Data management

Import Cascade Data (repeated measurements on several subjects) from the CascadeData package and turn them into a omics array object. The second line makes sure the CascadeData package is installed.

library(Patterns)
if(!require(CascadeData)){install.packages("CascadeData")}
data(micro_US)
micro_US<-as.omics_array(micro_US[1:100,],time=c(60,90,210,390),subject=6)
str(micro_US)

Get a summay and plots of the data:

summary(micro_US)
plot(micro_US)

Gene selection

There are several functions to carry out gene selection before the inference. They are detailed in the vignette of the package.

Data simulation

Let’s simulate some cascade data and then do some reverse engineering.

We first design the F matrix for T_i = 4 times and Ngrp = 4 groups. The Fmatobject is an array of sizes (T_i, T − i, Ngrp²) = (4, 4, 16).

Ti<-4
Ngrp<-4

Fmat=array(0,dim=c(Ti,Ti,Ngrp^2))

for(i in 1:(Ti^2)){
  if(((i-1) %% Ti) > (i-1) %/% Ti){
    Fmat[,,i][outer(1:Ti,1:Ti,function(x,y){0<(x-y) & (x-y)<2})]<-1
    }
}

The Patterns function CascadeFinit is an utility function to easily define such an F matrix.

Fbis=Patterns::CascadeFinit(Ti,Ngrp,low.trig=FALSE)
str(Fbis)

Check if the two matrices Fmat and Fbis are identical.

print(all(Fmat==Fbis))

End of F matrix definition.

Fmat[,,3]<-Fmat[,,3]*0.2
Fmat[3,1,3]<-1
Fmat[4,2,3]<-1
Fmat[,,4]<-Fmat[,,3]*0.3
Fmat[4,1,4]<-1
Fmat[,,8]<-Fmat[,,3]

We set the seed to make the results reproducible and draw a scale free random network.

set.seed(1)
Net<-Patterns::network_random(
  nb=100,
  time_label=rep(1:4,each=25),
  exp=1,
  init=1,
  regul=round(rexp(100,1))+1,
  min_expr=0.1,
  max_expr=2,
  casc.level=0.4
)
Net@F<-Fmat
str(Net)

Plot the simulated network.

Patterns::plot(Net, choice="network")

If a gene clustering is known, it can be used as a coloring scheme.

plot(Net, choice="network", gr=rep(1:4,each=25))

Plot the F matrix, for low dimensional F matrices.

plot(Net, choice="F")

Plot the F matrix using the pixmap package, for high dimensional F matrices.

plot(Net, choice="Fpixmap")

We simulate gene expression according to the network that was previously drawn

set.seed(1)
M <- Patterns::gene_expr_simulation(
  omics_network=Net,
  time_label=rep(1:4,each=25),
  subject=5,
  peak_level=200,
  act_time_group=1:4)
str(M)

Get a summay and plots of the simulated data:

summary(M)

plot(M)

Network inferrence

We infer the new network using subjectwise leave one out cross-validation (default setting): all measurements from the same subject are removed from the dataset). The inference is carried out with a general Fshape.

Net_inf_P <- Patterns::inference(M, cv.subjects=TRUE)

Plot of the inferred F matrix

plot(Net_inf_P, choice="F")

Heatmap of the inferred coefficients of the Omega matrix

stats::heatmap(Net_inf_P@omics_network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

Default values fot the F matrices. The Finit matrix (starting values for the algorithm). In our case, the Finitobject is an array of sizes (T_i, T − i, Ngrp²) = (4, 4, 16).

Ti<-4;
ngrp<-4
nF<-ngrp^2
Finit<-array(0,c(Ti,Ti,nF)) 
              for(ii in 1:nF){    
                if((ii%%(ngrp+1))==1){
                  Finit[,,ii]<-0
                } else {
                  Finit[,,ii]<-cbind(rbind(rep(0,Ti-1),diag(1,Ti-1)),rep(0,Ti))+rbind(cbind(rep(0,Ti-1),diag(1,Ti-1)),rep(0,Ti))
                }
              }

The Fshape matrix (default shape for F matrix the algorithm). Any interaction between groups and times are permitted except the retro-actions (a group on itself, or an action at the same time for an actor on another one).

Fshape<-array("0",c(Ti,Ti,nF)) 
for(ii in 1:nF){  
  if((ii%%(ngrp+1))==1){
    Fshape[,,ii]<-"0"
  } else {
    lchars <- paste("a",1:(2*Ti-1),sep="")
    tempFshape<-matrix("0",Ti,Ti)
    for(bb in (-Ti+1):(Ti-1)){
      tempFshape<-replaceUp(tempFshape,matrix(lchars[bb+Ti],Ti,Ti),-bb)
    }
    tempFshape <- replaceBand(tempFshape,matrix("0",Ti,Ti),0)
    Fshape[,,ii]<-tempFshape
  }
}

Any other form can be used. A “0” coefficient is missing from the model. It allows testing the best structure of an “F” matrix and even performing some significance tests of hypothses on the structure of the F matrix.

The IndicFshape function allows to design custom F matrix for cascade networks with equally spaced measurements by specifying the zero and non zero F_ij cells of the F matrix. It is useful for models featuring several clusters of actors that are activated at the time. Let’s define the following indicatrix matrix (action of all groups on each other, which is not a possible real modeling setting and is only used as an example):

TestIndic=matrix(!((1:(Ti^2))%%(ngrp+1)==1),byrow=TRUE,ngrp,ngrp)
TestIndic

For that choice, we get those init and shape F matrices.

IndicFinit(Ti,ngrp,TestIndic)
IndicFshape(Ti,ngrp,TestIndic)

Those F matrices are lower diagonal ones to enforce that an observed value at a given time can only be predicted by a value that was observed in the past only (i.e. neither at the same moment or in the future).

The plotF is convenient to display F matrices. Here are the the displays of the three F matrices we have just introduced.

plotF(Fshape,choice="Fshape")

plotF(CascadeFshape(4,4),choice="Fshape")

plotF(IndicFshape(Ti,ngrp,TestIndic),choice="Fshape")

We now fit the model with an F matrix that is designed for cascade networks.

Specific Fshape

Net_inf_P_S <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4))

Plot of the inferred F matrix

plot(Net_inf_P_S, choice="F")

Heatmap of the coefficients of the Omega matrix of the network. They reflect the use of a special F matrix. It is an example of an F matrix specifically designed to deal with cascade networks.

stats::heatmap(Net_inf_P_S@omics_network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

There are many fitting functions provided with the Patterns package in order to search for specific features for the inferred network such as sparsity, robust links, high confidence links or stable through resampling links. :

• LASSO, from the lars package
• LASSO2, from the glmnet package. An unweighted and a weighted version of the algorithm are available
• SPLS, from the spls package
• ELASTICNET, from the elasticnet package
• stability.c060, from the c060 package implementation of stability selection
• stability.c060.weighted, a new weighted version of the c060 package implementation of stability selection
• robust, lasso from the lars package with light random Gaussian noise added to the explanatory variables
• selectboost.weighted, a new weighted version of the selectboost package implementation of the selectboost algorithm to look for the more stable links against resampling that takes into account the correlated structure of the predictors. If no weights are provided, equal weigths are for all the variables (=non weighted case).

Net_inf_P_Lasso2 <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="LASSO2")

Plot of the inferred F matrix

plot(Net_inf_P_Lasso2, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_Lasso2@omics_network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

We create a weighting vector to perform weighted lasso inference.

Weights_Net=slot(Net,"omics_network")
Weights_Net[Net@omics_network!=0]=.1        
Weights_Net[Net@omics_network==0]=1000

Net_inf_P_Lasso2_Weighted <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="LASSO2", priors=Weights_Net)

Plot of the inferred F matrix

plot(Net_inf_P_Lasso2_Weighted, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_Lasso2_Weighted@omics_network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

Net_inf_P_SPLS <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="SPLS")

Plot of the inferred F matrix

plot(Net_inf_P_SPLS, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_SPLS@omics_network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

Net_inf_P_ELASTICNET <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="ELASTICNET")

Plot of the inferred F matrix

plot(Net_inf_P_ELASTICNET, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_ELASTICNET@omics_network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

Net_inf_P_stability <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="stability.c060")

Plot of the inferred F matrix

plot(Net_inf_P_stability, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_stability@omics_network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

Net_inf_P_StabWeight <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="stability.c060.weighted", priors=Weights_Net)

Plot of the inferred F matrix

plot(Net_inf_P_StabWeight, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_StabWeight@omics_network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

Net_inf_P_Robust <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="robust")

Plot of the inferred F matrix

plot(Net_inf_P_Robust, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_Robust@omics_network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

Weights_Net_1 <- Weights_Net
Weights_Net_1[,] <- 1
Net_inf_P_SelectBoost <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="selectboost.weighted",priors=Weights_Net_1)

detach("package:Patterns", unload=TRUE)
library(Patterns)

Plot of the inferred F matrix

plot(Net_inf_P_SelectBoost, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_SelectBoost@omics_network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

Net_inf_P_SelectBoostWeighted <- Patterns::inference(M, Finit=CascadeFinit(4,4), Fshape=CascadeFshape(4,4), fitfun="selectboost.weighted",priors=Weights_Net)

detach("package:Patterns", unload=TRUE)
library(Patterns)

Plot of the inferred F matrix

plot(Net_inf_P_SelectBoostWeighted, choice="F")

Heatmap of the coefficients of the Omega matrix of the network

stats::heatmap(Net_inf_P_SelectBoostWeighted@omics_network, Rowv = NA, Colv = NA, scale="none", revC=TRUE)

###Post inference network analysis Such an analysis is only required if the model was not fitted using the stability selection or the selectboost algorithm.

Create an animation of the network with increasing cutoffs with an animated .gif format or a html webpage. See the the webpage of the Patterns package for the animation results at the .gif and .html formats.

data(network)
sequence<-seq(0,0.2,length.out=20)
evolution(network,sequence,type.ani = "gif")
evolution(network,sequence,type.ani = "html")

Evolution of some properties of a reverse-engineered network with increasing cut-off values.

data(Net)
data(Net_inf_PL)

#Comparing true and inferred networks
Crit_values=NULL

#Here are the cutoff level tested
test.seq<-seq(0,max(abs(Net_inf_PL@omics_network*0.9)),length.out=200)
for(u in test.seq){
    Crit_values<-rbind(Crit_values,Patterns::compare(Net,Net_inf_PL,u))
}
matplot(test.seq,Crit_values,type="l",ylab="Criterion value",xlab="Cutoff level",lwd=2)
legend(x="topleft", legend=colnames(Crit_values), lty=1:5,col=1:5,ncol=2,cex=.9)

We switch to data that were derived from the inferrence of a real biological network and try to detect the optimal cutoff value: the best cutoff value for a network to fit a scale free network. The cutoff was validated only single group cascade networks (number of actors groups = number of timepoints) and for genes dataset. Instead of the cutoff function, manual curation or the stability selection or the selectboost algorithm should be used.

data("networkCascade")
set.seed(1)
cutoff(networkCascade)

Analyze the network with a cutoff set to the previouly found 0.133 optimal value.

analyze_network(networkCascade,nv=0.133)

data(Selection)
plot(networkCascade,nv=0.133, gr=Selection@group)

Perform gene selection

Import data.

library(Patterns)
library(CascadeData)
data(micro_S)
micro_S<-as.omics_array(micro_S,time=c(60,90,210,390),subject=6,gene_ID=rownames(micro_S))
data(micro_US)
micro_US<-as.omics_array(micro_US,time=c(60,90,210,390),subject=6,gene_ID=rownames(micro_US))

Select early genes (t1 or t2):

Selection1<-Patterns::geneSelection(x=micro_S,y=micro_US,20,wanted.patterns=rbind(c(0,1,0,0),c(1,0,0,0),c(1,1,0,0)))

Section genes with first significant differential expression at t1:

Selection2<-geneSelection(x=micro_S,y=micro_US,20,peak=1)

Section genes with first significant differential expression at t2:

Selection3<-geneSelection(x=micro_S,y=micro_US,20,peak=2)

Select later genes (t3 or t4)

Selection4<-geneSelection(x=micro_S,y=micro_US,50,
wanted.patterns=rbind(c(0,0,1,0),c(0,0,0,1),c(1,1,0,0)))

Merge those selections:

Selection<-unionOmics(Selection1,Selection2)
Selection<-unionOmics(Selection,Selection3)
Selection<-unionOmics(Selection,Selection4)
head(Selection)

Summarize the final selection:

summary(Selection)

Plot the final selection:

plot(Selection)

This process could be improved by retrieve a real gene_ID using the bitr function of the ClusterProfiler package or by performing independent filtering using jetset package to only keep at most only probeset (the best one, if there is one good enough) per gene_ID.