Steve Horvath

University of California, Los Angeles

Module preservation statistics

Module preservation is often an essential step in a 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 of interesting modulesRationale: 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

Motivational example: 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

Preservation of modules between human and chimpanzee brain networks

Standard cross-tabulation based statistics have severe disadvantages

Disadvantages1. only applicable for modules defined via a

clustering procedure2. ill suited for making the strong statement

that a module is not preserved

We argue that network based approaches are superior when it comes to studying module preservation

Is my network module preserved and reproducible?

Langfelder et al PloS Comp Biol. 7(1): e1001057.

Broad definition of a 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)

Networkof

cholesterol biosynthesis

genes

Message: female liver network (reference)Looks most similar to male liver network

Question

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

• Answer: network concepts aka statistics

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

iScaled connectivity=Kmax( )

i i ijj i

i

Connectivity k a

k

k

Density

• Density= mean adjacency• Highly related to mean connectivity

( )

( 1) 1

where is the number of network nodes.

iji j ia mean k

Densityn n n

n

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

• E.g. Density in the test set

• cor.k=cor(kref,ktest)

• cor(Aref,Atest)

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)

Module preservation in different types of networks

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?

Network-based module preservation statistics

Several connectivity preservation statisticsFor 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

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

Z=observed−mean permuted

sd permuted

Gene modules in AdiposePermutation test for estimating Z scores

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

Z=observed−mean permuted

sd permuted

Overview of Preservation statistics

Module preservation statistics are often closely related

Red=density statistics

Blue: connectivity statistics

Green: separability statistics

Cross-tabulation based statistics

Message: it makes sense to aggregate the statistics into “composite preservation statistics”Clustering module preservation statistics based on correlations across modules

Composite statistic in correlation networks based on Z statistics

( )( ) ( )

. ( )

Permutation test allows one to estimate Z version of each statistic

. ( . | )

( . | )

Composite connectivity based statistics for correlation networks

qq q

cor Cor q

connect

cor Cor E cor Cor nullZ

Var cor Cor null

Z

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

. . . .( , , , )

Composite density based statistics for correlation networks

( , , , )

Composit

q q q q q

q q q q q

ivity cor Cor cor kME cor A cor kIM

density meanCor meanAdj propVarExpl meanKME

median Z Z Z Z

Z median Z Z Z Z

( ) ( )

( )

e statistic of density and connectivity preservation

2

q q

q connectivity densitysummary

Z ZZ

Gene modules in AdiposeAnalogously define composite statistic:

medianRank

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

Summary 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)

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.

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

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

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).

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 combinations

Data provided by Jake Lusis

Networkof

cholesterol biosynthesis

genes

Message: female liver network (reference)Looks most similar to male liver network

Note that Zsummaryis highest in the male liver network

Jeremy Miller, et al Dan Geschwind (2010) Divergence of human and mouse brain

transcriptome highlights Alzheimer disease pathways.

PNAS 2010

Slide from Jeremy Miller

Why compare human and mouse brain transcription?

• 1) Module membership (kME) in conserved modules may be used to identify reliable markers for cell types and cellular components.

• 2) Studying differences in network organization could provide a basis for better understanding diseases enriched in human populations, such as Alzheimer’s Disease

Slide from J. Miller

Co-expression modules based on multiple human

and mouse gene expression data

Human Brain Modules

Mouse Brain Modules

Human modulesM7h and M9hwere enriched with AD genes

These modulescould not be found In mouse brains

Preservation of human network modules in mouse brains

Zsummary

Human specific modules M9h and M7h are related to AD

• Module preservation analysis identified two highly human-specific module, M9h and M7h– No clear functional annotation

• Guilt by association approaches show these modules are related to neurodegenerative dementias– M9H showed significant overlap with an Alzheimer’s

disease module that was identified using independent data sets run on different brain regions, on different platforms, and in different labs

– M7h contained two intramodular hub genes related to AD and frontotemporal dementia (FTD) in humans: GSK3β and tau

• These two modules provide key targets for furthering our understanding of neurodegenerative dementias

Genetic Programs in Human and Mouse Early Embryos Revealed

by Single-Cell RNA-Sequencing

Zhigang Xue, Kevin Huang, Xiaofei Ye,

et al

Guoping Fan

Background

• Mammalian preimplantation development is a complex process involving dramatic changes in the transcriptional architecture.

• Through single-cell RNA-sequencing (RNA-seq), we report here a comprehensive analysis of transcriptome dynamics from oocyte to morula in both human and mouse embryos.

PCA of RNA seq data reveals known trajectory

WGCNA analysis

Module eigengenes vs stages

Module preservation analysis

• General information on weighted correlation networks• Google search

– “WGCNA”– “weighted gene co-expression network”

R function modulePreservation is part of WGCNA package

Tutorials: preservation between human and chimp brains

www.genetics.ucla.edu/labs/horvath/CoexpressionNetwork/ModulePreservation

Implementation and R software tutorials, WGCNA R library

Network Methods for Describing Sample

Relationships in Genomic Datasets: Application to

Huntington's Disease

• Michael C Oldham et al BMC Syst Biol. 2012 PMID: 22691535

Rich but complex HD data• Affymetrix microarray data from “the HD study”

– Hodges et al: Regional and cellular gene expression changes in human Huntington’s disease brain. Hum Mol Genet 2006, 15(6):965-977

• Brain samples of patients with HD (n = 44 individuals) and unaffected controls (n = 36 individuals, matched for age and sex)

• caudate nucleus (CN), cerebellum (CB), primary motor cortex (Brodmann’s area 4; BA4), and prefrontal cortex (Brodmann’s area 9; BA9)

• across five grades using Vonsattel’s neuropath criteria• Further, age, sex, the country where the experiment was

performed (samples were processed in the United States and New Zealand) and the microarray hybridization batch

Defining sample adjacency

Why define this sample network adjacency measure?

• Our proposed sample adjacency measure (based on β = 2) also has several other advantages. – it preserves the sign of the correlation – while any other power β could be used, the

choice of β = 2 results in an adjacency measure that is close to the correlation when the correlation is large (e.g. larger than 0.6, which is often the case among samples in microarray data).

• The adjacency measure allows one to define network concepts.

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

– Scaled connectivity:

i ijjk a

Clustering Coefficient

Measures the cliquishness of a particular node« A node is cliquish if its neighbors know each other »

Clustering Coef of the black node = 0

Clustering Coef = 1

,

22

il lm mil i m i li

il ill i l i

a a aClusterCoef

a a

This generalizes directly to weightednetworks (Zhang and Horvath 2005)

C(k) curve is a plot of scaled clustering coefficient versus scaled connectivity

Sample network concepts reveal the profound effect of Huntington’s disease in caudate nucleus.

Summary sample network• Z.k is a very useful measure for finding array outliers.• The correlation cor(K,C) between the connectivity and the

clustering coefficient (two important network concepts) is a sensitive indicator of homogeneity among biological samples. – It can distinguish biologically meaningful relationships among

subgroups of samples. – Advantage: This measure can highlight differences that cannot be

found using differential expression– Disadvantage: It requires some work to figure out which genes

lead to this effect. – Here: effect is concentrated in specific modules of genes

• Sample network approach is implemented in an R function and tutorial

Acknowledgement

Current and former lab members: • Peter Langfelder first author on many related

articles• Jason Aten, Chaochao (Ricky) Cai, Jun Dong, Tova

Fuller, Ai Li, Wen Lin, Michael Mason, Jeremy Miller, Mike Oldham, Chris Plaisier, Anja Presson, Lin Song, Kellen Winden, Yafeng Zhang, Andy Yip, Bin Zhang

• Colleagues/Collaborators• Neuroscience: Dan Geschwind,

Giovanni Coppola, Jeremy Miller, Mike Oldham, Roel Ophoff

• Mouse: Jake Lusis, Tom Drake