sequence alignments and dynamic programming

Sequence AlignmentsSequence Alignmentsand Dynamic Programmingand Dynamic Programming

BIO/CS 471 – Algorithms for BioinformaticsBIO/CS 471 – Algorithms for Bioinformatics

Sequence Alignments 2

Module II: Sequence AlignmentsModule II: Sequence Alignments Finally a real bioinformatics problem! Problem: SequenceAlignment

• Input: Two or more strings of characters• Output: The optimal alignment of the input strings,

possibly including gaps, and an alignment score. Example

• Input: The calico cat was very sleepy. The cat was sleepy.

• Output: The calico cat was very sleepy.The ------ cat was ---- sleepy.


Biological Sequence AlignmentBiological Sequence Alignment Input: DNA or Amino Acid sequence Output: Optimal alignment Example

• Input: ACTATAGCTATAGCTATAGCGTATCG ACGATTACAGGTCGTGTCG

• Output: ACTATAGCTATAGCTATAGCGTATCG|| || ||*|| | |||*|||

ACGAT---TACAGGT----CGTGTCG Score: +8


Why align sequences?Why align sequences? Why would we want to align two sequences?

• Compare two genes/proteins, e.g. to infer function• Database searches• Protein structure prediction•

Why would we want to align multiple sequences?• Conservation analysis• Molecular evolution•


How do we decide on a scoring function?How do we decide on a scoring function? Objective: Sequences that are homologous or

evolutionarily related, should align with high scores. Less related sequences should score lower.

Question: How do homologous sequences arise?


TranscriptionTranscription

The Central Dogma

DNAtranscriptiontranscription

RNAtranslationtranslation

Proteins


DNA ReplicationDNA Replication Prior to cell division, all the

genetic instructions must be “copied” so that each new cell will have a complete set

DNA polymerase is the enzyme that copies DNA• Reads the old strand in the 3´ to 5´

direction

Over time, genes Over time, genes accumulate accumulate mutationsmutations

Environmental factors• Radiation• Oxidation

Mistakes in replication or repair Deletions, Duplications Insertions Inversions Point mutations


Codon deletion:ACG ATA GCG TAT GTA TAG CCG…• Effect depends on the protein, position, etc.• Almost always deleterious• Sometimes lethal

Frame shift mutation: ACG ATA GCG TAT GTA TAG CCG… ACG ATA GCG ATG TAT AGC CG?…• Almost always lethal

DeletionsDeletions


IndelsIndels Comparing two genes it is generally impossible

to tell if an indel is an insertion in one gene, or a deletion in another, unless ancestry is known:

ACGTCTGATACGCCGTATCGTCTATCTACGTCTGAT---CCGTATCGTCTATCT


The Genetic CodeThe Genetic Code

SubstitutionsSubstitutions are mutations accepted by natural selection.

Synonymous: CGC CGA

Non-synonymous: GAU GAA


Two kinds of Two kinds of homologshomologs Orthologs

• Species A has a particular gene

• Species A diverges into species B and C, each with the same gene

• The two genes accumulate substitutions independently


Orthologs and ParalogsOrthologs and Paralogs Paralogs

• A gene duplication event within a species results in two copies of a gene

• One of the copies is now under less selective constraint

• The copies can accumulate substitutions independently, before and after speciation


Scoring a sequence alignmentScoring a sequence alignment Match score: +1 Mismatch score:+0 Gap penalty: –1ACGTCTGATACGCCGTATAGTCTATCT ||||| ||| || ||||||||----CTGATTCGC---ATCGTCTATCT

Matches: 18 × (+1) Mismatches: 2 × 0 Gaps: 7 × (– 1)

Score = +11Score = +11


Origination and length penaltiesOrigination and length penalties We want to find alignments that are

evolutionarily likely. Which of the following alignments seems more

likely to you?ACGTCTGATACGCCGTATAGTCTATCTACGTCTGAT-------ATAGTCTATCTACGTCTGATACGCCGTATAGTCTATCTAC-T-TGA--CG-CGT-TA-TCTATCT

We can achieve this by penalizing more for a new gap, than for extending an existing gap


Scoring a sequence alignment (2)Scoring a sequence alignment (2) Match/mismatch score: +1/+0 Origination/length penalty: –2/–1ACGTCTGATACGCCGTATAGTCTATCT ||||| ||| || ||||||||----CTGATTCGC---ATCGTCTATCT

Matches: 18 × (+1) Mismatches: 2 × 0 Origination: 2 × (–2) Length: 7 × (–1)

Score = +7Score = +7


Optimal Substructure in AlignmentsOptimal Substructure in Alignments Consider the alignment:ACGTCTGATACGCCGTATAGTCTATCT ||||| ||| || ||||||||----CTGATTCGC---ATCGTCTATCT

Is it true that the alignment in the boxed region must be optimal?


A Greedy StrategyA Greedy Strategy Consider this pair of sequencesGAGCCAGC

Greedy Approach:G or G or -C - G

Leads toGAGC--- Better: GACG---CAGC CACG

GAP = 1

Match = +1

Mismatch = 2


Breaking apart the problemBreaking apart the problem Suppose we are aligning:ACTCGACAGTAG

First position choices:A +1 CTCGA CAGTAG

A -1 CTCG- ACAGTAG

- -1 ACTCGA CAGTAG


A Recursive Approach to AlignmentA Recursive Approach to Alignment Choose the best alignment based on these three

possibilities:align(seq1, seq2) {

if (both sequences empty) {return 0;}if (one string empty) {

return(gapscore * num chars in nonempty seq);else {

score1 = score(firstchar(seq1),firstchar(seq2)) + align(tail(seq1), tail(seq2));score2 = align(tail(seq1), seq2) + gapscore;score3 = align(seq1, tail(seq2) + gapscore;return(min(score1, score2, score3));

}}

}


Time Complexity of RecurseAlignTime Complexity of RecurseAlign What is the recurrence equation for the time

needed by RecurseAlign?3)1(3)( nTnT

3

3

3 3

3 3…

n

3

9

27

3n


RecurseAlign repeats its workRecurseAlign repeats its workA C G T A T C G C G T A T A

G

A

T

G

C

T

C

T

C

G


How can we find an optimal alignment?How can we find an optimal alignment? Finding the alignment is computationally hard:ACGTCTGATACGCCGTATAGTCTATCTCTGAT---TCG—CATCGTC--T-ATCT

C(27,7) gap positions = ~888,000 possibilities It’s possible, as long as we don’t repeat our

work! Dynamic programming: The Needleman &

Wunsch algorithm


What is the optimal alignment?What is the optimal alignment? ACTCGACAGTAG

Match: +1 Mismatch: 0 Gap: –1


Needleman-Wunsch: Step 1Needleman-Wunsch: Step 1 Each sequence along one axis Mismatch penalty multiples in first row/column 0 in [1,1] (or [0,0] for the CS-minded)

A C T C G0 -1 -2 -3 -4 -5

A -1 1C -2A -3G -4T -5A -6G -7


Needleman-Wunsch: Step 2Needleman-Wunsch: Step 2 Vertical/Horiz. move: Score + (simple) gap penalty Diagonal move: Score + match/mismatch score Take the MAX of the three possibilities

A C T C G0 -1 -2 -3 -4 -5

A -1 1C -2A -3G -4T -5A -6G -7


Needleman-Wunsch: Step 2 (cont’d)Needleman-Wunsch: Step 2 (cont’d) Fill out the rest of the table likewise…

a c t c g0 -1 -2 -3 -4 -5

a -1 1 0 -1 -2 -3c -2a -3g -4t -5a -6g -7


Needleman-Wunsch: Step 2 (cont’d)Needleman-Wunsch: Step 2 (cont’d) Fill out the rest of the table likewise…

The optimal alignment score is calculated in the lower-right corner

a c t c g0 -1 -2 -3 -4 -5

a -1 1 0 -1 -2 -3c -2 0 2 1 0 -1a -3 -1 1 2 1 0g -4 -2 0 1 2 2t -5 -3 -1 1 1 2a -6 -4 -2 0 1 1g -7 -5 -3 -1 0 2


a c t c g0 -1 -2 -3 -4 -5

a -1 1 0 -1 -2 -3c -2 0 2 1 0 -1a -3 -1 1 2 1 0g -4 -2 0 1 2 2t -5 -3 -1 1 1 2a -6 -4 -2 0 1 1g -7 -5 -3 -1 0 2

But what But what isis the optimal alignment the optimal alignment To reconstruct the optimal alignment, we must

determine of where the MAX at each step came from…


A path corresponds to an alignmentA path corresponds to an alignment = GAP in top sequence = GAP in left sequence = ALIGN both positions One path from the previous table: Corresponding alignment (start at the end):

AC--TCGACAGTAG

Score = +2


Practice ProblemPractice Problem Find an optimal alignment for these two

sequences:GCGGTTGCGT

Match: +1 Mismatch: 0 Gap: –1

g c g g t t0 -1 -2 -3 -4 -5 -6

g -1c -2g -3t -4


Practice ProblemPractice Problem Find an optimal alignment for these two

sequences:GCGGTTGCGT g c g g t t

0 -1 -2 -3 -4 -5 -6g -1 1 0 -1 -2 -3 -4c -2 0 2 1 0 -1 -2g -3 -1 1 3 2 1 0t -4 -2 0 2 3 3 2

GCGGTTGCG-T- Score = +2


g c g0 -1 -2 -3

g -1 1 0 -1g -2 0 1 1c -3 -1 1 1g -4 -2 0 2

Semi-global alignmentSemi-global alignment Suppose we are aligning:GCGGGCG

Which do you prefer?G-CG -GCGGGCG GGCG

Semi-global alignment allows gaps at the ends for free.


Semi-global alignmentSemi-global alignment

g c g0 0 0 0

g 0 1 0 1g 0 1 1 1c 0 0 2 1g 0 1 1 3

Semi-global alignment allows gaps at the ends for free.

Initialize first row and column to all 0’s Allow free horizontal/vertical moves in last

row and column


Local alignmentLocal alignment Global alignments – score the entire alignment Semi-global alignments – allow unscored gaps

at the beginning or end of either sequence Local alignment – find the best matching

subsequence CGATGAAATGGA

This is achieved by allowing a 4th alternative at each position in the table: zero.


c g a t g0 -1 -2 -3 -4 -5

a -1 0 0 0 0 0a -2 0 0 1 0 0a -3 0 0 1 0 0t -4 0 0 0 2 1g -5 0 1 0 1 3g -6 0 1 0 0 2a -7 0 0 2 1 1

Local alignmentLocal alignment Mismatch = –1 this time

CGATGAAATGGA


Food for thought…Food for thought… What is the asymptotic time complexity of

the Needleman & Wunsch algorithm?• Is this good or bad?

How about the space complexity?• Why might this be a problem?


Saving SpaceSaving Space Note that we can throw away the previous rows

of the table as we fill it in:a c t c g

0 -1 -2 -3 -4 -5a -1 1 0 -1 -2 -3c -2 0 2 1 0 -1a -3 -1 1 2 1 0g -4 -2 0 1 2 2t -5 -3 -1 1 1 2a -6 -4 -2 0 1 1g -7 -5 -3 -1 0 2

This row is based only on this one


Saving Space (2)Saving Space (2) Each row of the table contains the scores for

aligning a prefix of the left-hand sequence with all prefixes of the top sequence:

a c t c g0 -1 -2 -3 -4 -5

a -1 1 0 -1 -2 -3c -2 0 2 1 0 -1a -3 -1 1 2 1 0g -4 -2 0 1 2 2t -5 -3 -1 1 1 2a -6 -4 -2 0 1 1g -7 -5 -3 -1 0 2

Scores for aligning aca with

all prefixes of actcg


Divide and ConquerDivide and Conquer By using a recursive approach, we can use only

two rows of the matrix at a time:• Choose the middle character of the top sequence, i• Find out where i aligns to the bottom sequence

Needs two vectors of scores• Recursively align the sequences before and after the

fixed positions

ACGCTATGCTCATAGCGACGCTCATCG

i


Finding where Finding where ii lines up lines up Find out where i aligns to the bottom sequence

Needs two vectors of scores

Assuming i lines up with a character:alignscore = align(ACGCTAT, prefix(t)) + score(G, char from t)

+ align(CTCATAG, suffix(t)) Which character is best?

• Can quickly find out the score for aligning ACGCTAT with every prefix of t.

s: ACGCTATGCTCATAGt: CGACGCTCATCG

i


Finding where Finding where ii lines up lines up But, i may also line up with a gap

Assuming i lines up with a gap:

alignscore = align(ACGCTAT, prefix(t)) + gapscore+ align(CTCATAG, suffix(t))


i


Recursive CallRecursive Call Fix the best position for I Call align recursively for the prefixes and

suffixes:


i


ComplexityComplexity Let len(s) = m and len(t) = n Space: 2m Time:

• Each call to build similarity vector = m´n´

• First call + recursive call:


i

j

mnjnmmjmn

jnmTjmTmnmnnmT

2)(

,2

,222

,


General Gap PenaltiesGeneral Gap Penalties Suppose we are no longer using simple gap

penalties:• Origination = −2• Length = −1

Consider the last position of the alignment for ACGTA with ACG

We can’t determine the score for

unless we know the previous positions!

G-

-G

or


Scoring BlocksScoring Blocks Now we must score a block at a time

A block is a pair of characters, or a maximal group of gaps paired with characters

To score a position, we need to either start a new block or add it to a previous block

A A C --- A TATCCG A C T AC

A C T ACC T ------ C G C --


The AlgorithmThe Algorithm Three tables

• a – scores for alignments ending in char-char blocks• b – scores for alignments ending in gaps in the top

sequence (s)• c – scores for alignments ending in gaps in the left

sequence (t) Scores no longer depend on only three

positions, because we can put any number of gaps into the last block


The RecurrencesThe Recurrences

1,11,11,1

max,,jicjibjia

jipjia

jkkwkjicjkkwkjia

jib1for ,,1for ,,

max,

ikkwjkibikkwjkia

jic1for ,,1for ,,

max,


The Optimal AlignmentThe Optimal Alignment The optimal alignment is found by looking at

the maximal value in the lower right of all three arrays

The algorithm runs in O(n3) time• Uses O(n2) space

sequence alignments and dynamic programming

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