a multiple alignment A is the sum of the scores of all induced pairwise alignments

S(A) = i<j S(Aij)

Aij is the induced alignment of xi, xj

Dyn.Prog. Solution

s (NVNSNA )

s (NVN-NA )

V

S

A

s (N-NSNA)

s (N-NSN- )

s (NVN-N- )s (

NNN ) +δ (

−SA)

s (NVNSNA )=max {

s(NNN )+δ(

VSA )

s (NVN-N- )+δ(

−SA )

s(N-NSN- )+δ(

V−A )

s (N-N-NA )+δ(

VS−)

s (N-NSNA )+δ(

V−−)

s (NVN-NA )+δ(

−S−

)s (

NVNSN- )+δ (

−

−A)

k=3 2k –1=7

• 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 Dynamic Programming

• Space complexity: O(nk) for k sequences each n long.

• Computing at a cell: O(2k). cost of computing δ.

• Time complexity: O(2knk). cost of computing δ.

• Finding the optimal solution is exponential in k

• Proven to be NP-complete for a number of cost functions

Complexity

• Faster Dynamic Programming (SP)– Carrillo and Lipman 88 (MSA)– Pruning of hyperlattice in DP– Practical for about 6 sequences of length about 200.

• Star alignment (SP)• Progressive methods

– CLUSTALW– PILEUP

• Iterative algorithms• Sampling (Gibbs) based methods• Hidden Markov Model (HMM) based methods• Expectation Maximization Algorithm

Algorithms

• Find pairwise alignment• Trial multiple alignment produced by a tree, cost = d

• This provides a limit to the volume within which optimal alignments are found

• Specifics– Sequences x1,..,xr.

– Alignment A, cost = c(A)

– Optimal alignment A*

– Aij = induced alignment on xi,..,xj on account of A

– D(xi,xj) = cost of optimal pairwise alignment of xi,xj <= c(Aij )

Idea behind MSA algorithm

Progressive Alignment

• Multiple Alignment is NP-complete

• Most used heuristic: Progressive Alignment

Algorithm:

– Align two of the sequences xi, xj

– Fix that alignment

– Align a third sequence xk to the alignment xi,xj

– Repeat until all sequences are aligned

Running Time: O( N L2 )

Star Alignments

• Heuristic method for multiple sequence alignments

• Select a sequence c as the center of the star• For each sequence x1, …, xk such that index

i c, perform a Needleman-Wunsch global alignment

• Aggregate alignments with the principle “once a gap, always a gap.”

Star Alignments Example

s2

s1s3

s4

x1: MPEx2: MKEx3: MSKEx4: SKE

MPE

| |

MKE

MSKE

-||

MKE

SKE

||

MKE MPEMKE

-MPE-MKEMSKE

-MPE-MKEMSKE-SKE

Choosing a center

• Try them all and pick the one with the best score

• Calculate all O(k2) alignments, and pick the sequence xc that minimizes

D(xc,xi)

• D(xc,xi) = c(Aci), A is the multiple alignmenti > c

Analysis

• Assuming all sequences have length n• O(k2n2) to calculate center• Step i takes O((i.n).n) time

– two strings of length n and i.n

• O(k2n2) overall cost• Produces multiple sequence alignments

whose SP values are at most twice that of the optimal solutions, provided triangle inequality holds.

• Profile– Apply dynamic programming

– Score depends on the profile

• Consensus string– Apply dynamic programming

• Signature representations– Align to regular expressions / CFG/ …

Aligning to family representations

Progressive alignment(CLUSTALW)

• CLUSTALW is the 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• sequence to sequence• sequence to profile• profile to profile

+ a large number of heuristics

S1

S2

S3

S4

S1 S2 S3 S4

S1 4 9 4

S2 4 7S3 4

S4

All PairwiseAlignments

S1

S3

S2

S4

Distance

Cluster Analysis

Similarity Matrix Dendrogram

Multiple Alignment Step:1. Aligning S

1 and S

32. Aligning S

2 and S

43. Aligning (S

1,S

3) with (S

2,S

4).

From Higgins(1991) and Thompson(1994).

Problems with Progressive Alignments

• Depends on pairwise alignments

• If sequences are very distantly related, much higher likelihood of errors

• Care must be made in choosing scoring matrices and penalties

Progressive Alignment: CLUSTALW

CLUSTALW: most popular multiple protein alignment

Algorithm:

• Find all dij: alignment dist (xi, xj)

• Construct a tree

(Neighbor-joining hierarchical clustering)

• Align nodes in order of decreasing similarity

+ a large number of heuristics

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):

• Align most similar xi, xj

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

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

• Repeat 4 until convergence

Note: Guaranteed to converge

Other methods

• MEME (Expectation Maximization)

• GibbsDNA (Gibbs Sampling)

• HMMER (Hidden Markov Model)

• Random projections

• CONSENUS (greedy multiple alignment)

• WINNOWER (Clique finding in graphs)