fssemR: Fused Sparse Structural Equation Models to Jointly Infer Gene Regulatory Networks

In fssemR: Fused Sparse Structural Equation Models to Jointly Infer Gene Regulatory Network

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In this vignette, we introduce the functionality of the fssemR package to estimate the differential gene regulatory network by gene expression and genetic perturbation data. To meet the space and time constraints in building this vignette within the fssemR package, we are going to simulate gene expression and genetic perturbation data instead of using a real dataset. For this purpose, we will use function randomFSSEMdata2 in fssemR to generate simulated data, and then apply fused sparse structural equation model (FSSEM) to estimate the GRNs under two different conditions and their differential GRN. Also, please go to https://github.com/Ivis4ml/fssemR/tree/master/inst for more large dataset analysis. In conlcusion, this vignette is composed by three sections as follow,

• Simulating two GRNs and their eQTL effects under two different conditions
• Estimating GRNs from the simulated gene expression data and genetic perturbation data
• Differential GRN Visualization

For user using package fssemR, please cite the following article:

Xin Zhou and Xiaodong Cai. Inference of Differential Gene Regulatory Networks Based on Gene Expression and Genetic Perturbation Data. Bioinformatics, submitted.

Simulating two GRNs and their eQTL effects under two different conditions (Acyclic example)

We are going to simulate two GRNs and their corresponding gene expression and genetic perturbation data in the following steps:

1. Load the necessary packages

library(fssemR)
library(network)
library(ggnetwork)
library(Matrix)

1. Simulate 20 genes expression data from a directed acyclic networks (DAGs) under two conditions, and each gene is simulated having average 3 cis-eQTLs. Also, the genotypes of corresponding eQTLs are generated from F2-cross.

n = c(100, 100)    # number of observations in two conditions
p = 20             # number of genes in our simulation
k = 3              # each gene has nonzero 3 cis-eQTL effect
sigma2 = 0.01      # simulated noise variance
prob = 3           # average number of edges connected to each gene
type = "DG"        # `fssemR` also offers simulated ER and directed graph (DG) network
dag  = TRUE        # if DG is simulated, user can select to simulate DAG or DCG
## seed = as.numeric(Sys.time())  # any seed acceptable
seed = 1234        # set.seed(100)
set.seed(seed)
data = randomFSSEMdata2(n = n, p = p, k = p * k, sparse = prob / 2, df = 0.3, 
                        sigma2 = sigma2, type = type, dag = T)

# genes 1 to 20 are named as g1, g2, ..., g20
rownames(data$Vars$B[[1]]) = colnames(data$Vars$B[[1]]) = paste("g", seq(1, p), sep = "")
rownames(data$Vars$B[[2]]) = colnames(data$Vars$B[[2]]) = paste("g", seq(1, p), sep = "")
rownames(data$Data$Y[[1]]) = rownames(data$Data$Y[[2]]) = paste("g", seq(1, p), sep = "")
names(data$Data$Sk) = paste("g", seq(1, p), sep = "")
# qtl 1 to qtl 60 are named as rs1, rs2, ..., rs60
rownames(data$Vars$F) = paste("g", seq(1, p), sep = "")
colnames(data$Vars$F) = paste("rs", seq(1, p * k), sep = "")
rownames(data$Data$X[[1]]) = rownames(data$Data$X[[2]]) = paste("rs", seq(1, p * k), sep = "")

• Summary of simulated GRNs under two conditions, for simplicity, we named our simulated genes as g{%d} and eQTLs as rs{%d}.

# data$Vars$B[[1]]    ## simulated GRN under condition 1
GRN_1 = network(t(data$Vars$B[[1]]) != 0, matrix.type = "adjacency", directed = TRUE)
plot(GRN_1, displaylabels = TRUE, label = network.vertex.names(GRN_1), label.cex = 0.5)

# data$Vars$B[[2]]    ## simulated GRN under condition 2
GRN_2 = network(t(data$Vars$B[[2]]) != 0, matrix.type = "adjacency", directed = TRUE)
plot(GRN_2, displaylabels = TRUE, label = network.vertex.names(GRN_2), label.cex = 0.5)

# data$Vars$B[[2]]    ## simulated GRN under condition 2
diffGRN = network(t(data$Vars$B[[2]] - data$Vars$B[[1]]) != 0, matrix.type = "adjacency", directed = TRUE)
ecol = 3 - sign(t(data$Vars$B[[2]] - data$Vars$B[[1]]))
plot(diffGRN, displaylabels = TRUE, label = network.vertex.names(GRN_2), label.cex = 0.5, edge.col = ecol)

• Simulated eQTLs's effect for 20 genes.

library(Matrix)
print(Matrix(data$Vars$F, sparse = TRUE))

Therefore, the $B$ matrices and $F$ matrix in data$Vars are the true values in our simulated model. We then need to estimated the $\hat{B}$ and $\hat{F}$ by the FSSEM algorithm.

Estimating GRNs from the simulated gene expression data and genetic perturbation data

We need to input the gene expression and corresponding genotype data of two conditions into the FSSEM algorithm. They are stored in the data$Data.

1. 20 simulated gene expression under two conditions

head(data$Data$Y[[1]])
head(data$Data$Y[[2]])

1. 60 corresponding cis-eQTLs' genotype under two conditions

head(data$Data$X[[1]] - 1)
head(data$Data$X[[2]] - 1)

1. data$Data$Sk stores each gene's cis-eQTL's indices. In real data application, we recommend to use package MatrixEQTL to search the significant cis-eQTLs for genes of interested and build Sk for your research

head(data$Data$Sk)

Initialization of fssemR by ridge regression

We implement our fssemR by the observed gene expression data and genetic perturbations data that stored in data$Data, and it is initialized by ridge regression, the $l_2$ norm penalty's hyperparameter $\gamma$ is selected by 5-fold cross-validation.

Xs  = data$Data$X     ## eQTL's genotype data
Ys  = data$Data$Y     ## gene expression data
Sk  = data$Data$Sk    ## cis-eQTL indices
gamma = cv.multiRegression(Xs, Ys, Sk, ngamma = 50, nfold = 5, n = data$Vars$n, 
                           p = data$Vars$p, k = data$Vars$k)
fit0   = multiRegression(data$Data$X, data$Data$Y, data$Data$Sk, gamma, trans = FALSE,
                         n = data$Vars$n, p = data$Vars$p, k = data$Vars$k)

Run fssemR algorithm for data

Then, we chose the fit0 object from ridge regression as intialization, and implement the fssemR algorithm, BIC is used to select optimal hyperparameters $\lambda, \rho$, where nlambda is the number of candidate lambda values for $l_1$ regularized term, and nrho is the number of candidate rho values for fused lasso regularized term.

fitOpt <- opt.multiFSSEMiPALM2(Xs = Xs, Ys = Ys, Bs = fit0$Bs, Fs = fit0$Fs, Sk = Sk,
                               sigma2 = fit0$sigma2, nlambda = 10, nrho = 10,
                               p = data$Vars$p, q = data$Vars$k, wt = TRUE)

fit <- fitOpt$fit

Comparing our estimated GRNs and differential GRN with ground truth

cat("Power of two estimated GRNs = ", 
    (TPR(fit$Bs[[1]], data$Vars$B[[1]]) + TPR(fit$Bs[[2]], data$Vars$B[[2]])) / 2)
cat("FDR of two estimated GRNs = ", 
    (FDR(fit$Bs[[1]], data$Vars$B[[1]]) + FDR(fit$Bs[[2]], data$Vars$B[[2]])) / 2)
cat("Power of estimated differential GRN = ", 
    TPR(fit$Bs[[1]] - fit$Bs[[2]], data$Vars$B[[1]] - data$Vars$B[[2]]))
cat("FDR of estimated differential GRN = ", 
    FDR(fit$Bs[[1]] - fit$Bs[[2]], data$Vars$B[[1]] - data$Vars$B[[2]]))

From these 4 metrics, we can get the performance of our fssemR algorithm comparing to the ground truth (if we know)

Differential GRN Visualization

# data$Vars$B[[2]]    ## simulated GRN under condition 2
diffGRN = network(t(fit$Bs[[2]] - fit$Bs[[1]]) != 0, matrix.type = "adjacency", directed = TRUE)
# up-regulated edges are colored by `red` and down-regulated edges are colored by `blue`
ecol = 3 - sign(t(fit$Bs[[2]] - fit$Bs[[1]]))
plot(diffGRN, displaylabels = TRUE, label = network.vertex.names(GRN_2), label.cex = 0.5, edge.col = ecol)

Additionally, the differeitial effect of two GRN are also estimated. Therefore, we can tell how the interactions in two GRNs change.

diffGRN = Matrix::Matrix(fit$Bs[[1]] - fit$Bs[[2]], sparse = TRUE)
rownames(diffGRN) = colnames(diffGRN) = rownames(data$Vars$B[[1]])
diffGRN

From the diffGRN, we can determined how the gene-gene interactions in GRN changes across two conditions, then, we can find out the key genes for condition-specific gene regulatory network.

Additionally, for more applications and the replications of our real data analysis, please go to the https://github.com/Ivis4ml/fssemR/tree/master/inst for more cases.

Session Information

sessionInfo()

fssemR documentation built on March 18, 2022, 7:24 p.m.