Extended Overview of Weighted Gene Co-Expression

Network Analysis (WGCNA)Steve Horvath

University of California, Los Angeles

Book on weighted networks

E-book is often freely accessible if the library has a subscription to Springer books

Contents• How to construct a weighted gene co-expression

network?• Why use soft thresholding?• How to detect network modules?• How to relate modules to an external clinical trait?• What is intramodular connectivity?• How to use networks for gene screening?• How to integrate networks with genetic marker

data?• What is weighted gene co-expression network

analysis (WGCNA)?

Standard microarray analyses seek to identify ‘differentially expressed’ genes

• Each gene is treated as an individual entity

• Often misses the forest for the trees: Fails to recognize that thousands of genes can be organized into relatively few modules

Control Experimental

Philosophy of Weighted Gene Co-Expression Network Analysis

• Understand the “system” instead of reporting a list of individual parts– Describe the functioning of the engine instead

of enumerating individual nuts and bolts• Focus on modules as opposed to individual

genes– this greatly alleviates multiple testing problem

• Network terminology is intuitive to biologists

How to construct a weighted gene co-expression network?

Bin Zhang and Steve Horvath (2005) "A General Framework for Weighted Gene Co-Expression Network Analysis", Statistical Applications in Genetics and Molecular Biology: Vol. 4: No. 1,

Article 17.

Network=Adjacency Matrix

• A network can be represented by an adjacency matrix, A=[aij], that encodes whether/how a pair of nodes is connected. – A is a symmetric matrix with entries in

[0,1] – For unweighted network, entries are 1 or

0 depending on whether or not 2 nodes are adjacent (connected)

– For weighted networks, the adjacency matrix reports the connection strength between gene pairs

Steps for constructing a

co-expression network

A) Microarray gene expression data

B) Measure concordance of gene expression with a Pearson correlation

C) The Pearson correlation matrix is either dichotomized to arrive at an adjacency matrix unweighted network

Or transformed continuously with the power adjacency function weighted network

Weighted Network View Unweighted View

•All genes are connected Some genes are connected•Connection Widths=Connection strenghts All connections are equal

Our `holistic’ view….

Hard thresholding may lead to an information loss. If two genes are correlated with r=0.79, they are deemed unconnected with regard to a hard threshold of tau=0.8

Power adjacency function for constructing unsigned and signed weighted gene co-expr. networks

Unsigned network, absolute value

| ( , ) |

Signed network preserves sign info

| 0.5 0.5 ( , ) |

ij i j

ij i j

a cor x x

a cor x x

Default values: beta=6 for unsigned and beta=12 for signed networks.Alternatively, use the “scale free topology criterion” described in Zhang and Horvath 2005.

Comparing adjacency functions for transforming the correlation into a

measure of connection strengthUnsigned Network Signed Network

Question 1:Should network construction account for the sign of the co-expression relationship?

Answer: Overall, recent applications have convinced me that signed networks are preferable.

• For example, signed networks were critical in a recent stem cell application

• Michael J Mason, Kathrin Plath, Qing Zhou, SH (2009) Signed Gene Co-expression Networks for Analyzing Transcriptional Regulation in Murine Embryonic Stem Cells. BMC Genomics 2009, 10:327

Why construct a co-expression network based on the correlation coefficient ?

1. Intuitive2. Measuring linear relationships avoids the pitfall

of overfitting3. Because many studies have limited numbers

of arrays hard to estimate non-linear relationships

4. Works well in practice5. Computationally fast6. Leads to reproducible research

Relationship between Correlation and Mutual Information

• Standardized mutual information represents soft-thresholding of correlation.

Why soft thresholding as opposed to hard thresholding?

1. Preserves the continuous information of the co-expression information

2. Results tend to be more robust with regard to different threshold choices

But hard thresholding has its own advantages: In particular, graph theoretic algorithms from the computer science community can be applied to the resulting networks

Questions:How should we choose the power beta or a hard threshold?Or more generally the parameters of an adjacency function?

IDEA: use properties of the connectivity distribution

Generalized Connectivity

• Gene connectivity = row sum of the adjacency matrix– For unweighted networks=number of direct neighbors– For weighted networks= sum of connection strengths to

other nodes

i ijjk a

Approximate scale free topology is a fundamental property of such networks

(Barabasi et al)• It entails the presence of hub nodes that are

connected to a large number of other nodes• Such networks are robust with respect to the

random deletion of nodes but are sensitive to the targeted attack on hub nodes

• It has been demonstrated that metabolic networks exhibit scale free topology at least approximately.

P(k) vs k in scale free networks

Frequency Distribution of Connectivity

Connectivity k

Freq

uenc

y

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

010

020

030

040

050

060

070

0P(k) •Scale Free Topology refers to the frequency distribution of the connectivity k •p(k)=proportion of nodes that have connectivity k•p(k)=Freq(discretize(k,nobins))

How to check Scale Free Topology?

Idea: Log transformation p(k) and k and look at scatter plots

Linear model fitting R^2 index can be used to quantify goodness of fit

Generalizing the notion of scale free topology

Barabasi (1999)Csanyi-Szendroi (2004)Horvath, Dong (2005)

0 1

0 1 2

0 1 2

ScaleFree Topology means log( ( )) log( )ExponentiallyTrunated SFT means log( ( )) log( )LogLogSFT means log( ( )) log( ) log(log( ))

p k c c kp k c c k c k

p k c c k c k

Motivation of generalizations: using weak general assumptions, we have proven that gene co-expression networks satisfy these distributions approximately.

Checking Scale Free Topology in the Yeast Network

• Black=Scale Free• Red=Exp. Truncated• Green=Log Log SFT

0.4 0.6 0.8 1.0 1.2 1.4 1.6

-2.5

-2.0

-1.5

-1.0

log10(k)

log1

0(p(

k))

power=6 , slope= -1.6 , scaleR2= 0.73 , loglogR2= 0.95 , trunc.R^2= 0.9

The scale free topology criterion for choosing the parameter

values of an adjacency function.A) CONSIDER ONLY THOSE PARAMETER VALUES IN THE

ADJACENCY FUNCTION THAT RESULT IN APPROXIMATE SCALE FREE TOPOLOGY, i.e. high scale free topology fitting index R^2

B) SELECT THE PARAMETERS THAT RESULT IN THE HIGHEST MEAN NUMBER OF CONNECTIONS

• Criterion A is motivated by the finding that most metabolic networks (including gene co-expression networks, protein-protein interaction networks and cellular networks) have been found to exhibit a scale free topology

• Criterion B leads to high power for detecting modules (clusters of genes) and hub genes.

Criterion A is measured by the linear model fitting index R2 Step AF (tau) Power AF (b)tau= b=

Trade-off between criterion A (R2) and criterion B (mean no. of

connections) when varying the power b

Power AF(s)=sb

criterion A: SFT model fit R^2 criterion B: mean connectivity

Trade-off between criterion A and B when varying tau

criterion A criterion B

Step Function: I(s>tau)

How to detect network modules(clusters) ?

How to cut branches off a tree?

Langfelder P, Zhang B et al (2007) Defining clusters from a hierarchical cluster tree: the Dynamic Tree Cut library for R. Bioinformatics 2008 24(5):719-720

Module=branch of a cluster tree

Dynamic hybrid branch cutting method combinesadvantages of hierarchical clustering and pam clustering

Cluster Dendrogram & Module Definition

Module Definition• Numerous methods have been developed • Here, we use average linkage hierarchical

clustering coupled with the topological overlap dissimilarity measure.

• Once a dendrogram is obtained from a hierarchical clustering method, we choose a height cutoff to arrive at a clustering.

• Modules correspond to branches of the dendrogram

The topological overlap dissimilarity is used as input of hierarchical clustering

• Generalized in Zhang and Horvath (2005) to the case of weighted networks

• Generalized in Yip and Horvath (2006) to higher order interactions

min( , ) 1

iu uj iju

iji j ij

a a a

TOMk k a

1ij ijDistTOM TOM

Using the topological overlap matrix (TOM) to cluster genes

– Here modules correspond to branches of the dendrogram

TOM plot

Hierarchical clustering dendrogram

TOM matrix

Module:Correspond to branches

Genes correspond to rows and columns

Topological Overlap Plot Gene Functions

Multi Dimensional Scaling Traditional View

Different Ways of Depicting Gene Modules

1) Rows and columns correspond to genes2) Red boxes along diagonal are modules3) Color bands=modules

Idea:Use network distance in MDS

Heatmap view of module

Rows=Genes Color band indicates module membership

Columns= tissue samples

Message: characteristic vertical bands indicate tight co-expression of module genes

brown

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brown

-0.1

0.0

0.1

0.2

0.3

0.4

Module Eigengene= measure of over-expression=average redness

Rows,=genes, Columns=microarray

The brown module eigengenes across samples

Using the singular value decomposition to define (module) eigngenes

1 2

1 2

1 2

1

(q)

Scale the gene expressions profiles (columns)( )

( )( )

(| |,| |, ,| |)Message: u is the (first) eigengene E

If datX corresponds to the q-th module then

T

m

m

m

datX scale datX

datX UDVU u u uV v v vD diag d d d

(q)E is the q-th module eigengene.

Martingale.Res

-0.2 0.2 -0.1 0.2 -0.1 0.2

-2.0

0.0

-0.2

0.2

0.08

ME.blue

0.19 0.22

ME.brow n

-0.2

0.2

-0.1

0.2

0.14 0.27 0.42ME.green

0.09 0.78 0.09 0.55ME.grey

-0.2

0.1

-0.1

0.2

0.12 0.39 0.41 0.67 0.72ME.turquoise

-2.0 0.0

0.01 0.07

-0.2 0.2

0.13 0.08

-0.2 0.1

0.04 0.34

-0.3 0.0-0

.30.

0

ME.yellow

Module eigengenes can be used to determine whether 2 modulesare correlated. If correlation of MEs is high-> consider merging.

Eigengene networks Langfelder, Horvath (2007) BMC Systems Biology

How to relate modules to external data?

Clinical trait (e.g. case-control status) gives rise to a gene significance

measure• Abstract definition of a gene significance

measure– GS(i) is non-negative, – the bigger, the more *biologically* significant for the i-

th gene Equivalent definitions• GS.ClinicalTrait(i) = |cor(x(i),ClinicalTrait)| where

x(i) is the gene expression profile of the i-th gene• GS(i)=|T-test(i)| of differential expression

between groups defined by the trait• GS(i)=-log(p-value)

A SNP marker naturally gives rise to a measure of gene

significance

• Additive SNP marker coding: AA->2, AB->1, BB->0

• Absolute value of the correlation ensures that this is equivalent to AA->0, AB->1, BB->2– Dominant or recessive coding may be more

appropriate in some situations– Conceptually related to a LOD score at the SNP

marker for the i-th gene expression trait

GS.SNP(i) = |cor(x(i), SNP)|.

A gene significance naturally gives rise to a module significance measure

• Define module significance as mean gene significance

• Often highly related to the correlation between module eigengene and trait

Important Task in Many Genomic Applications:Given a network (pathway) of interacting genes how to

find the central players?

Which of the following mathematicians had the biggest

influence on others?

Connectivity can be an important variable for identifying important nodes

**Slide courtesy of A Barabasi

Flight connections and hub airports

The nodes with the largest number of links (connections) are most important!

What is intramodular connectivity?

Intramodular Connectivity

• Intramodular connectivity kIN with respect to a given module (say the Blue module) is defined as the sum of adjacencies with the members of the module.– For unweighted networks=number of direct links to

intramodular nodes– For weighted networks= sum of connection strengths to

intramodular nodes

{ }BlueModulei ijj BlueModule

kIN a

Gene significance versus intramodular connectivity kIN

How to use networks for gene screening?

Intramodular connectivity kIN versus gene significance GS

• Note the relatively high correlation between gene significance and intramodular connectivity in some modules

• In general, kIN is a more reliable measure than GS

• In practice, a combination of GS and k should be used

• Module eigengene turns out to be the most highly connected gene (under mild assumptions)

What is weighted gene co-expression network analysis?

Construct a networkRationale: make use of interaction patterns between genes

Identify modulesRationale: module (pathway) based analysis

Relate modules to external informationArray Information: Clinical data, SNPs, proteomicsGene Information: gene ontology, EASE, IPARationale: find biologically interesting modules

Find the key drivers in interesting modulesTools: intramodular connectivity, causality testingRationale: experimental validation, therapeutics, biomarkers

Study Module Preservation across different data Rationale: • Same data: to check robustness of module definition• Different data: to find interesting modules.

What is different from other analyses?• Emphasis on modules (pathways) instead of individual

genes– Greatly alleviates the problem of multiple comparisons

• Less than 20 comparisons versus 20000 comparisons• Use of intramodular connectivity to find key drivers

– Quantifies module membership (centrality)– Highly connected genes have an increased chance of validation

• Module definition is based on gene expression data– No prior pathway information is used for module definition– Two module (eigengenes) can be highly correlated

• Emphasis on a unified approach for relating variables– Default: power of a correlation– Rationale:

• puts different data sets on the same mathematical footing • Considers effect size estimates (cor) and significance level• p-values are highly affected by sample sizes (cor=0.01 is highly

significant when dealing with 100000 observations)• Technical Details: soft thresholding with the power adjacency

function, topological overlap matrix to measure interconnectedness

Case Study 1:Finding brain cancer genes

Horvath S, Zhang B, Carlson M, Lu KV, Zhu S, Felciano RM, Laurance MF, Zhao W, Shu, Q, Lee Y, Scheck AC, Liau LM, Wu H, Geschwind DH, Febbo PG, Kornblum HI, Cloughesy TF, Nelson SF, Mischel PS (2006) "Analysis of Oncogenic Signaling Networks in Glioblastoma

Identifies ASPM as a Novel Molecular Target", PNAS | November 14, 2006 | vol. 103 | no. 46

Topological Overlap Plot Gene Functions

Multi Dimensional Scaling Traditional View

Different Ways of Depicting Gene Modules

1) Rows and columns correspond to genes2) Red boxes along diagonal are modules3) Color bands=modules

55 Brain Tumors VALIDATION DATA: 65 Brain Tumors

Normal brain (adult + fetal)

Normal non-CNS tissues

Comparing the Module Structure in Cancer and Normal tissues

Messages: 1)Cancer modules can be independently validated2) Modules in brain cancer tissue can also be found in normal, non-brain tissue.--> Insights into the biology of cancer

Mean Prognostic Significance of Module Genes

Message: Focus the attention on the brown module genes

Module hub genes predict cancer survival

1. Cox model to regress survival on gene expression levels

2. Defined prognostic significance as –log10(Cox-p-value) the survival association between each gene and glioblastoma patient survival

3. A module-based measure of gene connectivity significantly and reproducibly identifies the genes that most strongly predict patient survival

Test set – 55 gbmsr = 0.56; p-2.2 x 10-16

Validation set – 65 gbmsr = 0.55; p-2.2 x 10-16

The fact that genes with high intramodular connectivity are more likely to be prognostically significant facilitates a novel

screening strategy for finding prognostic genes

• Focus on those genes with significant Cox regression p-value AND high intramodular connectivity. – It is essential to to take a module centric view: focus on

intramodular connectivity of disease related module

• Validation success rate= proportion of genes with independent test set Cox regression p-value<0.05.

• Validation success rate of network based screening approach (68%)

• Standard approach involving top 300 most significant genes: 26%

Validation success rate of gene expressions in independent data

26%

67%

300 most significant genes Network based screening (Cox p-value<1.3*10-3) p<0.05 and

high intramodular connectivity

The network-based approach uncovers novel therapeutic

targetsFive of the top six hub genes in the mitosis module are already known cancer targets: topoisomerase II, Rac1, TPX2, EZH2 and KIF14.We hypothesized that the 6-th gene ASPM gene is novel therapeutic target. ASPM encodes the human ortholog of a drosophila mitotic spindle protein.

Biological validation: siRNA mediated inhibition of ASPM

MC Oldham, S Horvath, DH Geschwind (2006) Conservation and evolution of gene

co-expression networks in human and chimpanzee brain. PNAS

Case Study 2

What changed?

Image courtesy of Todd Preuss (Yerkes National Primate Research Center)

1 Cheng, Z. et al. Nature 437, 88-93 (2005)

• Despite pronounced phenotypic differences, genomic similarity is ~96% (including single-base substitutions and indels)1

– Similarity is even higher in protein-coding regions

Assessing the contribution of regulatory changes to human evolution

• Hypothesis: Changes in the regulation of gene expression were critical during recent human evolution (King & Wilson, 1975)

• Microarrays are ideally suited to test this hypothesis by comparing expression levels for thousands of genes simultaneously

Raw data from Khaitovich et al., 2004

Mike Oldham

Gene expression is more strongly preserved than gene connectivity

Human Expression Human Connectivity

Chimp Chimp Expression

Cor=0.93 Cor=0.60

Hypothesis: molecular wiring makes us human

A B

Human Chimp

p = 1.33x10-4 p = 8.93x10-4 p = 1.35x10-6 p = 1.33x10-4

Connectivity diverges across brain regions whereas expression does not

Conclusions: chimp/human• Gene expression is highly preserved across species

brains • Gene co-expression is less preserved• Some modules are highly preserved• Gene modules correspond roughly to brain

architecture• Species-specific hubs can be validated in silico

using sequence comparisons

Software and Data Availability

• Sample data and R software tutorials can be found at the following webpage

• http://www.genetics.ucla.edu/labs/horvath/CoexpressionNetwork

AcknowledgementJun Dong, Sud Doss, Tom Drake, Tova Fuller, Anatole

Ghazalpour, Dan Geschwind, Peter Langfelder, Ai Li, Wen Lin, Jake Lusis, Michael Mason, Paul Mischel, Nicole MacLennan, Ed McCabe, Atila Van Nas, Stan Nelson, Mike Oldham, Roel Ophoff, Anja Presson, Lin Wang, Bin Zhang, Wei Zhao

A short methodological summary of the publications.• How to construct a gene co-expression network using the scale free topology criterion? Robustness of network results. Relating a gene

significance measure and the clustering coefficient to intramodular connectivity: – Zhang B, Horvath S (2005) "A General Framework for Weighted Gene Co-Expression Network Analysis", Statistical Applications in

Genetics and Molecular Biology: Vol. 4: No. 1, Article 17• Theory of module networks (both co-expression and protein-protein interaction modules):

– Dong J, Horvath S (2007) Understanding Network Concepts in Modules, BMC Systems Biology 2007, 1:24• What is the topological overlap measure? Empirical studies of the robustness of the topological overlap measure:

– Yip A, Horvath S (2007) Gene network interconnectedness and the generalized topological overlap measure. BMC Bioinformatics 2007, 8:22

• Software for carrying out neighborhood analysis based on topological overlap. The paper shows that an initial seed neighborhood comprised of 2 or more highly interconnected genes (high TOM, high connectivity) yields superior results. It also shows that topological overlap is superior to correlation when dealing with expression data.

– Li A, Horvath S (2006) Network Neighborhood Analysis with the multi-node topological overlap measure. Bioinformatics. doi:10.1093/bioinformatics/btl581

• Gene screening based on intramodular connectivity identifies brain cancer genes that validate. This paper shows that WGCNA greatly alleviates the multiple comparison problem and leads to reproducible findings.

– Horvath S, Zhang B, Carlson M, Lu KV, Zhu S, Felciano RM, Laurance MF, Zhao W, Shu, Q, Lee Y, Scheck AC, Liau LM, Wu H, Geschwind DH, Febbo PG, Kornblum HI, Cloughesy TF, Nelson SF, Mischel PS (2006) "Analysis of Oncogenic Signaling Networks in Glioblastoma Identifies ASPM as a Novel Molecular Target", PNAS | November 14, 2006 | vol. 103 | no. 46 | 17402-17407

• The relationship between connectivity and knock-out essentiality is dependent on the module under consideration. Hub genes in some modules may be non-essential. This study shows that intramodular connectivity is much more meaningful than whole network connectivity:

– "Gene Connectivity, Function, and Sequence Conservation: Predictions from Modular Yeast Co-Expression Networks" (2006) by Carlson MRJ, Zhang B, Fang Z, Mischel PS, Horvath S, and Nelson SF, BMC Genomics 2006, 7:40

• How to integrate SNP markers into weighted gene co-expression network analysis? The following 2 papers outline how SNP markers and co-expression networks can be used to screen for gene expressions underlying a complex trait. They also illustrate the use of the module eigengene based connectivity measure kME.

– Single network analysis: Ghazalpour A, Doss S, Zhang B, Wang S, Plaisier C, Castellanos R, Brozell A, Schadt EE, Drake TA, Lusis AJ, Horvath S (2006) "Integrating Genetic and Network Analysis to Characterize Genes Related to Mouse Weight". PLoS Genetics. Volume 2 | Issue 8 | AUGUST 2006

– Differential network analysis: Fuller TF, Ghazalpour A, Aten JE, Drake TA, Lusis AJ, Horvath S (2007) "Weighted Gene Co-expression Network Analysis Strategies Applied to Mouse Weight", Mammalian Genome. In Press

• The following application presents a `supervised’ gene co-expression network analysis. In general, we prefer to construct a co-expression network and associated modules without regard to an external microarray sample trait (unsupervised WGCNA). But if thousands of genes are differentially expressed, one can construct a network on the basis of differentially expressed genes (supervised WGCNA):

– Gargalovic PS, Imura M, Zhang B, Gharavi NM, Clark MJ, Pagnon J, Yang W, He A, Truong A, Patel S, Nelson SF, Horvath S, Berliner J, Kirchgessner T, Lusis AJ (2006) Identification of Inflammatory Gene Modules based on Variations of Human Endothelial Cell Responses to Oxidized Lipids. PNAS 22;103(34):12741-6

• The following paper presents a differential co-expression network analysis. It studies module preservation between two networks. By screening for genes with differential topological overlap, we identify biologically interesting genes. The paper also shows the value of summarizing a module by its module eigengene.

– Oldham M, Horvath S, Geschwind D (2006) Conservation and Evolution of Gene Co-expression Networks in Human and Chimpanzee Brains. 2006 Nov 21;103(47):17973-8

THE END