Reverse engineering gene and protein regulatory networks using Graphical Models.

Reverse engineering gene and protein regulatory networks

using Graphical Models. A comparative evaluation study.

Marco Grzegorczyk

Dirk Husmeier

Adriano Werhli

Systems biology

Learning signalling pathways and regulatory networks from

postgenomic data

possibly completely unknown


E.g.: Flow cytometry

experiments

data

Here: Concentrations of (phosphorylated) proteins


E.g.: Flow cytometry

experiments

data data

Machine Learning

statistical methods

true network extracted network

Is the extracted network a good prediction of the real relationships?

true network extracted network

biological knowledge

(gold standard network)

Evaluation of

learning

performance

Reverse Engineering of Regulatory Networks

• Can we learn network structures from postgenomic data themselves?

• Are there statistical methods to distinguish between direct and indirect correlations?

• Is it worth applying time-consuming Bayesian network approaches although computationally cheaper methods are available?

• Do active interventions improve the learning performances?– Gene knockouts (VIGs, RNAi)

direct

interaction

common

regulator

indirect

interaction

co-regulation

Reverse Engineering of Regulatory Networks

• Can we learn network structures from postgenomic data themselves?

• Are there statistical methods to distinguish between direct and indirect correlations?

• Is it worth applying time-consuming Bayesian network approaches although computationally cheaper methods are available?

• Do active interventions improve the learning performances?– Gene knockouts (VIGs, RNAi)

• Relevance networks

• Graphical Gaussian models

• Bayesian networks

Three widely applied methodologies:

Relevance networks(Butte and Kohane, 2000)

1. Choose a measure of association A(.,.)

2. Define a threshold value tA

3. For all pairs of domain variables (X,Y) compute their association A(X,Y)

4. Connect those variables (X,Y) by an undirected edge whose association A(X,Y) exceeds the predefined threshold value tA


1. Choose a measure of association A(.,.)

2. Define a threshold value tA

3. For all pairs of domain variables (X,Y) compute their association A(X,Y)

4. Connect those variables (X,Y) by an undirected edge whose association A(X,Y) exceeds the predefined threshold value tA

direct

interaction

common

regulator

indirect

interaction

co-regulation

Pairwise associations without taking the context of the

system into consideration

direct

interaction

pseudo-correlations

between A and B

E.g.:Correlation between A and C is

disturbed (weakend) by the

influence of B

1 2

X

21

X

21

‘direct interaction’

‘common regulator’

‘indirect interaction’X

21

1 2

strong

correlation σ12

Graphical Gaussian Models

jjii

ijij

)()(

)(111

1

2

2

1

1

direct interaction

Partial correlation, i.e. correlation

conditional on all other domain variables

Corr(X1,X2|X3,…,Xn)

But usually: #observations < #variables

strong partial

correlation π12

Shrinkage estimation of the covariance matrix

(Schäfer and Strimmer, 2005)

TMLMLS T |000ˆˆ)1()|(ˆ

]||)|(ˆ[||))|(ˆ( 2FSS TETMSE

))|(ˆ())|(ˆ( 0 TMSETMSE SS with

where

0<λ0<1 estimated (optimal) shrinkage intensity,

is guaranteed0)|(ˆ0 TS

nn

TML

00

0

0

00

ˆ 22

11

|

direct

interaction

common

regulator

indirect

interaction

co-regulation

Graphical Gaussian Models

direct

interaction

common

regulator

indirect

interaction

P(A,B)=P(A)·P(B)

But: P(A,B|C)≠P(A|C)·P(B|C)

Further drawbacks

• Relevance networks and Graphical Gaussian models can extract undirected edges only.

• Bayesian networks promise to extract at least some directed edges. But can we trust in these edge directions?

It may be better to learn undirected edges than learning directed edges with false orientations.

Bayesian networks

A

CB

D

E F

NODES

EDGES

•Marriage between graph theory and probability theory.

•Directed acyclic graph (DAG) represents conditional independence relations.

•Markov assumption leads to a factorization of the joint probability distribution:

),|()|(),|()|()|()(

),,,,,(

DCFPDEPCBDPACPABPAP

FEDCBAP

Bayesian networks versus causal networks

Bayesian networks represent conditional (in)dependency relations - not necessarily causal interactions.

Bayesian networks versus causal networks

A

CB

A

CB

True causal graph

Node A unknown

Bayesian networks

A

CB

D

E F

NODES

EDGES

•Marriage between graph theory and probability theory.

•Directed acyclic graph (DAG) represents conditional independence relations.

•Markov assumption leads to a factorization of the joint probability distribution:

),|()|(),|()|()|()(

),,,,,(

DCFPDEPCBDPACPABPAP

FEDCBAP

Bayesian networks

n

jijjijii xbNX

1

2 )),((~ ijij XXb 0

)()|()(

)()|()|( graphPgraphdataP

dataP

graphPgraphdataPdatagraphP

Parameterisation: Gaussian BGe scoring metric:

data~N(μ,Σ)

with normal-Wishart distribution of the (unknown) parameters,

i.e.:

μ~N(μ*,(vW)-1) and W~Wishart(T0)

)()|)(,()( graphdgraphgraphdataPgraphP

Learning the network structure

n 4 6 8 10

#DAGs 543 3,7· 106 7,8 ·1011 4,2 ·1018

Idea: Heuristically searching for the graph M* that is most supported by the data P(M*|data)>P(graph|data), e.g. greedy

search

graph → scoreBGe(graph)

MCMC sampling of Bayesian networks

• Better idea: Bayesian model averaging via Markov Chain Monte Carlo (MCMC) simulations

• Construct and simulate a Markov Chain (Mt)t in the space of DAGs {graph} whose distribution converges to the graph posterior distribution as stationary distribution, i.e.:

P(Mt=graph|data) → P(graph|data) t → ∞

to generate a DAG sample: G1,G2,G3,…GT

Order MCMC(Friedman and Koller, 2003)

• Order MCMC generates a sample of node orders from which in a second step DAGs can be sampled:

A B D E FC

A B D E FC

A F D E BC

old

new

A F D E AC → DAG

)|(

)|(,1min)|(

old

newoldnew DP

DPA

Acceptance probability (Metropolis Hastings)

G1,G2,G3,…GT

DAG sample

Equivalence classes of BNs

)|()()|(

)()|()()()|( 1

BCPBPCAP

CPCAPCPBPBCP

11 )(),()(),()(

)|()|()(

APACPCPCBPAP

ACPCBPAP

),|()()( BACPBPAP

A

B

C

A

B

A

B

A

B

C

C

C

)()|()|(

),()|(

CPCBPCAP

CBPCAP

A

B

C

completed partially directed graphs (CPDAGs)

A

C

B

v-structure

P(A,B)=P(A)·P(B)

P(A,B|C)≠P(A|C)·P(B|C)

P(A,B)≠P(A)·P(B)

P(A,B|C)=P(A|C)·P(B|C)

CPDAG representationsA

B C

D

E F

DAGs CPDAGs

Utilise the CPDAG sample for estimating the posterior probability of edge relation

features:

T

iiGIT

BAP1

)(1

)(ˆ

where I(Gi) is 1 if the CPDAG Gi contains the directed edge A→B, and 0 otherwise

A

B C

D

E F

CPDAG representationsA

B C

D

E F

A

B C

D

E F

DAGs CPDAGs

Utilise the DAG (CPDAG) sample for estimating the posterior probability of edge relation

features:

T

iiGIT

BAP1

)(1

)(ˆ

where I(Gi) is 1 if the CPDAG of Gi contains the directed edge A→B, and 0 otherwise

A

B C

D

E F

interpretation

superposition

Interventional data

A B

A B A B

inhibition of A

A B

n

iXpaiii i

DXpaDXPMDP1

][ )][|()|(

n

i

iXpai

iii i

DXpaDXP1

}{][

}{ )][|(

down-regulation of B

no effect on B

A and B are correlated

Evaluation of Performance

• Relevance networks and Graphical Gaussian models extract undirected edges (scores = (partial) correlations)

• Bayesian networks extract undirected as well as directed edges (scores = posterior probabilities of edges)

• Undirected edges can be interpreted as superposition of two directed edges with opposite direction.

• How to cross-compare the learning performances when the true regulatory network is known?

• Distinguish between DGE (directed graph evaluation) and UGE (undirected graph evaluation)

data

Probabilistic inference - DGE

true regulatory network

Thresholding

edge scores

TP:1/2

FP:0/4

TP:2/2

FP:1/4

concrete networkpredictions

lowhigh

data

Probabilistic inference - UGE

undirected edge scores

add up scores of directed edges with opposite direction

skeleton of


data

Probabilistic inference - UGE

skeleton of



add up scores of directed edges with opposite direction

+

++

data

Probabilistic inference

skeleton of



data

Probabilistic inference

skeleton of


Thresholding


TP:1/2

FP:0/1

TP:2/2

FP:1/1

high low

concrete network(skeleton) predictions

Evaluation 1: AUC scoresArea under Receiver Operator Characteristic

(ROC) curve

AUC=0.5 AUC=1 0.5<AUC≤1

sen

siti

vit

y

inverse specificity

Evaluation 2: TP scores

We set the threshold such that we obtained 5 spurious edges (5 FPs) and counted the corresponding number of true edges (TP count).


5 FP counts


5 FP counts

BN

GGM

RN

Evaluation

• On real experimental cytometric from the RAF signalling pathway for which a gold standard network is known

• On synthetic data simulated from this gold-standard network topology

Evaluation

• On real experimental cytometric from the RAF signalling pathway for which a gold standard network is known

• On synthetic data simulated from the gold-standard network topology

Evaluation: Raf signalling pathway

• Cellular signalling cascade which consists of 11 phosphorylated proteins and phospholipids in human immune systems cell

• Deregulation carcinogenesis

• Extensively studied in the literature gold standard network

‘gold standard RAF pathway‘ according to Sachs et al. (2004)

Raf pathway

11 nodes (proteins) and 20 directed edges

Data

• Intracellular multicolour flow cytometry experiments concentrations of 11 proteins

• 5400 cells have been measured under 9 different cellular conditions (cues)

• We decided to downsample our test data sets to 100 instances - indicative of microarray experiments

Two types of experiments

Evaluation

• On real experimental data, using the gold standard network from the literature

• On synthetic data simulated from this gold-standard network

Raf pathway

Gaussian simulated data

Netbuilder simulated data

DNA + TF`s→ DNA●TF → mRNA → protein

011

mRNAkk

t

mRNA

0][Pr][

Pr22

oteinkmRNAkt

otein

)1()1(0

1

ABBAk

A

A

K

TF ][

B

B

K

TF ][

DNA + TFA DNA●TFA

DNA + TFB DNA●TFB


Steady-state approximation

KA

KB

Generating data using Netbuilder tool

The main idea of Netbuilder is instead of solving the steady state approximation to ODEs explicitly, we approximate them with a qualitatively equivalent

combination of multiplications and sums of sigmoidal transfer functions.


Experimental Results

Synthetic data, observations

Synthetic data, interventions

Cytometry data, observations

Cytometry data, interventions

How can we explain the difference between synthetic

and real data ?

Raf pathway

Regulation of Raf-1 by Direct Feedback Phosphorylation. Molecular Cell, Vol. 17, 2005 Dougherty et al

Disputed structure of the gold-standard network

Complications with real data

• Interventions are not “ideal” owing to negative feedback loops.

• Putative negative feedback loops: Can we trust our gold-standard network?

Stabilisationthrough negative feedback loops inhibition

Conclusions 1

• BNs and GGMs outperform RNs, most notably on Gaussian data.

• No significant difference between BNs and GGMs on observational data.

• For interventional data, BNs clearly outperform GGMs and RNs, especially when taking the edge direction (DGE score) rather than just the skeleton (UGE score) into account.

Conclusions 2

Performance on synthetic data better than on real data.

• Real data: more complex• Real interventions are not ideal• Errors in the gold-standard

network

Additional analysis I: Raf pathway

CPDAGs of networks

3/20 directed edges 13/16 directed edges

ORIGINALMODIFIED

ORIGINAL MODIFIEDG

au

ssi

an

Ne

tbu

ilde

r

Some additional analysis II

Thank you

References:

Butte, A.S. and Kohane, I.S. (2003): Relevance networks: A first step toward finding genetic regulatory networks within microarray data. In Parmigiani, G., Garett, E.S., Irizarry, R.A. und Zeger, S.L. editors, The analysisof Gene Expression Data, pages 428-446. Springer.

Doughtery, M.K. et al. (2005): Regulation of Raf-1 by Direct Feedback Phosphorylation. Molecular Cell, 17, 215-227.

Friedman, N. and Koller, D. (2003): Being Bayesian about network structure.Machine Learning, 50:95-126.

Madigan, D. and York, J. (1995): Bayesian graphical models for discrete data. International Statistical Review, 63:215-232.

Sachs, K., Perez, O., Pe`er, D., Lauffenburger, D.A., Nolan, G.P. (2004):Protein-signaling networks derived from multiparameter single-cell data. Science, 308:523-529.

Schäfer, J. and Strimmer, K. (2005): A shrinkage approach to large-scalecovariance matrix estimation and implications for functional genomics. Statistical Applications in Genetics and Molecular Biology, 4(1): Article 32.

Reverse engineering gene and protein regulatory networks using Graphical Models.

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