Gene regulatory network inference using fused LASSO on multiple data sets: application to Escherichia coli

Implementation Notes

To implement the modified fused LASSO approach for reconstruction of gene regulatory networks over different data sets, we defined a penalty matrix D composed of two blocks: The first block D₁ is a diagonal kP × kP matrix for the LASSO penalty which includes W in diagonal, while the second block D₂ corresponds to the fusion penalty, expressed as:

To solve the proposed extension to the fused LASSO, we used the lqa package in R [1], with remarkable computational performance due to the Cholesky decomposition [2] which requires ~p + n ~p² /2 operations, in which n is the number of observations and ~p=p+1 while p is the number of regressors. We also slightly changed the function fused.lasso in lqa package to allow inclusion of the penalty matrices defined in Eq. 1. The regression coefficients were robustly estimated by 10-fold cross validation based on the optimum values for l₁ and l₂ from the sets {0.05,0.1,0.5,1,1.5} and {0.1,0.5,1,1.5,2}, respectively. To further speed up the algorithm, we used mclapply function from R package parallel [3].

In addition, we used minet [4] and pROC [5] packages in R to draw ROC curves and estimate statistics for them.

• Proposed approach

The code for regression based approach which simultaneously infer gene regulatory networks from multiple data sets can be download from the folowing link.

Download



• Global Silencing

The code is provided by Barzel et al. (2013)


function [S] = SMatrix(G)
clear S;

[N cols] = size(G);
D = (G - eye(N)) * G;
for i = 1:N
   for j = i + 1:N
        D(i,j) = 0;
        D(j,i) = 0;
   end
end

S = (G - eye(N) + D) * inv(G);

for i = 1:N
    S(i,i) = 0;
end

S = abs(S)/max(max(abs(S)));


Barzel B, Barabási AL (2013) Network link prediction by global silencing of indirect correlations. Nat Biotechnology 31: 720–725.



• Network Deconvolution

The Network Deconvolution code is available at http://compbio.mit.edu/nd/code/ND_regulatory.m.

Feizi S, Marbach D, Mdard M, Kellis M (2013) Network deconvolution as a general method to distinguish direct dependencies in networks. Nat Biotechnology 31: 726–733.



• Graphical Gaussian Models (GGM)

We used the package GeneNet to infer gene regulatory networks based on the GGM proposed in Schäfer et al.



inferred.pcor <- ggm.estimate.pcor(data)

net <- ggm.test.edges(inferred.pcor)


Schäfer J, Strimmer K (2005) An empirical bayes approach to inferring large-scale gene association networks. Bioinformatics 21: 754–764.



• GENIE3

We used online tool available at GP-DREAM in order to infer gene regulatory networks.

Reich, M., Liefeld, T., Gould, J., Lerner, J., Tamayo, P., and Mesirov, J. P. (2006).Genepattern 2.0. Nature Genetics, 38(5),500501.

Marbach, D., Costello, J. C., Kffner, R., Vega, N. M., Prill, R. J.,Camacho, D. M., Allison, K. R., , D. R. E. A. M. C., Kellis, M.,Collins, J. J., and Stolovitzky, G. (2012). Wisdom of crowds for robust gene network inference. Nat Methods, 9(8), 796–804.



• The Algorithm for the Reconstruction of Accurate Cellular Networks (ARACNE)

We used the package minet in order to infer gene regulatory networks based on mutual information.



mim <- build.mim(data,estimator="spearman")

net <- aracne(mim)


Margolin AA, Wang K, Lim WK, Kustagi M, Nemenman I, et al. (2006) Reverse engineering cellular networks. Nat Protoc 1: 662–671.



• Context Likelihood of Relatedness (CLR)

We used the package minet in order to infer gene regulatory networks based on mutual information.



mim <- build.mim(data,estimator="spearman")

net <- clr(mim)


Margolin AA, Wang K, Lim WK, Kustagi M, Nemenman I, et al. (2006) Reverse engineering cellular networks. Nat Protoc 1: 662–671.



• Categorical Bayesian Networks

We used the package catnet in order to infer gene regulatory networks rely on Bayesian Network.



parentlist = vector(dim(data)[1], mode = "list")

parentlist = lapply(parentlist,function(parentlist)c("rpod", "rpoe" , "rpoh" , "rpos"))

net <- cnSearchSA(data,parentsPool=parentlist,maxParentSet=4,fixedParents=parentlist,selectMode="BIC")

best = cnFindBIC(net, data)

cnMatEdges(best)


Balov N. (2013) A categorical network approach for discovering differentially expressed regulations in cancer BMC medical genomics. BMC medical genomics 6: S1.