Bayesian integration of biological prior knowledge into the reconstruction of gene regulatory networks Dirk Husmeier Adriano V. Werhli

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Learning Bayesian networks from data and prior knowledge

Bayesian networks A CB D EF NODES EDGES Marriage between graph theory and probability theory. Directed acyclic graph (DAG) representing conditional independence relations. It is possible to score a network in light of the data. We can infer how well a particular network explains the observed data.

Bayesian networks versus causal networks A CB A CB True causal graph Node A unknown

Bayesian networks versus causal networks A CB Equivalence classes: networks with the same scores. Equivalent networks cannot be distinguished in light of the data. We can only learn the undirected graph. Unless… we use interventions or prior knowledge. A CB A CB A CB

Learning Bayesian networks from data P(M|D) = P(D|M) P(M) / Z

Use TF binding motifs in promoter sequences

Biological prior knowledge matrix Biological Prior Knowledge Indicates some knowledge about the relationship between genes i and j

Biological prior knowledge matrix Biological Prior Knowledge Define the energy of a Graph G Indicates some knowledge about the relationship between genes i and j

Prior distribution over networks Energy of a network

Sample networks and hyperparameters from the posterior distribution Capture intrinsic inference uncertainty Learn the trade-off parameters automatically P(M|D) = P(D|M) P(M) / Z

Prior distribution over networks Energy of a network

Rewriting the energy Energy of a network

Approximation of the partition function

Multiple sources of prior knowledge

Rewriting the energy Energy of a network

Approximation of the partition function

MCMC sampling scheme

Sample networks and hyperparameters from the posterior distribution Metropolis-Hastings scheme Proposal probabilities

MCMC with one prior Sample graph and the parameter . Separate in two samples to improve the acceptance: 1.Sample graph with  fixed. 2.Sample  with graph fixed.

Sample graph and the parameter . BGe BDe MCMC with one prior Separate in two samples to improve the acceptance: 1.Sample graph with  fixed. 2.Sample  with graph fixed.

Sample graph and the parameter . BGe BDe MCMC with one prior Separate in two samples to improve the acceptance: 1.Sample graph with  fixed. 2.Sample  with graph fixed.

Sample graph and the parameter . BGe BDe MCMC with one prior Separate in two samples to improve the acceptance: 1.Sample graph with  fixed. 2.Sample  with graph fixed.

Sample graph and the parameter . BGe BDe MCMC with one prior Separate in two samples to improve the acceptance: 1.Sample graph with  fixed. 2.Sample  with graph fixed.

Approximation of the partition function

MCMC with two priors Sample graph and the parameters    and  2 Separate in three samples to improve the acceptance: 1.Sample graph with  1 and  2 fixed. 2.Sample  1 with graph and  2 fixed. 3.Sample  2 with graph and  1 fixed.

Bayesian networks with biological prior knowledge Biological prior knowledge: Information about the interactions between the nodes. We use two distinct sources of biological prior knowledge. Each source of biological prior knowledge is associated with its own trade-off parameter:  1 and  2. The trade off parameter indicates how much biological prior information is used. The trade-off parameters are inferred. They are not set by the user!

Bayesian networks with two sources of prior Data BNs + MCMC Recovered Networks and trade off parameters Source 1 Source 2 11 22

Bayesian networks with two sources of prior Data BNs + MCMC Source 1 Source 2 11 22 Recovered Networks and trade off parameters

Bayesian networks with two sources of prior Data BNs + MCMC Source 1 Source 2 11 22 Recovered Networks and trade off parameters

Evaluation Can the method automatically evaluate how useful the different sources of prior knowledge are? Do we get an improvement in the regulatory network reconstruction? Is this improvement optimal?

Application to the Raf regulatory network

Raf regulatory network From Sachs et al Science 2005

Evaluation: Raf signalling pathway Cellular signalling network of 11 phosphorylated proteins and phospholipids in human immune systems cell Deregulation  carcinogenesis Extensively studied in the literature  gold standard network

Data Prior knowledge

Intracellular multicolour flow cytometry. Measured protein concentrations. 11 proteins: 1200 concentration profiles. We sample 5 separate subsets with 100 concentration profiles each. Flow cytometry data and KEGG

Microarray example Spellman et al (1998) Cell cycle 73 samples Tu et al (2005) Metabolic cycle 36 samples Genes time

Data Prior knowledge

KEGG PATHWAYS are a collection of manually drawn pathway maps representing our knowledge of molecular interactions and reaction networks. Flow cytometry data and KEGG

Prior knowledge from KEGG

Prior distribution

The data and the priors + KEGG + Random

Evaluation Can the method automatically evaluate how useful the different sources of prior knowledge are? Do we get an improvement in the regulatory network reconstruction? Is this improvement optimal?

Bayesian networks with two sources of prior Data BNs + MCMC Recovered Networks and trade off parameters Source 1 Source 2 11 22

Bayesian networks with two sources of prior Data BNs + MCMC Source 1 Source 2 11 22 Recovered Networks and trade off parameters

Sampled values of the hyperparameters

Evaluation Can the method automatically evaluate how useful the different sources of prior knowledge are? Do we get an improvement in the regulatory network reconstruction? Is this improvement optimal?

How to compare the recovered networks?

True networkPredicted network compare Thresholding Performance evaluation

True networkPredicted network compare True Positives False Positives Counting Performance evaluation

True networkPredicted network compare True Positives False Positives Counting Performance evaluation DGE – Consider edge directions UGE – Discard the edge directions

Performance evaluation: ROC curves

We use the Area Under the Receiver Operating Characteristic Curve (AUC). AUC=0.75 AUC=1 AUC=0.5 Performance evaluation: ROC curves

Evaluation 2: TP scores We set the threshold such that we obtain 5 spurious edges (5 FPs) and count the corresponding number of true edges (TP count).

Flow cytometry data and KEGG

Evaluation Can the method automatically evaluate how useful the different sources of prior knowledge are? Do we get an improvement in the regulatory network reconstruction? Is this improvement optimal?

Learning the trade-off hyperparameter Repeat MCMC simulations for large set of fixed hyperparameters β Obtain AUC scores for each value of β Compare with the proposed scheme in which β is automatically inferred. Mean and standard deviation of the sampled trade off parameter

Learning the trade-off hyperparameters on simulated data Mean and standard deviation of the sampled trade-off parameter Repeat MCMC simulations for large set of fixed hyperparameters β Obtain AUC scores for each value of β Compare with the proposed scheme in which β is automatically inferred.

Regulation of Raf-1 by Direct Feedback Phosphorylation. Molecular Cell, Vol. 17, 2005 Dougherty et al New evidence for the accepted network

Conclusion Bayesian scheme for the systematic integration of different sources of biological prior knowledge. The method can automatically evaluate how useful the different sources of prior knowledge are. We get an improvement in the regulatory network reconstruction. This improvement is close to optimal.

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