2.1 Data

The Genotype-Tissue Expression (GTEx) portal provides open-access tissue-specific gene expression data [77]. For the analyses in the present work, we use the gene transcripts per million (TPM) data from release V8 for four exocrine glands: pancreas, minor salivary gland, mammary gland, and skin (not sun exposed). In this pilot study, we limit our analysis to four tissues. We chose these tissues because they are all tissues that interact with the outside world and may have adaptively evolved to different environmental conditions. Specifically, the pancreas plays a vital role in the digestive system, secreting digestive enzymes [78]. The salivary gland is the main gatekeeper of our body and contributes to the oral proteome [79]. The mammary gland produces milk containing immunologic agents to nourish and protect young offspring [80]. The skin protects the body against pathogens, regulates body temperature, and has changed most drastically in human lineage [81, 82]. Consequently, we hypothesized that these tissues would retain a high level of variation in gene expression levels.

There are 328 samples from the pancreas, 162 samples from the minor salivary gland, 459 samples from the mammary gland, and 604 samples from the skin (not sun exposed) in this TPM data. Each sample contains gene expression data for 56, 200 different genes.

The number of genes is much larger than the number of samples for all tissues. Therefore, we focused on a subset of genes for our analysis around the same size as the number of samples in our data, as in [39, 83]. To subset the genes, we identified the top 75 genes with the highest variance of TPM across all samples [22], separately for each tissue. We chose the number 75 because the union of the top 75 genes in terms of the variance of TPM across the four tissues contains 203 genes, which is not much larger than the smallest number of samples (162 samples). It is well known that estimation of correlation matrices from data is unreliable if the number of elements (i.e., genes in the present case) is comparable with or larger than the number of samples [84]. Nevertheless, to further validate our choice for the number of genes, we repeated some analysis on an expanded network with 371 genes. We found that the expanded network produces a similar type of partition as the original network, supporting the robustness of our analysis with respect to the number of genes selected for our analysis (see Text A in S1 Text).

We looked at the most variable genes because, again, our goal is to understand the underlying genetic and environmental bases of gene expression variation. In fact, most of the highly variably expressed genes are also highly expressed genes. To show this, for each tissue, we calculate the Jaccard index between the top 75 genes in terms of average TPM and the top 75 genes in terms of variance of TPM. The Jaccard index is defined as the size of the intersection of two finite sets A and B divided by the size of the union of A and B [85]. The range of the Jaccard index is 0 to 1, and a larger Jaccard index implies a greater overlap between the two sets of genes. We also examine the average rank of the top 75 genes in variance among all 56, 200 genes. We compute the rank in terms of the average TPM. Therefore, if the average rank is high (i.e., a low number), then the highly variable genes are also relatively highly expressed. We show in Table 1 the Jaccard index and the average rank of the top 75 genes for each tissue. The table indicates that the Jaccard index is at least 0.402 and the average rank is at most 167.8. These results suggest that the top 75 genes in terms of variance of TPM are overall highly expressed genes as well because we have calculated these indices for 75 genes in comparison to the 56, 200 genes. This finding is consistent with an established understanding that sequence read count data follows a negative binomial distribution [86–88].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Similarity between the highly variable genes and the highly expressed genes in each tissue.

https://doi.org/10.1371/journal.pcbi.1011616.t001

We analyze four-layer networks composed of the 203 genes in the union of the top 75 genes in terms of the variance of TPM across the four tissues. We note that the number of nodes must be the same in each layer for our multilayer community detection method described in section 2.4.

2.2 Multilayer network construction

For each of the four tissues, we generate a 203 × 203 gene co-expression matrix in which the (i, j)-th entry is the Pearson correlation coefficient between the log-normalized TPM of gene i and the log-normalized TPM of gene j across all samples from that tissue. We take the logarithm of TPM before calculating the Pearson correlation coefficient to suppress the effect of outliers; TPM is extremely large for some samples. Let S denote the number of samples from tissue α. We denote by x_i,α,s and x_{j,α, s} the TPM value for gene i and j, respectively, for sample s ∈ {1, 2, …, S} in tissue α. Then, we calculated the Pearson correlation coefficient between log(x_i,α,s + 1) and log(x_{j,α, s} + 1) across the S samples as the co-expression between gene i and gene j in tissue α. In other words, we calculate (1) where (2) and (3) We took the logarithm of x_i,α,s + 1 because, in this manner, x_i,α,s = 0 is mapped to 0.

To compare the gene co-expression patterns across the different tissues, we view the four correlation matrices as a four-layer correlation matrix, or categorical layers of a multilayer gene co-expression network. Because the set of genes is the same in the four layers, we place an interlayer edge between the same gene in each pair of layers (i.e., tissues) as shown by the dashed lines in Fig 1. Therefore, our network is a multiplex network with diagonal and categorical interlayer couplings, where, by definition, the interlayer edges connect each gene with itself in each other layer [89, 90].

We denote the strength of the interlayer coupling that connects node i in layer α to node i in layer β as ω_iαβ [43]. One typically assumes that ω_iαβ takes binary values {0, ω}, where ω is a parameter indicating the absence (i.e., 0) or presence (i.e., ω) of interlayer edges [43]. However, how to set and interpret the ω value is not straightforward [91]. In this work, we use the empirical co-expression (i.e., Pearson correlation coefficient) of gene i between tissues α and β as ω_iαβ. Specifically, ω_iαβ is equal to the right-hand side of Eq (1) with x_j,α,s and m_j,α being replaced by x_i,β,s and m_i,β, respectively, and with S being interpreted as the number of samples common to tissues α and β. Since the majority of studies on multilayer modularity maximization assume non-negative interlayer edge weights, if the obtained ω_iαβ is negative, we force ω_iαβ = 0. However, note that some studies do include negative interlayer edge weights [92].

2.3 Community detection in conventional multilayer networks

We are interested in detecting communities (also called modules and gene sets) in our multilayer networks to find sets of genes that are similarly expressed across individuals and therefore potentially involved in related biological processes. Some algorithms can detect communities that span between multiple layers as well as communities that lie within just one layer. We are interested in these different types of communities and their biological implications. A common method to find such communities in multilayer networks is to maximize an objective function called the multilayer modularity [43]. However, our multilayer gene networks are based on correlation. Therefore, we develop multilayer modularity for multilayer correlation matrices. In this section, we review multilayer modularity for usual multilayer networks as a primer to the multilayer modularity for correlation matrices.

The modularity for single-layer undirected networks, which may be weighted, is given by [93, 94] (4) where N is the number of nodes in the given network; A_ij is the (i, j)-th entry of the adjacency matrix and we assume A_ii = 0 ∀i ∈ {1, …, N}; is the number of edges in the case of unweighted networks and the total weight of all edges in the case of weighted networks; γ is the resolution parameter controlling the size of typical communities found by modularity maximization [95]; a large γ tends to lead to relatively many small communities; k_ik_j/2M is equal to the probability that an edge exists, or alternatively the expected edge weight, between nodes i and j under the configuration model; is the (weighted) degree of node i; g_i is the community to which node i belongs; δ(g_i, g_j) = 1 if g_i = g_j and δ(g_i, g_j) = 0 otherwise.

To generalize the modularity to the case of multilayer networks, let be the number of layers in the multilayer network. We let A_ijα be the (i, j)-th entry of the intralayer adjacency matrix, which may be weighted, in network layer α. We assume A_iiα = 0 ∀i ∈ {1, …, N} and . We remind that ω_iαβ is the weight of the interlayer coupling between node i in layer α and node i itself in layer β. The multilayer modularity is given by [43] (5) where is the strength (i.e., weighted degree) of node i in layer α, and is the total edge weight in layer α. We set , which is equal to twice of the total edge weight. Let γ_α be the resolution parameter in layer α; δ_αβ = 1 if α = β and δ_αβ = 0 otherwise; δ_ij is defined in the same manner; and g_iα is the community to which node i in layer α belongs. Eq (5) implies that communities that contain interlayer edges are rewarded with higher modularity values.

We will discuss the selection of γ_α in section 2.5. We use the Louvain algorithm for multilayer modularity maximization. Specifically, we use the iterated GenLouvain function from GenLouvain version 2.2, which repeatedly implements GenLouvain until convergence to an output partition (i.e., until the output partition does not change between two successive iterations) [96, 97].

The modularity function Q typically has many local maxima [98]. Reflecting this fact, most modularity maximization algorithms are stochastic and do not output a unique answer. A common approach to combine the results from multiple partitions of nodes is consensus clustering to obtain a consensus partition [99]. We use the consensus clustering algorithm described in [100] and implemented in the Python package netneurotools version 0.2.3 [101].

2.4 Community detection in multilayer correlation matrices

In this section, we expand modularity maximization for correlation matrices [72, 73] to the case of multilayer correlation matrices.

Let ρ = (ρ_ij) be an N × N correlation matrix and 〈ρ〉 be a null model of the correlation matrix of the same size. The modularity for a single correlation matrix is given by (6) where is a normalization constant. One can use a modularity maximization algorithm to maximize Q given 〈ρ〉.

We generalize Eq (6) to the case of a multilayer correlation matrix by writing down an equation in the same form as Eq (5). We will use the term node to refer to a gene in a specific layer of the four-layer correlation matrix. Let ρ_ijα be the empirical Pearson correlation coefficient between nodes i and j in layer α, and let 〈ρ_ijα〉 be the correlation between nodes i and j in layer α in the null model of the correlation matrix. Then, the modularity of a multilayer correlation matrix is (7) where . Parameter γ_α represents the resolution in layer α [95], and we will discuss the selection of γ_α in section 2.5. We remind that ω_iαβ is the empirical co-expression of gene i between tissues α and β. We double-count (i, j) and (j, i), with i ≠ j, in Eq (7) following previous literature [72, 73].

We use a configuration model for correlation matrices [74] as the null model, while other null models are also possible, such as the H-Q-S algorithm [102] and those derived from random matrix theory [72]. The configuration model [74], implemented in the configcorr package [103], generates the correlation matrix maximizing the entropy under the constraint that the strength (i.e., weighted degree) of each node of the input correlation matrix is conserved. The model assumes normality of the input data. While the algorithm accepts a covariance matrix or a correlation matrix as input, if the input is a covariance matrix, it is first transformed to the correlation matrix before being fed to the configuration model. To maximize Q given by Eq (7), we feed the supra-modularity matrix B, where B_iαjβ = (ρ_ijα − γ_α〈ρ_ijα〉)δ_αβ + ω_iαβ δ_ij, to GenLouvain. Again, we use the iterated GenLouvain function [97] and a consensus clustering technique to obtain a final partition [100] but by inputting 200 partitions of the same network.

Prior studies developed methods to assess statistical significance of the detected communities in single-layer networks [104–106]. Here, we extend this approach to the case of multilayer correlation matrices and multilayer networks. We do this by comparing a detected community to the same set of nodes in a random graph (or null model) in terms of some quality measure. For each detected community and given quality measure, we calculated the Z score defined by (8) where x is the quality measure calculated for the empirical community, and μ and σ are the expected value and the standard deviation, respectively, of the same quality measure for the same community but under a null model. In the following text, we explain this method for multilayer correlation matrices, which we primarily use for our gene data analysis. We show the details of our methods for general multilayer networks in Text B in S1 Text.

We introduce a quality measure of a community that is analogous to the total weight of the intralayer edges within the community. Let W be the total weight of intralayer edges within the set of nodes S in a multilayer correlation matrix. In the remainder of this section, we use the covariance matrices instead of correlation matrices for analytical tractability. This assumption is not detrimental to the application of our methods to multilayer correlation matrix data because a correlation matrix is a covariance matrix in general. Let be the (i, j)-th element of , an empirical covariance matrix for layer α. Then, we have (9) where (i, α) represents gene i in layer α, and the summation is over all node pairs ((i, α), (j, α)) in S. We exclude the diagonal elements, i.e., in Eq (9) because they are equal to 1 for correlation matrices.

Let be a sample covariance matrix for layer α generated by the configuration model for correlation matrices [74]. Let be the (i, j)-th element of . Using , where C_α is the covariance matrix for the estimated multivariate normal distribution for layer α [74], we obtain (10) We obtain (11)

We show the derivation of Eq (11) in Text C in S1 Text. Note that (12) which is consistent with the central limit theorem.

2.5 Determining a resolution parameter value

For simplicity, we assume γ_α to be common for all layers and denote the common value by γ. We use the Convex Hull of Admissible Modularity Partitions (CHAMP) algorithm version 2.1.0 [107, 108] to determine the γ value. The CHAMP algorithm takes a set of partitions generated by any community detection method as input and identifies the parameter regions in which each partition attains the largest modularity among all the partitions. The algorithm then obtains a pruned subset of admissible partitions and allows one to select parameter values corresponding to more robust community structures, which are large parameter regions in which the same partition maximizes the modularity.

Because we inform the interlayer coupling strength values by the empirical data as we described in section 2.2, we only need to tune the γ value. Therefore, using 15 evenly spaced γ values ranging from γ = 1 to γ = 4, we run a multilayer community detection method to obtain 15 partitions, one for each γ value, for a given multilayer network. Then, we employ the one-dimensional CHAMP on the 15 corresponding partitions to identify the ranges of γ in which the same partition maximizes the modularity. The wider ranges of γ correspond to more robust ranges of γ, so we choose a γ value in the two widest ranges according to CHAMP.

2.6 Specialist and generalist communities

The communities in multilayer correlation matrices and multilayer networks determined by the maximization of multilayer modularity may span multiple layers. We refer to a community containing genes belonging to various layers, i.e., tissues, as a generalist community. We refer to a community that contains genes in mostly just one tissue as a specialist community. The genes in a generalist community are general in the sense that they are co-expressed similarly across multiple tissues, whereas the genes in a specialist community are specialist in the sense that they are uniquely co-expressed in a single tissue. We will give the precise definitions of a generalist community and a specialist community in the following text. These different types of communities occur due to the similarity or difference between gene co-expression patterns across different tissues. In particular, some pairs of genes show co-expression across individuals in only specific tissues and others in multiple tissues. We are interested in whether our community detection method can detect these different types of communities. Therefore, we need a measure to classify each detected community as a generalist community or a specialist community.

We define a measure called the specialist fraction to quantify how specialized any multilayer community is as follows. For a given community, we first find the number of genes unique to each tissue α, i.e., the genes i for which node (i, α) belongs to the community and node (i, β) does not for any β ≠ α. Second, we define the specialist tissue of the community as the tissue that has the largest number of unique genes. The specialist fraction is the number of genes unique to the specialist tissue divided by the total number of nodes in the community. If the community lies within one layer, the specialist fraction is equal to 1. A large value of the specialist fraction suggests that the community is a specialist community. Genes unique to a specialist community may have functions specific to the tissue. In contrast, if all genes belong to at least two tissues, the specialist fraction is equal to 0. If many genes belong to different tissues in the community, the specialist fraction is low, suggesting that the community is relatively a generalist community. Genes in a generalist community may have functions expressed across various tissues.

2.7 Gene set enrichment analysis

To explore the biological processes associated with the set of genes constituting a detected community, we carried out a gene set enrichment analysis. It is a standard method for detecting statistically significant enriched biological processes, pathways, regulatory motifs, protein complexes, and disease phenotypes in the given gene set. We use g:Profiler (version e109_eg56_p17_1d3191d) for this purpose [109] and restrict our analysis to the Gene Ontology biological process (GO:BP) release 2023–03-06 [110, 111] and Human phenotype ontology (HP) release 2023–01-27 [112] results. We use a Benjamini-Hochberg FDR significance threshold [113] of 0.05.

2.8 Localization of genes on chromosomes

We developed statistical methods to investigate whether the genes in a community detected by our community detection method are physically clustered across the genome. To this end, we first ask whether a group of genes are more frequently located on the same chromosome than a control. Consider a group of genes, denoted by c. Let n be the number of genes in group c. We define the fraction of pairs of genes on the same chromosome as (13) The denominator of x_c is equal to the number of pairs of genes in group c and gives the normalization. For the control, we uniformly randomly shuffle the association between the N = 203 genes that we initially selected for our analysis and the chromosome to which each of the N genes belongs. After this random shuffling, the n genes are randomly distributed on various chromosomes as the N = 203 genes are distributed on those chromosomes. Then, we calculate according to this random distribution of the n genes using Eq (13). We repeat this randomization 100 times and calculate the average and standard deviation of , and then the Z score. If the Z score is significantly positive, then we say that the group of genes c has more pairs of genes on the same chromosome than the control.

Second, we tested whether the genes in c are located closer to each other on the chromosome than a control, given the number of genes in c on each chromosome. To this end, we define the physical distance measured in base pairs between gene i and gene j on the same chromosome, d(i, j), as follows. Without loss of generality, assume that the end position of gene i is less than the start position of gene j. Then, we set (14) Furthermore, we define the average distance between genes in group c as (15) Denote by n_k the number of genes in group c that are on chromosome k. Note that the denominator in Eq (15) is equal to ∑_k n_k(n_k − 1)/2. For the control, for each k, we choose n_k genes uniformly at random out of all genes on chromosome k from the N genes. We carry out this procedure for all chromosomes k on which there are at least two genes in group c (i.e., n_k ≥ 2). Then, we calculate d_c for this random distribution of genes, which we refer to as . We repeat this randomization 100 times and calculate the average and standard deviation of , and then the Z score. If the Z score is significantly negative, then we say that the genes in group c are localized on the chromosomes.

Third, we test whether the genes in c are located closer to each other than a control on a given chromosome. We define the average distance between genes in group c on chromosome k as (16) For the control, we choose n_k genes uniformly at random out of all genes that are among the N genes and on chromosome k. Then, we calculate for this random distribution of genes, which we refer to as . We repeat this randomization 100 times and calculate the average and standard deviation of , and then the Z score. We carry out this procedure for each chromosome k on which there are at least two genes in group c (i.e., n_k ≥ 2). We apply the Bonferroni correction [114] separately to each c to determine which communities have a significantly smaller average distance between pairs of genes on a specific chromosome than the control. We chose to apply the Bonferroni correction because it is a more conservative statistical method than others, such as FDR.

2.9 Pancreas-specific cis-eQTL analysis

Expression quantitative trait loci (eQTL) analysis identifies variants that have significant associations with expression levels of specific genes. We hypothesize that changes in expression levels of a pair of co-expressed genes are associated with the same set of variants. If true, we expect to identify variants that are associated with the expression of both genes in the pair. To investigate gene pairs with shared eQTL single nucleotide polymorphisms (SNPs) in the pancreas, we downloaded the cis-eQTL data set from GTEx release V8. This data set involves SNP-gene pairs with association significance indicated with a nominal p value. The changes in the expression levels of a given gene may be associated with one or multiple SNPs. Alternatively, it may have no eQTLs, meaning that no SNPs are associated with its gene expression. Using this data set, we searched for SNPs that were associated with both of the genes in a given gene pair of interest. Given that we are interested in whether co-expressed genes share common SNPs, we only investigate gene pairs with co-expression (as defined by Eq (1)) greater than 0.5 in the pancreas.

3.1 Communities in the multilayer correlation matrix

We compare the gene communities obtained from the multilayer correlation matrix and those obtained from multilayer gene networks constructed using graphical lasso. For a brief review of graphical lasso, see Text D in S1 Text.

We run iterated GenLouvain [97] on the multilayer correlation matrix to approximately maximize the multilayer modularity at each value of the resolution parameter, γ, which we assume to be common for all layers. We then use the CHAMP algorithm to determine optimal values of γ [107, 108]. We show the results of CHAMP in Fig 2(a). The figure indicates that robust ranges of γ, which are relatively wide ranges of γ in which the optimal partition is the same and correspond to relatively long straight line segments in the figure, are approximately 0 < γ < 1.2 or 2.9 < γ < 4.4. Therefore, we examine the node partitions with one arbitrary γ value from each of these two stable regions of γ, i.e., γ = 1 and γ = 3.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Determination of the resolution parameter value by CHAMP.

(a) Multilayer correlation matrix. (b) Multilayer network obtained by graphical lasso. The convex hull of the lines in the (γ, Q) plane, each of which corresponds to a node partitioning, is a piecewise linear curve with the transition values indicated by a cross and change in the line color. Each line segment corresponds to the optimal node partitioning in the corresponding range of γ.

https://doi.org/10.1371/journal.pcbi.1011616.g002

We show the composition of the resulting node partitions with γ = 1 and γ = 3 in Fig 3(a) and 3(b), respectively. As expected, the number of communities increases when γ increases. We show the Z score for the total intralayer weight within each community detected with γ = 1 and γ = 3 in Table 2. With γ = 3, communities 8 through 12 contain no intralayer edges such that one cannot run the randomization, leading to a null Z score. These communities contain only one gene; communities 8, 9, 10, and 11 detected with γ = 3 contain two nodes representing the same gene in two different tissues, and community 12 contains only one node. We omitted these trivial communities in Table 2. The table indicates that all the communities detected with γ = 1 and all the communities containing at least two genes detected with γ = 3 (i.e., communities 1 through 7) are statistically significant.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Composition of each community by layer, i.e., tissue.

(a) Multilayer correlation matrix, γ = 1. (b) Multilayer correlation matrix, γ = 3. (c) Unweighted multilayer network obtained by graphical lasso, γ = 1. (d) Unweighted multilayer network obtained by graphical lasso, γ = 3. (e) Weighted multilayer network obtained by graphical lasso, γ = 1. (f) Weighted multilayer network obtained by graphical lasso, γ = 3. The darker shades indicate nodes corresponding to genes that only appear in one layer in the given community. The lighter shades indicate nodes corresponding to genes that appear in multiple layers in the community.

https://doi.org/10.1371/journal.pcbi.1011616.g003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Z scores for the total intralayer weight within each community detected in the multilayer correlation matrix.

https://doi.org/10.1371/journal.pcbi.1011616.t002

Both node partitions contain some communities that appear to be generalist communities and other communities that appear to be specialist communities. We remind that a generalist community indicates genes that are similarly co-expressed in multiple tissues and that a specialist community indicates genes that are uniquely co-expressed in one tissue. To quantify these findings, we show in Table 3 the specialist fraction for each community in the partition with γ = 1. Communities 3, 4, and 5 have specialist fractions greater than 0.5, so we regard them as specialist communities. In contrast, because communities 1 and 2 have specialist fractions substantially less than 0.5, we regard them as generalist communities. The same table also shows the specialist fraction for each significant community found with γ = 3. Communities 5 and 7 have specialist fractions greater than 0.5. Both of them are pancreas specialist communities. We regard communities 1, 2, 3, 4, and 6, whose specialist fraction is substantially less than 0.5, as generalist communities.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Specialist fraction and the corresponding tissue for each community detected in the multilayer correlation matrix and for each community detected in the multilayer networks obtained by graphical lasso.

https://doi.org/10.1371/journal.pcbi.1011616.t003

3.2 Communities in the multilayer networks obtained by graphical lasso

For comparison purposes, we run the iterated GenLouvain on the multilayer networks that we constructed using graphical lasso (see Text D in S1 Text for the methods). The results of CHAMP on the detected node partition of the unweighted network, shown in Fig 2(b), indicate that the optimal ranges of γ are approximately 0.7 < γ < 1.7 or 1.7 < γ < 3.2. Therefore, we use the same γ values as those for our multilayer correlation matrix, i.e., γ = 1 and γ = 3.

We show the composition of the resulting node partitions of the unweighted network obtained using graphical lasso with γ = 1 and γ = 3 in Fig 3(c) and 3(d), respectively. With γ = 1, we find eleven communities, nine of which are significant. With γ = 3, we find fourteen communities, ten of which are significant. See Text B in S1 Text for the statistical results. We also show the composition of the node partitions of the weighted multilayer network obtained using graphical lasso with γ = 1 and γ = 3 in Fig 3(e) and 3(f), respectively.

Fig 3(c)–3(f) suggests that these partitions apparently contain generalist communities only. Table 3 shows the specialist fraction for each significant community in the unweighted network and each community in the weighted network. Note that we have not evaluated the significance of the communities detected for the weighted multilayer network because the configuration model for weighted networks, which is necessary for constructing a significance test, is not a straightforward concept [115, 116]. For the unweighted network, with both γ = 1 and γ = 3, all the significant communities have specialist fractions at most 0.211. For the weighted network, with both γ = 1 and γ = 3, all the communities with more than one gene have specialist fractions at most 0.133. Therefore, we conclude that there are no specialist communities for either the unweighted or weighted network and with either γ = 1 or γ = 3.

In sum, our community detection method on correlation matrices finds tissue-specific gene co-expression patterns, evident by the detection of specialist communities, whereas the graphical lasso does not. Because we are interested in comparing the biological implications of specialist communities versus generalist communities, in the following sections, we only analyze the communities detected for our multilayer correlation matrix. In particular, we will carry out tissue-specific analysis to investigate the specialist communities detected by our method.

3.3 Localization of genes on chromosomes

To investigate the possible localization of genes in the detected communities on the chromosomes, we first analyze whether the N = 203 among the 56, 200 genes that we are analyzing in the GTEx data set are already localized in the genome. The Z score for a fraction of pairs of genes on the same chromosome is 6.735 (p < 10⁻⁶), which suggests that the N = 203 genes are distributed on different chromosomes in a highly biased manner relative to how all the 56, 200 genes are distributed. The Z score for the average distance between pairs of genes on the same chromosome is −6.059 (p < 10⁻⁶). Therefore, the average distance between pairs of genes among the N = 203 genes is significantly smaller than by chance. This result is expected given that highly expressed genes in glandular tissues cluster in specific loci [40]. We show the Z scores for the average distance between pairs of genes on each chromosome, analyzed separately, in Table 4. At a significance level of p = 0.05, there is significant localization of genes on chromosomes 2 (p = 0.0088; Bonferroni corrected; same for the following p values), 4 (p < 10⁻⁴), 12 (p = 0.0098), and 17 (p < 10⁻⁴).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 4. Z score for the average distance between pairs of genes on each chromosome for the N = 203 genes.

https://doi.org/10.1371/journal.pcbi.1011616.t004

Next, we run the same localization analysis for each community in the multilayer correlation matrix detected with γ = 1 and γ = 3. For a generalist community, we only included the genes in the community that appear in at least three out of the four tissues in this analysis. This is because such genes may play functional roles, which the generalist community represents, across many types of tissues. With this restriction, each gene is present in at most one generalist community. Note that, without this restriction, a gene may appear in multiple generalist communities because the four nodes in the multilayer network representing the same gene may belong to different communities. We exclude this case for simplicity.

For each community, we show in Table 5 the Z score for the fraction of pairs of genes in the community that are on the same chromosome. With γ = 1, communities 2 (p < 10⁻⁴) and 3 (p = 4.05 ⋅ 10⁻⁴) have significantly more genes among the N = 203 genes on the same chromosome than by chance. The same table also shows the Z score for the average distance on the chromosome between pairs of genes in the same community for each community. We find that, with γ = 1, community 2 has a significantly smaller average gene-to-gene distance than by chance (p = 0.0336). With γ = 3, communities 1 (p < 10⁻⁴), 5 (p < 10⁻⁴), and 6 (p < 10⁻⁴) have significantly more pairs of genes on the same chromosome than by chance, and community 1 (p = 0.0069) has a significantly smaller average gene-to-gene distance than by chance (see Table 5).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 5. Analysis of localization of genes in each community detected in the multilayer correlation matrix.

https://doi.org/10.1371/journal.pcbi.1011616.t005

We then compute the Z score for the average distance between pairs of genes separately for each chromosome in addition to each community. We exclude the community-chromosome pairs that have less than three genes from this analysis. With both γ = 1 and γ = 3, no group of genes on a specific chromosome in a specific community is significantly clustered when we impose the Bonferroni correction over all the community-chromosome pairs (45 and 23 pairs with γ = 1 and γ = 3, respectively; see Tables B and C in S1 Text for the Z scores). With the Bonferroni correction applied to each community separately, there are still no significant clusters in the partition with γ = 1. However, with γ = 3, we find that the genes in community 1 on chromosome 1 (p = 0.0199) and those in community 5 on chromosome 11 (p = 0.0371) are significantly clustered.

3.4 Functional analysis of selected communities

For the communities detected for our multilayer correlation matrix, we found clusters of physically localized genes within two communities with γ = 3 but none with γ = 1. Because we are interested in exploring biological implications of localized clusters of genes, we carry out further analysis on the node partition with γ = 3 in this section. A table showing which nodes (i.e., genes) belong to which communities in this partition is available on GitHub [76].

First, we conducted an enrichment analysis of the communities identified with γ = 3. We started with an enrichment analysis for the top 50 genes that have the highest expression out of the 203 genes in the network in each tissue. We find that, in all tissues, the top 50 highly expressed genes are enriched significantly in well-established housekeeping categories, such as oxidative phosphorylation and aerobic electron transport chain (FDR <0.05; see Table D in S1 Text). Echoing this finding, one of the modules that we identified (community 1) shows similar enrichment for mitochondrial function, such as aerobic electron transport chain (p = 1.05 ⋅ 10⁻¹⁰) and oxidative phosphorylation (p = 5.90 ⋅ 10⁻¹¹) (see Table E in S1 Text). However, in the other six communities, our network approach identifies novel gene modules with functional enrichments in epidermis development (community 2, p = 1.90 ⋅ 10⁻²⁴), keratinization (community 5, p = 1.95 ⋅ 10⁻¹⁹), positive regulation of respiratory burst (community 6, p = 5.36 ⋅ 10⁻⁸), and adaptive thermogenesis (community 7, p = 1.73 ⋅ 10⁻²). Furthermore, these modules are enriched with diseases relevant to the tissues examined, such as hyperkeratosis (community 2, p = 3.05 ⋅ 10⁻⁷) and recurrent pancreatitis (community 1, p = 1.78 ⋅ 10⁻¹⁹). In addition, we analyzed the top 50 highly connected genes (i.e., top 50 genes in terms of the weighted degree, or in other words, top 50 hub genes) in each of the single-layered networks for each tissue. Not surprisingly, this analysis identified genes that are enriched for functions and diseases that are specific to each tissue (see Table F in S1 Text). However, we found that most of the genes that are identified in our multilayer network approach are different from those identified with single-layer analysis (see Text G in S1 Text). We also found that the functional enrichments of these two network approaches were different (see Table E versus Table F in S1 Text). Overall, our method provides additional biological insights than simple expression-level filtering and single-layer network analysis.

Our multilayer network analysis allowed us to investigate genes that are co-expressed in multiple tissues. We surmised that membership of genes in the same community can be facilitated by shared regulatory sequences affecting multiple genes at the same time. Given that regulatory regions affect gene expression in cis (i.e., nearby regions), we hypothesize that genes in the same multilayer community may be physically close to each other. To investigate this, we visualize in Fig 4 the location of the genes in the different communities on the chromosomes. As in the localization analysis presented in section 3.3, for a generalist community, we only show in Fig 4 the genes in the community that are present in at least three tissues. In Fig 4, a color of the circles represents a community. Note that a gene can belong to more than one community, denoted by multiple colored circles next to each other horizontally pointing to the same gene. It happens to be the case that a gene is associated with a maximum of two different communities, hence a maximum of two colored circles pointing to the same gene. Visually, Fig 4 suggests some tight clusters of genes, especially in community 5.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Location of genes on chromosomes, colored by community.

There is a colored circle for each associated community pointing to each gene. Note that a gene can belong to more than one community, denoted by multiple colored circles next to each other horizontally pointing to the same gene. This figure allows us to visually see clusters of genes on specific chromosomes and their associated community.

https://doi.org/10.1371/journal.pcbi.1011616.g004

In section 3.3, we found significantly localized clusters of genes in community 1 on chromosome 1 and in community 5 on chromosome 11 in the partition with γ = 3. It is somewhat surprising that only these two community-chromosome pair gene sets are significantly localized because there appear to be more localized clusters in Fig 4. A possible reason for this discrepancy is that, besides the genes in the community-chromosome pair of interest, there are so few other genes on the chromosome that the random shuffling of gene associations does not provide sufficient randomization. In this case, the empirical average distance between genes in the community-chromosome pair will not be statistically different from the average distance for the randomized data. Therefore, here we directly compared the average distance between pairs of genes on each community-chromosome pair, as defined by Eq (16), to that for community 5 on chromosome 11. We decided to analyze community 5 because it is a pancreas specialist community while community 1 is a generalist community, as we discussed in regards to functional enrichment earlier in this section.

We denote the average distance between the pairs of genes among the three genes in community 5 on chromosome 11 by , calculated using Eq (16). We looked for any community-chromosome pair, containing all the genes in the selected community on the selected chromosome, with at least three genes whose average distance between genes is less than . There are five such additional gene clusters: community 1 on chromosome M, which contains 15 genes, community 5 on chromosome 4, which contains 6 genes, community 5 on chromosome 17, which contains 9 genes, community 6 on chromosome 2, which contains 6 genes, and community 6 on chromosome 14, which contains 3 genes. Among all these community-chromosome pairs, we focused on the three gene clusters in community 5, including the gene cluster on chromosome 11. We opted to do so because community 5 is a pancreas specialist community, whereas communities 1 and 6 are generalist communities.

After initial investigation of the three gene clusters in community 5, i.e., one each on chromosome 4, 11, and 17, we further analyzed the one on chromosome 17, because keratin loci have been discussed in the context of human evolution [117, 118]. We show in Fig 5A and 5B the expression of each gene in this gene cluster in the skin and pancreas, respectively. We found that gene expression trends vary between the two tissues. Specifically, our method identified community 5 because of co-expression trends in the pancreas. However, in terms of the sheer expression level, the present gene cluster is expressed multiple folds higher in the skin than pancreas. Further, we found that the co-expression patterns for some gene pairs within this gene cluster are common between the skin and pancreas but differ for other gene pairs. The physical clustering of the genes that are co-expressed implicates genetic variation in shared gene regulatory factors as the main basis for co-expression. For example, a search of the GTEx eQTL database showed that the common single nucleotide polymorphism rs12450846 is significantly (p < 10⁻¹⁸) associated with lower expression of KRT31 in the skin but higher expression of this gene in ovaries (p < 0.005). Unfortunately, this analysis was not conducted in the pancreas. Regardless, this polymorphism and the haplotype linked to it regulate multiple other keratins and keratin-associated protein genes in this particular locus in a tissue and gene-specific manner according to the GTEx database. Thus, genetic variation that affects the efficacy of regulatory regions (Fig 5C) or the formation of topologically associated domains (Fig 5D) in a tissue-specific manner may underly the co-expression of the genes in community 5 on chromosome 17. Indeed, we found several topologically associated domains, enhancers, transcription factor binding sites, and open chromatins within this region, affecting co-expressed genes in a similar fashion (Fig 5). Overall, our analysis provides several exciting hypotheses for future work to investigate regulatory regions that target multiple nearby genes and explain tissue-specific co-expression trends.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Expression and co-expression analysis of a cluster of genes in community 5 on chromosome 17.

The co-expression matrices for these genes in (A) skin and in (B) pancreas are shown. The average expression for each gene in these tissues is shown in the bar graphs. The location of these genes on chromosome 17 is shown in (C), with arrows (colored according to the associated tissue) pointing from putative regulatory elements to highly co-expressed genes. (D) The panel shows different measures of the regulatory potential of this genome section. From top to bottom: 1. H3K27AC modification to histone H3 within the region, which often correlates with activation of transcription and is associated with active enhancers in a given tissue available through ENCODE database [119]. 2. DNAse1 hypersensitivity sites. They are sections of the genome that are cut by DNAse1 enzyme. Given that the chromatin has to be “open” for the DNAse to access the sequence, the sequences that are cut by DNAse indicate open chromatin, which is in turn associated with regulatory activity. Data are available through ENCODE database [119]. 3. Enhancer/promoters. These are sequences that are predicted as enhancers (gray) and promoters (red) from the GeneHancer database [120]. 4. Established interactions between regulatory regions and genes as documented by GeneHancer database [120]. These data sets combined with our co-expression analysis provide a novel outlook into potential topologically associated domains that may be regulated by specific sequences in a tissue-specific manner.

https://doi.org/10.1371/journal.pcbi.1011616.g005

Another interesting community we identified is community 7. The genes in this community are located on different chromosomes and are enriched for response to temperature change (adaptive thermogenesis; see Table E in S1 Text). Because they exist on different chromosomes, it is unlikely that these genes share any common regulatory sequences or topologically associating domains. Instead, their co-expression may be due to environmental stimuli that are shared among the samples at the time of sampling (e.g., warm or cold environments). If true, the co-expression is due to a response to environmental stimuli that is controlled by specific regulatory sequences with broad effects across the genome, such as transcription factors. Thus, our network analysis may be useful for identifying gene clusters that respond to different environments.

3.5 Gene pairs with shared associated SNPs in pancreas

As described in section 2.9, we hypothesized that genetic variation that affects gene expression in a tissue-specific manner can explain some of the co-expression trends we observed. Identifying such variation is challenging because of the huge amount of combinations that are possible between genetic variants and gene expression levels. To overcome this challenge and identify examples of where genetic variation may explain the co-expression trends and chromosomal clustering, we conduct an eQTL analysis considering only cis variants that are physically close to genes of interest. This analysis provides a list of variants (SNPs in this case) that are significantly associated with expression levels of nearby genes. We will refer to these SNPs as eQTLs. Using this approach, we identified three gene pairs (i.e., CELA3B and CELA3A; AMY2B and AMY2A; REG3G and REG1B) that share associated eQTLs in the pancreas out of all the gene pairs in the network of 203 genes with co-expression greater than 0.5.

Notably, out of these three gene pairs, two pairs, i.e., the CELA3B-CELA3A and AMY2B-AMY2A pairs, are not composed of hub genes within the pancreas single-layered network and are only identified through our multilayer network approach. Both pairs are within community 5. For example, if we searched for the top 86 genes in terms of the weighted degree in the pancreas to match the number of genes in community 5, we were not able to identify the CELA3B-CELA3A or AMY2B-AMY2A pairs. In contrast, the other gene pair with shared eQTLs (i.e., REG3G and REG1B) consists of two hub genes in the single-layered pancreas co-expression network. Therefore, we would have missed two out of three gene pairs that may be biologically interesting if we simply investigated hub genes in the pancreas.

Next, to identify the biological relevance of this putatively genetically determined co-expression pattern, we investigated the CELA3 locus. We identified a set of 96 variants from statistically significant eQTLs for both CELA3A and CELA3B in the pancreas. CELA3A and CELA3B, which are proteases, are produced as zymogens in the pancreas. They then perform their digestive function in the intestine once they have been transported there. It has previously been speculated that the presence of two CELA3 copies provides a functional substitute for the lack of pancreatic expression of CELA1 in humans relative to pigs [121]. The 96 variants are present in the genomic region spanned by HSPG2, CELA3A, and CELA3B. The minor allele for each of these 96 variants is associated with a decreased expression of CELA3A and an increased expression of CELA3B in the pancreas. This observation may hint at a possible constraint on the combined expression level of CELA3A and CELA3B in the pancreas, further supporting the idea that CELA genes may have compensatory roles for the functions of other members in this gene family. To understand the population genetics trends affecting the regulatory variants that we identified, we analyzed 83 SNPs that are associated with gene expression of CELA3A and CELA3B and genotyped in the 1000 Genomes Project Phase-3 data set. We found that these variants form a single linkage-disequilibrium (LD) group in Europeans at an r² threshold of 0.6 [122]. The minor alleles of 10 of these variants are associated with a decreased blood phosphate concentration [123, 124] (see Fig 6). In order to identify putative causal variants in the LD group, we investigated whether any of these variants lie in a regulatory region. We find that four (rs57030248, rs59134693, rs113385886, and rs111651468) of these variants lie in an enhancer (ENSR00000350171), identified by ENSEMBL’s variant effect predictor [125] (Fig 6B), which is active in the pancreas. Three of these four variants (rs57030248, rs59134693, and rs113385886) are both present in the enhancer region and associated with decreased blood phosphate levels. It is likely that one or more of these three variants are causal in the context of differences in the expression levels of CELA3A (p = 1.1 ⋅ 10⁻⁸, normalized effect size = −0.43) and CELA3B (p = 4.5 ⋅ 10⁻¹⁰, normalized effect size = +0.43). Our results allowed us to construct a hypothetical model (Fig 6). Our multilayer network approach facilitated the narrowing down of putatively causal genetic variants that affect the expression levels of negatively co-expressed gene pairs within the context of protein and phosphate metabolism.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. A schematic of SNPs in an enhancer region (gray box) that affect the expression of CELA3A (blue box) and CELA3B (orange box) in the pancreas and are associated with blood phosphate concentration.

(A) Expression levels of CELA3A and CELA3B, and blood phosphate concentration when the derived alleles for the putatively causal SNPs are absent. (B) The presence of the derived alleles for the putatively causal SNPs decreases the expression level of CELA3A, increases the expression level of CELA3B, and decreases the blood phosphate concentration.

https://doi.org/10.1371/journal.pcbi.1011616.g006