Using selective pressure to improve protein tridimensional structure prediction

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improve protein tridimensional structure prediction

Aude GRELAUD

Context Markov random fields Parameter posterior distribution Model choice Simulations Conclusion

Using selective pressure to improve protein tridimensional structure prediction

Aude GRELAUD ¹ ^, ² Jean-Michel MARIN ³ , Christian P.

ROBERT ¹ , François RODOLPHE ²

1

Cérémade, Université Paris Dauphine et Laboratoire de statistique, CREST-INSEE

2

Unite Mathématique, Informatique et Génome, INRA

3

INRIA Saclay

MIEP

Hameau de l’Etoile, june 2008

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1 Context

2 Markov random fields

3 Parameter posterior distribution

4 Model choice

5 Simulations

6 Conclusion

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Aim

Predict the tridimensional structure of the protein

Knowning amino acid sequence

Met Thr Gln Cys · · · ·

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Existing methods

• Experimental methods :

• X-ray cristallography

• Nuclear magnetic resonance spectroscopy

• Cryomicroscopy

# Expensive and slow, but provide the exact 3D structure

• Computational methods :

• Based on homologies with proteins of known structure : methods based on sequence similarity, protein threading

• De novo prediction

# _Gives several possible 3D structures, with no criterion of

choice

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Purpose : build a ranking method based on a phylogenetic stability criterion

• In a 3D structure, amino acids in contact frequently have similar modification tolerances

• Criterion : selective pressure sequence

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Data

• First step : Estimate the selective pressure sequence

ω 1 , ...ω

n

on a multiple aligment of homologs

Seq : caa agg tgc tta H1 : cat agg tgc gta H2 : cat tgg tgc cta H3 : aat tgg tgc ctg

↓

ω

1

ω

2

ω

3

ω

4

• m folding candidates

or

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Statistical tools

• Markov random fields

• ABC (Approximate Bayesian Computation)

• Bayesian model choice

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1 Context

2 Markov random fields

3 Parameter posterior distribution

4 Model choice

5 Simulations

6 Conclusion

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Definition

• Markov chain :

• Markov random field : Markov chain generalisation

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Definition (2)

• State at a point i only depends on the state of its neighbours n ( i ) :

π( ^x

i

= j | x ₋

_i

) = π( ^x

i

= j | z

_n

₍

_i

₎ )

• Hammersley-Clifford theorem :

P ( X = x ) = ¹

Z exp (− U ( x )) with

• U ( x ) : potential

U ( x ) = ∑

c

∈

C

V

c

( x )

U ( x ) = −θ ∑

(

i

,

j

):

i

∼

^sj

1 _{

_x_i

₌

_x_j

_}

• Z : normalizing constant Z = ∑

x

exp (− U ( x ))

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1 Context

2 Markov random fields

3 Parameter posterior distribution

4 Model choice

5 Simulations

6 Conclusion

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Context Markov random fields Parameter posterior distribution

Model choice Simulations Conclusion

Bayesian modelisation

• Prior distribution :

• (θ) ∼ π(θ)

• Likelihood : ( X |θ) ∼ MRF (θ)

f ( x |θ) = ¹

Z _θ exp (θ ∑

(

i

,

j

):

i

∼

j

1 _{

_x_i

₌

_x_j

_} )

# Target : Posterior distribution of θ

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Parameter posterior distribution

MCMC methods :

Hastings-Metropolis algorithm :

• Proposal : θ ⁰ ∼ p (θ ⁰ |θ ⁽

^t

⁾ )

• θ ⁽

^t

⁺

¹

⁾ = θ ⁰ with probability min { ¹ ,

1 Z_θ₀

q _θ

0

( X )

1

Zθ(t)

q _θ

(t)

( X )

p (θ ⁽

^t

⁾ |θ ⁰ ) p (θ ⁰ |θ ⁽

^t

⁾ )

π(θ ⁰ ) π(θ ⁽

^t

⁾ ) }

# Ratio involves intractable normalizing constants Z _θ

⁰

and Z _θ

(t)

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ABC : Approximate Bayesian Computation

• Bayesian inference without using likelihood

• Idea : Data sufficiently close provide similar parameter posterior distribution

• What we need :

• Simulate data given parameter values

• Summary statistics (sufficient)

• Calculate closeness between our data (X

⁰

) and simulated

data (X

ⁱ^∗

) : distance between summary statistics

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ABC Algorithm

• Sufficient statistic : S ( X ) = ∑ (

i

,

j

):

i

∼

j

1 _{

_x_i

₌

_x_j

_}

• Distance : d ( ^S ( ^X

⁰

), ^S ( ^X

ⁱ

^∗ )) = ( ^S ( ^X

⁰

) − ^S ( ^X

ⁱ

^∗ ))

²

• Algorithm :

• Generate θ

ⁱ

^∗ ∼ π(θ

ⁱ

^∗ )

• Generate ( X |θ

ⁱ

^∗ ) ∼ MRF (θ

ⁱ

^∗ )

• Calculate d

_i

= d ( S ( X

⁰

), S ( X

ⁱ

^∗ ))

• Accept θ

ⁱ

^∗ if d

i

< ε

• Result : sample of independent draws from f (θ| d < ε)

# Good approximation of f (θ| X ⁰ )

• In practice, ε is a 1 % quantile of d

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1 Context

2 Markov random fields

3 Parameter posterior distribution

4 Model choice

5 Simulations

6 Conclusion

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Context Markov random fields Parameter posterior distribution Model choice

Simulations Conclusion

Bayesian hierarchical modelisation

1 model ←→ 1 neighborhood / 3D structure

• Prior distributions :

• s ∼ π( ^s )

• (θ

s

| ^s ) ∼ π

s

(θ

s

)

• Likelihood :

( X |θ

s

, s ) ∼ MRF (θ

s

, s ) f

_s

( x |θ

s

) = ¹

Z _θ

_s

_,

_s

exp (θ

s

∑

(

i

,

j

):

i

∼

^sj

1 _{

_x_i

₌

_x_j

_} )

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Bayes factor definition

• BF ₀ _/ ₁ =

P

(

s

= 0 |

X

)

P

(

s

= 1 |

X

)

P

(

s

= 0 )

P

(

s

= 1 )

=

R f 0 ( X |θ 0 )π 0 (θ 0 ) d θ 0

R f ₁ ( X |θ 1 )π 1 (θ 1 ) d θ 1

• Interpretation :

• BF > 1 : Model 0

• BF < 1 : Model 1

• Jeffreys scale :

< 10

⁻²

[ 10

⁻²

, 10

⁻³^/²

] [ 10

⁻³^/²

, 10

⁻¹

] [ 10

⁻¹

, 10

⁻¹^/²

]

> 10

²

[ 10

³^/²

, 10

²

] [ 10

¹

, 10

³^/²

] [ 10

¹^/²

, 10

¹

]

decisive very hard hard substantial

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Another way to write the Bayes factor

P ( S

i

( X ) = s |θ

i

) = ∑

X

:

Si

(

X

)=

s

f

i

( X |θ

i

)

= ¹

Z _θ

_i

_,

_i

exp (θ

i

s ) ^card { ^X : ^S

i

( ^X ) = ^s }

BF

₀

_/

₁

= ^card { X : ^S

1

( X ) = s

₁

} card { ^X : ^S

0

( ^X ) = ^s

0

}

R P ( S

₀

( X ) = s

₀

|θ

0

)π

0

(θ

0

) d θ

0

R P ( ^S

1

( ^X ) = ^s

1

|θ

1

)π

1

(θ

1

) ^d θ

1

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ABC algorithm

• Vector of summary statistics : S ( X ) = S

1

( X ), .. S

m

( X ) avec S

_s

( ^X ) = ∑ ₍

_i

_,

_j

_):

_i

_∼

^s_j

¹ {

xi

=

xj

}

• Distance : d ( ^S ( ^X

⁰

), ^S ( ^X

ⁱ

^∗ )) = ∑

s

( ^S

s

( ^X

⁰

) − ^S

s

( ^X

ⁱ

^∗ ))

²

• Algorithm :

• Generate s

ⁱ

^∗ ∼ π( ^s )

• Generate (θ

sⁱ^∗

| ^s

ⁱ

^∗ ) ∼ π

sⁱ^∗

(θ ^∗

_si∗

)

• Generate ( X |θ ^∗

_s_i∗

_,

_i

_∗ ) ∼ MRF (θ ^∗

_s_i∗

_,

_i

_∗ )

• Calculate d

_i

= d ( S ( X

⁰

), S ( X

ⁱ

^∗ ))

• Accept ( s

_i

_∗ ,θ ^∗

s_i∗

) if d

_i

< ε

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• Result : (( s

i

∗ , θ ^∗

_s_i∗

)

i

)

# ^card ⁽ ^s

ⁱ

^∗ ⁼ ⁰ ⁾

card ( s

_i

_∗ = 1 ) estimate of

R P ( S

0

( X ) = s

0

|θ

0

)π

0

(θ

0

) d θ

0

R P ( S

₁

( X ) = s

₁

|θ

1

)π

1

(θ

1

) d θ

1

• Calculate card { X : S 1 ( X ) = s 1 }

card { X : ^S 0 ( X ) = s ₀ } ^{to obtain} BF c

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1 Context

2 Markov random fields

3 Parameter posterior distribution

4 Model choice

5 Simulations

6 Conclusion

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Models

• M0 : iid case, Bernouilli ( p )

• f

0

( X |θ, m = 0 ) =

_Z¹

θ,0

exp ( θ∑

i

1 _{

_x_i

₌

₁

_} )

• p =

₁

₊

^exp_exp

^(θ) _(θ)

• M1 : Markov chain with transition matrix P

• f ( X |θ, 1 ) =

^exp

⁽

²

^{θ ∑}

n−1

i=11_{_xi₌_xi+1}

)

2

(

1

+

exp

(

2

θ))

ⁿ⁻¹

• P =

exp

(

2

θ)

1

+

exp

(

2

θ)

1

+

exp¹

(

2

θ)

1

+

exp

(

2

θ)

^exp

(

2

θ)

1

+

exp

(

2

θ)

!

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Parameter estimation

• • Comparison ML/ABC estimates : | b θ

abc

− b θ

mv

|

1stQu .

Median Mean

3rdQu .

Var

1 . 68e − 03 3 . 28e − 03 3 . 27e − 03 5 . 001e − 03 3 . 89e − 06

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Model choice

• BF c = ^C

s1 n

− 1

C

n^s⁰

card ( s

i

∗ = 0 ) card ( s

_i

_∗ = 1 )

• BF =

R

exp

(θ

0S0

(

X

))

( 1 +

exp

(θ

0

))

ⁿ

π 0 (θ 0 ) d θ 0

R

exp

( 2 θ

1S1

(

X

))

( 1 +

exp

( 2 θ

1

))

ⁿ⁻¹

π 1 (θ 1 ) d θ 1

• Comparison on 10.000 simulated data : BF

BF c

M0 ? M1

M0 4684 2 30

? 158 339 53

M1 5 261 4464

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1 Context

2 Markov random fields

3 Parameter posterior distribution

4 Model choice

5 Simulations

6 Conclusion

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Conclusion

• Conclusion

• Parameter estimation in MRF is not too expensive using ABC

• BF is a good way to choose between some neighborhoods

• Perspectives :

• Estimate / calculate the number of configurations given a value of S

• Apply on biological data

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Using selective pressure to improve protein tridimensional structure prediction