Weighted Gene Co-Expression Network Analysis of Multiple Independent Lung Cancer Data Sets

1. Weighted Gene Co-Expression Network Analysis of Multiple Independent Lung Cancer Data Sets Steve Horvath University of California, Los Angeles

2. Contents • Mini review of weighted correlation network analysis (WGCNA) • Module preservation statistics • Application to multiple adenocarcinoma

3. 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 node pairs • Our convention: diagonal elements of A are all 1.

4. Connectivity (aka degree) • Node connectivity = row sum of the adjacency matrix • For unweighted networks=number of direct neighbors • For weighted networks= sum of connection strengths to other nodes

5. Density • Density= mean adjacency • Highly related to mean connectivity

6. How to construct a weighted gene co-expression network?

7. Use power β for soft thresholding a correlation coefficient Default values: β=6 for unsigned and β=12 for signed networks. Zhang et al SAGMB Vol. 4: No. 1, Article 17.

8. Comparing adjacency functions for transforming the correlation into a measure of connection strength Unsigned Network Signed Network

9. Advantages of soft thresholding with the power function • Robustness: Network results are highly robust with respect to the choice of the power β (Zhang et al 2005) • Calibrating different networks becomes straightforward, which facilitates consensus module analysis • Math reason: Geometric Interpretation of Gene Co-Expression Network Analysis. PloS Computational Biology. 4(8): e1000117 • Module preservation statistics are particularly sensitive for measuring connectivity preservation in weighted networks

10. How to detect network modules?

11. Module Definition • Numerous methods have been developed • We often 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

12. 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 combines advantages of hierarchical clustering and pam clustering

13. Question: How does one summarize the expression profiles in a module?Answer: This has been solved.Math answer: module eigengene= first principal componentNetwork answer: the most highly connected intramodular hub geneBoth turn out to be equivalent

14. Module Eigengene= measure of over-expression=average redness Rows,=genes, Columns=microarray The brown module eigengenes across samples

15. Module eigengene is defined by the singular value decomposition of X • X=gene expression data of a module gene expressions (rows) have been standardized across samples (columns)

16. Module detection in very large data sets • Large may mean >25k variables R function blockwiseModules (in WGCNA library) implements 3 steps: • Variant of k-means to cluster variables into blocks • Hierarchical clustering and branch cutting in each block • Merge modules across blocks (based on correlations between module eigengenes)

17. Define 2 alternative measures of intramodular connectivity and describe their relationship.

18. 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 this module. • For unweighted networks=number of direct links to intramodular nodes • For weighted networks= sum of connection strengths to intramodular nodes

19. Eigengene based connectivity, also known as kME or module membership measure kME(i) is simply the correlation between the i-th gene expression profile and the module eigengene. Very useful measure for annotating genes with regard to modules. Module eigengene turns out to be the most highly connected gene

21. Question • How to measure relationships between different networks (e.g. how similar is the female liver network to the male network).

22. Networkof cholesterol biosynthesis genes Message: female liver network (reference) Looks most similar to male liver network

23. Network concepts to measure relationships between networks Numerous network concepts can be used to measure the preservation of network connectivity patterns between a reference network and a test network • cor.k=cor(kref,ktest) • cor(Aref,Atest) • Cor(ClusterCoefref,ClusterCoeftest)

24. Is my network module preserved and reproducible?Langfelder et al PloS Comp Biol. 7(1): e1001057.

25. Network module • Abstract definition of module=subset of nodes in a network. • Thus, a module forms a sub-network in a larger network • Example: module (set of genes or proteins) defined using external knowledge: KEGG pathway, GO ontology category • Example: modules defined as clusters resulting from clustering the nodes in a network • Module preservation statistics can be used to evaluate whether a given module defined in one data set (reference network) can also be found in another data set (test network)

26. In general, studying module preservation is different from studying cluster preservation. • Many statistics for assessing cluster preservation e.g.Kapp AV, Tibshirani R (2007) Are clusters found in one dataset present in another dataset? Biostatistics (2007), 8, 1, pp. 9–31 • But in general network modules are different from clusters (e.g. KEGG pathways may not correspond to clusters in the network). • However, many module preservation statistics lend themselves as cluster preservation statistics and vice versa

27. Module preservation is often an essential step in a network analysis

28. Construct a network Rationale: make use of interaction patterns between genes Identify modules Rationale: module (pathway) based analysis Relate modules to external information Array Information: Clinical data, SNPs, proteomics Gene Information: gene ontology, EASE, IPA Rationale: find biologically interesting modules • Study Module Preservation across different data • Rationale: • Same data: to check robustness of module definition • Different data: to find interesting modules Find the key drivers of interesting modules Rationale: experimental validation, therapeutics, biomarkers

29. Module preservation in different types of networks • One can study module preservation in general networks specified by an adjacency matrix, e.g. protein-protein interaction networks. • However, particularly powerful statistics are available for correlation networks • weighted correlation networks are particularly useful for detecting subtle changes in connectivity patterns. But the methods are also applicable to unweighted networks (i.e. graphs)

30. Network-based module preservation statistics • Input: module assignment in reference data. • Adjacency matrices in reference Aref and test data Atest • Network preservation statistics assess preservation of • 1. network density: Does the module remain densely connected in the test network? • 2. connectivity: Is hub gene status preserved between reference and test networks? • 3. separability of modules: Does the module remain distinct in the test data?

31. Several connectivity preservation statistics For general networks, i.e. input adjacency matrices • cor.kIM=cor(kIMref,kIMtest) • correlation of intramodular connectivity across module nodes • cor.ADJ=cor(Aref,Atest) • correlation of adjacency across module nodes For correlation networks, i.e. input sets are variable measurements • cor.Cor=cor(corref,cortest) • cor.kME=cor(kMEref,kMEtest) One can derive relationships among these statistics in case of weighted correlation network

32. Choosing thresholds for preservation statistics based on permutation test For correlation networks, we study 4 density and 4 connectivity preservation statistics that take on values <= 1 Challenge: Thresholds could depend on many factors (number of genes, number of samples, biology, expression platform, etc.) Solution: Permutation test. Repeatedly permute the gene labels in the test network to estimate the mean and standard deviation under the null hypothesis of no preservation. Next we calculate a Z statistic

33. Permutation test for estimating Z scores Gene modules in Adipose • For each preservation measure we report the observed value and the permutation Z score to measure significance. • Each Z score provides answer to “Is the module significantly better than a random sample of genes?” • Summarize the individual Z scores into a composite measure called Z.summary • Zsummary < 2 indicates no preservation, 2<Zsummary<10 weak to moderate evidence of preservation, Zsummary>10 strong evidence

34. Details are provided below and in the paper…

35. Module preservation statistics are often closely related Message: it makes sense to aggregate the statistics into “composite preservation statistics” Clustering module preservation statistics based on correlations across modules Red=density statistics Blue: connectivity statistics Green: separability statistics Cross-tabulation based statistics

36. Composite statistic in correlation networks based on Z statistics

37. Analogously define composite statistic: medianRank Gene modules in Adipose • Based on the ranks of the observed preservation statistics • Does not require a permutation test • Very fast calculation • Typically, it shows no dependence on the module size

38. Summary preservation • Standard cross-tabulation based statistics are intuitive • Disadvantages: i) only applicable for modules defined via a module detection procedure, ii) ill suited for ruling out module preservation • Network based preservation statistics measure different aspects of module preservation • Density-, connectivity-, separability preservation • Two types of composite statistics: Zsummary and medianRank. • Composite statistic Zsummary based on a permutation test • Advantages: thresholds can be defined, R function also calculates corresponding permutation test p-values • Example: Zsummary<2 indicates that the module is *not* preserved • Disadvantages: i) Zsummary is computationally intensive since it is based on a permutation test, ii) often depends on module size • Composite statistic medianRank • Advantages: i) fast computation (no need for permutations), ii) no dependence on module size. • Disadvantage: only applicable for ranking modules (i.e. relative preservation)

39. Application:Modules defined as KEGG pathways.Connectivity patterns (adjacency matrix) is defined as signed weighted co-expression network.Comparison of human brain (reference) versus chimp brain (test) gene expression data.

40. Preservation of KEGG pathwaysmeasured using the composite preservation statistics Zsummary and medianRank • Humans versus chimp brain co-expression modules Apoptosis module is least preserved according to both composite preservation statistics

41. Apoptosis module has low value of cor.kME=0.066

42. Visually inspect connectivity patterns of the apoptosis module in humans and chimpanzees Weighted gene co-expression module. Red lines=positive correlations, Green lines=negative cor Note that the connectivity patterns look very different. Preservation statistics are ideally suited to measure differences in connectivity preservation

43. Literature validation:Neuron apoptosis is known to differ between humans and chimpanzees • It has been hypothesized that natural selection for increased cognitive ability in humans led to a reduced level of neuron apoptosis in the human brain: • Arora et al (2009) Did natural selection for increased cognitive ability in humans lead to an elevated risk of cancer? Med Hypotheses 73: 453–456. • Chimpanzee tumors are extremely rare and biologically different from human cancers • A scan for positively selected genes in the genomes of humans and chimpanzees found that a large number of genes involved in apoptosis show strong evidence for positive selection (Nielsen et al 2005 PloS Biol).

44. Application:Studying the preservation of human brain co-expression modules in chimpanzee brain expression data. Modules defined as clusters(branches of a cluster tree)Data from Oldam et al 2006

45. Preservation of modules between human and chimpanzee brain networks

46. 2 composite preservation statistics Zsummary is above the threshold of 10 (green dashed line), i.e. all modules are preserved. Zsummary often shows a dependence on module size which may or may not be attractive (discussion in paper) In contrast, the median rank statistic is not dependent on module size. It indicates that the yellow module is most preserved

47. Application: Studying the preservation of a female mouse liver module in different tissue/gender combinations.Module: genes of cholesterol biosynthesis pathway Network: signed weighted co-expression networkReference set: female mouse liverTest sets: other tissue/gender combinationsData provided by Jake Lusis

48. Networkof cholesterol biosynthesis genes Message: female liver network (reference) Looks most similar to male liver network

49. Note that Zsummary is highest in the male liver network

50. Application:Modules defined as KEGG pathways.Comparison of human brain (reference) versus chimp brain (test) gene expression data.Connectivity patterns (adjacency matrix) is defined as signed weighted co-expression network.