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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 strength
Unsigned 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. Intuitive
2. Measuring linear relationships avoids the pitfall of overfitting
3. Because many studies have limited numbers of arrays hard to estimate non-linear relationships
4. Works well in practice
5. Computationally fast
6. 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
Fre
qu
en
cy
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035
01
00
20
03
00
40
05
00
60
07
00P(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 k
p 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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101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185
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 UDV
U u u u
V v v v
D 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 0.42
ME.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
targets
Five 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
Acknowledgement
Jun 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