wTO: an R package for computing weighted topological overlap and consensus networks with an integrated visualization tool

11/13/2017 ∙ by Deisy Morselli Gysi, et al. ∙ 0

Background: Gene co-expression network analyses have become a central approach for the systems-level analysis 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 some genes are able to up- or down-regulate other genes, it is important to explicitly consider both positive and negative correlations when constructing a gene co-expression network. Additionally, there has been a growing interest in the systematic comparison of multiple networks to identify deferentially changed links. Typically, such analyses are focused on the comparison of networks or data from two different conditions. Results: Here, we present an R package for calculating the weighted topological overlap (wTO), that explicitly addresses the sign of wTO values. The package includes the calculation of p-values (raw and adjusted) for each pairwise gene score. Our package also allows the calculation of networks from time series, without replicates. Additionally, our R package incorporates a novel method for calculating a consensus network (CN) from two or more networks. To visualize 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 20,000 genes in under two hours. Conclusion: In this work, we developed an R 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.r-project.org/web/packages/wTO/).

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Background

Recent applications of complex network analysis methods have provided important new knowledge of the functioning and interactions of genes at the systems level 7; 6; 29; 22. Within the area of biological network analyses, co-expression networks have received much attention 72; 61. For the co-expression 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 “presence” and “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 gene-pair relationship, and the sign as indicative of the type of associated gene interaction: positive if the genes are co-regulated, negative if they are oppositely controlled 64.

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  5; 40. For these situations, oftentimes the choice is made to only take into account direct interactions between the gene-subset 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 non-binary weights, the method of weighted topological () network analysis 55 has been found very useful. In a -analysis, a new link-weight for a pair of connected nodes is determined through an averaging process that accounts for all common network neighbors 55. Thus, is a method that implicitly includes correlations among nodes that are going to be exempt from further analysis. The method 55; 73; 16 can be used to determine the overlap among classes of transcripts, for example TFs and non-coding RNAs (ncRNAs). The resulting network provides a more robust representation of the connections and interactions among the node-set of interest than a simple correlation network analysis focused only on the node-set of interest 47.

The packages WGCNA 33; 34 and ARACNe 39; 38 are widely used for weighted gene co-expression network analysis studies. The former provides functions for the calculation of the adjacency matrix for all pairs of genes as the n-th 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 2; 64.

Previously, Nowick and collaborators 47 developed a mathematical method to calculate the for a set of nodes that explicitly takes into account both positive and negative correlations. This version of the wTO-measure 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 predator-prey 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 -calculation methodology is implemented in the R package presented here. In order to avoid confusion, we will refer to the method for calculating a pair-wise link score as 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 9. 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 ().

Here, we present wTO, an R package that is capable of computing both signed and unsigned networks as well as the , thus providing methods for assigning -values 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.

Figure 1: The wTO package workflow Gray boxes refer to inputs, red boxes refer to content of the wTO package, yellow boxes are functions included in the package, blue boxes are outputs of those functions, and green boxes refer to methods internal to the package. Our package can deal with multiple kinds of data, for example RNA-seq counts or normalized values, microarray expression data, abundance data coming from metagenomic studies, and many more. All input data should be pre-processed with the quality control and normalization methods recommended for each respective type of data. The function wTO.Complete calculates the values, as many times as desired. As output, the user will obtain an object containing the signed and absolute values for each pair of nodes, -values and  values for multiple testing. This output can be used for the construction of a from independent networks using the function wTO.Consensus. Outputs from the and networks can be used as an input for NetVis, which is an integrated tool for plotting networks. As an interactive tool it also allows the user to modify the network.

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 genome-wide expression studies of healthy human pre-frontal cortex samples and an analysis of a time-series 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 RNA-Seq 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 co-occurrence 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 based on correlations between a pair of nodes and as

(1)

with being a correlation measure. Assuming that nodes and represent a sub-set of factors (e.g genes) of particular interest selected from the nodes, we calculate the weighted topological overlap (  47, ) between node and node as

(2)

where

(3)

Note that, this expression explicitly includes both positive and negative correlations, and thus allows for to take both positive and negative values. Other software packages calculating the have implemented definitions of the method that do not allow for negative values 33, 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 , we need to be aware of the limits of this expression. Consider three nodes , and , and assume that . All the terms in the numerator of Eq. (2) will be negative if for all nodes . However, if , then at least some contributions to the sum will cancel out. The same rationale applies for the case of .

To systematically assess the potential effect of term cancellation in Eq. (2), we calculate the absolute weighted topological overlap, which uses the absolute value of the correlations () as input for Eq. (2). In this case, the sign of the correlation is excluded from the analysis and only the magnitude of the link-strength is taken into account. Consequently, by generating a scatter plot of the signed and unsigned weights, it is possible to assess at which -values term cancellations start affecting the results. Thus, for values of interest, the closer the plot of vs. is to , the better.

However, by just computing the

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

(4)

of the null hypothesis (

) of no association against the two-sided alternative () of non-zero association. This can be computed by using bootstrap 25 or permutation resampling methods 47. 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 as the maximal distance between the calculated with each bootstrap and the of the real dataset. This means that, the smaller is, the stronger is our confidence in a particular . By default, is set to 0.2.

One advantage of the wTO package is its application to analyze and make networks out of time-series data. Therefore, we are interested in the implementation of blocked bootstrap resampling 25 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 time-series 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 time-series, 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 for a time series where the observations are not independent of each other.

A method for determining a Consensus Network

Berto and collaborators 9 described a consensus network based on gene-expression 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) values, to be combined into a single . Our approach has the advantage of penalizing links with opposite signs. According to the same rationale, links with the same sign among the multiple networks, will have their values closer to the largest of a link among the 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 networks, we calculate a using the following approach:

If we have replicated networks (note that means the index of the networks, not the exponent of nor ), then we define the consensus network as

(5)

where

(6)

A threshold can be used to remove links with 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 p values are combined using the Fisher method.

Results and discussion

The representation of interactions between a set of nodes by the method  55; 73; 16 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 network contains more robust information about the connections among nodes than what would result from simply taking direct correlations into account 73; 47. The

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 under-report 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 

4; 41; 44; 10.

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 function.

To test whether the calculated 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 is calculated a number of times, 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’’) 47. 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 computer threads to be used in the calculations. To implement the parallel function, we used the R package parallel 52.

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 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 values (signed and unsigned), the p-values and the adjusted p-values 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

    values, when it is not desired to visualize low 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 vs , as previously discussed. The scatter plot of p-values against the and is also plotted along with suggested threshold values that are the empirical quantiles.

Computing of the 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 p-value. The output is a data.table containing the two nodes’ names and the consensus weight, and the combined p-value. 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 is close to zero. When all links agree to the same value or show little deviation, the strength of the resulting value is closer to the determined value. If a node is absent in at least one network, it is removed.

Figure 2: A schematic example of the method: Panel I shows four independent networks to be combined into one . Note that the rightmost network does not include the ’A’ node. Blue links indicates negative sign, while orange, positive. The can be seen on Panel II. Note that the missing node from Panel I is not present in the . Also, only links that are constant in its sign among networks are present in the final network. For example, the link between D and E is removed since it has a different signal in the last network.

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 60 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 Consensus both generate network data-structures (edge list) that can be visualized with this function. The user needs to choose a relevant -threshold (the quantiles resulting from the bootstrap), or -value cut-off, 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 21 package. By default, the -threshold value is set to , and the network layout-style is set to layout_nicely. To avoid false positives, we recommend to filter the data according to the desired significance  value and to choose the -threshold according to the computed empirical quantiles. The size of the nodes is relative to their degree. Our package further includes an option for making clusters 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 co-expressions, as in the WGCNA package. The width of a link is relative to the , 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 non-protein 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 , 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 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 for a full network with 20,000 nodes in 20 miliseconds per link. This shows that it is quite feasible to compute the full for a realistic gene expression network.

Figure 3: Computational time for the calculation of wTO for each link for different sizes of networks and proportions of sets of nodes of interest: The run time of the calculation increases with increasing proportion of nodes of interest. The graph presented here shows the time for computing each link for different sizes of nodes and proportions of subsets of nodes of interest.

Comparison with existing methods

A variety of methods currently exist to analyze gene co-expression networks, in particular ARACNe 39; 38, SPACE 49 and WGCNA  33; 34. These methods rest on a multitude of different mathematical principles, particularly with respect to how co-expression 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 co-expression at the gene-pair level. But in WGCNA, the final edge weight corresponds to the absolute value of 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 co-expression has previously been shown to compare favourably with other methods 2.

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 pair-wise possible connections, while ARACNe will only compute the pair-wise information if their information is not independent.

Relative to WGCNA, wTO provides three major additions: the determination of p-values (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 node-and-edge type view of the co-expression 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 (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 inter-set wTO score).

Another minor difference resides in how wTO is determined for each gene with itself. From Eq. 2, we see that (assuming and ):

(7)

For an unweighted network, where or for all , this approximates to for large . However, this is not the case for weighted networks. WGCNA differs from the wTO package in that is explicitly set for all , while our package retains the score as defined by Eq. 2.

Method wTO WGCNA ARACNe
Topological overlap Yes Yes No
Signed topological overlap Optional No No
Consensus topological overlap Weighted sum Minimum weight (strict) No
Pairwise p-values Yes No Used to filter MI
Network view Native Exported to Cytoscape Exported to Cytoscape
Soft thresholding No Optional (on by default) No
Correlation choices Spearman, Pearson Bicor, Pearson Spearman, Pearson, Kendall
Able to deal with time-series Yes No No
Table 1: Comparison of key differences between wTO, WGCNA and ARACNe

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 35; 26; 27; 20. 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 TF-TF interactions, we used the RegulonDB 30 database, which contains experimental data from E. coli, as a reference. We defined as True-Positive interactions those that are described in RegulonDB, and as True-Negatives 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 73; 32. We used the functions pickSoftThreshold and pickHardThreshold for defining the power of the soft-threshold and for choosing the hard-threshold, respectively. The power was defined as 4 and the hard-threshold 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 43. The wTO networks were built using 100 simulations, Pearson correlation and filtered for p values 0.01 and the 90% quantile. One ARACNe network was constructed using a of 0.2, the default of the package, and another network was built using a 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 57 (see Fig. 4).

Figure 4: ROC curves for the comparison of methods. Overall, our wTO method performs better than ARACNe, WGCNA and raw Pearson correlations. ARACNe is better in finding true positives, while WGCNA is more conservative, and therefore better in finding true negatives but identifies fewer true positives.

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 , 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 method further reduces false effects coming from incorrectly assigned linked genes (false positives) when compared to ARACNe and raw correlations.

ReactomeDB
(Total)
Pearson
Correlation
ARACNe WGCNA
wTO
(delta 0.1)
wTO
(delta 0.2)
True Negative 7234 2259 2633 7092 6520 5235
False Negative 0 216 245 321 318 288
False Positive 0 4975 4601 142 714 1999
True Positive 328 112 83 7 10 40
Total 7562 7562 7562 7562 7562 7562
Table 2: Accuracy of the 3 methods and correlation

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 24. From the study GSE20168 74; 75, we used data from a total of 15 postmortem brain samples. From the study GSE2164 66, we used a total of 26 samples from post mortem brains. And finally, from the study GSE54568 17 we used all the 15 controls. All individuals were older than 5 years and died without any neuro-pathological phenotypes. We chose the TFs to be our genes of interest and calculated a TF-wTO network for each of the three datasets. Subsequently, we computed the consensus network for the three TF wTO networks.

The downloaded data were pre-processed and normalized by ourselves independently, using the R environment 51, and the affy 31 package from the Bioconductor set. The probe expression levels (RMA expression values) and MAS5 detection  values were computed, and only probesets significantly detected in at least one sample ( value 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 co-expressed in brain networks. We used a set of 3229 unique TF symbols from the TF-Catalog (Perdomo-Sabogal 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 GRF inventories 42; 65; 54; 48; 19; 63; 69; 70 that are associated with gene ontology terms for regulation of transcription, DNA-depending transcription, RNA polymerase II transcription co-factor and co-repressor 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 wTO R 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  value of . The Consensus Network was built based on the calculated signed wTO values of significant links. Weights for links with in-significant wTO were set to zero. Fig. 5 shows the distributions and the networks for our three datasets.

Figure 5: Comparison of the three networks used to compute the CN. The first row shows the distribution of significant wTO values ( value ). Note that the wTO range of the second network is larger than of the other two networks. The second row show the wTO network for each method. The third and forth row refer to the CN. Note that now the distribution of the wTO values does not include the wTO values close to zero, and retains only values that show a high correlation between the TFs. In the histograms, the presence of negative wTO values is visible, indicating that there are TFs that downregulate other genes.

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 with at least one TF of the cluster were used for GO enrichment analysis for that cluster, using the R package topGO 1. 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).

Figure 6: GO terms enriched within each cluster. Enriched GO terms of the category ”biological process” are clustered by REVIGO [76] with the SimRel measurement and allowed similarity of 0.5. The size of the circle represents the frequency of the GO term in the database, i.e. GO groups with many members are represented by larger circles. The color code refers to the log(p-value) of the GO enrichment analysis: the closer to 0, the more red, the lower this value, the greener the bubble is. After removing redundancies, the remaining terms are visualized in semantic similarity-based scatter-plots, where the axes correspond to semantic distance. Brain related functions were detected, for instance in Clusters 1 and 3, that are involved with cognition.
Cluster TFs Genes correlated to TFs GO.ID Term
1 589 58 GO:0042775 mitochondrial ATP synthesis coupled
GO:0010498 proteasomal protein catabolic process
GO:0050890 cognition
GO:0033238 regulation of cellular amine metabolic pathway
GO:0008090 retrograde axonal transport
GO:0070050 neuron cellular homeostasis
GO:0090168 Golgi reassembly
GO:0006099 tricarboxylic acid cycle
GO:0051443 positive regulation of ubiquitin-protein
GO:0061418 regulation of transcription from RNA polimerase
GO:0047496 vesicle transport along microtubule
GO:0061640 cytoskeleton-dependent cytokinesis
GO:0043488 regulation of mRNA stability
GO:0000086 G2/M transition of mitotic cell cycle
GO:0038061 NIK/NF-kappaB signaling
GO:0000209 protein polyubiquitination
GO:0007052 mitotic spindle organization
GO:0031333 negative regulation of protein complex
GO:0002223 stimulatory C-type lectin receptor signal
GO:0016486 peptide hormone processing
GO:0034314 Arp2/3 complex-mediated actin nucleation
GO:1900271 regulation of long-term synaptic potential
GO:0000715 nucleotide-excision repair, DNA damage
GO:1901983 regulation of protein acetylation
GO:0016082 synaptic vesicle priming
GO:0043243 positive regulation of protein complex
GO:2000637 positive regulation of gene silencing
GO:0021902 commitment of neuronal cell
GO:0051683 establishment of Golgi localization
GO:0060013 righting reflex
GO:0061732 mitochondrial acetyl-CoA biosynthetic pr…
2 647 77 GO:0035773 insulin secretion involved in cellular
GO:0098930 axonal transport
GO:0000086 G2/M transition of mitotic cell cycle
GO:0061640 cytoskeleton-dependent cytokinesis
GO:0090083 regulation of inclusion body assembly
GO:0034112 positive regulation of homotypic
GO:1902750 negative regulation of cell cycle G2/M
GO:0031146 SCF-dependent proteasomal ubiquitin-dependent
GO:0061003 positive regulation of dendritic spine
GO:0032922 circadian regulation of gene expression
GO:0072600 establishment of protein localization
GO:0061077 chaperone-mediated protein folding
GO:0016191 synaptic vesicle uncoating
GO:1902309 negative regulation of peptidyl-serine
GO:0048024 regulation of mRNA splicing, via spliceosome
GO:0016486 peptide hormone processing
GO:0048268 clathrin coat assembly
GO:0000209 protein polyubiquitination
GO:0035902 response to immobilization stress
GO:2000757 negative regulation of peptidyl-lysine
3 402 17 GO:0043687 post-translational protein modification
GO:0050851 antigen receptor-mediated signaling pathway
GO:0002479 antigen processing and presentation
GO:0090199 regulation of release of cytochrome c
GO:1905323 telomerase holoenzyme complex assembly
GO:0050890 cognition
GO:0043248 proteasome assembly
GO:0030177 positive regulation of Wnt signaling pat…
GO:0047496 vesicle transport along microtubule
GO:0042775 mitochondrial ATP synthesis
GO:0035773 insulin secretion involved in cellular
GO:0045116 protein neddylation
GO:0090141 positive regulation of mitochondrial
GO:0060071 Wnt signaling pathway, planar cell
GO:0010635 regulation of mitochondrial fusion
GO:0016579 protein deubiquitination
GO:0090090 negative regulation of canonical Wnt signal
GO:0051131 chaperone-mediated protein complex
GO:0051560 mitochondrial calcium ion homeostasis
GO:0008090 retrograde axonal transport
GO:0032700 negative regulation of interleukin-17
GO:0048170 positive regulation of long-term neuronal
GO:0051036 regulation of endosome size
GO:0061588 calcium activated phospholipid
GO:0090149 mitochondrial membrane fission
GO:0097112 gamma-aminobutyric acid receptor
GO:0097332 response to antipsychotic drug
GO:0097338 response to clozapine
GO:1902683 regulation of receptor localization
GO:0060052 neurofilament cytoskeleton organization
GO:0048678 response to axon injury
Table 3: GO terms associated with each one of the CN Clusters
Cluster TFs Genes correlated to TFs GO.ID Term
4 677 39 GO:0007612 learning
GO:0000209 protein polyubiquitination
GO:0070646 protein modification by small protein
GO:0035567 non-canonical Wnt signaling pathway
GO:0038061 NIK/NF-kappaB signaling
GO:0090313 regulation of protein targeting to membrane
GO:0016339 calcium-dependent cell-cell adhesion
GO:0002223 stimulatory C-type lectin receptor signal
GO:0043687 post-translational protein modification
GO:0008090 retrograde axonal transport
GO:0061732 mitochondrial acetyl-CoA biosynthetic
GO:0070050 neuron cellular homeostasis
GO:0016236 macroautophagy
GO:0043488 regulation of mRNA stability
GO:0061178 regulation of insulin secretion involved…
GO:0016486 peptide hormone processing
GO:0035493 SNARE complex assembly
GO:0034112 positive regulation of homotypic
GO:1902260 negative regulation of delayed rectifier…
GO:1902267 regulation of polyamine transmembrane
GO:2000574 regulation of microtubule motor activity
GO:0016082 synaptic vesicle priming
GO:0051560 mitochondrial calcium ion homeostasis
GO:0006596 polyamine biosynthetic process
GO:0060052 neurofilament cytoskeleton organization
GO:1903608 protein localization to cytoplasmic stress
GO:0000715 nucleotide-excision repair, DNA damage
GO:0047496 vesicle transport along microtubule
GO:1990542 mitochondrial transmembrane transport
GO:0031333 negative regulation of protein complex
GO:0046826 negative regulation of protein export
5 18 4 GO:0072369 regulation of lipid transport
GO:1901379 regulation of potassium ion transmembrane
GO:0032700 negative regulation of interleukin-17
GO:0051036 regulation of endosome size
GO:1904219 positive regulation of CDP-diacylglycerol
GO:1904222 positive regulation of serine C-palmitoyl
GO:1905664 regulation of calcium ion import
GO:2000286 receptor internalization
GO:0021769 orbitofrontal cortex development
GO:0045716 positive regulation of low-density lipo.
GO:0060430 lung saccule development
GO:0070885 negative regulation of calcineurin-NFAT
GO:1900272 negative regulation of long-term synaptic
GO:1902951 negative regulation of dendritic spine
Table 4: Continuation: GO terms associated with each one of the CN Clusters

Time series: Metagenomics data from the ocean

Only about 1% of marine bacteria can be easily studied using standard laboratory procedures 37. 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 co-occurrence in that environment (Fig. 7). We found that 61 out of 67 OTUs had at least one significant interaction ( value ). Positive correlations in co-occurence networks may represent symbiotic or commensal relationships, while negative correlations may represent predator-prey 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 36 and to live in commensality with other bacteria from the same order 8.

Figure 7: OTUs analysis using the Time-Series method of the wTO package. In this network, the sizes of the nodes are proportional to a node’s degree, and the width of a link is proportional to its wTO-absolute value. The link color refers to its sign, with green links being negative and purple ones positive. Nodes belonging to the same cluster are shown in the same color. There are four distinct clusters of bacteria. The orange cluster contains only negative interactions (green links), suggesting that the bacterial species in this cluster do not co-exist. We also notice, that many of the bacteria belonging to the same order are well connected by purple links, indicating that they co-exist and share interactions. However, the number of interactions among non-related bacteria demonstrate that interactions are not intra-order specific.

Conclusion

This new wTO package allows 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 p-values 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 , 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, networks and , for efficient further custom analysis.

We qualitatively and quantitatively compared our new package to state-of-the-art 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 genome-wide expression datasets of human pre-frontal cortex samples, and one on wTO co-occurence 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

wTO relies on the following packages: som 71, plyr 68, stringr 67, network 14; 15, igraph 21, visNetwork 3, data.table 23 and the standard packages stats and parallel 52. The visualization tool implemented in our package was built using a combination of the packages network 14; 15, igraph 21 and visNetwork 3. The MakeGroups parameter, passed to the function NetVis for constructing the network, allows the user to choose clustering algorithms from: “walktrap” 50, “optimal”13, “spinglass” 56; 46; 62, “edge.betweenness’ 28; 12, ‘fast_greedy” 18, “infomap’ 58; 59, “louvain” 11, “label_prop’ 53 and “leading_eigen” 45. All those algorithms are implemented in the igraph package21. It is publicly available on CRAN repositories under the GPL-2 Open Source License https://cran.r-project.org/web/packages/wTO/. It is platform independent.

Declarations

Abbreviations

wTO: Weighted topological overlap; CN: Consensus Network; TF: Transcription Factor; ncRNA: non coding RNA; miRNA: micro RNA. OTU: Operational Taxonomic Unit. acf: autocorrelation function. GEO: Gene Expression Omnibus. PFC: Pre-frontal Cortex. AUC: Area under the curve. ROC: Receiver operating characteristic. TOM: Topological Overlap Matrix. ARACNe: An Algorithm for the Reconstruction of Gene Regulatory Networks. WGCNA: Weighted Correlation Network Analysis. MI: Mutual Information. DPI: data processing inequality.

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.

Author’s contributions

DG implemented the code in R. DG and TM conceived the idea of p-values for the edges. KN and EA generalized the 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.

Availability of data and materials

wTO is open source and freely available from CRAN https://cran.r-project.org/web/packages/wTO/ under the the GPL-2 Open Source License. It is platform independent.

Competing interests

The authors declare that they have no competing interests.

Consent for publication

Not applicable.

Ethics approval and consent to participate

Not applicable.

Funding

This work was supported partially by a doctoral grant from the Brazilian government’s Science without Borders program (GDE 204111/2014-5).

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