简介

WGCNA网络分析以后，我们可以对一些sample构建出共表达的网络，那么这里有一个问题，那就是不同处理下的sample（比方说treat和control），它们的共表达网络是一样的吗？如果不一样，那些产生差异的gene pair是否与这些处理有关系，是否是一种比较重要的核心基因呢？

本篇推文老药新用，这是一篇2010年发表的文章：《DiffCoEx: a simple and sensitive method to find differentially coexpressed gene modules》
这篇文章基于传统的WGCNA构建的网络，来寻找两种处理的差异共表达网络

数学原理

构建差分网络大致分为5个步骤：
step 1：
建立邻接矩阵C，邻接矩阵是由两个gene pair的相关性来表征的，即两个基因之间的权重

step 2：
计算gene pair在两个组别权重的变化程度，并构建邻接变化矩阵 D；处理组为 [1]，对照组为 [2]

其中:

1. C^[1]_ij 代表处理组的邻接矩阵；C^[2]_ij 代表对照组的邻接矩阵
2. β 为WGCNA的计算邻接矩阵的指数 β
3. sign() 为符号函数

如果 d_ij 越高，代表在 A 处理中 gene i 和 gene j 权重（相关性，调控关系）与 在 B 处理中 gene i 和 gene j 权重（相关性，调控关系）的差值越大，即 gene i 与 gene j 的权重（相关性，调控关系）在两个处理组之间具有显著差异

step 3：
由邻接变化矩阵 D 计算相异矩阵 T

理解：假设C^[1]_ij很高，C^[2]_ij很低，则 d_ij 很高，代表在 A 处理中 gene i 和 gene j 权重（相关性，调控关系）与 在 B 处理中 gene i 和 gene j 权重（相关性，调控关系）的差值越大，即 gene i 与 gene j 的权重（相关性，调控关系）在两个处理组之间具有显著差异，此时，t_ij 值就越低

step 4：
进行树聚类，由step3中可知，t_ij 值小代表在 A 处理中 gene i 和 gene j 权重（相关性，调控关系）与 在 B 处理中 gene i 和 gene j 权重（相关性，调控关系）有显著差异；即代表在 A 处理中 gene i 和 gene j 权重（相关性，调控关系）与 在 B 处理中 gene i 和 gene j 权重（相关性，调控关系）的差值越大，

step 5：
显著性统计

代码：

作者以一套RNA探针表达谱的数据为例子：
数据在这里下载：http://www.ncbi.nlm.nih.gov/sites/GDSbrowser?acc=GDS2901

#required libraries
#WGCNA package can be found at
#http://www.genetics.ucla.edu/labs/horvath/CoexpressionNetwork/Rpackages/WGCNA
library(WGCNA)          ###used for topological overlap calculation and clustering steps
library(RColorBrewer)   ###used to create nicer colour palettes
library(preprocessCore) ###used by the quantile normalization function
library(flashClust)

#Note: the data can be downloaded from the Gene Expression Omnibus
# http://www.ncbi.nlm.nih.gov/sites/GDSbrowser?acc=GDS2901
data<-as.matrix(read.csv(file="GDS2901.soft",skip=166,row.names=1,sep="\t",header=T))
data<-data[-15924,]
rawData<-matrix(as.numeric(data[,-1]),nrow=15923)
dimnames(rawData)<-dimnames(data[,-1])
#we create an annotation matrix containing the matches between probesets and gene names
anno<-as.matrix(data[-2475,1]) 
normData<-normalize.quantiles(log2(rawData))
dimnames(normData)<-dimnames(rawData)

#we remove the probeset at index 2475 because
#after quantile normalization it has zero variance
#(the probeset has the highest signal of all samples)
normData<-normData[-2475,]  
# 定义组别 1 的邻接矩阵
datC1<-t(normData[,c(1:12,25:36,37:48)]) ### these samples correspond to the Eker mutants.
# Note that since the Eker mutants have two sets of 12 control samples (13:24 and 37:48)
# we discard one to have a symmetric perturbation (carcinogenic vs control) between the two conditions (Eker mutants vs wild-types)
# 定义组别 2 的邻接矩阵
datC2<-t(normData[,49:84]) ###those samples correspond to the wild-types

# 定义 β 值
beta1=6 #user defined parameter for soft thresholding
# 可参考step 2数学原理
AdjMatC1<-sign(cor(datC1,method="spearman"))*(cor(datC1,method="spearman"))^2
AdjMatC2<-sign(cor(datC2,method="spearman"))*(cor(datC2,method="spearman"))^2
diag(AdjMatC1)<-0
diag(AdjMatC2)<-0
collectGarbage()
# 计算相异矩阵 T
dissTOMC1C2=TOMdist((abs(AdjMatC1-AdjMatC2)/2)^(beta1/2))
collectGarbage()

# 对相异矩阵 T 进行聚类
#Hierarchical clustering is performed using the Topological Overlap of the adjacency difference as input distance matrix
geneTreeC1C2 = flashClust(as.dist(dissTOMC1C2), method = "average");

# Plot the resulting clustering tree (dendrogram)
png(file="hierarchicalTree.png",height=1000,width=1000)
plot(geneTreeC1C2, xlab="", sub="", main = "Gene clustering on TOM-based dissimilarity",labels = FALSE, hang = 0.04);
dev.off()

#We now extract modules from the hierarchical tree. This is done using cutreeDynamic. Please refer to WGCNA package documentation for details
dynamicModsHybridC1C2 = cutreeDynamic(dendro = geneTreeC1C2, distM = dissTOMC1C2,method="hybrid",cutHeight=.996,deepSplit = T, pamRespectsDendro = FALSE,minClusterSize = 20);

#Every module is assigned a color. Note that GREY is reserved for genes which do not belong to any differentially coexpressed module
dynamicColorsHybridC1C2 = labels2colors(dynamicModsHybridC1C2)

#the next step merges clusters which are close (see WGCNA package documentation)
mergedColorC1C2<-mergeCloseModules(rbind(datC1,datC2),dynamicColorsHybridC1C2,cutHeight=.2)$color
colorh1C1C2<-mergedColorC1C2

#reassign better colors
colorh1C1C2[which(colorh1C1C2 =="midnightblue")]<-"red"
colorh1C1C2[which(colorh1C1C2 =="lightgreen")]<-"yellow"

colorh1C1C2[which(colorh1C1C2 =="cyan")]<-"orange"
colorh1C1C2[which(colorh1C1C2 =="lightcyan")]<-"green"
# Plot the dendrogram and colors underneath
png(file="module_assignment.png",width=1000,height=1000)
plotDendroAndColors(geneTreeC1C2, colorh1C1C2, "Hybrid Tree Cut",dendroLabels = FALSE, hang = 0.03,addGuide = TRUE, guideHang = 0.05,main = "Gene dendrogram and module colors cells")
dev.off()

#We write each module to an individual file containing affymetrix probeset IDs
modulesC1C2Merged<-extractModules(colorh1C1C2,datC1,anno,dir="modules",file_prefix=paste("Output","Specific_module",sep=''),write=T)
write.table(colorh1C1C2,file="module_assignment.txt",row.names=F,col.names=F,quote=F)

#We plot to a file the comparative heatmap showing correlation changes in the modules
#The code for the function plotC1C2Heatmap and others can be found below under the Supporting Functions section

plotC1C2Heatmap(colorh1C1C2,AdjMatC1,AdjMatC2, datC1, datC2)
png(file="exprChange.png",height=500,width=500)
plotExprChange(datC1,datC2,colorh1C1C2)
dev.off()

进行显著性分析

#This function computes the dispersion value that
#quantifies the change in correlation between two conditions
#for pair of genes drawn from module c1 and module c2
# in case c1 = c2, the function quantifies the differential coexpression in c1.
#cf Choi and Kendziorski 2009
dispersionModule2Module<-function(c1,c2,datC1,datC2,colorh1C1C2)
{
    if (c1==c2)
    {
       difCor<-(cor(datC1[,which(colorh1C1C2 == c1)],method="spearman")-
       cor(datC2[,which(colorh1C1C2 == c1)],method="spearman"))^2
       n<-length(which(colorh1C1C2  ==c1))
      (1/((n^2 -n)/2)*(sum(difCor)/2))^(.5)
    }
    else if (c1!=c2)
    {
      difCor<-(cor(datC1[,which(colorh1C1C2 == c1)],datC1[,which(colorh1C1C2==c2)],method="spearman")-
              cor(datC2[,which(colorh1C1C2 == c1)],datC2[,which(colorh1C1C2==c2)],method="spearman"))^2
      n1<-length(which(colorh1C1C2  ==c1))
      n2<-length(which(colorh1C1C2  ==c2))
      (1/((n1*n2))*(sum(difCor)))^(.5)
    }
}

# we generate a set of 1000 permuted indexes
permutations<-NULL
for (i in 1:1000)
{
   permutations<-rbind(permutations,sample(1:(nrow(datC1)+nrow(datC2)),nrow(datC1)))
}

# we scale the data in both conditions to mean 0 and variance 1.
d<-rbind(scale(datC1),scale(datC2))

# This function calculates the dispersion value of a module to module coexpression change on permuted data
permutationProcedureModule2Module<-function(permutation,d,c1,c2,colorh1C1C2)
{
  d1<-d[permutation,]
  d2<-d[-permutation,]
  dispersionModule2Module(c1,c2,d1,d2,colorh1C1C2)
}

#We compute all pairwise module to module dispersion values, and generate a null distribution from permuted scaled data
dispersionMatrix<-matrix(nrow=length(unique(colorh1C1C2))-1,ncol=length(unique(colorh1C1C2))-1)
nullDistrib<-list()
i<-j<-0
for (c1 in setdiff(unique(colorh1C1C2),"grey"))
{
  i<-i+1
  j<-0
  nullDistrib[[c1]]<-list()
  for (c2 in setdiff(unique(colorh1C1C2),"grey"))
  {
    j<-j+1
    dispersionMatrix[i,j]<-dispersionModule2Module(c1,c2,datC1,datC2,colorh1C1C2)
    nullDistrib[[c1]][[c2]]<-apply(permutations,1,permutationProcedureModule2Module,d,c2,c1,colorh1C1C2)
  }
}


#We create a summary matrix indicating for each module to module 
#differential coexpression the number of permuted data yielding 
#an equal or higher dispersion.
permutationSummary<-matrix(nrow=8,ncol=8)
colnames(permutationSummary)<-setdiff(unique(colorh1C1C2),"grey")
rownames(permutationSummary)<-setdiff(unique(colorh1C1C2),"grey")
for (i in 1:8) { 
  for (j in 1:8) {
     permutationSummary[i,j]<-length(which(nullDistrib[[i]][[j]] >= dispersionMatrix[i,j]))
                  }
}

#We plot the result (cf supplementary figure 1)
plotMatrix(permutationSummary)

关于相异矩阵 dissTOMC1C2 的理解：

gene_1 与 gene_2 的系数为 0.9731268，代表在 A 处理中 gene 1 和 gene 2 权重（相关性，调控关系）与 在 B 处理中 gene i 和 gene j 权重（相关性，调控关系）的相异系数 t_ij 为 0.9731268（相异系数 t_ij 越大代表：在 A 处理中 gene i 和 gene j 权重（相关性，调控关系）与 在 B 处理中 gene i 和 gene j 权重（相关性，调控关系）差异越小；反之越大）