Nested Effects Models at Work Prof. Dr. Holger Frhlich Algorithmic - - PowerPoint PPT Presentation

• Prof. Dr. Holger Fröhlich

Algorithmic Bioinformatics Bonn-Aachen International Center for Information Technology (B-IT)

30/09/2010

Nested Effects Models at Work

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S1 S2 S3 S4

Principle Idea of Nested Effects Models

E E E E E S1 S3 S2 S4 E E E E

Distinguish between: Perturbed genes (hidden variables) Observed effects Measure downstream effects of each knock- down Network reconstruction is based on observed effects under different perturbations

Observed effects Perturbed genes Markowetz et al., 2005

Φ θ

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Nested Effects Models (NEMs) are transitively closed causal networks explaining the nested structure of downstream effects.

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Likelihood of the Signaling Graph (Φ) Two different approaches:  Bayesian: Integrate over effects linkage graphs Θ assuming :  Take MAP/ML estimator for Θ:

( | ) ( | , ) ( ) P D P D P

Θ

Φ = Φ Θ Θ

∫

ˆ arg max ( | , ) ( ) ˆ ( | , ) ( ) ˆ ( | , ) ( ) P D P P D P P D P D

Θ

Θ = Φ Θ Θ Φ Θ Φ Φ Θ =

Markowetz et al., 2005; Fröhlich et al., 2007, 2008 Tresch & Markowetz., 2008

( | ) ( ) P P Θ Φ = Θ

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Calculation of Effect Likelihoods  Factorization of the likelihood under i.i.d. assumption:

• 1. Model for binary data D with fixed error probabilities α and β:

( | ) ( | , ) ( ) ( | , 1) ( 1) ~ ( | )

tk sk sk s S k t S tk tk ts s S k t S

P D P D P P D P P D m

ε ε Θ ∈ ∈ ∈ ∈ ∈ ∈

Φ = Φ Θ Θ = Φ Θ = Θ = = Φ

∫ ∑ ∏ ∏ ∑ ∏ ∏

1 if 1 ( | ) 1 if 1

tk tk tk tk tk tk

D D m P D m m α α β β = =   = = −   = − 

Markowetz et al., 2005

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Modeling Continuous Data

• 2. Data D are computed as p-values for significant change, when

comparing interventions to non-interventions.

 Under the null hypothesis (i.e. expecting no effect) p-values are distributed uniformly  Under the alternative hypothesis (i.e. expecting an effect) there is a high density for small p-values and a strong decrease for increasing p-values [Pounds et al., 2003].  -> fit via EM algorithm

Fröhlich et al., 2008

1 2 3

( ) Beta( , ,1) Beta( ,1, )

tk k k tk t k tk t

f D D D π π α π β = + +

( ) (1) 1 (1)

if 1 ( | ) if 1

tk

f D f f tk tk tk tk

m P D m m

− −

 =  =  =  

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Bioconductor Package ”nem” library(nem) load(“raw_pvaluesBoutros2002.rda“) D = getDensityMatrix(pvalues)

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How to Infer the Network Structure?  Choose candidate graph  Calculate score, e.g. using Bayesian statistics (average over E-Gene positions)  Propose different topology

S1 S3 S2 E E E E E S4 E E E E

Likelihood model

Markowetz et al., 2005

• Complete enumeration of

all topologies

Com

• mbi

bina natorial al ex expl plos

• sion:

n = 4: 355 possible networks n = 10: ~1027 possible networks

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Heuristics for Large Networks (> 4 S-Genes).

 Sampling Based (MCMC, Simulated Annealing)  SA: Fröhlich et al., BMC Bioinformatics, 2007  time consuming  neighborhood relation in transitively closed graphs difficult  Greedy hill climbing Fröhlich et al., Bioinformatics, 2008  Module networks Fröhlich et al., BMC Bioinformatics, 2007 Fröhlich et al., Bioinformatics, 2008  Triplets inference Markowetz et al., Bioinformatics, 2007  Alternating MAP optimization over Φ and θ Tresch and Markowetz, Stat. Appl. Mol. Biol., 2008

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Large Scale Networks: Module Networks

• Problem:

complete enumeration of all network hypotheses

• nly possible for small

networks (< 5 S-genes)

• Solution:

Divide and conquer 1. Highest scoring subnetworks for modules of S-Genes 2. Estimate connections between modules

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Large Scale Networks: Module Networks

E S2 Network Log-likelihood S3 S5 S4 S9 E S8 E S6 S7 S1 S10

10

E E Fröhlich et al., 2007, 2008

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Network Inference with the nem-Package control=set.default.paramet ers(unique(colnames(D)), type="CONTmLLBayes") mynem = nem(D, inference=“ModuleNetwork “, control=control, verbose=FALSE) plot.nem(mynem, SCC=FALSE, D=D, draw.lines=TRUE)

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Automated Selection of Relevant E-Genes (Feature Selection)  Motivation: Irrelevant E-genes can degrade network estimation accuracy 1. Select E-Genes having a positive contribution to the model’s log-likelihood

• nly.

2. Re-estimate the network with the new set of E-Genes 3. Iterate the process until convergence

Fröhlich et al., 2008

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Network Inference with the nem-Package D2 = BoutrosRNAiDiscrete[,9:16] control=set.default.parameters (unique(colnames(D2)), selEGenes=TRUE) mynem2 = nem(D2, inference=“triples“, control=control, verbose=FALSE) plot.nem(mynem2, D=D2, draw.lines=TRUE)

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Incorporation of Prior Knowledge

,

( ) ( ) | | 1 ( | ) exp 2

ij i j ij ij ij

P P P ν ν ν Φ = Φ   − Φ − Φ Φ =      

∏



• Bias scoring such that known interactions are considered
• Bayesian prior on network structure

( )

2

( ) ( | ) ( ) ~ (1,0.5) 1 ( ) 1 2 | |

ij ij ij ij ij

P P P d InvGamma P ν ν ν ν

∞

Φ = Φ Φ = + Φ − Φ

∫



ν (scale of prior) Complete trust in prior Complete trust in data

Φ= Signaling Graph Φ‘ ‘ = Prior Belief ν = Hyperparameter of Laplace Distribution

Fröhlich et al., 2008

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Using Prior Knowledge with the nem-Package control=set.default.parameters (unique(colnames(D)), selEGenes=TRUE, type=“CONTmLLMAP“, Pm=diag(4)) mynem3 = nem(D, control=control, verbose=FALSE) plot.nem(mynem3, SCC=FALSE, D=D, draw.lines=TRUE)

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Statistical Stability and Significance  How stable the inferred network?  Do small changes of E-genes lead to different network hypotheses?   Use non-parametric bootstrap  Is the inferred network better than random?  Randomly permute node labels and look, whether random network has a higher likelihood.

Sample n E-genes with replacement

R Q S P

0.8 0.9 0.7 repeat

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Statistical Stability and Significance  How stable the inferred network?  Do small changes of E-genes lead to different network hypotheses?   Use non-parametric bootstrap  Is the inferred network better than random?  Randomly permute node labels and look, whether random network has a higher likelihood.

Sample n E-genes with replacement

S P R Q

0.8 0.9 0.7 repeat

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Statistical Stability and Significance  How stable the inferred network?  Do small changes of E-genes lead to different network hypotheses?   Use non-parametric bootstrap  Is the inferred network better than random?  Randomly permute node labels and look, whether random network has a higher likelihood.

Sample n E-genes with replacement

P R S Q

0.8 0.9 0.7 repeat

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Bootstrapping and Significance Calculation with the nem-Package control=set.default.parameters (unique(colnames(D)), type=“CONTmLLBayes“, Pm=diag(4)) mynem.boot = nem.bootstrap(D, nboot=100, control=control) plot.nem(mynem.boot, SCC=FALSE, plot.probs=TRUE) nem.calcSignificance(D, N=1000, mynem.boot)

p = 0.037 (label permutation test)

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Summary: nem-package  Inference of features of signaling pathways from high dimensional, targeted perturbation effects  Different likelihood models  Discretized data  P-value log-densities  Algorithms for inference of large networks  Module Networks  Triplets  Greedy hillclimbing  ...  Possibility to integrate prior knowledge  Automatic selection of relevant E-genes  Various plotting and analysis methods  Non-parametric bootstrap  Label permutation p-values

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Fast and Efficient Learning of Dynamic Nested Effects Models Please come to our poster (G28, Tue)!

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Acknowledgements  German Cancer Research Cancer (DKFZ)  Holger Sültmann  Marc Fellmann (alumnus)  Sabrina Belauger  University of Göttingen  Tim Beißbarth  University of Regensburg  Rainer Spang  University of Munich (LMU)  Achim Tresch  Cancer Research UK  Florian Markowetz  Bonn-Aachen International Center for IT (B-IT) Algorithmic Bioinformatics  Paurush Praveen  Khalid Abnaof  Yupeng Cun  Jan Plitschka  Afshin Sadeghi  Amit Kawalia  Sohil Anand