Pairwise Sequence Alignment and Scoring Matrices Xiaole Shirley Liu And Jun Liu Stat 115 Lecture 3.

Pairwise Sequence Alignmentand Scoring Matrices

Xiaole Shirley Liu

And

Jun Liu

Stat 115 Lecture 3

STAT 1152

Mol Bio Quick Facts

• Building block of DNA is deoxyribonucleic acid

• Building block of protein is amino acid– Protein is a peptide, a long peptide

• Only exons can code for functional proteins or RNAs– Introns are spliced out

STAT 1153

Outline

• Motivation and introduction• Dynamic programming

– Global sequence alignment• Needleman-Wunsch• 3 steps: Initialize, fill matrix, trace back• Gap penalties

– Local sequence alignment, Smith-Waterman

• Scoring matrices– PAM– BLOSUM

STAT 1154

Pairwise Sequence Alignment

• Given: two sequences, scoring for match/mismatch/gap

• Goal: find pairing of letters in the two sequences that optimize the total scoreThis is a hard example.

That is another easy example.

This is a --hard---- example.

|| ||||| | | |||||||||


gap

match

mismatch

STAT 1155

Align Biological Sequences

• DNA (4 nt + gap)TTGATCAC

TTTA-CAC

• Protein (20 aa + gap)RKVA--GMAKPNM

RKIAVAAASKPAV

• Sometimes > 4 nt for DNA and > 20 aa for proteins– A word on IUPAC

STAT 1156

IUPAC for DNA

A adenosine

C cytidine

G guanine

T thymidine

U uridine

R G A (purine)

Y T C (pyrimidine)

K G T (keto)

M A C (amino)

S G C (strong)

W A T (weak)

B C G T (not A)

D A G T (not C)

H A C T (not G)

V A C G (not T)

N A C G T (any)

– gap

STAT 1157

IUPAC for ProteinA AlaB Asp or AsnC CysD AspE GluF PheG GlyH HisI IsoK LysL LeuM MetN Asn

P ProG GlnR ArgS SerT ThrU SelV ValW TryY TyrZ Glu or GlnX Any* Translation stop– Gap

STAT 1158

Why Align Two Sequences

• If two sequences are similar, they might share the same ancestor

• If two sequences are similar, they may share the same structure, therefore similar function

• In genome sequencing assembly, if two sequences have overlapping similar regions, they might be connected to represent longer sequenced region.

STAT 1159

Scoring Schemes• Match, mismatch, gap score determines the final

alignmentThis is a --hard---- example.

|| ||||| | | |||||||||


• Match OK, mismatch costly, gap cheap.This is a-- h-ard---- example.

Th--at is anothe-r easy example.

• Match cheap, mismatch cheap, gap costly.This is a hard example.------


STAT 11510

Dot Matrix Approach

• Naïve algorithm

• Dot – match, find diagonal lines

• Can’t afford more complex scoring

• Visual analysis,

hard to find

optimal alignments

STAT 11511

Dynamic Programming

• Essence of dynamic programming:– Store the sub-problem solutions for later use– Best alignment at (i,j) is the best alignment

previous to (i,j) plus aligning these two

• Earliest method, Needleman & Wunsch 1969

• Still the best (sensitive and optimal) algorithm for pair-wise alignment

STAT 11512

Dynamic Programming

• Best alignment at (i,j) is the best alignment previous to (i,j) plus aligning these two

i

j

Best previous alignment

STAT 11513

Dynamic Programming

• Best alignment at (i,j) is the best alignment previous to (i,j) plus aligning these two

• Repeat the process until reaching the two sequences’ ends

i

j

New best alignment = Best previous alignment + align (i,j)

STAT 11514

Dynamic Programming Steps

• Initialize NxM matrix– N, M are the length of the two sequences

• Fill the matrix– Each element record the current best score,

and pointer to the previous best alignment– Always search the previous column and row

for the best previous alignment

• Trace back to obtain optimal alignment

STAT 11515

Fill the Matrix• BestScore[i, j] = max{BS[i-1, j] - ; BS[i, j-1] - ;

BS[i-1,j-1]+match(i,j)}

A A T G C

0 -1 -2 -3 -4 -5

A

G

G

C

1, say.

STAT 11516



A A T G C

0 -1 -2 -3 -4 -5

A -1 1 0 -1 -2 -3

G

G

C

1, say.

STAT 11517



A A T G C

0 -1 -2 -3 -4 -5

A -1 1 0 -1 -2 -3

G -2 0 1 0 0 -1

G

C

1, say.

STAT 11518



A A T G C

0 -1 -2 -3 -4 -5

A -1 1 0 -1 -2 -3

G -2 0 1 0 0 -1

G -3 -1 0 1 1 0

C

1, say.

STAT 11519



A A T G C

0 -1 -2 -3 -4 -5

A -1 1 0 -1 -2 -3

G -2 0 1 0 0 -1

G -3 -1 0 1 1 0

C -4 -2 -1 0 1 2

1, say.

STAT 11520



A A T G C

0 -1 -2 -3 -4 -5

A -1 1 0 -1 -2 -3

G -2 0 1 0 0 -1

G -3 -1 0 1 1 0

C -4 -2 -1 0 1 2

1, say.

Trace back

AATGC-AGGC

AATGCAG-GC

STAT 11521

Alignment Recursion(linear gap penalty)

F(i,j) F(i-1,j)

F(i,j-1)F(i-1,j-1)

( 1, 1) ( , )

( , ) max ( 1, )

( , 1)

i jF i j s x y

F i j F i j

F i j

C A T T G

TCATG

0 -1 -2 -3 -4 -5-1

1-2-3-4-5

E.g., { }( , ) 2 , 1x ys x y I 0-1-2

0 -10332

554

0-12 1

46

STAT 11522

Affine Gap Penalty

• Gap penalty function: – Typically a>b (e.g., a=12; b=2)

• An order O(nm) algorithm.

( ) ( 1)g a g b

F(i,j)

( , ) max{ ( 1, 1) ( , ); ( , ); ( , )}i j h vF i j F i j s x y F i j F i j

( , ) max{ ( , 1) , ( , 1) }h hF i j F i j a F i j b

( , ) max{ ( 1, ) , ( 1, ) }v vF i j F i j a F i j b

Key: keep 3 functions, each recording a directional optimum.

STAT 11523

Gap Penalty

• Gap penalty = g + l e

A A T G C

A 1 1 0 0 0

G 0 1 1 1.4 0.3

G

C

g – gap startl – gap lengthe – gap extend

e.g. g = -0.5 e = -0.1

STAT 11524

Gap Penalty

• Gap penalty = g + l e

A A T G C

A 1 1 0 0 0

G 0 1 1 1.4 0.3

G 0 0.4 1 2 1.4

C 0 0.3 0.4 1 3

g – gap startl – gap lengthe – gap extend

e.g. g = -0.5 e = -0.1

STAT 11525

End Gaps

• Should we penalize gaps at the ends?

ATCCGCATACCGGA

--CCGCATAC----• If two sequences similar length and entire

sequences are supposed to be similar, penalize.

• If two sequences have very different length, do not penalize (most of the time, ignore end gap penalties)

STAT 11526

Global vs. Local Alignment

• Global: Needleman-Wunsch– Find best alignment for the whole 2 sequences– Could have no penalty for mismatches/gaps– Trace back from lower right corner to upper left

corner

• Local: Smith-Waterman– Find high scoring subsequences– E.g. Two proteins only share one similar

functional domain– Can be achieved by modifying global

alignment method

STAT 11527

Local Alignment Modifications

1. Use negative mismatch and gap penalties2. The minimum score for [i, j] is 0

– If S[i,j] < 0, rewrite it to 0, point to self– If previous col/row is all 0, S[i,j] point to self

3. The best score is sought anywhere in the matrix

– Not just last column and last row (should keep a global pointer to the best score)

– Trace back until a cell pointing to itself (not necessary to the beginning of the two sequences)

STAT 11529

Matrix Filling in Smith-Waterman

S(i,j)S(i-3,j)

S(i,j-2)

g(3)

g(2)

S(i-1,j-1)

max {k < j} S(i,j-k) + g(k) S(i,j) = max S(i-1,j-1) + m(i,j) max {l < i} S(i-l,j) + g(l) 0

Example

STAT 11530

Finding suboptimal alignments

STAT 11531

STAT 11532

Smith-Waterman

• Negative mis-match & negative gaps• Scoring matrix >= 0• Trace from maximum

Seq. T j j+1 … … … … nM A T C H E S

Seq. S 0 0 0 0 0 0 0 0i T 0 0 0 5 0 0 0 2i+1 H 0 0 0 0 2 10 2 0… A 0 0 5 0 0 2 9 3… T 0 0 0 10 2 0 9 3… C 0 0 0 2 23 15 7 3… H 0 0 0 0 15 33 25 17… E 0 0 0 0 7 25 39 31m R 0 0 0 0 0 17 31 38

A T C H EA T C H E

STAT 11533

Scoring Matrices

• For DNA, match + 5, mismatch – 4

• For proteins, different amino acid pairs receive different scores– Consideration: size, shape, electric charge, van

der Waals interaction, ability to form salt, hydrophobic, and hydrogen bonds

– Substitution matrices • Often symmetrical

• + / – scores:

functional similarity

STAT 11534

PAM Matrices

• MO Dayhoff 1978

• PAM: percent accepted mutations– Database of 1572 changes in 71 groups of

closely related proteins (> 85% similar)– Construct phylogenetic tree of each group,

tabulate probability of amino acid changes between every pairs of aa

– For statistician: Markov chain transition b/w aa pairs

STAT 11535

PAM Matrix Family

• PAM-N– PAM-0: 1 on diagonal and 0 all the rest– PAM-N: what would happen if N out of 100 aa

mutate – For statisticians: matrix multiplication N times– Bigger N, more diverged substitution matrice

• Final matrix

102/]})(

)([log]

)(

)([{log

1

1210

2

2110

aaFreq

aatoaaPb

aaFreq

aatoaaPb

STAT 11536

PAM250 • 250 substitutions in 100 residues, only 1/5 residue

remain unchanged

STAT 11537

BLOSUM Matrices

• BLOcks amino acid SUbstitution Matrices

• Henikoff and Henikoff, PNAS. 1992, 89:10915-9– Check >500 protein families in the Prosite

database (Bairoch 1991)– Find ~2000 blocks of aligned segments

• BLOCKS database

• Characterize ungapped patterns 3-60 aa long

• Check aligned columns for observed substitutions

STAT 11538

BLOSUM Matrix Entry

• How frequently do aa appear

• How often do we expect to see i, j together

• How often do we actually see them together in all the alignments

• BLOSUM entry

ji ff ,

jiij ffe 2

ijq

)/(log2 2 ijijij eqs

STAT 11539

BLOSUM62 Matrix

STAT 11540

BLOSUM Matrices

• Blocks are grouped before looking at aa substitutions– BLOSUMN: if sequences > N% identical, their

contributions are weighted to sum to 1

• Most widely used: BLOSUM62 and PAM250

STAT 11541

More About Dynamic Programming• Example 1: Suppose I have x0 amount of savings at retirement,

and also receive st amount of social security payment every year. Annual interest rate is t, what is an optimal spending plan if I want to leave zero dollars at the end (say, year 5)?

Year 0 Year 1 Year 2 Year 3 Year 4

x0 x1 x2 x3 x4

s0 s1 s2 s3 s4

u0 u1 u2 u3 u4

1 0

0 0 0

(1 )( )

xx s u

2 1

1 1 1

(1 )( )xx s u

5 4

4 4 4

(1 )( )xx s u

4 3

3 3 3

(1 )( )xx s u

3 2

2 2 2

(1 )( )xx s u

4 4 4u s x

Maximizing total spending? 0 1 4u u u

3 4 3 4

3 3 3 3(1 )( )

u u u s

x s u

3 0u

40 1 4u u u

STAT 11542

Example 2: Secretary Problem

• We get to observe the “qualities” of m secretaries: X1,…, Xm sequentially according to a random order. Our goal is to maximize the probability of finding the “best” candidate with no looking back!

• Heuristic: We start our reasoning backwards. Suppose we have seen X1,…, Xj, should we stop or go on?

STAT 11543

• What if we wait till the last person?

• What if we wait till having two people left?– Strategy: if m-1st person is better than previous

ones, take her; otherwise wait till the last one.

• Get a recursion? If we let go j-1 people, and take the best-person-so-far starting from jth person ...

1

1 11

1m mP Pm m

1

1 11j jP P

m j

Let’s start reasoning ...1

mPm

1 1 1

1 1

j

m j m

Pj maximized at

2.718...

m mj

e

STAT 11544

Final “answer”

• We should reject the first 37% of the candidates and start to look: recruit the first person who is the best among all that have been interviewed.

• The chance of getting the best one is ~37% !

STAT 11545

Summary

• Dynamic programming finds optimal alignment between two sequences– Keep subproblem solution for later use– Needleman & Wunsch, global– Smith & Waterman, local

• Scoring sensitive to gap penalty and substitution matrices used– Substitution matrices capture aa similarity– PAM and BLOSUM matrices

STAT 11546

Question for Thoughts

• Given a string of integers (both positive and negative)– E.g. 3, -1, -5, 2, 4, -3, 6, -4, 2, 5, -8, 3, 1, -7, 6

• Can you read each number only ONCE, and tell from which number to which number you will get the largest sum?– 3, -1, -5, 2, 4, -3, 6, -4, 2, 5, -8, 3, 1, -7, 6, -2– Largest sum = 12

• Hint: dynamic programming

STAT 11547

Acknowledgement

• Russ Altman

• Theodor Hanekamp

• Eric Rouchka

Pairwise Sequence Alignment and Scoring Matrices Xiaole Shirley Liu And Jun Liu Stat 115 Lecture 3.

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