.

Class 8:

Pair HMMs

FSA HHMs: Why?

Advantages:

Obtain reliability of alignment

Explore alternative (sub-optimal) alignments Score similarity of sequences independent of

any specific alignment

FSA HHMs

B(+1,+0)

A(+1,+1)

C(+0,+1)

Ws

Wg+W

s

Wg+Ws

Ws

s(si,tj)

s(si,tj)

s(si,tj)

Bqsi

Apsitj

Cqtj

ε

ε1-ε

1-ε

δ

δ

1-2δ

Affine gap alignment: the full probabilistic model

Bqsi

Apsitj

Cqtj

ε

1-ε-τ

δ

δ1-2δ-τ1-ε-τ

δ

τ

τ

τ

τ

Begin End

1-2δ-τ

δε

Affine Weight Model – DP

( 1, 1) ( , )

( , ) max ( 1, 1) ( , )

( 1, 1) ( , )

i j

i j

i j

A i j s t

A i j B i j s t

C i j s t

( , 1)( , ) max

( , 1)g s

s

A i j W WB i j

B i j W

( 1, )( , ) max

( 1, )g s

s

A i j W WC i j

C i j W

B(+1,+0)

A(+1,+1)

C(+0,+1)

Ws

Wg+W

s

Wg+Ws

Ws

s(si,tj)

s(si,tj)

s(si,tj)

Viterbi in Pair-HMM

Finding the most probable sequence of hidden states is exactly the global sequence alignment

1, 1

, 1, 1

1, 1

(1 2 ) ( )

( ) max (1 ) ( )

(1 ) ( )i j

i j

i j s t i j

i j

v A

V A p v B

v C

1,

,1,

( )( ) max

( )i

i j

i j si j

v AV B q

v B

, 1,

, 1

( )( ) max

( )j

i ji j t

i j

v AV C q

v C

Bqsi

Apsitj

Cqtj ε

1-ε-τ

δ

δ1-2δ-τ1-ε-τ

δ

τ

τ

τ

τ

Begin End

1-2δ-τ

δε

Viterbi in Pair-HMM

Initial condition:

Optimal alignment:

Bqsi

Apsitj

Cqtj ε

1-ε-τ

δ

δ1-2δ-τ1-ε-τ

δ

τ

τ

τ

τ

Begin End

1-2δ-τ

δε

0,0

,0 0,

( ) 1

all other (*) (*) set to 0i j

V A

V V

,

,

,

( )

( ) max ( )

( )

n m

n m

n m

V A

V End V B

V C

Pair-HMM for random model

sqsi

tqtj

η

η1-η

1-η η

1-η η

1-η

Begin End

1 1

2

1 1

( , | ) (1 ) (1 )

(1 )

i j

i j

n mn m

s ti j

n mn m

s ti j

p s t R q q

q q

Pair-HMM for local alignment

Rs1qsi

Rt1qtj

1-η

1-η

1-η

1-η

ηη

η

η

Begin

Bqsi

Apsitj

Cqtj ε

1-ε-τ

δ

δ1-2δ-τ1-ε-τ

δ

τ

τ

τ

τ

1-2δ-τ

δε

1-η

Rs2qsi

Rt2qtj

1-η

1-η

1-η

ηη

η

η

End

The full probability: P(s,t)

alignments

( , ) )P s t P(s,t,

Use the “forward” algorithm:

The posterior probability:

,( , ) ( )n mP s t f End

( , , )( | , )

( , )

P s tP s t

P s t

Suboptimal alignments

Suboptimal alignments: alignments with nearly the same score as the best alignment

Only slightly different from the optimal alignment

Substantially or completely different

Probabilistic sampling

From the forward algorithm:

Choose the next step to be:

, 1, 1 1, 1 1, 1( ) [(1 2 ) ( ) (1 ) ( ) ( )]i ji j s t i j i j i jf A p f A f B f C

1, 1

,

1, 1

,

1, 1

,

(1 2 ) ( )( 1, 1) with prob.

( )

(1 ) ( )( 1, 1) with prob.

( )

(1 ) ( )( 1, 1) with prob.

( )

i j

i j

i j

s t i j

i j

s t i j

i j

s t i j

i j

p f AA i j

f A

p f BB i j

f A

p f CC i j

f A

Probabilistic sampling – example

s HEAGAWGHEE

t PAWHEAE

Distinct suboptimal alignments

Waterman and Eggert [1987]

1... 1... 1... 1... 1... 1...

1... 1... 1... 1...

, ,

( , , )( | , )

( , )

( , , )

( , , ) ( , | , , )

( , , ) ( , | )

( ) ( )

i ji j

i j

i j i j i n j m i j i j

i j i j i n j m i j

i j i j

P s t s tP s t s t

P s t

P s t s t

P s t s t P s t s t s t

P s t s t P s t s t

f A b A

( | , )i jP s t s tThe posterior probability