CS 5263 Bioinformatics

Lecture 6: Sequence Alignment Statistics

Review of last lecture

• How to map gaps more accurately?

GACGCCGAACG||||| |||GACGC---ACG

GACGCCGAACG|||| | | ||GACG-C-A-CG

Score = 8 x m – 3 x d Score = 8 x m – 3 x d

Gaps usually occur in bunches

- During evolution, chunks of DNA may be lost or inserted entirely

- Aligning genomic sequences vs. cDNAs: cDNAs are spliced versions of the genomic seqs

Model gaps more accurately

• Previous model:– Gap of length n incurs penalty nd

• General: – Convex function– E.g. (n) = c * sqrt (n)

F(i-1, j-1) + s(xi, yj)

F(i, j) = max maxk=0…i-1F(k,j) – (i-k)

maxk=0…j-1F(i,k) – (j-k)

– Running Time: O((M+N)MN) (cubic)– Space: O(NM)

n

n

Compromise: affine gaps

(n) = d + (n – 1)e | |gap gapopen extension

de

(n)

Match: 2

Gap open: -5

Gap extension: -1

GACGCCGAACG||||| |||GACGC---ACG

GACGCCGAACG|||| | | ||GACG-C-A-CG

8x2-5-2 = 9 8x2-3x5 = 1

• We want to find the optimal alignment with affine gap penalty in

• O(MN) time

• O(MN) or better O(M+N) memory

Dynamic programming

• Consider three sub-problems when aligning x1..xi and y1..yj

– F(i,j): best alignment (score) of x1..xi & y1..yj if xi aligns to yj

– Ix(i,j): best alignment of x1..xi & y1..yj if yj aligns to gap– Iy(i,j): best alignment of x1..xi & y1..yj if xi aligns to gap

xi

yj

xi

yj

xi

yj

F(i, j) Ix(i, j) Iy(i, j)

F

Ix

Iy

(xi,yj) /

(xi,yj) /

(xi,yj) /

(xi,-) / d

(xi,-) / e

(-, yj) / d

(-, yj) / eInput Output

Start state

Current state Input Output Next state

F (xi,yj) F

F (-,yj) d Ix

F (xi,-) d Iy

Ix (-,yj) e Ix

… … … …

AAC

ACT

F-F-F-F

AAC

|||

ACT

F-Iy-F-F-Ix

AAC-

||

-ACT

F-F-Iy-F-Ix

AAC-

| |

A-CT

F

Ix

Iy

(xi,yj) /

(xi,yj) /

(xi,yj) /

(xi,-) / d

(xi,-) / e

(-, yj) / d

(-, yj) / e

startstate

Given a pair of sequences, an alignment (not necessarily optimal) corresponds to a state path in the FSM.

Optimal alignment: a state path to read the two sequences such that the total output score is the highest

F

Ix

Iy

(xi,yj) /

(xi,yj) /

(xi,yj) /

(xi,-) /d

(xi,-)/e

(-, yj) /d

(-, yj)/e

F(i-1, j-1) + (xi, yj)

F(i, j) = max Ix(i-1, j-1) + (xi, yj)

Iy(i-1, j-1) + (xi, yj)

xi

yj

F

Ix

Iy

(xi,yj) /

(xi,yj) /

(xi,yj) /

(xi,-) /d

(xi,-)/e

(-, yj) /d

(-, yj)/e

F(i, j-1) + d

Ix(i, j) = max

Ix(i, j-1) + e

xi

yj

Ix(i, j)

F

Ix

Iy

(xi,yj) /

(xi,yj) /

(xi,yj) /

(xi,-) /d

(xi,-)/e

(-, yj) /d

(-, yj)/e

F(i-1, j) + d

Iy(i, j) = max

Iy(i-1, j) + e

xi

yj

Iy(i, j)

F(i – 1, j – 1)F(i, j) = (xi, yj) + max Ix(i – 1, j – 1)

Iy(i – 1, j – 1)

F(i, j – 1) + d Ix(i, j) = max

Ix(i, j – 1) + e

F(i – 1, j) + d Iy(i, j) = max

Iy(i – 1, j) + e

Continuing alignment

Closing gaps in x

Closing gaps in y

Opening a gap in x

Gap extension in x

Opening a gap in y

Gap extension in y

0 - - -

-

-

-

-

- - - -

-5

-6

-7

-8

- -5 -6 -7

-

-

-

-

F: aligned on both Iy: Insertion on y

F(i, j)

F(i-1, j-1)

Ix(i-1, j-1)

Iy(i-1, j-1)

Ix(i,j)Ix(i,j-1)

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

Iy(i-1,j)

F(i-1,j)

G C C

G

C

A

C

G

C

A

C

G

C

A

C

G C C

G C C

x =

y =

x =

y =

x =

y =

Ix: Insertion on x

(xi, yj)

d

e

de

m = 2s = -2d = -5e = -1

0 - - -

- 2

-

-

-

- - - -

-5

-6

-7

-8

- -5 -6 -7

-

-

-

-

F Iy

Ix

G C C

G

C

A

C

G

C

A

C

G

C

A

C

G C C

G C C

x =

y =

x =

y =

x =

y =

F(i, j)

F(i-1, j-1)

Ix(i-1, j-1)

Iy(i-1, j-1)

(xi, yj) = 2

m = 2s = -2d = -5e = -1

0 - - -

- 2 -7

-

-

-

- - - -

-5

-6

-7

-8

- -5 -6 -7

-

-

-

-

F Iy

Ix

G C C

G

C

A

C

G

C

A

C

G

C

A

C

G C C

G C C

x =

y =

x =

y =

x =

y =

F(i, j)

F(i-1, j-1)

Ix(i-1, j-1)

Iy(i-1, j-1)

(xi, yj) = -2

m = 2s = -2d = -5e = -1

0 - - -

- 2 -7 -8

-

-

-

- - -

-5

-6

-7

-8

-5 -6 -7

- - -3 -4

-

-

-

F Iy

Ix

G C C

G

C

A

C

G

C

A

C

G

C

A

C

G C C

G C C

x =

y =

x =

y =

x =

y =

Ix(i,j)Ix(i,j-1)

F(i,j-1)d = -5

e = -1

m = 2s = -2d = -5e = -1

0 - - -

- 2 -7 -8

- -7

-

-

- - -

-5 - - -

-6

-7

-8

-5 -6 -7

- - -3 -4

-

-

-

F Iy

Ix

G C C

G

C

A

C

G

C

A

C

G

C

A

C

G C C

G C C

x =

y =

x =

y =

x =

y =

F(i, j)

F(i-1, j-1)

Ix(i-1, j-1)

Iy(i-1, j-1)

(xi, yj) = -2

m = 2s = -2d = -5e = -1

0 - - -

- 2 -7 -8

- -7 4 -1

-

-

- - -

-5 - - -

-6

-7

-8

-5 -6 -7

- - -3 -4

-

-

-

F Iy

Ix

G C C

G

C

A

C

G

C

A

C

G

C

A

C

G C C

G C C

x =

y =

x =

y =

x =

y =

F(i, j)

F(i-1, j-1)

Ix(i-1, j-1)

Iy(i-1, j-1)

(xi, yj) = 2

m = 2s = -2d = -5e = -1

0 - - -

- 2 -7 -8

- -7 4 -1

-

-

- - -

-5 - - -

-6

-7

-8

-5 -6 -7

- - -3 -4

- - -12 -1

-

-

F Iy

Ix

G C C

G

C

A

C

G

C

A

C

G

C

A

C

G C C

G C C

x =

y =

x =

y =

x =

y =

Ix(i,j)Ix(i,j-1)

F(i,j-1)d = -5

e = -1

m = 2s = -2d = -5e = -1

0 - - -

- 2 -7 -8

- -7 4 -1

-

-

- - -

-5 - - -

-6 -3

-7

-8

-5 -6 -7

- - -3 -4

- - -12 -1

-

-

F Iy

Ix

G C C

G

C

A

C

G

C

A

C

G

C

A

C

G C C

G C C

x =

y =

x =

y =

x =

y =

Iy(i,j)

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

d=-5e=-1

m = 2s = -2d = -5e = -1

0 - - -

- 2 -7 -8

- -7 4 -1

- -8 -5 2

- -9 -6 1

- - -

-5 - - -

-6 -3 -12 -13

-7 -8 -1 -6

-8 -13 -2 -3

-5 -6 -7

- - -3 -4

- - -12 -1

- - -13 -10

- - -14 -11

F Iy

Ix

G C C

G

C

A

C

G

C

A

C

G

C

A

C

G C C

G C C

x =

y =

x =

y =

x =

y =

F(i, j)

F(i-1, j-1)

Ix(i-1, j-1)

Iy(i-1, j-1)

Ix(i,j)Ix(i,j-1)

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

Iy(i-1,j)

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

d

e

de

m = 2s = -2d = -5e = -1

0 - - -

- 2 -7 -8

- -7 4 -1

- -8 -5 2

- -9 -6 1

- - -

-5 - - -

-6 -3 -12 -13

-7 -8 -1 -6

-8 -13 -2 -3

-5 -6 -7

- - -3 -4

- - -12 -1

- - -13 -10

- - -14 -11

F Iy

Ix

G C C

G

C

A

C

G

C

A

C

G

C

A

C

G C C

G C C

GCAC

|| |

GC-C

x =

y =

x =

y =

x =

y =

x

y

G

C

A

C

G C C

x =

y =

m = 2s = -2d = -5e = -1

Today: statistics of alignment

Where does (xi, yj) come from?

Are two aligned sequences actually related?

Probabilistic model of alignments

• We’ll first focus on protein alignments without gaps

• Given an alignment, we can consider two possible models– R: the sequences are related by evolution– U: the sequences are unrelated

• How can we distinguish these two models?• How is this view related to amino-acid

substitution matrix?

Model for unrelated sequences

• Assume each position of the alignment is independently sampled from some distribution of amino acids

• ps: probability of amino acid s in the sequences

• Probability of seeing an amino acid s aligned to an amino acid t by chance is– Pr(s, t | U) = ps * pt

• Probability of seeing an ungapped alignment between x = x1…xn and y = y1…yn randomly is

i

Model for related sequences

• Assume each pair of aligned amino acids evolved from a common ancestor

• Let qst be the probability that amino acid s in one sequence is related to t in another sequence

• The probability of an alignment of x and y is give by

Probabilistic model of Alignments

• How can we decide which model (U or R) is more likely?

• One principled way is to consider the relative likelihood of the two models (the odd ratios)– A higher ratio means that R is more likely than U

Log odds ratio

• Taking logarithm, we get

• Recall that the score of an alignment is given by

• Therefore, if we define

• We are actually defining the alignment score as the log odds ratio between the two models R and U

How to get the probabilities?

• ps can be counted from the available protein sequences

• But how do we get qst? (the probability that s and t have a common ancestor)

• Counted from trusted alignments of related sequences

Protein Substitution Matrices

• Two popular sets of matrices for protein sequences– PAM matrices [Dayhoff et al, 1978]

• Better for aligning closely related sequences

– BLOSUM matrices [Henikoff & Henikoff, 1992]• For both closely or remotely related sequences

BLOSUM-N matrices

• Constructed from a database called BLOCKS• Contain many closely related sequences

– Conserved amino acids may be over-counted

• N = 62: the probabilities qst were computed using trusted alignments with no more than 62% identity– identity: % of matched columns

• Using this matrix, the Smith-Waterman algorithm is most effective in detecting real alignments with a similar identity level (i.e. ~62%)

Positive for chemically similar substitution

Common amino acids get low weights

Rare amino acids get high weights

: Scaling factor to convert score to integer.Important: when you are told that ascoring matrix is in half-bits => = ½ ln2

BLOSUM-N matrices

• If you want to detect homologous genes with high identity, you may want a BLOSUM matrix with higher N. say BLOSUM75

• On the other hand, if you want to detect remote homology, you may want to use lower N, say BLOSUM50

• BLOSUM-62: good for most purposes

45 62 90

Weak homology Strong homology

For DNAs

• No database of trusted alignments to start with

• Specify the percentage identity you would like to detect

• You can then get the substitution matrix by some calculation

For example

• Suppose pA = pC = pT = pG = 0.25

• We want 88% identity

• qAA = qCC = qTT = qGG = 0.22, the rest = 0.12/12 = 0.01

(A, A) = (C, C) = (G, G) = (T, T)

= log (0.22 / (0.25*0.25)) = 1.26(s, t) = log (0.01 / (0.25*0.25)) = -1.83 for

s ≠ t.

Substitution matrix

A C G T

A 1.26 -1.83 -1.83 -1.83

C -1.83 1.26 -1.83 -1.83

G -1.83 -1.83 1.26 -1.83

T -1.83 -1.83 -1.83 1.26

• Scale won’t change the alignment• Multiply by 4 and then round off to get integers

A C G T

A 5 -7 -7 -7

C -7 5 -7 -7

G -7 -7 5 -7

T -7 -7 -7 5

Arbitrary substitution matrix

• Say you have a substitution matrix provided by someone

• It’s important to know what you are actually looking for when you use the matrix

• What’s the difference? • Which one should I use for my sequences?

A C G T

A 1 -2 -2 -2

C -2 1 -2 -2

G -2 -2 1 -2

T -2 -2 -2 1

A C G T

A 5 -4 -4 -4

C -4 5 -4 -4

G -4 -4 5 -4

T -4 -4 -4 5

NCBI-BLAST WU-BLAST

• We had

• Scale it, so that

• Reorganize:

• Since all probabilities must sum to 1,

• We have

• Suppose again ps = 0.25 for any s

• We know (s, t) from the substitution matrix

• We can solve the equation for λ

• Plug λ into to get qst

A C G T

A 1 -2 -2 -2

C -2 1 -2 -2

G -2 -2 1 -2

T -2 -2 -2 1

A C G T

A 5 -4 -4 -4

C -4 5 -4 -4

G -4 -4 5 -4

T -4 -4 -4 5

= 1.33

qst = 0.24 for s = t, and 0.004 for s ≠ t

Translate: 95% identity

= 0.19

qst = 0.16 for s = t, and 0.03 for s ≠ t

Translate: 65% identity

NCBI-BLAST WU-BLAST

Details for solving

Known: (s,t) = 1 for s=t, and (s,t) = -2 for s t.Since

and s,t qst = 1, we have

12 * ¼ * ¼ * e-2 + 4 * ¼ * ¼ * e = 1 Let e = x, we have¾ x-2 + ¼ x = 1. Hence,x3 – 4x2 + 3 = 0;• X has three solutions: 3.8, 1, -0.8• Only the first leads to a positive = ln (3.8) = 1.33

A C G T

A 1 -2 -2 -2

C -2 1 -2 -2

G -2 -2 1 -2

T -2 -2 -2 1

Today: statistics of alignment

Where does (xi, yj) come from?

Are two aligned sequences actually related?

Statistics of Alignment Scores

• Q: How do we assess whether an alignment provides good evidence for homology (i.e., the two sequences are evolutionarily related)?– Is a score 82 good? What about 180?

• A: determine how likely it is that such an alignment score would result from chance

P-value of alignment

• p-value– The probability that the alignment score can

be obtained from aligning random sequences– Small p-value means the score is unlikely to

happen by chance

• The most common thresholds are 0.01 and 0.05– Also depend on purpose of comparison and

cost of misclaim

Statistics of global seq alignment

• Theory only applies to local alignment• For global alignment, your best bet is to do Monte-Carlo

simulation– What’s the chance you can get a score as high as the real

alignment by aligning two random sequences?

• Procedure– Given sequence X, Y– Compute a global alignment (score = S)– Randomly shuffle sequence X (or Y) N times, obtain

X1, X2, …, XN

– Align each Xi with Y, (score = Ri)– P-value: the fraction of Ri >= S

Human HEXA

Fly HEXO1

Score = -74

-95 -90 -85 -80 -75 -70 -65 -60 -55 -500

5

10

15

20

25

30

35

40

45

Alignment Score

Num

ber

of S

eque

nces

-74

Distribution of the alignment scores between fly HEXO1 and 200 randomly shuffled human HEXA sequences

There are 88 random sequences with alignment score >= -74. So: p-value = 88 / 200 = 0.44 => alignment is not significant

……………………………………………………

Mouse HEXA

Human HEXA

Score = 732

-200 -100 0 100 200 300 400 500 600 700 8000

5

10

15

20

25

30

35

40

45

Alignment Score

Num

ber

of S

eque

nces

732

Distribution of the alignment scores between mouse HEXA and 200 randomly shuffled human HEXA sequences

-230 -220 -210 -200 -190 -180 -170 -160 -1500

5

10

15

20

25

30

35

40

45

Alignment Score

Num

ber

of S

eque

nces

• No random sequences with alignment score >= 732– So: the P-value is less than 1 / 200 = 0.05

• To get smaller p-value, have to align more random sequences– Very slow

• Unless we can fit a distribution (e.g. normal distribution)– Such distribution may not be generalizable– No theory exists for global alignment score distribution

Statistics for local alignment

• Elegant theory exists• Score for ungapped local alignment follows extreme value

distribution (Gumbel distribution)

Normal distribution

Extreme value distribution

An example extreme value distribution:

• Randomly sample 100 numbers from a normal distribution, and compute max

• Repeat 100 times.

• The max values will follow extreme value distribution

Statistics for local alignment

• Given two unrelated sequences of lengths M, N• Expected number of ungapped local alignments

with score at least S can be calculated by– E(S) = KMN exp[-S]– Known as E-value : scaling factor as computed in last lecture– K: empirical parameter ~ 0.1

• Depend on sequence composition and substitution matrix

P-value for local alignment score

• P-value for a local alignment with score S

)(

exp1)(exp1

SE

SKMNeSESxP

when P is small.

Example

• You are aligning two sequences, each has 1000 bases

• m = 1, s = -1, d = -inf (ungapped alignment)

• You obtain a score 20

• Is this score significant?

= ln3 = 1.1 (computed as discussed on slide #41)• E(S) = K MN exp{- S}• E(20) = 0.1 * 1000 * 1000 * 3-20 = 3 x 10-5

• P-value = 3 x 10-5 << 0.05• The alignment is significant

9 10 11 12 13 14 15 16 17 180

50

100

150

200

250

300

350

400

Alignment Score

Num

ber

of S

eque

nces

20

Distribution of 1000 random sequence pairs

Multiple-testing problem

• Searching a 1000-base sequence against a database of 106 sequences (each of length 1000)

• How significant is a score 20 now?• You are essentially comparing 1000 bases with 1000x106

= 109 bases (ignore edge effect)• E(20) = 0.1 * 1000 * 109 * 3-20 = 30• By chance we would expect to see 30 matches

– The P-value (probability of seeing at least one match with score >= 30) is 1 – e-30 = 0.9999999999

– The alignment is not significant– Caution: it does NOT mean that the two sequences are unrelated.

Rather, it simply means that you have NO confidence to say whether the two sequences are related.

Score threshold to determine significance

• You want a p-value that is very small (even after taking into consideration multiple-testing)

• What S will guarantee you a significant p-value?

E(S) P(S) << 1

=> KMN exp[-S] << 1

=> log(KMN) -S < 0=> S > T + log(MN) / (T = log(K) / , usually small)

Score threshold to determine significance

• In the previous example– m = 1, s = -1, d = -inf => = 1.1

• Aligning 1000bp vs 1000bpS > log(106) / 1.1 = 13.

So 20 is significant.

• Searching 1000bp against 106 x 1000bpS > log(1012) / 1.1 = 25.

so 20 is not significant.

Statistics for gapped local alignment

• Theory not well developed• Extreme value distribution works well

empirically• Need to estimate K and empirically

– Given the database and substitution matrix, generate some random sequence pairs

– Do local alignment– Fit an extreme value distribution to obtain K

and

In summary

• How to obtain a substitution matrix?– Obtain qst and ps from established alignments (for DNA: from

your knowledge)– Computing score:

• How to understand arbitrary substitution matrix?– Solve function to obtain and target qst

– Which tells you what percent identity you are expecting

• How to understand alignment score?– probability that a score can be expected from chance.– Global alignment: Monte-Carlo simulation– Local alignment: Extreme Value Distribution

• Estimate p-value from a score• Determine a score threshold without computing a p-value