Space-log: a novel approach to inferring gene-gene net-works using SPACE model with log penalty

Gene expression data have been used to infer gene-gene networks (GGN) where an edge between two genes implies the conditional dependence of these two genes given all the other genes. Such gene-gene networks are of-ten referred to as gene regulatory networks since it may reveal expression regulation. Most of existing methods for identifying GGN employ penalized regression with L1 (lasso), L2 (ridge), or elastic net penalty, which spans the range of L1 to L2 penalty. However, for high dimensional gene expression data, a penalty that spans the range of L0 and L1 penalty, such as the log penalty, is often needed for variable selection consistency. Thus, we develop a novel method that em-ploys log penalty within the framework of an earlier network identification method space (Sparse PArtial Correlation Estimation), and implement it into a R package space-log. We show that the space-log is computationally efficient (source code implemented in C), and has good performance comparing with other methods, particularly for networks with hubs. Space-log is open source and available at GitHub, https://github.com/wuqian77/SpaceLog


Introduction
Complex diseases, such as colorectal cancer (CRC), are caused by a combination of genetic, environmental and lifestyle factors, most of which have not yet been identified and explained. In the era of precision medicine, human genome project, especially breakthroughs in high throughput technologies, provides information based on patient's genetic and genomic data and have changed how researchers explain complex disease through personalized profiles. There is an urgent need to develop efficient statistical and computational tools to integrating genetic and genomic data to identify their contributions to complex diseases, such as gene expression and genetic regulatory networks. Analysis of gene expression data has led to the identification of novel disease-causing gene networks 1 , thus contributing important new insights into understanding of these complex diseases. The objective of this paper is to introduce a novel method that constructs gene-gene network (GGN) based on high dimensional gene expression data. Popular methods for GGN include neighborhood selection 2 , graphical Lasso 3 , and space (Sparse PArtial Correlation Estimation) 4 . Neighborhood selection consistently estimates the non-zero entries of the partial correlation matrix, and provide an approximation of the maximum likelihood estimate of partial correlation matrix. Graphical Lasso improves on neighborhood selection by providing a maximum likelihood estimate of the partial correlation matrix. The space method exploits the symmetry of partial correlation matrix to improve the estimation accuracy. It also avoids potential conflicts in neighborhood selection, that is, Y i is selected as a neighbor of Y j but Y j is not selected as a neighbor of Y i , and one has to make a post-hoc decision for whether Y i and Y j are connected. Furthermore, those available methods employ L 1 , L 2 or elastic net penalty. However, penalties in the range of L 0 to L 1 is often needed to improve the accuracy of variable selection for high-dimensional gene expression data 5 . In this paper, we propose a new statistical method to estimate GGN by implementing the log penalty for the space approach, which enhances sparsity by reweighted L 1 minimization, and we refer to our method as space-log.
We compared space-log with the space algorithm through extensive simulations, as well as the comparison with neighborhood selection methods using lasso or log penalty below. Peng et al. (2009) 4 have compared the space approach with gLasso and showed space outperformed gLasso in different simulation settings, thus we didn't include gLasso methods in our simulation studies here.

Methods
Suppose that we have data on n independent individuals and m genes. Assume the expression of m genes, after appropriate normalization, follow a multivariate Gaussian distribution N(0, ∑).
Neighborhood selection using lasso or log penalty: NSlasso, NS-log The neighborhood selection (NS) approach considers each gene separately. Let Y i be the gene expression value for the ith gene and i j j i i j i j sign ρ β β β = Thus, the problem is translated into partial correlation matrix estimation. Specifically 4 , proposed to minimize a penalized loss function 2 , 1 where w i ≥ 0 is the weight, e.g., uniform weights w i = 1 for space-no, residual variance based weights w i = σ jj for space-res, and degree based weights w i = number of genes that {j : and we call it as space-lasso.
New algorithm space-log: joint modeling space using log penalty ; , The goal is to estimate  ρ = argmin ρ f(ρ) for a given λ and τ. We implement the penalized estimation using space and Log penalties by Local Linear Approximation (LLA) 10 .
, ; , ; , is the estimate of regression coefficient ρ i,j at the k-th iteration. After applying LLA for the penalty part, we can minimize loss function at the (k + 1)-th step, while solving By letting we can find the solution for ρ i,j as follows: ,; , .

Active-shooting
We adapted the same idea active-shooting algorithm from 4 to update the coefficient estimation iteratively in space-log. Without loss of generality, we kept most notation from 4 but tailored with space-log. The details are included in the Supplementary Materials.

Simulation studies
In this section, we present Monte Carlo simulation to evaluate the performance of the space-log, space-lasso, NS-log, and NS-lasso. Following 8 , we studied two types of graphs: the traditional random graphs (ER model) where all the genes have the same expected number of neighbors 11,12 , and hubs graphs where a few genes may have a large number of neighbors (BA model), and BA model is more frequently observed in gene networks 13 .
We simulated GGN of m genes under both the BA and ER models, respectively. The initial graph had one gene and no edge.
In the (k+1)th step, we added e edges between a new gene and e old genes. Under the BA model, there is a greater probability for the new gene to connect to an existing hub gene that has larger number of edges with the probability where ( ) t i v number of edges connected with the ith gene at the tth step. For the ER model, each edge of any gene pair (G i , G j ) was added randomly in the GGN with probability pE independent from all other edges. After constructing the bone of GGN, we simulated gene expression based on multivariate Gaussian. Without loss of generality, we simulated data sets with n = 400 individuals, which is similar to the sample size in our real data examples. As shown in Table 1, we considered different number of genes m = 100, 200, 300 with various sparsity level determined by p E = 1=m or 2=m for the ER model and e = 1 or e = 2 for the BA model.
We evaluated the performance of the methods by the following metrics: number of false positives (FP), false negatives (FN), FP+FN, F1 score, FDR, true positive rate (power). Note that there are three different weights used in joint modeling setting (space-log, space-lasso): (1) uniform weights; (2) residual variance based weights; and (3) degree freedom based weights. The corresponding methods are referred to as sp_no, sp_res, and sp_df with/without log respectively. Under the BA model with m=100 and e=1 (Figure 1), we can see that space-log has smallest Errors (FP+FN), smallest FDR, and highest F1 score than other approaches, indicating that space-log controls overall false positive and false negative rates well. Under the ER model ( Figure 2) with m=100 and e=1, space-log is slightly better than space, and NS-log shows lower Errors and higher F1 score than other approaches including space-log. Under both models, the log penalty has less false positives but slightly more false negatives compared to lasso penalty. We note that although log penalty performs well for both the ER and BA models, space-log is particularly powerful in identifying hub networks (such as BA models).
In the Extended data, Figures S3 and S4 show the results under the BA model for m=100,200,300 with low number of connections (e=1) and high number of connections (e=2), respectively. Figures S5 and S6 show the results under the ER model with low and high numbers of connections, respectively. Comparing with Figure 1, a similar pattern was noted with the increase of number of genes (m increases from 100, 200, to 300). In BA with low connections (Figure S3), space-log showed smallest FP+FN error and largest F1 score, which outperform all other methods. In BA with high connections (Figure S4), NS-log showed smallest FP+FN error and largest F1 score. For ER model with low and high connections, NS-log outperforms other methods in terms of FP+FN and F1 scores. It's in line with our understanding that space-log is powerful at identifying hubs network, and NS-log is powerful at dealing with complex network with high number of gene-gene interactions and random networks.
We showed a simulated graph for the BA model with 400 subjects, 100 genes and each gene has only 1 connection (Figure 3). The GGN was estimated by 8 different approaches. In Figure 3, the true edges were indicated by black color, false positive (FP) edges by red color, and false negative (FN) edges by grey color. It's clear that space-log identified far fewer false positive edges (red line) comparing with space-lasso and NS approaches, while clearly indicating the hub structures. We observed that the FP edges by two NS approaches were quite randomly identified, and the FP edges by two space approaches were mostly within a hub and not between hubs.
We also compared the run-time under different simulation scenarios in Figure S7 and Table 2 under the BA model and low number of connections (e=1). Here the computing time for the space with no weight is presented. The computational time of the methods using log penalty is higher than that using lasso penalty, and the computational time for the space methods is much less than NS methods. Similar to what was observed in 8, the runtime increases approximately linearly with m. The difference of the run-time between the ER and BA model, the low and high numbers of connections or different weights of space methods are not significant ( Figure S7).
In summary, the log penalty generally has better performance than the lasso penalty, and both space-log and NS-log control false positive and false negative rate well. For random networks, i.e., no hub, NS-log performs better than other methods. space-log performs best for hub-like gene networks ( Figure 1 and Figure 3) with higher F1 score and less false positive edges. Identifying hub networks is generally considered of great interest in the GGN analysis, because a few of hubs connecting with a large proportion of genes, and those hub genes are thought to be master regulators and play a critical role in a biological system 14 .

Application to GTEx and TCGA data TCGA data
We applied both proposed space-log and existing methods (space-lasso, NS-log, and NS-lasso) to identify    We considered gene sets C6 curated oncogenic pathways by MSigDB from the Broad Institute and inferred the GGN within each gene set. There were 189 gene pathways with a total of 8,737 unique genes for which TCGA have expression data. The sizes of gene sets ranged from 9 to 338 genes. Since we don't know the true GGN, we downloaded the common pathway version 10 from www.pathwaycommons.org to provide a partial "gold standard". The observed GGN by different methods were compared with the known edges from common pathway and calculated FP, FN, FP+FN, number of total discovery, F1 score, and true positive rate (Figure 4). The NS-based approach with both LASSO and log penalty discovered much more edges than space-based approach and space-log had fewer false positive (fewer FN+FP too) than space-lasso. There is almost no difference on number of false negative between different methods, as well as F1 score (Figure 4). Furthermore, in order to show the performance of these methods on the hub networks, we identified 17 pathways with hub-like genes (each hub gene set has < 50 genes and variance of the number of identified edges for each gene in the gene set > the first quartile of all 189 gene sets) and re-calculated the summary metrics in Figure 5. We noted that space-log approach has smallest Errors and slightly higher F1 than other approaches, which is in line with our finding in simulation that space-log is powerful in identifying hub networks, (such as BA models).
To further understand whether our proposed methods can better handle high dimensional data with larger number of genes, we generated k=71 larger network by combining gene sets with overlapping genes. The sizes of this new larger gene sets list ranged from 476 to 1253 genes, which is larger than the sample size n=450 (m > n). We applied spacelog, space-lasso, NS-log, and NS-lasso approaches on this new larger list, and calculated similar metrics (FP, FN, FP+FN, etc in Figure S8). Comparing with TCGA data with smaller gene sets (Figure 4 and Figure 5), we observed that the NS-based approach with both lasso and log penalty discovered much more false positive edges than space-based approaches for this new larger gene sets setting, which is consistent with our finding in simulation that NS-based approach has more false positive in general.

GTEx data
The Genotype Tissue Expression (GTEx) project 16 aims to study tissue-specific gene expression and regulation in normal individuals. In this paper, we used gene expression data (RNA-seq) from blood tissue of 451 patients to identify GGN. We pre-processed gene expression data using the same procedure as for TCGA data. We mapped genes to gene pathways by MSigDB (https://www.gsea-msigdb.org/gsea/msigdb/index.jsp). A total of 189 gene pathways were represented with a total of 8097 unique genes. The size of gene sets ranged from 8 to 306 genes.
Again, we applied space-log, space-lasso, NS-log, and NS-lasso approaches to identify GGN. Using the same common pathway file used for the TCGA analysis as gold standard, we calculated FP, FN, FP+FN, # of discovery, F1, TPR ( Figure 6). We obtained very similar results to the TCGA data. The NS-based approach with both LASSO and log penalty discovered much more edges than the space-based approach and space-log has fewer false positive (fewer FN+FP too) than space-lasso. There is almost no difference in the number of false negative between different methods, as well as F1 score. A similar sensitivity analysis was conducted to a subset of hub-type genes (Figure 7), where 30 pathways were selected to be in the first quartile of the variance of the number of identified genes with < 50. It also showed the space-log approach has smallest Errors (F1 is similar to other approaches).

Conclusions
In this paper, we proposed a new joint modeling method with log penalty, space-log, to identify gene-gene network. An assumption of the GGN analysis is that most of gene pairs do not directly interact with each other, and there are a few of master genes (hubs) for network that connect with many other genes, which are thought to play a critical role in a biological system 1,14 . Both simulation and real data analyses showed that space-log is particularly powerful in identifying hub networks and master genes, which is considered of great interest in gene-gene network analysis. In the Extended data, we compared several tuning parameter selection approaches, such as BIC Zou et al. 17 , extBIC 9,18 , and oracle 5 , and showed that extBIC outperforms other methods in simulation. The R package "SpaceLog" on GitHub includes algorithms, simulation, and real data examples: https://github.com/wuqian77/SpaceLog.

Data availability
Underlying data Simulation data. We used barabasi.game function from igraph R package to generate the skeleton of a BA model. The paper introduces an extension of the SPACE (Sparse Partial Correlation Estimation) method, which is used to estimate a gene-gene network (GNN). The proposed framework, space-log, relies on the log penalty, which delivers better variable selection performance than LASSO, especially for GGN with hubs. The authors also have created a very efficient R package for the proposed method. The paper is clearly written, and the proposed method showed promising results for real data analyses. I list my questions below: In both TCGA and GTEx analyses, the dimensions m is smaller than the sample size n. Since log penalty can handle high dimension low sample size data, can you also include a real application with m larger than n? 1.
The numerical results are all based on the tuned model using extended BIC, which is oneshot of the result. It is better to also include ROC curves reflecting the whole spectrum of the results for a range of tuning parameters.

Are sufficient details provided to allow replication of the method development and its use by others? Yes
If any results are presented, are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions about the method and its performance adequately supported by the findings presented in the article? Yes