Needleman-Wunsch with affine gaps

Gap scores: (g)=-d-(g-1)e where d=2, e=1

Precedence: M, Ix, Iy

€

M(i, j) = max

M(i −1, j −1) + s(x i,y j )

Ix (i −1, j −1) + s(x i,y j )

Iy (i −1, j −1) + s(x i,y j )

⎧

⎨ ⎪

⎩ ⎪

Ix (i, j) = maxM(i −1, j) − d

Ix (i −1, j) − e

⎧ ⎨ ⎩

Iy (i, j) = maxM(i, j −1) − d

Iy (i, j −1) − e

⎧ ⎨ ⎩

PAM 250

A C D

A 2 -2 0

C 12 -5

D 4

Align the sequences: CA and DC

Multiple sequence alignment

Biology 162 Computational Genetics

Todd Vision2 Sep 2004

Preview• How to score a multiple alignment

– Sum of pairs scores– Weighting

• Generalizing pairwise alignment algorithms– Full dynamic programming– Carillo-Lipman

• Practical methods– Progressive– Iterative– Stochastic– Probabilistic

• Some final thoughts

Multiple sequence alignment (MSA)

Mind the gaps

GCGGCCCA TCAGGTAGTT GGTGGGCGGCCCA TCAGGTAGTT GGTGGGCGTTCCA TCAGCTGGTT GGTGGGCGTCCCA TCAGCTAGTT GGTGGGCGGCGCA TTAGCTAGTT GGTGA******** ********** *****

TTGACATG CCGGGG---A AACCGTTGACATG CCGGTG--GT AAGCCTTGACATG -CTAGG---A ACGCGTTGACATG -CTAGGGAAC ACGCGTTGACATC -CTCTG---A ACGCG******** ?????????? *****

Trivial

Difficult

Natural score• Tree score

• Even with a known tree, finding an MSA to optimize the tree score is NP-hard

D

E

A

B

C

SAE

SDESBD

SCD

€

S=Sij

i,j are adjacent

∑

Star-tree scores

• Assume an unresolved phylogeny• Sum-of-pairs ()

• Entropy

• Consistency– Weighs agreement with external

evidence

€

S = s(ai,b j )i< j

∑

€

Scolumn = − qa ln2 qa( )a

∑

Entropy as used in a sequence logo

SP scores: pros and cons

• Pros– Easy, intuitive, work OK

• Cons– Substitution scores based on pairs of residues

– Inconsistent behavior with k• One mismatch matters more when k is large than

when k is small

– Gap penalties undefined for s(-,-)€

log(pab /qaqb ) + log(pac /qaqc ) + log(pbc /qbqc )

log(pabc /qaqbqc )

Natural gap penalties

• Gap costs in multiple alignment should be equal to sum of gap costs in induced pairwise alignments

• Computationally prohibitive to compute for most algorithms

• Instead, quasi-natural gap costs are computed– They are almost always identical

Weighted SP scores

• Scores are not independent due to (unaccounted for) shared ancestry

• To correct this, sum-of-pairs scores from related sequences can be down-weighted

• Variety of weighting schemes exist• Tree-based weighting is simplest

– Assign weights proportional to sum of branch lengths on a phylogenetic tree

– Obviously requires a tree (but we have an approximate tree in some algorithms)

Full dynamic programming

• We have k sequences of length n– Recursion equations are similar to pairwise case– We can use a simple extension of pairwise

scoring– As before, we can guarantee an optimal

alignment

• The problem is we must fill out a k-dimensional hypercube– Time and space grow exponentially in k– At least O(k22knk)– Computationally prohibitive even for a moderate

number of short sequences

Carillo-Lipman algorithm• Reduce volume of hypercube that is searched• Upper bound on score

– Score of optimal MSA is less than or equal to sum of scores of optimal pairwise alignments

• Lower bound on score– Score of optimal MSA must be greater or equal to

score of heuristic MSA

• Projections in each dimension defined by optimal pairwise alignments and induced heuristic alignments

• Optimum path is bounded by projections in all dimensions

Carillo-Lipman algorithm

Carillo-Lipman algorithm

• Only works for SP scoring function• Implemented in MSA software

– Can still only tackle small cases (eg 15 sequences of length 300)

Practical global alignment methods

• Progressive– Uses a guide tree to reduce the problem to

multiple pairwise alignments

• Iterative– Initialized with a fast multiple alignment– Sequences are randomly partitioned and

pairwise aligned until convergence

• Stochastic– Genetic algorithms as an example

• Probabilistic– Hidden Markov models

Progressive Alignment• Fast, but no guarantee of finding the

optimum • Implementations: Feng-Doolittle,

ClustalW, Pileup• Steps

– Compute all k(k-1)/2 pairwise alignments– Use alignment scores to construct guide tree– Perform pairwise alignments beginning at the

leaves of the guide tree and working toward the root

Pairwise score matrix

Sequence 1

Sequence 2

Sequence 3

Sequence 4

Sequence 5

Sequence 1

S12 S13 S14 S15

Sequence 2

S23 S24 S25

Sequence 3

S34 S35

Sequence 4

S45

Sequence 5

Sequence 2

Sequence 3

Sequence 4

Sequence 5

Sequence 1

Guide Tree

2

4

31

New Problem

• How to align a sequence to an alignment?• Or two alignments to each other?

• Feng-Doolittle solution– Choose highest scoring pair of sequences

between the two groups to guide the alignment

• ClustalW solution– Profile alignment: compute generalized sum

of pairs score

Profiles

Profile I

1 2 3 4 ---------- a w w w wpos c w w w w g w w w w t w w w w 1 1 1 1

Profile II

1 2 3 4 ---------- a w w w wpos c w w w w g w w w w t w w w w 1 1 1 1

ClustalW- ad hoc improvements

• Variable substitution matrix• Encourage gaps preferentially in structural loops

– Residue-specific gap penalties– Reduced penalties in hydrophilic regions

• Reduced gap penalties in positions already containing gaps

• Increased gap opening penalties in flanking sequence of gap

Progressive alignment: major weakness

• Errors introduced in the alignment of subgroups are propagated through all subsequent steps

• There is no provision for correcting such errors once they happen

• Local optimum versus global optimum

Iterative alignment

• Again capitalizes on the ease of pairwise alignment between groups of sequences

• Allows for gaps to be removed and positions to be shifted in each iteration

• Some algorithms guarantee convergence given long enough

• Can be several orders of magnitude slower than progressive methods

• Most successful implementation: PRRN

Iterative alignment

CGA-TAGAGACCGA-TACAGAC

ACGATAGACATACG-TACAGATACGATAGACAT

ACG-TACAGATCGA-TAGAGACCGA-TACAGAC

ACGATAGACATACG-TACAGAT-CGATAGAGAC-CGATACAGAC

T-COFFEE• Uses consistency as an objective

function– Evaluates consistency with pairs of residues

found in optimal local alignments and heuristic global alignment

• The consistency function can also incorporate extraneous information (such as structural constraints)

• Among the most successful of approaches when % identity is moderate to good

Dialign

• A multiple local alignment algorithm• Informally, it works by chaining

together ungapped segments from dotplots

• Does not explicitly score gaps at all• May contain unaligned regions

flanked by aligned regions

Stochastic methods

• Genetic algorithms (eg SAGA)– Initalize with population of heuristic alignments– Evaluate ‘fitness’ of individual alignments

• Can employ computationally intensive scoring functions

– Create new generation of alignments• Select parents according to fitness• ‘Cross-over’ attributes of parents• Apply mutation to perturb progeny alignments

– Return to ‘evaluate fitness’ step– Stopping rule

Probabilistic methods

• Hidden Markov Models– Models that generate MSAs– Many parameters to fit

• Probability of each residue in each column• Probability of entering gap states between columns

– Perform poorly on unaligned sequences– But are commonly used in signature databases

• Perform well for finding matches to already aligned sequences

• Efficient algorithms exist for aligning sequences to HMMs

Hidden Markov model

How do you know when you’ve got the right

answer?• Short answer: you don’t.• Structural superposition typically

used to evaluate methodologies• BAliBASE: database of curated

reference alignments

Comparison of test and reference alignments

• Modified SP score– Frequency with which pairs of

residues aligned in test are aligned in reference

• Column score– Frequency with which entire columns

of residues are aligned in both test and reference

Be skeptical!

• MSA is a hard problem– Computationally– Biologically

• There is no ‘one size fits all’ algorithm

• No two algorithms need agree

The future of MSA

• Chances are your new sequence matches something already in the database

• It may soon be a rarity to generate an MSA from scratch– Signature databases currently allow

local alignment of a query to a pre-existing local multiple alignment (eg InterProScan)

Summary

• Challenges in MSA– Even bounded dynamic programming is

impractical – Appropriate scoring is not obvious

• How MSA is achieved in practice– Fastest

• Progressive pairwise alignment

– Slower• Iterative alignment• Stochastic alignment

• Automated MSAs require manual scrutiny

Reading Assignment

• Pertsemlidis A, Fondon JW (2002) Having a BLAST with bioinformatics (and avoiding BLASTphemy), 10 pgs.http://genomebiology.com/2001/2/10/reviews/2002.1