 Software
 Open Access
 Published:
wTO: an R package for computing weighted topological overlap and a consensus network with integrated visualization tool
BMC Bioinformatics volume 19, Article number: 392 (2018)
Abstract
Background
Network analyses, such as of gene coexpression networks, metabolic networks and ecological networks have become a central approach for the systemslevel study of biological data. Several software packages exist for generating and analyzing such networks, either from correlation scores or the absolute value of a transformed score called weighted topological overlap (wTO). However, since gene regulatory processes can up or downregulate genes, it is of great interest to explicitly consider both positive and negative correlations when constructing a gene coexpression network.
Results
Here, we present an R package for calculating the weighted topological overlap (wTO), that, in contrast to existing packages, explicitly addresses the sign of the wTO values, and is thus especially valuable for the analysis of gene regulatory networks. The package includes the calculation of pvalues (raw and adjusted) for each pairwise gene score. Our package also allows the calculation of networks from time series (without replicates). Since networks from independent datasets (biological repeats or related studies) are not the same due to technical and biological noise in the data, we additionally, incorporated a novel method for calculating a consensus network (CN) from two or more networks into our R package. To graphically inspect the resulting networks, the R package contains a visualization tool, which allows for the direct network manipulation and access of node and link information. When testing the package on a standard laptop computer, we can conduct all calculations for systems of more than 20,000 genes in under two hours. We compare our new wTO package to state of art packages and demonstrate the application of the wTO and CN functions using 3 independently derived datasets from healthy human prefrontal cortex samples. To showcase an example for the time series application we utilized a metagenomics data set.
Conclusion
In this work, we developed a software package that allows the computation of wTO networks, CNs and a visualization tool in the R statistical environment. It is publicly available on CRAN repositories under the GPL −2 Open Source License (https://cran.rproject.org/web/packages/wTO/).
Background
Recent applications of complex network analysis methods have provided important new knowledge of the functioning and interactions of genes at the systems level [1–4]. Within the area of biological network analyses, coexpression networks have received much attention [5, 6]. For the coexpression networks, a pair of nodes are typically connected by a link if the genes they represent show a significantly correlated expression pattern. In the network, this link may be represented as a binary relationship, where 1= “presence” and 0= “absence” of the link, or alternatively, the link may have a numeric value (often called weight). The magnitude of the weight is typically interpreted as representing the strength of a genepair relationship, and the sign as indicative of the type of associated gene interaction: positive if the genes are coregulated, negative if they are oppositely controlled [7].
In many implementations of network analyses, we may primarily be interested in an a priori defined subset of genes with a specific set of properties. Examples include transcription factors (TFs), genes with known orthologs in a set of organisms of interest, or disease associated genes [8, 9]. For these situations, oftentimes the choice is made to only take into account direct interactions between the genesubset of interest, instead of including the full set of correlations. A major drawback with such an approach, is that relevant information contained in interaction patterns among excluded genes that would affect network topology and link strength values, is not incorporated in the network. The loss of such information is not only undesirable, but may also lead to biased results.
When analyzing networks in which the links have nonbinary weights, the method of weighted topological (wTO) network analysis [10] has been found very useful. In a wTOanalysis, a new linkweight for a pair of connected nodes is determined through an averaging process that accounts for all common network neighbors [10]. Thus, wTO is a method that implicitly includes correlations among nodes that are going to be exempt from further analysis. The wTO method [10–12] can be used to determine the overlap among classes of transcripts, for example TFs and noncoding RNAs (ncRNAs). The resulting wTO network provides a more robust representation of the connections and interactions among the nodeset of interest than a simple correlation network analysis focused only on the nodeset of interest [13].
The packages WGCNA [14, 15] and ARACNe [16, 17] are widely used for weighted gene coexpression network analysis studies. The former provides functions for the calculation of the adjacency matrix for all pairs of genes as the nth power of absolute correlations, resulting in an unsigned network. Network modules can be defined with this package by unsupervised clustering. The latter uses the mutual information (MI) of the expression in order to build the networks. These methods have received much attention in the literature [7, 18].
Previously, Nowick and collaborators [13] developed a mathematical method to calculate the wTO for a set of nodes that explicitly takes into account both positive and negative correlations. This version of the wTOmeasure is especially valuable for investigating networks, in which it matters whether an interaction is activating or inhibiting/repressing. For instance, in gene regulatory networks the effect of a transcription factor or a ncRNA on its target genes can be activating or repressing. In metabolic networks, the increase of a substance can lead to an increase or decrease of another substance. Or in ecological networks, species interactions can be positive or negative, for instance in symbiotic or predatorprey relationships. In such cases, a distinction between positive and negative correlations for the calculation of the wTO is necessary and using the absolute correlations would falsify the biological insights. This wTOcalculation methodology is implemented in the R package presented here. In order to avoid confusion, we will refer to the method for calculating a pairwise link score as wTO and to the package as wTO.
When analyzing similar datasets, e.g. from a repeated experiment or independent studies on a similar subject, the resulting networks are usually different [19]. These differences may arise from several sources: (A) technical differences, such as the platform on which the expression data was measured, the facility where data was collected and prepared, or how data was processed. (B) Another cause may be biological differences from confounding factors, such as sex, age, and geographic origin of the individuals measured. It is thus desirable to obtain an integrated network that considers all independently derived networks as biological replicates and systematically identifies their commonalities. We developed a novel method to compute the network that captures all this information; we call this the consensus network (CN).
Here, we present wTO, an R package that is capable of computing both signed and unsigned wTO networks as well as the CN, thus providing methods for assigning pvalues to each link. The package also comes with an integrated tool to visualize the resulting networks and allows for nine different methods for network clustering to aid in module identification. The workflow of the package is shown in Fig. 1.
We compare our method to other state of art methods. To exemplify the usage of our package, we show here results from the calculation of wTO and CN networks from three independent genomewide expression studies of healthy human prefrontal cortex samples and an analysis of a timeseries dataset from a metagenomics study.
Implementation
Input data
Our package can handle a wide range of input data. Data can be discrete or continuous values. We recommend performing all commonly used steps for quality control and normalization before passing on the data to our package. For RNASeq data, our package can handle normalized quantification, for example RPKM (Reads Per Kilobase Million), FPKM (Fragments Per Kilobase Million), and TPM (Transcripts Per Kilobase Million). For microarray data, rma or mas5 values can be used. If our package is used with metagenomics data, for instance for analyzing cooccurrence networks, we recommend the abundance data to be normalized per day/ sample.
Weighted topological overlap calculation
For a system of N nodes (e.g. genes or species), we define the adjacency matrix A= [a_{i,j}] based on correlations between a pair of nodes i and j as
with ρ_{i,j} being a correlation measure. Assuming that nodes i and j represent a subset of factors (e.g genes) of particular interest selected from the N nodes, we calculate the weighted topological overlap (wTO [13], ω_{i,j}) between node i and node j as
where
Note that, this expression explicitly includes both positive and negative correlations, and thus allows for ω_{i,j} to take both positive and negative values. Other software packages calculating the ω_{i,j} have implemented definitions of the wTO method that do not allow for negative values [14], making this version valuable for gene regulatory network analysis. The wTO package also calculates the unsigned network, and for that, it takes as an input the absolute values of the correlation.
Since Eq. (2) explicitly allows a_{i,j}≤0, we need to be aware of the limits of this expression. Consider three nodes i, j and u, and assume that a_{ij}≤0. All the terms in the numerator of Eq. (2) will be negative if a_{iu}a_{uj}≤0 for all nodes u. However, if a_{iu}a_{uj}>0, then at least some contributions to the sum will cancel out. The same rationale applies for the case of a_{ij}≥0.
To systematically assess the potential effect of term cancellation in Eq. (2), we calculate the absolute weighted topological overlap, ω which uses the absolute value of the correlations (a_{i,j}=a_{i,j}) as input for Eq. (2). In this case, the sign of the correlation is excluded from the analysis and only the magnitude of the linkstrength is taken into account. Consequently, by generating a scatter plot of the signed and unsigned weights, it is possible to assess at which ω_{i,j}values term cancellations start affecting the results. Thus, for wTO values of interest, the closer the plot of ω vs. ω is to y=x, the better.
However, by just computing the wTO network we do not avoid all spurious correlations. A way to detect them is to compute a probability of each one of the link scores being zero using the hypothesis test
of the null hypothesis (H_{0}) of no association against the twosided alternative (H_{a}) of nonzero association. This can be computed by using bootstrap [20] or permutation resampling methods [13]. In the former, one resamples individuals, thus approximating the weights’ empirical distribution and calculating the probability that an observed weight is sufficiently distant from zero. In the latter, one operates under the null hypothesis of no dependence among genes and permutes the gene labels, obtaining the weights’ distribution under the null hypothesis, which is rejected if the observed weight is sufficiently extreme. We define δ as the maximal distance between the ω_{i,j} calculated with each bootstrap and the ω_{i,j} of the real dataset. This means that, the smaller δ is, the stronger is our confidence in a particular ω_{i,j}. By default, δ is set to 0.2.
One advantage of the wTO package is its application to analyze and make networks out of timeseries data. Therefore, we are interested in the implementation of blocked bootstrap resampling [20] that can be used for temporal data without sample replicates for each time point. This type of resampling is necessary once there are two correlation components in those samples: The correlation inside the factors of each sample and the correlation across the time of different samples. For this situation, the use of a lag is required. Lags are particularly helpful in timeseries analyses as autocorrelations are often present: a tendency of consecutive values to be correlated. An important benefit of the presence of autocorrelations is that we may be able to identify patterns inside a timeseries, such as seasonality (patterns that repeat themselves at a periodic frequency). Therefore, the lag can be chosen using a partial correlation of the time per sample. This is followed by calculating the wTO for a time series where the observations are not independent of each other.
A method for determining a consensus network
Berto and collaborators [19] described a consensus network based on geneexpression data from primates’ frontal lobes by applying a Wilcoxon test on the links. Our proposed methodology allows the use of two or more datasets, each generating different (and significant) wTO values, to be combined into a single CN. Our approach has the advantage of penalizing links with opposite signs. According to the same rationale, links with the same sign among the multiple wTO networks, will have their CN_{i,j} values closer to the largest ω_{i,j} of a link among the k networks. Our first step is to remove nodes that do not exist in all networks. Consequently, if a node is absent in at least one network, we are not able to compute a consensus of the links that belong to that node. It is particularly important not to associate factors that were not measured in a particular condition.
In order to obtain a single integrated network derived from multiple independent wTO networks, we calculate a CN using the following approach:
If we have k=1,…,n replicated networks (note that n means the index of the networks, not the exponent of α nor ω), then we define the consensus network wTO_{CN}=[Ω_{i,j}] as
where
A threshold can be used to remove links with Ω_{i,j} values close to zero, thus should not be included in the consensus network. To join networks that were generated with the proposed wTO method into the consensus network, the pvalues are combined using the Fisher’s method.
Results and discussion
The representation of interactions between a set of nodes by the wTO method [10–12] takes into account the overall commonality of all the links a node has, instead of basing the analysis only on calculating raw correlations among the nodes. It thus provides a more comprehensive understanding of how two nodes are related. Therefore, it is expected that a wTO network contains more robust information about the connections among nodes than what would result from simply taking direct correlations into account [11, 13]. The wTO can be computed based on a similarity matrix, where the link weights are calculated using Pearson’s product moment correlation coefficient or the Spearman Rank correlation. The first one measures the linear relationship between two genes. Note that, the Pearson’s correlation coefficient is sensitive to extreme values, and therefore it can exaggerate or underreport the strength of a relationship. The Spearman Rank Correlation is recommended when data is monotonically correlated, skewed or ordinal, and it is less sensitive to extreme outliers than the Pearson coefficient [21–24].
Package functions
The function wTO calculates the weights for all links according to Eq. (2) between a set of nodes for a given input data set. If the user is not interested in the resampling option, one may simply run this wTO function.
To test whether the calculated wTO is different from random expectation and to decide on a suitable threshold value for including link weights, we implemented the function wTO.Complete. Here, the wTO is calculated a number of times, n specified by the user, by using either the 1) Bootstrapping (method_resampling = “Bootstrap”), or (method_resampling = “BlockBootstrap”) for time series data or 2) Permuting the expression values for each individual (method_resampling = “Reshuffle”) [13]. The user may specify the correlation method that this function should use, Pearson correlation is the default choice.
Because bootstrapping and permutation tests can be computationally expensive, the wTO.Complete can also run in parallel over multiple cores to reduce the wall clock time. For running in parallel, the user may specify a given number of k computer threads to be used in the calculations. To implement the parallel function, we used the R package parallel [25].
The execution of the wTO.Complete function returns two outputs; a diagnosis set of plots and a list consisting of the following three objects:

$Correlation is a data.table containing the Pearson or Spearman correlations between all the nodes, not only the set of interest. The wTO links for the set of nodes of interest are based on these correlations. The default of this output is set to FALSE.

$wTO is a data.table containing the nodes, the wTO values (signed and unsigned), the pvalues and the adjusted pvalues computed using both signed and unsigned correlations.

$Quantile is a table containing the quantiles for the empirical distribution, computed using the bootstrap and the quantiles for the real data: 0.1%, 2.5%, 10%, 90%, 97.5% and 99.9%. Those empirical values can be used as a threshold for the wTO values, when it is not desired to visualize low wTO scores.
The set of plots indicate the quality of the resample: the closer the density of the resampled data is to the real data, the better. Another generated plot is the scatter plot of the ω_{i,j} vs ω_{i,j}, as previously discussed. The scatter plot of pvalues against the ω_{i,j} and ω_{i,j} is also plotted along with suggested threshold values that are the empirical quantiles.
Computing of the CN is done using the function wTO.Consensus. This function allows the user to give a list of networks in the format of data.frames with: Node 1, Node 2, the link weight and the pvalue. The output is a data.table containing the two nodes’ names and the consensus weight, and the combined pvalue. This allows the user to filter out the links that were not significant in part of the networks. A visual representation of the Consensus Network methodology is shown in Fig. 2. The thicker the link between two nodes is, the stronger the correlation between them. The signs are represented by the colors blue and orange, respectively. If a link has different signs in the networks, the strength of the link in the CN is close to zero. When all links agree to the same value or show little deviation, the strength of the resulting CN value is closer to the determined maximum value. If a node is absent in at least one network, it is removed.
The output data.frames (from both, wTO.Complete and wTO.Consensus) can be easily exported using the function export.wTO. This allows, for instance, to pass on the results of our package to Cytoscape [26] for further analysis.
Our R package also includes options to visualize the resulting networks. The function NetVis generates an interactive graph using as input a list of links and their corresponding weights. The analysis functions wTO.Complete and wTO.Consensus both generate network datastructures (edge list) that can be visualized with this function. The user needs to choose a relevant wTOthreshold (the quantiles resulting from the bootstrap), or pvalue cutoff, to select the set of links to be plotted. Additionally, the user may choose a layout for the network visualization from those available in the igraph [27] package. By default, the wTOthreshold value is set to 0.5, and the network layoutstyle is set to layout_nicely. To avoid false positives, we recommend to filter the data according to the desired significance pvalue and to choose the wTOthreshold according to the computed empirical quantiles. The size of the nodes is relative to their degree. Our package further includes an option for MakeCluster from the nodes; if allowed, nodes are colored according to the cluster they belong to. The user can choose the method to create the clusters.
One important difference between our package and the WGCNA package, is that we only use significant links for cluster (modules) network representation instead of the full set of coexpressions, as in the WGCNA package. The width of a link is relative to the wTO_{i,j}, and its color is respective to its sign (if a signed network was calculated). Nodes can have different shapes, allowing for labeling nodes of different classes, for example target genes or protein coding and nonprotein coding genes. Furthermore, the user may also zoom in and out of the network visualization, drag nodes and links, edit nodes and links, and export the image as html or png. The package provides example datasets and an example of nodes of interest as well.
Algorithm compute time with varying system size
Normally, when running the wTO, the interest lies on a subset of nodes of interest. In Fig. 3 we show the runtime for different network sizes, and different proportions of nodes of interest. When running the wTO for all expressed genes coding for transcription factors (TFs) being the genes of interest, we have around 14% of nodes of interest. Using a standard laptop computer, it’s possible to compute the wTO for a full network with 20,000 nodes in 20 miliseconds per link. This shows that it is quite feasible to compute the full wTO for a realistic gene expression network.
Comparison with existing methods
A variety of methods currently exist to analyze gene coexpression networks, in particular ARACNe [16, 17], SPACE [28] and WGCNA [14, 15]. These methods rest on a multitude of different mathematical principles, particularly with respect to how coexpression is quantified. Of particular interest is WGCNA, which shares notable similarities with our wTO package in heuristic terms, but with some substantial differences in functionality. In particular, WGCNA also uses the weighted topological overlap (in their nomenclature, the “topological overlap matrix”, or TOM) to quantify coexpression at the genepair level. But in WGCNA, the final edge weight corresponds to the absolute value of ω_{i,j} as defined in Eq. 2, or the absolute value of the terms in the numerator of Eq. 2. These are referred to as signed or unsigned, respectively. Topological overlap as a measure of coexpression has previously been shown to compare favourably with other methods [18].
While wTO and WGCNA construct the networks based on overlaping topologies, the ARACNe method builds the network using the mutual information (MI) and removing links that are indirect interactions using data processing inequality (DPI). Another important difference between the methods is that wTO and WGCNA will compute a link for all pairwise possible connections, while ARACNe will only compute the pairwise information if their information is not independent (Table 1).
Relative to WGCNA, wTO provides three major additions: the determination of pvalues (determined by bootstrapping) for each pairwise wTO value; the calculation of a consensus network, and the ability to visualize the topological overlap network (along with node grouping according to a choice of nine algorithms). While WGCNA provides a variety of tools for visualizing the hierarchical tree forming the network, as well as for rendering the correlation matrix in heatmap form, it does not provide a nodeandedge type view of the coexpression network (but does allow for exporting networks into Cytoscape, in which network views are possible). Additionally, the consensus network as defined in Eq. 6 differs from the consensus TOM defined in WGCNA, which simply assigns to each edge of the consensus network the minimal value of the topological overlap across the input conditions. This is a strict version of consensus (unanimity), in that it will discard any gene pair if the overlap is weak in even a single network. In contrast, while Eq. 6 will remove contributions from networks where the topological overlap is weak (or where the sign of the wTO score is in conflict with the other networks), an edge may still be included if it is sufficiently present across the other networks.
Further additions in wTO include the possibility of choosing the Spearman correlation as the basis of a_{i,j} (while WGCNA provides biweight midcorrelation, or bicor for short; both provide Pearson), as well as reducing computation time by the option of restricting the calculation of wTO scores to a set of genes of interest (while still including the adjacency to genes outside this set in each interset wTO score).
Another minor difference resides in how wTO is determined for each gene with itself. From Eq. 2, we see that (assuming a_{i,i}=0 and a_{i,j}=a_{j,i}):
For an unweighted network, where a_{i,j}=0 or a_{i,j}=1 for all (i,j), this approximates to \(\omega _{i_{i}} \approx 1\) for large k_{i}. However, this is not the case for weighted networks. WGCNA differs from the wTO package in that w_{i,i}=1 is explicitly set for all i, while our package retains the score as defined by Eq. 2.
Comparing wTO, WGCNA and ARACNe using an E. coli transcription factor network
In order to quantitatively compare the performance of wTO, WGCNA and ARACNe, we downloaded a gene expression dataset from E. coli from http://systemsbiology.ucsd.edu/InSilicoOrganisms/Ecoli/EcoliExpression2 [29–32]. The data consists of 213 Affymetrix microarray gene expression profiles, corresponding to multiple different strains under different growth conditions, and contains gene expression data for 7312 distinct probes. Gene expressions were calculated as the mean of probes corresponding to the same gene. To assess the capability of the three tools in identifying true TFTF interactions, we used the RegulonDB [33] database, which contains experimental data from E. coli, as a reference. We defined as TruePositive interactions those that are described in RegulonDB, and as TrueNegatives all interactions that could not be experimentally validated in that dataset. For comparison, we also calculated networks using only the raw Pearson correlation. We generated the network for WGCNA following the steps described by the authors in the Tutorial [11, 34]. We used the functions pickSoftThreshold and pickHardThreshold for defining the power of the softthreshold and for choosing the hardthreshold, respectively. The power was defined as 4 and the hardthreshold was set to 0.3.
The ARACNe network was built using the Pearson correlation with build.mim and ARACNe functions in the minet R package [35]. The wTO networks were built using 1000 simulations, Pearson correlation and filtered for p_{adj}values ≤0.01 and the 90% quantile. One wTO network was constructed using a δ of 0.2, the default of the package, and another network was built using a δ of 0.1. All networks were filtered to only contain TFs with information in the RegulonDB. We calculated the Receiver operating characteristic (ROC)curve using the pROC R package [36] (see Fig. 4).
ARACNe was able to better identify the amount of true positives compared to WGCNA and wTO, but performs worse when finding true negatives and also has a larger number of false positives (Fig. 4, Table 2). WGCNA is better at finding true negatives, but does not identify many true links. Our proposed wTO method performs better than WGCNA in finding true positives and better than ARACNe in finding true negatives. It also finds fewer false positives than ARACNe. In general, even when using a large δ, wTO performs better than the two other methods, as seen in the Area Under the Curve (AUC; the closer it is to unity, the better). This demonstrates that the use of the wTO method further reduces false effects coming from incorrectly assigned linked genes (false positives) when compared to ARACNe and raw correlations.
Examples of wTO networks using the wTO R package
wTO and CN networks for TFs of the human prefrontal cortex
To exemplify the usage and results of our package, we analyzed three independent datasets of microarray data from human prefrontal cortex. Data sets were downloaded as raw data from Gene Expression Omnibus (GEO) website [37]. From the study GSE20168 [38, 39], we used data from a total of 15 postmortem brain samples. From the study GSE2164 [40], we used a total of 26 samples from post mortem brains. And finally, from the study GSE54568 [41] we used all the 15 controls. All individuals were older than 5 years and died without any neuropathological phenotypes. We chose the TFs to be our genes of interest and calculated a TFwTO network for each of the three datasets. Subsequently, we computed the consensus network for the three TF wTO networks.
The downloaded data were preprocessed and normalized by ourselves independently, using the R environment [42], and the affy [43] package from the Bioconductor set. The probe expression levels (RMA expression values) and MAS5 detection pvalues were computed, and only probesets significantly detected in at least one sample (pvalue <0.05) were considered. After the Quality Control and normalization of the data, the probes that were not specific for only one gene were deleted. If one gene was bound by more than one probeset, the average expression was computed.
Here, we will focus on how TFs are coexpressed in brain networks. We used a set of 3229 unique TF symbols from the TFCatalog (PerdomoSabogal et al. (in preparation)) with ENSEMBL protein IDs. The construction of this catalog contains the information for TF proteins sourced from the most influential studies in the field of human Gene Regulatory Factors (GRF) inventories [44–51] that are associated with gene ontology terms for regulation of transcription, DNAdepending transcription, RNA polymerase II transcription cofactor and corepressor activity, chromatin binding, modification, remodeling, or silencing, among others.
Signed wTO networks were calculated for each dataset separately using the function wTO.Complete of our wTOR package and then merged with the function wTO.Consensus into the consensus. Significance of all networks was evaluated using 1000 bootstraps, Pearson correlation and filtered for p_{adj}value of <0.01. The Consensus Network was built based on the calculated signed wTO values of significant links. Weights for links with insignificant wTO were set to zero. Figure 5 shows the distributions and the networks for our three datasets.
TFs were clustered using the Louvain algorithm with the NetVis function, which identified 5 clusters in the CN. When considering each network independently, we had 18, 8 and 16 clusters. This shows that the CN detects fewer clusters of genes, which are more densely connected, compared to the clusters detected in the individual wTO networks. In order to investigate the function of each one of the 5 CN clusters, we calculated the correlation of each TF of a cluster with all other expressed genes using Pearson correlation. Genes with a correlation of at least 0.80 with at least one TF of the cluster were used for GO enrichment analysis for that cluster, using the R package topGO [52]. The enrichment analysis revealed many brain related functions, for instance, clusters 1 and 3 show overrepresentation of groups related to cognition (Table 3 and Fig. 6).
Time series: Metagenomics data from the ocean
Only about 1% of marine bacteria can be easily studied using standard laboratory procedures [53]. This is a major drawback for the understanding of how those microorganisms interact. Systems biology methods can provide helpful insights to shed light on species interactions.
To demonstrate an application of our wTO package for time series data with no replicates, we use as an example metagenomics data from The USC Microbial Observatory. The data is public available at https://www.ebi.ac.uk/metagenomics/projects/ERP013549.
The sampling site is located between Los Angeles and the USC Wrigley Marine Laboratory on Santa Catalina and spans approximately 900 m of water. Over the course of 98 months, samples were taken once a month. Operational Taxonomic Unity (OTUs) were determined using 16S ribosomal RNA (rRNA). The authors found 67 OTUs that will be used in our analysis. In order to find the correct lag for the blocked bootstrap, we used the autocorrelation function (acf) for all OTUs and chose a median lag of 2. This allowed us to define the blocks with high autocorrelation in the same sample, meaning that for them the abundance of the OTU on each specific time point is correlated to the following next 2 time points.
Based on that, we built the network of bacteria cooccurrence in that environment (Fig. 7). We found that 61 out of 67 OTUs had at least one significant interaction (p_{adj}value <0.01). Positive correlations in cooccurence networks may represent symbiotic or commensal relationships, while negative correlations may represent predatorprey interactions, allelopathy or competition for limited resources. Using the community detection method for defining clusters we identified four distinct clusters of bacteria. We did not find any association of the phylogeny with clusters, which is in agreement with previous studies. However, we can clearly see (Fig. 7) that the blue group is rich in negative relationships, while both, the purple and orange groups, possess many positive relationships. These positive relationships are formed mostly by Flavobacteriales, bacteria that are known to infect fishes [54] and to live in commensality with other bacteria from the same order [55].
Conclusion
This new wTO package allows wTO network calculation for both, positive and negative correlations, which is not provided in any other published R package. With this feature it becomes valuable for the analysis of gene regulatory network, metabolic networks, ecological networks and other networks, in which the biological interpretation strongly depends on distinguishing between activating and inhibiting/repressing interactions.
Another novel feature is the computation of pvalues for each link based on its empirical distribution, which allows for the reduction of false positive links in wTO networks. With our package, networks can also be calculated from time series data. In addition, our package includes the computation of a CN, which enables integrating networks derived from different studies or datasets to determine links that consistently appear in these networks.
By focusing on what these independently derived networks have in common, the CN should be of higher biological confidence than each individual network is. We also provide an interactive visualization tool that can be used to visualize both, wTO networks and CN, for efficient further custom analysis.
We qualitatively and quantitatively compared our new package to stateoftheart methods and demonstrated that it performs better in identifying true positives and false negatives.
We provide two use cases for our package, one on wTO and CN calculation from three independent genomewide expression datasets of human prefrontal cortex samples, and one on wTO cooccurence networks calculated from time series data of a metagenomics abundance dataset from the ocean. Here, we demonstrated that clusters and GO enrichment in the CN are more defined than in individual wTO networks, highlighting the benefits of our package for analyzing and interpreting large biological datasets.
Availability and requirements
Project name: wTO: Computing Weighted Topological Overlaps (wTO) & Consensus wTO Network
Project home page: https://CRAN.Rproject.org/package=wTO
Operating system: Platform independent
Programming language: R
Other requirements: wTO relies on the following packages: som [56], plyr [57], stringr [58], network [59, 60], igraph [27], visNetwork [61], data.table [62] and the standard packages stats and parallel [25]. The visualization tool implemented in our package was built using a combination of the packages network [59, 60], igraph [27] and visNetwork [61]. The MakeGroups parameter, passed to the function NetVis for constructing the network, allows the user to choose clustering algorithms from: “walktrap” [63], “optimal”[64], “spinglass” [65–67], “edge.betweenness’ [68, 69], ‘fast_greedy” [70], “infomap’ [71, 72], “louvain” [73], “label_prop’ [74] and “leading_eigen” [75]. All those algorithms are implemented in the igraph package[27].
License: GLP2
Abbreviations
 acf:

Autocorrelation function
 ARACNe:

An algorithm for the reconstruction of gene regulatory networks
 AUC:

Area under the curve
 CN:

Consensus network
 DPI:

data processing inequality
 GEO:

Gene expression Omnibus
 MI:

Mutual information
 miRNA:

Micro RNA
 ncRNA:

Non coding RNA
 OTU:

Operational taxonomic unit
 PFC:

Prefrontal cortex
 ROC:

Receiver operating characteristic
 TF:

Transcription factor
 TOM:

Topological overlap matrix
 WGCNA:

Weighted correlation network analysis
 wTO:

Weighted topological overlap
References
 1
Barabasi AL, Oltvai ZN. Network biology: understanding the cell’s functional organization. Nat Rev Genet. 2004; 5(2):101–13.
 2
Bansal M, Belcastro V, AmbesiImpiombato A, Di Bernardo D. How to infer gene networks from expression profiles. Mol Syst Biol. 2007; 3(1):78.
 3
Furlong LI. Human diseases through the lens of network biology. Trends Genet. 2013; 29(3):150–59.
 4
Dempsey K, Thapa I, Cortes C, Eriksen Z, Bastola DK, Ali H. On Mining Biological Signals Using Correlation Networks. In: 2013 IEEE 13th International Conference on Data Mining Workshops: 2013. p. 327–334. https://doi.org/10.1109/ICDMW.2013.125.
 5
Yang Y, Han L, Yuan Y, Li J, Hei N, Liang H. Gene coexpression network analysis reveals common systemlevel properties of prognostic genes across cancer types. Nat Commun. 2014; 5:3231.
 6
Taylor IW, Linding R, WardeFarley D, Liu Y, Pesquita C, Faria D, Bull S, Pawson T, Morris Q, Wrana JL. Dynamic modularity in protein interaction networks predicts breast cancer outcome. Nat Biotechnol. 2009; 27(2):199–04.
 7
van Dam S, Võsa U, van der Graaf A, Franke L, de Magalhães JP. Gene coexpression analysis for functional classification and gene–disease predictions.Brief Bioinform. 2017;139.
 8
Babu MM, Luscombe NM, Aravind L, Gerstein M, Teichmann SA. Structure and evolution of transcriptional regulatory networks. Curr Opin Struct Biol. 2004; 14(3):283–91.
 9
Mason MJ, Fan G, Plath K, Zhou Q, Horvath S. Signed weighted gene coexpression network analysis of transcriptional regulation in murine embryonic stem cells. BMC genomics. 2009; 10(1):327.
 10
Ravasz E, Somera AL, Mongru DA, Oltvai ZN, Barabási AL. Hierarchical organization of modularity in metabolic networks. Science. 2002; 297(5586):1551–55.
 11
Zhang B, Horvath S. A general framework for weighted gene coexpression network analysis. Stat Appl Genet Mol Biol. 2005; 4:17.
 12
Carlson MR, Zhang B, Fang Z, Mischel PS, Horvath S, Nelson SF. Gene connectivity, function, and sequence conservation: predictions from modular yeast coexpression networks. BMC Genomics. 2006; 7(1):40.
 13
Nowick K, Gernat T, Almaas E, Stubbs L. Differences in human and chimpanzee gene expression patterns define an evolving network of transcription factors in brain. Proc Natl Acad Sci. 2009; 106(52):22358–363.
 14
Langfelder P, Horvath S. Wgcna: an r package for weighted correlation network analysis. BMC Bioinf. 2008; 9(1):559.
 15
Langfelder P, Horvath S. Fast R functions for robust correlations and hierarchical clustering. J Stat Softw. 2012; 46(11):1–17.
 16
Margolin AA, Wang K, Lim WK, Kustagi M, Nemenman I, Califano A. Reverse engineering cellular networks. Nat Protoc. 2006; 1(2):662.
 17
Margolin AA, Nemenman I, Basso K, Wiggins C, Stolovitzky G, Dalla Favera R, Califano A. ARACNE: an algorithm for the reconstruction of gene regulatory networks in a mammalian cellular context. BMC Bioinf. 2006; 7(Suppl 1):S7.
 18
Allen JD, Xie Y, Chen M, Girard L, Xiao G. Comparing statistical methods for constructing large scale gene networks. PLoS ONE. 2012; 7(1):1–9. https://doi.org/10.1371/journal.pone.0029348.
 19
Berto S, PerdomoSabogal A, Gerighausen D, Qin J, Nowick K. A consensus network of gene regulatory factors in the human frontal lobe. Front Genet. 2016; 7:31.
 20
Efron B, Tibshirani RJ. An introduction to the bootstrap. Ed. Chapman & Hall. 1994; 1:31–103. New York.
 21
Altman DG. Practical statistics for medical research. 1990; 624:277–321.
 22
McCrumGardner E. Which is the correct statistical test to use?Br J Oral Maxillofac Surg. 2008; 46(1):38–41.
 23
Mukaka M. A guide to appropriate use of correlation coefficient in medical research. Malawi Med J. 2012; 24(3):69–71.
 24
Bishara AJ, Hittner JB. Testing the significance of a correlation with nonnormal data: comparison of pearson, spearman, transformation, and resampling approaches. Psychol Methods. 2012; 17(3):399.
 25
R Core Team. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing; 2017. https://www.Rproject.org/.
 26
Shannon P, Markiel A, Ozier O, Baliga NS, Wang JT, Ramage D, Amin N, Schwikowski B, Ideker T. Cytoscape: a software environment for integrated models of biomolecular interaction networks. Genome Res. 2003; 13(11):2498–04.
 27
Csardi G, Nepusz T. The igraph software package for complex network research. Inter J, Complex Sys. 2006; 1695(5):1–9.
 28
Peng J, Wang P, Zhou N, Zhu J. Partial correlation estimation by joint sparse regression models. J Am Stat Assoc. 2009; 104(486):735–46. https://doi.org/10.1198/jasa.2009.0126.
 29
Lewis NE, Cho BK, Knight EM, Palsson BO. Gene Expression Profiling and the Use of GenomeScale In Silico Models of Escherichia coli for Analysis: Providing Context for Content. J Bacteriol. 2009; 191(11):3437–44. https://doi.org/10.1128/JB.0003409.
 30
Fong SS, Joyce AR, Palsson BO. Parallel adaptive evolution cultures of Escherichia coli lead to convergent growth phenotypes with different gene expression states. Genome Res. 2005; 15(10):1365–72. https://doi.org/10.1101/gr.3832305.
 31
Fong SS, Nanchen A, Palsson BO, Sauer U. Latent Pathway Activation and Increased Pathway Capacity Enable <i>Escherichia coli</i> Adaptation to Loss of Key Metabolic Enzymes. J Biol Chem. 2006; 281(12):8024–33. https://doi.org/10.1074/jbc.M510016200.
 32
Covert MW, Knight EM, Reed JL, Herrgard MJ, Palsson BO. Integrating highthroughput and computational data elucidates bacterial networks. Nature. 2004; 429(6987):92–6. https://doi.org/10.1038/nature02456.
 33
GamaCastro S, Salgado H, SantosZavaleta A, LedezmaTejeida D, MuñizRascado L, GarcíaSotelo JS, AlquiciraHernández K, MartínezFlores I, Pannier L, CastroMondragón JA, MedinaRivera A, SolanoLira H, BonavidesMartínez C, PérezRueda E, AlquiciraHernández S, PorrónSotelo L, LópezFuentes A, HernándezKoutoucheva A, MoralChávez VD, Rinaldi F, ColladoVides J. RegulonDB version 9.0: highlevel integration of gene regulation, coexpression, motif clustering and beyond. Nucleic Acids Res. 2016; 44(D1):133–43. https://doi.org/10.1093/nar/gkv1156.
 34
Horvath S, Zhang B, Carlson M, Lu KV, Zhu S, Felciano RM, Laurance MF, Zhao W, Qi S, Chen Z, Lee Y, Scheck AC, Liau LM, Wu H, Geschwind DH, Febbo PG, Kornblum HI, Cloughesy TF, Nelson SF, Mischel PS. Analysis of oncogenic signaling networks in glioblastoma identifies ASPM as a molecular target. Proc Natl Acad Sci. 2006; 103(46):17402–07. https://doi.org/10.1073/pnas.0608396103.
 35
Meyer PE, Lafitte F, Bontempi G. MINET: An open source R/Bioconductor Package for Mutual Information based Network Inference. BMC Bioinf. 2008;9. http://www.biomedcentral.com/14712105/9/461.
 36
Robin X, Turck N, Hainard A, Tiberti N, Lisacek F, Sanchez JC, Müller M. proc: an opensource package for r and s+ to analyze and compare roc curves. BMC Bioinf. 2011; 12:77.
 37
Edgar R, Domrachev M, Lash AE. Gene expression omnibus: Ncbi gene expression and hybridization array data repository. Nucleic acids res. 2002; 30(1):207–10.
 38
Zhang Y, James M, Middleton FA, Davis RL. Transcriptional analysis of multiple brain regions in parkinson’s disease supports the involvement of specific protein processing, energy metabolism, and signaling pathways, and suggests novel disease mechanisms. Am J Med Genet B Neuropsychiatr Genet. 2005; 137(1):5–16.
 39
Zheng B, Liao Z, Locascio JJ, Lesniak KA, Roderick SS, Watt ML, Eklund AC, ZhangJames Y, Kim PD, Hauser MA, et al. Pgc1 α, a potential therapeutic target for early intervention in parkinson’s disease. Sci Transl Med. 2010; 2(52):52–735273.
 40
Vawter MP, Evans S, Choudary P, Tomita H, MeadorWoodruff J, Molnar M, Li J, Lopez JF, Myers R, Cox D, et al. Genderspecific gene expression in postmortem human brain: localization to sex chromosomes. Neuropsychopharmacology. 2004; 29(2):373.
 41
Chang LC, Jamain S, Lin CW, Rujescu D, Tseng GC, Sibille E. A conserved bdnf, glutamateand gabaenriched gene module related to human depression identified by coexpression metaanalysis and dna variant genomewide association studies. PloS ONE. 2014; 9(3):90980.
 42
R Core Team. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing; 2017. https://www.Rproject.org/.
 43
Gautier L, Cope L, Bolstad BM, Irizarry RA. affy—analysis of affymetrix genechip data at the probe level. Bioinformatics. 2004; 20(3):307–15. https://doi.org/10.1093/bioinformatics/btg405.
 44
Messina DN, Glasscock J, Gish W, Lovett M. An orfeomebased analysis of human transcription factor genes and the construction of a microarray to interrogate their expression. Genome Res. 2004; 14(10b):2041–47.
 45
Vaquerizas JM, Kummerfeld SK, Teichmann SA, Luscombe NM. A census of human transcription factors: function, expression and evolution. Nat Rev Genet. 2009; 10(4):252.
 46
Ravasi T, Suzuki H, Cannistraci CV, Katayama S, Bajic VB, Tan K, Akalin A, Schmeier S, KanamoriKatayama M, Bertin N, et al. An atlas of combinatorial transcriptional regulation in mouse and man. Cell. 2010; 140(5):744–52.
 47
Nowick K, Fields C, Gernat T, CaetanoAnolles D, Kholina N, Stubbs L. Gain, loss and divergence in primate zincfinger genes: a rich resource for evolution of gene regulatory differences between species. PLoS ONE. 2011; 6(6):21553.
 48
Corsinotti A, Kapopoulou A, Gubelmann C, Imbeault M, de Sio FRS, Rowe HM, Mouscaz Y, Deplancke B, Trono D. Global and stage specific patterns of krüppelassociatedbox zinc finger protein gene expression in murine early embryonic cells. PloS ONE. 2013; 8(2):56721.
 49
Tripathi S, Christie KR, Balakrishnan R, Huntley R, Hill DP, Thommesen L, Blake JA, Kuiper M, Lægreid A. Gene ontology annotation of sequencespecific dna binding transcription factors: setting the stage for a largescale curation effort. Database. 2013; 2013:062.
 50
Wingender E, Schoeps T, Dönitz J. Tfclass: an expandable hierarchical classification of human transcription factors. Nucleic Acids Res. 2012; 41(D1):165–70.
 51
Wingender E, Schoeps T, Haubrock M, Dönitz J. Tfclass: a classification of human transcription factors and their rodent orthologs. Nucleic Acids Res. 2014; 43(D1):97–02.
 52
Alexa A, Rahnenführer J. Gene set enrichment analysis with topGO. Bioconductor Improv. 2009;27.
 53
Mac Rygaard A, Thøgersen MS, Nielsen KF, Gram L, BentzonTilia M. Effects of gelling agent and extracellular signaling molecules on the culturability of marine bacteria. Appl Environ Microbiol. 2017; 83(9):00243–17.
 54
Loch TP, Faisal M. Emerging flavobacterial infections in fish: A review. J Adv Res. 2015; 6(3):283–300. Editors and International Board Member collection.
 55
Bernardet JF. Bergey’s Manual of Systematic Bacteriology, 2nd ed., vol. 1 (The Archaea and the deeply branching and phototrophic Bacteria) (D.R. Boone and R.W. Castenholz, eds.)New York: SpringerVerlag; 2001. pp. 465–466.
 56
Yan J. Som: SelfOrganizing Map. 2016. R package version 0.35.1. https://CRAN.Rproject.org/package=som.
 57
Wickham H. The splitapplycombine strategy for data analysis. J Stat Softw. 2011; 40(1):1–29.
 58
Wickham H. stringr: modern, consistent string processing. The R J. 2010; 2(2):38–40.
 59
Butts C. T.network: a package for managing relational data in r. J Stat Softw. 2008;24(2).
 60
Butts CT. Network: Classes for Relational Data. 2015. The Statnet Project (http://statnet.org). R package version 1.13.0. http://CRAN.Rproject.org/package=network.
 61
Almende BV, Thieurmel B.visNetwork: Network Visualization Using ’vis.js’ Library. 2016. R package version 1.0.3. https://CRAN.Rproject.org/package=visNetwork.
 62
Dowle M, Srinivasan A. Data table: Extension of data frame. 2017. R package version 1.10.4. https://CRAN.Rproject.org/package=data.table.
 63
Pons P, Latapy M. Computing communities in large networks using random walks. J Graph Algorithms Appl. 2006; 10(2):191–18.
 64
Brandes U, Delling D, Gaertler M, Gorke R, Hoefer M, Nikoloski Z, Wagner D.On modularity clustering. IEEE Trans Knowl Data Eng. 2008; 20(2):172–88.
 65
Reichardt J, Bornholdt S. Statistical mechanics of community detection. Physical Review E. 2006; 74(1):016110.
 66
Newman ME, Girvan M. Finding and evaluating community structure in networks. Physical Rev E. 2004; 69(2):026113.
 67
Traag VA, Bruggeman J. Community detection in networks with positive and negative links. Phys Rev E. 2009; 80(3):036115.
 68
Freeman LC. Centrality in social networks conceptual clarification. Soc Netw. 1978; 1(3):215–39.
 69
Brandes U. A faster algorithm for betweenness centrality. J Math Sociol. 2001; 25(2):163–77.
 70
Clauset A, Newman ME, Moore C. Finding community structure in very large networks. Phys Rev E. 2004; 70(6):066111.
 71
Rosvall M, Axelsson D, Bergstrom CT. The map equation. Eur Phys J Spec Top. 2009; 178(1):13–23. https://doi.org/10.1140/epjst/e2010011791. Springer.
 72
Rosvall M, Axelsson D, Bergstrom CT. The map equation. Eur Phys JSpecial Topics. 2009; 178(1):13–23. Springer https://doi.org/10.1140/epjst/e2010011791.
 73
Blondel VD, Guillaume JL, Lambiotte R, Lefebvre E. Fast unfolding of communities in large networks. J stat mech: theory and experiment. 2008; 2008(10):10008.
 74
Raghavan UN, Albert R, Kumara S. Near linear time algorithm to detect community structures in largescale networks. Phys Rev E. 2007; 76(3):036106.
 75
Newman ME. Finding community structure in networks using the eigenvectors of matrices. Phys Rev E. 2006; 74(3):036104.
Acknowledgements
We thank Professor Martin Middendorf, Martina Hall and Marlis Reich for fruitful discussions on the methodology and suggestions on the package. We thank Alvaro Perdomo Sabogal for providing us the Transcription Factors list used to build the PFC networks. We thank Daniel Gerighausen for discussions. We acknowledge support by the German Research Foundation and the Open Access Publication Fund of the Freie Universität Berlin.
Funding
This work was supported partially by a doctoral grant from the Brazilian government’s Science without Borders program (GDE 204111/20145).
Availability of data and materials
wTO is open source and freely available from CRANhttps://cran.rproject.org/web/packages/wTO/under the the GPL2 Open Source License. It is platform independent.
Author information
Affiliations
Contributions
DG implemented the code in R. DG and TM conceived the idea of pvalues for the edges. KN and EA generalized the wTO for signed values. DG and AV compared the wTO method to other methods. DG run the example analysis. DG wrote the draft of manuscript. All authors discussed the manuscript, read and approved the final version of the manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
About this article
Cite this article
Gysi, D., Voigt, A., Fragoso, T. et al. wTO: an R package for computing weighted topological overlap and a consensus network with integrated visualization tool. BMC Bioinformatics 19, 392 (2018). https://doi.org/10.1186/s1285901823517
Received:
Accepted:
Published:
Keywords
 Coexpression network
 Network
 Expression
 R package
 Software
 Consensus Network
 wTO
 Meta analysis
 Cooccurrence network
 Metagenomics