Protein and gene model inference based on statistical modeling in k -partite graphs

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Protein and gene model inference based on statistical modeling in k-partite graphsSarah Gester, Ermir Qeli, Christian H. Ahrens, and Peter Buhlmann

+Problem Description

Given peptides and scores/probabilities, infer the set of proteins present in the sample.

PERFGKLMQK

MLLTDFSSAWCR

FFRDESQINNR

TGYIPPPLJMGKR

Protein A

Protein B

Protein C

+Previous Approaches

N-peptides rule ProteinProphet (Nesvizhskii et al. 2003. Anal Chem)

Assumes peptide scores are correct.

Nested mixture model (Li et al. 2010. Ann Appl Statist) Rescores peptides while doing the protein inference Does not allow shared peptides Peptide scores are independent

Hierarchical statistical model (Shen et al. 2008. Bioinformatics) Allows for shared peptides Assume PSM scores for the same peptide are independent Impractical on normal datasets

MSBayesPro (Li et al. 2009. J Comput Biol) Uses peptide detectabilities to determine peptide priors.

+Markovian Inference of Proteins and Gene Models (MIPGEM) Inclusion of shared/degenerate peptides in the model. Treats peptide scores/probabilities as random values Model allows dependence of peptide scores. Inference of gene models

+Why scores as random values?

PERFGKLMQK

MLLTDFSSAWCR

FFRDESQINNR

TGYIPPPLJMGKR

Protein A

Protein B

Protein C

+Building the bipartite graph

+Shared peptides

+Definitions

Let pi be the score/probabilitiy of peptide i. I is the set of all peptides.

Let Zj be the indicator variable for protein j. J is the set of all proteins.

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P[Z j =1 |{pi;i ∈ I}]

+Simple Probability Rules

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P(A | B) = P(A,B)P(B)

= P(B | A)P(A)P(B)

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P(A,B) = P(A | B)P(B) = P(B | A)P(A)

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P(A) = P(A,B = b)b∑

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P(A = a | B)a∑ =1

+Bayes Rule

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P[Z j =1 |{pi;i ∈ I}] =P[Z j =1,{pi;i ∈ I}]

P[{pi;i ∈ I}]

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=P[{pi;i ∈ I} | Z j =1]⋅P[Z j =1]

P[{pi;i ∈ I}]

Prior probability on

the protein being present

Joint probability of seeing these peptide scores

Probability of observing these peptide scores given that the protein is present

+Assumptions

Prior probabilities of proteins are independent

Dependencies can be included with a little more effort. This does not mean that proteins are independent.€

P[{Z j ; j ∈ J}] = P(Z j )j∈J∏

+Assumptions

Connected components are independent

+Assumptions

Peptide scores are independent given their neighboring proteins. Ne(i) is the set of proteins connected to peptide i in the

graph. Ir is the set of peptides belonging to the rth connected

component R(Ir) is the set of proteins connected to peptides in Ir

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P[{pi;i ∈ I} |{Z j; j ∈ R(Ir )}] = P[ pi |{Z j ; j ∈ Ne(i)}]i∈I r

∏

+Assumptions

Conditional peptide probabilities are modeled by a mixture model. The specific mixture model they use is based on the

peptide scores used (from PeptideProphet).

+Bayes Rule

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P[Z j =1 |{pi;i ∈ I}] =P[Z j =1,{pi;i ∈ I}]

P[{pi;i ∈ I}]

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=P[{pi;i ∈ I} | Z j =1]⋅P[Z j =1]

P[{pi;i ∈ I}]

Prior probability on

the protein being present

Joint probability of seeing these peptide scores

Probability of observing these peptide scores given that the protein is present

+Joint peptide score distribution

Assumption: peptides in different components are independent

Ir is the set of peptides in component r

R(Ir) is the set of proteins connected to peptides in Ir

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P({pi;i ∈ I}) = P({pi;i ∈ Ir})r=1

R

∏

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P({pi;i ∈ Ir}) = P({pi;i ∈ Ir} |{Z j; j ∈ R(Ir )})P({Z j; j ∈ R(Ir )})Z j ∈{0,1}j∈R (I r )

∑

+Conditional Probability

Mixture model

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P(pi |{Z j ; j ∈ Ne(i)}) ≈

1u − l

if Z j = 0j∈Ne( i)∑

f1(pi) if Z j > 0j∈Ne( i)∑

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

l =i

min(pi)

m =i

median (pi)

u =i

max(pi)

+Conditional Probability

Mixture model

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f1(x) =b1(x − l) l ≤ x ≤ m

(b1 + b2)(x − m) + b1(m − l) m < x ≤ u ⎧ ⎨ ⎩

l =i

min(pi)

m =i

median (pi)

u =i

max(pi)

€

f1(x)dx =1l

u

∫

+f1(x) – pdf of P(pi|{zj})

0.8 0.82

0.84

0.86

0.88 0.9 0.9

20.9

40.9

60.9

80

0.050.1

0.150.2

0.250.3

0.350.4

f(x)

f(x)

median

+Choosing b1 and b2

Seek to maximize the log likelihood of observing the peptide scores.

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l= log(P({pi;i ∈ I})) = log P({pi;i ∈ Irr=1

R

∏ }) ⎛

⎝ ⎜

⎞

⎠ ⎟

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l= log(P({pi;i ∈ Ir}))r=1

R

∑

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l= log P(pi |{Z j = z; j ∈ Ne(i)}i∈I r

∏z∈{0,1}j∈R (I r )

∑ ) ⋅ P(Z j = z)j∈R (I r )∏

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟r=1

R

∑

+Choosing b1 and b2

It turns out:

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ˆ b 1 =b1

argmin − l (b1)

ˆ b 2 = 2 − ˆ b 1(u − l)2

(u − m)2

+Conditional Protein Probabilities

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P[Z j =1 |{pi;i ∈ I}] =P[Z j =1,{pi;i ∈ I}]

P[{pi;i ∈ I}]

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=P[{pi;i ∈ I} | Z j =1]⋅P[Z j =1]

P[{pi;i ∈ I}]

+Conditional Protein Probabilities

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P(Z j =1 |{pi;i ∈ I}) =P[{pi;i ∈ I} | Z j =1]⋅P[Z j =1]

P[{pi;i ∈ I}]

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=

P[{pi;i ∈ I} | Z j =1,Zk = z]⋅P[Z j =1,Zk = z]k∈R(I d ( j ) )

∑P[{pi;i ∈ I}]

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=A(1)

P[{pi;i ∈ I}]

+Conditional Protein Probabilities(NEC Correction)

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P(Z j =1 |{pi;i ∈ I}) =P[{pi;i ∈ I} | Z j =1]⋅P[Z j =1]

P[{pi;i ∈ I}]

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=

P[{pi;i ∈ I} | Z j =1,{Zk;k ≠ j}) ⋅P[Z j =1,{Zk;k ≠ j}]zk ∈{0,1}∑

P[{pi;i ∈ I}]

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=A(1)

P[{pi;i ∈ I}]

+Conditional Protein Probabilities

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P(Z j =1 |{pi;i ∈ I}) = A(1)P[{pi;i ∈ I}]

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P(Z j = 0 |{pi;i ∈ I}) = A(0)P[{pi;i ∈ I}]

+Conditional Protein Probabilities

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P(Z j =1 |{pi;i ∈ I}) + P(Z j = 0 |{pi;i ∈ I}) =1

€

A(0)P({pi;i ∈ I})

+ A(1)P({pi;i ∈ I})

= A(1) + A(0)P({pi;i ∈ I})

=1

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A(1) + A(0) = P({pi;i ∈ I})

+Conditional Protein Probabilities

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P(Z j =1 |{pi;i ∈ I}) = A(1)P[{pi;i ∈ I}]

= A(1)A(0) + A(1)

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P(Z j = 0 |{pi;i ∈ I}) = A(0)P[{pi;i ∈ I}]

= A(0)A(0) + A(1)

+Shared Peptides

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Aunshared (1) = P[{pi;i ∈ IA} | Z1 =1,Zk = z]⋅P[Z1 =1,Zk = z]k∈R (I A )∑

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Aunshared (1) = P[{pi;i ∈ IA} | Z1 =1]⋅P[Z1 =1]

+

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Ashared (1) = P[{pi;i ∈ IB} | Z1 =1,Zk = z]⋅P[Z1 =1,Zk = z]k∈R (I B )∑

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Ashared (1) = P[{pi;i ∈ IB} | Z1 =1,Z2 =1]⋅P[Z1 =1]⋅P[Z2 =1] + P[{pi;i ∈ IB} | Z1 =1,Z2 = 0]⋅P[Z1 =1]⋅P[Z2 = 0]

Shared Peptides

+Shared Peptides

If the shared peptide has pi ≥ median

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Punshared[Z1 =1 |{pi;i ∈ I}] ≥ Pshared[Z1 =1 |{pi;i ∈ I}]

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Punshared[Z1 = 0 |{pi;i ∈ I}] ≤ Pshared[Z1 = 0 |{pi;i ∈ I}]

+Shared Peptides

If the shared peptide has pi < median

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Punshared[Z1 =1 |{pi;i ∈ I}] < Pshared[Z1 =1 |{pi;i ∈ I}]

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Punshared[Z1 = 0 |{pi;i ∈ I}] > Pshared[Z1 = 0 |{pi;i ∈ I}]

+Gene Model Inference

+Gene Model Inference

Assume a gene model, X, has only protein sequences which belong to the same connected component.

Peptide 1

Peptide 2

Peptide 3

Peptide 4

Protein A

Protein B

Gene X

+Gene Model Inference

Assume a gene model, X, has only protein sequences which belong to the same connected component.

R(X) is the set of proteins with edges to X. Ir(X) is the set of peptides with edges to proteins with edges

to X

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P[X =1 |{pi;i ∈ I}] =1− P {Z j = 0} |{pi;i ∈ Ir(X )}j∈R (X )I

⎡

⎣ ⎢

⎤

⎦ ⎥

+Gene Model Inference

Gene model, X, has proteins from different connected components of the peptide-protein graph.

Peptide 1

Peptide 2

Peptide 3

Peptide 4

Protein A

Protein B

Gene X

+Gene Model Inference

Gene model, X, has proteins from different connected components of the peptide-protein graph.

Rl(X) is the set of proteins with edges to X in component l.

Il(X) is the set of peptides with edges to proteins with edges to X in component l.

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P {Z j = 0} |{pi;i ∈ Ir(X )}j∈R(X )I

⎡

⎣ ⎢

⎤

⎦ ⎥= P {Z j = 0} |{pi;i ∈ Il (X )}

j∈R l (X )I

⎡

⎣ ⎢

⎤

⎦ ⎥

l =1

m

∏

+Datasets

Mixture of 18 purified proteins Mixture of 49 proteins (Sigma49) Drosophila melanogaster Saccharomyces cerevisiae (~4200 proteins) Arabidopis thaliana (~4580 gene models)

+Comparisons with other tools

Small datasets with a known answerMix of 18 proteins

Sigma49

+

Sigma49

Comparisons with other tools

One hit wonders

Sigma49 no one hit wonders

+Comparison with other tools

Arabidopsis thaliana dataset has many proteins with high sequence similarity.

+Splice isoforms

+Conclusion +Criticism

Developed a model for protein and gene model inference.

Comparisons with other tools do not justify complexity: Value of a small FP rate at the expense of many FN is not

shared for all applications.

Discard some useful information such as #spectra/peptide

Assumptions of parsimony from pruning may be too aggressive.