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

30/09/2010 Nested Effects Models at Work Prof. Dr. Holger Fröhlich Algorithmic Bioinformatics Bonn-Aachen International Center for Information Technology (B-IT)

Principle Idea of Nested Effects Models  Distinguish between:  Perturbed genes Φ S 1 S 2 S 3 S 4 (hidden variables) θ  Observed effects E E E E E E E E E Perturbed genes S1 S2 S3 S4  Measure downstream effects of each knock- Observed effects down  Network reconstruction is based on observed effects under different perturbations Markowetz et al., 2005 Page 5 Holger Fröhlich Algorithmic Bioinformatics

Nested Effects Models (NEMs) are transitively closed causal networks explaining the nested structure of downstream effects. Page 6 Holger Fröhlich Algorithmic Bioinformatics

Likelihood of the Signaling Graph ( Φ ) Two different approaches: Θ Φ = Θ  Bayesian: Integrate over effects linkage graphs Θ assuming : ( | ) ( ) P P ∫ Φ = Φ Θ Θ ( | ) ( | , ) ( ) P D P D P Θ Markowetz et al., 2005; Fröhlich et al., 2007, 2008  Take MAP/ML estimator for Θ : Θ = ˆ Φ Θ Θ arg max ( | , ) ( ) P D P Θ Tresch & Markowetz., 2008 Φ Θ ˆ Φ ( | , ) ( ) P D P Φ Θ = ˆ ( | , ) P D ( ) P D Page 7 Holger Fröhlich Algorithmic Bioinformatics

Calculation of Effect Likelihoods  Factorization of the likelihood under i.i.d. assumption: ∫ Φ = Φ Θ Θ ( | ) ( | , ) ( ) P D P D P Θ ∏ ∑ ∏ = Φ Θ = Θ = ( | , 1) ( 1) P D P tk sk sk ∈ ε ∈ ∈ k s S t S ∏ ∑ ∏ = Φ ~ ( | ) P D m tk tk ts ∈ ε ∈ ∈ k s S t S 1. Model for binary data D with fixed error probabilities α and β : = =  1 0 D D tk tk  = = α − α if 1 m  ( | ) 1 P D m tk tk tk  = β − β if 0 1 m  tk Markowetz et al., 2005 Page 8 Holger Fröhlich Algorithmic Bioinformatics

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]. = π + π α + π β ( ) Beta( , ,1) Beta( ,1, ) f D D D tk 1 k 2 k tk t 3 k tk t  -> fit via EM algorithm  − ( ) (1) f D f =  tk if 1 m − =  1 (1) f ( | ) tk P D m tk tk =   if 0 m 1 tk Fröhlich et al., 2008 Page 9 Holger Fröhlich Algorithmic Bioinformatics

Bioconductor Package ”nem” library(nem) load(“raw_pvaluesBoutros2002.rda“) D = getDensityMatrix(pvalues) Page 10 Holger Fröhlich Algorithmic Bioinformatics

How to Infer the Network Structure?  Choose candidate graph S 1 S 2 S 3 S 4  Calculate score, e.g. using E E E E E Bayesian Com ombi bina natorial al ex expl plos osion: E E statistics E E (average over n = 4: 355 possible networks E-Gene n = 10: ~10 27 possible networks positions) Likelihood model  Propose different Complete enumeration of • topology all topologies Markowetz et al., 2005 Page 11 Holger Fröhlich Algorithmic Bioinformatics

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 Page 12 Holger Fröhlich Algorithmic Bioinformatics

Large Scale Networks: Module Networks • Problem : complete enumeration of all network hypotheses only 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 Page 13 Holger Fröhlich Algorithmic Bioinformatics

Large Scale Networks: Module Networks S 3 S 4 S 2 S 5 Log-likelihood S 9 E E S 6 S 7 S 1 S 10 10 Network E E S 8 E Fröhlich et al., 2007, 2008 Page 14 Holger Fröhlich Algorithmic Bioinformatics

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) Page 15 Holger Fröhlich Algorithmic Bioinformatics

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 only. 2. Re-estimate the network with the new set of E-Genes 3. Iterate the process until convergence Fröhlich et al., 2008 Page 16 Holger Fröhlich Algorithmic Bioinformatics

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) Page 17 Holger Fröhlich Algorithmic Bioinformatics

Incorporation of Prior Knowledge • Bias scoring such that known interactions are considered • Bayesian prior on network structure ∏ Φ = Φ ( ) ( ) P P Φ = Signaling Graph ij , i j Φ ‘ ‘ = Prior Belief    − Φ − Φ | | 1 Φ ν =  ij ij  ν = Hyperparameter of ( | ) exp P   ν ν ij 2   Laplace Distribution Complete Complete trust in prior trust in data ν (scale of prior) ∞ ∫ Φ = Φ ν ν ν ( ) ( | ) ( ) P P P d ij ij 0 ν ~ (1,0.5) InvGamma 1 Φ = ( ) P  ( ) ij 2 + Φ − Φ 1 2 | | Fröhlich et al., 2008 ij ij Page 18 Holger Fröhlich Algorithmic Bioinformatics

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) Page 19 Holger Fröhlich Algorithmic Bioinformatics

Statistical Stability and Significance  How stable the inferred network?  Do small changes of E-genes lead to different network hypotheses?   Use non-parametric bootstrap Sample n E-genes with repeat replacement  Is the inferred network better than 0.9 R Q random? 0.8  Randomly permute node labels 0.7 and look, whether random P S network has a higher likelihood. Page 20 Holger Fröhlich Algorithmic Bioinformatics

Statistical Stability and Significance  How stable the inferred network?  Do small changes of E-genes lead to different network hypotheses?   Use non-parametric bootstrap Sample n E-genes with repeat replacement  Is the inferred network better than 0.9 S P random? 0.8  Randomly permute node labels 0.7 and look, whether random Q R network has a higher likelihood. Page 21 Holger Fröhlich Algorithmic Bioinformatics

Statistical Stability and Significance  How stable the inferred network?  Do small changes of E-genes lead to different network hypotheses?   Use non-parametric bootstrap Sample n E-genes with repeat replacement  Is the inferred network better than 0.9 P R random? 0.8  Randomly permute node labels 0.7 and look, whether random Q S network has a higher likelihood. Page 22 Holger Fröhlich Algorithmic Bioinformatics

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, p = 0.037 (label N=1000, mynem.boot) permutation test) Page 23 Holger Fröhlich Algorithmic Bioinformatics

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 Page 24 Holger Fröhlich Algorithmic Bioinformatics

Fast and Efficient Learning of Dynamic Nested Effects Models Please come to our poster (G28, Tue)! Page 25 Holger Fröhlich Algorithmic Bioinformatics