Position Weight Matrices for Representing Signals in Sequences Triinu Tasa, Koke 04.02.05.

Position Weight Matrices for Representing Signals in

Sequences

Triinu Tasa, Koke 04.02.05

Definitions

• Sequence, string – ordered arrangement of letters {'A', 'C', 'G', 'T'}

• Pattern – simplified regular expression, alphabet {'A', 'C', 'G', 'T', '.'}, where '.' - wild-card of length 1 ('A', 'C', 'G' or 'T')


What is a weight matrix?

GATGAG

GATGAT

TGATAT

GATGATor

[GT][AG][TA][GT]A[GT]



Alignment matrix C:A 0 2 1 0 3 0C 0 0 0 0 0 0G 2 1 0 2 0 1T 1 0 2 1 0 2

Frequency matrix F:A 0 0.7 0.3 0 1 0 C 0 0 0 0 0 0 G 0.7 0.3 0 0.7 0 0.3T 0.3 0 0.7 0.3 0 0.7

Better: GATGAG

GATGAT

TGATAT



Or weight matrix W:

where

N – number of sequences used

- a priori probability of letter iip

1.09 1.10 0.51 1.09 1.47 1.09

1.09 1.09 1.09 1.09 1.09 1.09

1.10 0.51 1.09 1.10 1.09 0.51

0.51 1.09 1.10 0.51 1.09 1.10

A

C

G

T

, ,, ln ~ ln

( 1)i j i i j

i ji i

c p fw

N p p



Importance matrix I:

I(i, j) = *,i jc ,i jf

A 0 1.4 0.3 03 0

C 0 0 0 00 0

G 1.4 0.3 0 1.40 0.3

T 0.3 0 1.4 0.30 1.4



Applications

• Pattern clustering1. G.GATGAG.T 62/75 1:39/49 2:23/26 R:17.3026 BP:1.12008e-37

2. G.GATGAG 89/110 1:45/60 2:44/50 R:10.436 BP:1.61764e-34

3. GATGAG.T 124/148 1:52/70 2:72/78 R:7.36961 BP:2.79148e-33

4. TG.AAA.TTT 132/145 1:53/61 2:79/84 R:6.84578 BP:1.83509e-32

5. AAAATTTT 200/231 1:63/77 2:137/154 R:4.69239 BP:1.19109e-30

6. TGAAAA.TTT 104/114 1:45/53 2:59/61 R:7.78277 BP:3.86086e-29

7. AAA.TTTT 343/537 1:79/145 2:264/392 R:3.05349 BP:5.66833e-29

8. G.AAA.TTTT 135/156 1:51/62 2:84/94 R:6.19534 BP:5.69933e-29

9. TG.GATGAG 49/57 1:30/35 2:19/22 R:16.1117 BP:9.35765e-28

10. TG.AAA.TTTT 86/91 1:40/43 2:46/48 R:8.87311 BP:1.1124e-27

...


Applications - Clustering

G.GATGAG.T:

GAGATGAGAT

GTGATGAGAT

GAGATGAGGT

...

A -6.9 0.98 -6.9 1.38 -6.9 -6.9 1.38 -6.9 0.98 -6.9

C -6.9 -6.9 -6.9 -6.9 -6.9 -6.9 -6.9 -6.9 -6.9 -6.9

G 1.38 -6.9 1.38 -6.9 -6.9 1.38 -6.9 1.38 0.29 -6.9

T -6.9 0.29 -6.9 -6.9 1.38 -6.9 -6.9 -6.9 -6.91.38



Compare matrices with each other using the dynamic programming approach

:

where

A, B – matrices

i, j - columns

If D(m,n) > threshold => matrices are different

( 1, ) _ cos ,

( , ) min ( , 1) _ cos ,

( 1, 1) ( , )

D i j deletion t

D i j D i j insertion t

D i j d i j

2, ,

1

( , ) ( )m

i j k i k jk

d A B A B



G.GATGAG.T TG.AAA.TTT AAAATTTT

G.GATGAG TGAAAA.TTT AAA.TTTT

GATGAG.T TG.AAA.TTTT

We want to represent the clusters by

logos:

We need to align the patterns first – position the similar parts of the patterns above each other:

G.GATGAG.T

G.GATGAG--

--GATGAG.T

or the logo will look like this:



•Multiple Alignment

Importance matrix I – represents the aligned patterns.

Example:

G.GATGAG.T

GATGAG.T

G.GATGAG

1. Insert the first pattern into I: ('.' gives 0.25 to each)

A 0 0.25 0 1 0 0 1 0 0.25 0

C 0 0.25 0 0 0 0 0 0 0.25 0

G 1 0.25 1 0 0 1 0 1 0.25 0

T 0 0.25 0 0 1 0 0 0 0.25 1

2. Align the second pattern with I using a dynamic programming approach:

, : ( ,0) 0, (0, ) 0i j v i v j

,

, ,

0, 0 :

0, 0,( , ) max

( 1, 1) , 0i

i i

S j

S j S j

i j

Iv i j

v i j I I


Applications – Multiple alignment

Dynamic programming matrix:

G . G A T G A G . T

G 0.00 0.10 0.01 0.10 0.00 0.00 0.10 0.00 0.10 0.01 0.00

A 0.00 0.00 0.11 0.00 0.20 0.00 0.00 0.20 0.00 0.11 0.00

T 0.00 0.00 0.01 0.00 0.00 0.30 0.00 0.00 0.00 0.01 0.21

G 0.00 0.10 0.01 0.11 0.00 0.00 0.40 0.00 0.10 0.01 0.00

A 0.00 0.00 0.11 0.00 0.21 0.00 0.00 0.50 0.00 0.11 0.00

G 0.00 0.10 0.01 0.21 0.00 0.00 0.10 0.00 0.60 0.01 0.00

. 0.00 0.00 0.10 0.01 0.21 0.00 0.00 0.10 0.00 0.60 0.01

T 0.00 0.00 0.01 0.00 0.00 0.31 0.00 0.00 0.00 0.01 0.70

G.GATGAG.T

--GATGAG.T



3. Add the pattern '--GATGAG.T' to I, if necessary add columns to the matrix.

4. Repeat the procedure for every pattern.

Output:

G.GATGAG.TG.GATGAG----GATGAG.T

Why importance matrix?



Example:

Pattern: GATG

So far aligned:

GATGATGTA-- - - GATGTGG

We want: w(G, 4) > w(G, 1) > w(G, 9)

Solution – importance matrix



● Weight Matrix Matching

Purpose: find the sequences that the weight matrix describes best in a given text file

...CATAGGAAATTCCACCTCTTTGGCTTTGCCCAGTCTTCCCTTGAGGATGCCTACGTTC...

1. Calculate the score for each position

2. if score > threshold => signal

Problem: finding a good threshold

● Threshold – 99.5% quantile

1.09 1.10 0.51 1.09 1.47 1.09

1.09 1.09 1.09 1.09 1.09 1.09

1.10 0.51 1.09 1.10 1.09 0.51

0.51 1.09 1.10 0.51 1.09 1.10

A

C

G

T


Applications – Weight matrix matching

Questions?


Position Weight Matrices for Representing Signals in Sequences Triinu Tasa, Koke 04.02.05.

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