Investigating perturbed pathway modules from gene expression data via structural equation models

Illustration of the proposed pipeline

The pipeline, similar to that described by [12], is illustrated in Figure 1. In step-1, DEGs were obtained by Significance Analysis of Microarray (SAM) [16] and the perturbed pathways by Signaling Pathway Impact Analysis (SPIA) [17] using KEGG database [18]; in step-2, the pathway models were generated by network analysis and evaluated with SEM in step-3 for: 1) improving the models generated by the biological pathways found; 2) testing if the pathway models differ across groups by multiple group analysis; 3) screening of single differences in expression (gene nodes) and in regulation (gene-gene edges) across groups. We used the implementation provided by the R packages samr [19] and SPIA [20] for SAM/SPIA procedures.

Structural equations models (SEM)

SEM is a statistical procedure for confirmatory causal inference originated from path analysis proposed in 1921 by the American geneticist Sewall Wright [21]. It is based on multivariate linear regression equations, where the response variable in one regression equation may appear as a predictor in another equation. Indeed, variables may influence one-another reciprocally, either directly or through other variables as intermediaries. Additionally, correlated or uncorrelated unmeasured variables may indicate the presence of unobserved factors that influences observed variables.

The system of linear equations affirms that every node is characterized by the relationships with his parents, while the covariance structure describes the relationships between unobserved nodes.

They encode two distinct causal assumptions: (1) a “weak” assumption on the possible existence of (direct) casual influences of explanatory variables on Y _i , and (bi-directed) correlated unmeasured variables U _i , quantified by the regression (path) coefficients β _ij , and the covariances ψ _ij , respectively; and 2) a “strong” assumption based on the absence of (direct) causal influences or (bi-directed) correlations of any observed/unobserved variables neither in the “parents” set pa(i) nor in the “siblings” set sib(i). In other terms, a weak assumption excludes some values for a parameter (the null value zero), but permits a range of other values; while, strong assumptions assume that parameters take specific values (null value zero or a fixed a priori value). The linear equations and the covariance structure can be encoded and visualized in a “path diagram”, that is, a mixed graph G = (V,E) featuring both directed (→) and bi-directed (↔) edges. The vertex set V includes the genes and the edge set E represents relations or reactions among vertexes. The “activity” of a given gene is embedded in a path diagram: the actions performed by a gene on downstream molecules, and the signals that it receives from upstream regulators. “Directed edges” between two genes (j→i, if and only if j ∈ pa(i)), measured by path coefficients (ranging usually from -1 to 1, if genes are standardized), represent expected change in the activity of the downstream gene, given a unit of change in the upstream gene while the values of the other genes remain constant. Considering that paths reflect a direct influence of one gene on another, negative path coefficients indicate ensemble inhibition (negative control) and positive paths measure net activation (positive control). “Bi-directed edges” between two genes (j↔i if and only if j ∈ sib(i), or equivalently, if and only if i ∈ sib(j)) encode a hidden common cause that may be interpreted as latent or unobserved measurement of upstream regulators that could account for the observed covariances (correlations) between the two genes.

One important feature of SEM is that direct and indirect effects can be computed and compared. “Directed paths” between two genes are the sequence of all the directed edges (j → k₁,… → …, k_m → i) from genes Y_j to Y_i. Each directed path is a channel along which information (gene’s activities) can flow, and so a “total effect” (TE) of gene Y_j on gene Y_i is defined as the total sum of the products of the sequence of arrows (edges) along all directed paths from Y_j to Y_i. Accordingly, a “indirect effect” (IE) of gene Y_j on gene Y_i represents the portion of the total effects not considering the directed edge effect (DE), i.e. TE = DE + IE.

The well-known SEM analysis consists of four steps [22]: a) definition and identification of an initial path model, b) estimation of parameters, c) evaluation of the fitting, and d) model modification.

Initial model building

Specification of initial pathway models (step a) was obtained taking the perturbed pathways and converting them in directed graphs or gene networks. Generalizing, each pathway can be seen as a mixed graph. The idea is to understand how DEGs are connected in the perturbed pathways by other microarray genes. A natural way to solve this problem is to identify the shortest paths (geodesic distance) between DEGs. The geodesic distance d_geo(y_i, y_j) between two DEGs, y_i and y_j, is defined as the minimum distance between these two genes. The function get.shortest.paths( ) of the R package igraph was used to compute all the shortest paths [24]. Define the microarray genes, DEGs, and not DEGs in the following way: MG = {mg ₁ , mg ₂ , …, mg _m }; DEG = {deg ₁ , deg ₂ ,…, deg _n } and NDEG = {ndeg ₁ , ndeg ₂ ,…,ndeg _m-n }, where MG = DEG ∪ NDEG and DEG ∩ NDEG = {∅}. Each shortest path could be represented as a list of nodes Y _k = (y _i , y _i+1 ,..., y _j-1, y _j ) and a list of the corresponding edges E_k = (e_i(i+1), ..., e_(j-1)j) where (y _i , y _j ) ∈ DEG; (y _i+1 ,…, y _j-1 ) ∈ (DEG ∨ NDEG); Y_k ⊆ Y and E_k ⊆ E. The shortest paths for each pathway constitute k (k = 1, …, K) subgraphs G_k = {Y_k,E_k} of the original pathway, G = {Y, E}. Not all DEGs and NDEGs will be included in the shortest paths. Therefore, we define two new sets: DEG(s) and NDEG(s) respectively, the sets of DEGs and the set of not DEGs that include all genes in shortest paths, where DEG(s) ⊆ DEG and NDEG(s) ⊆ NDEG.

To reach the final model the NDEG(s) that connect DEG(s) are grouped in basis to their PIR superfamily (PIRSF). Based on the evolutionary relationships of whole proteins, this classification system allows annotation of both specific biological and generic biochemical functions. The PIRSF can be represented as SUPF = {supf ₁ , supf ₂ ,…, supf _g }, where ∀ supf _i  ⊆ NDEG(s) and ∀_i≠j supf _i ∩ supf _j  = {∅}. Using this information, each original shorthest path G_k = {Y_k,E_k} is transformed in a new shortest paths, G*_k = {Y*_k = (y*_i, y*_i+1,…, y*_j-1, y*_j), E*_k = (e*_i(i+1),…, e*_(j-1)j) }, where (y*_i+1,…, y*_j-1) ∈ (DEG(s) ∨ SUPF ∨ NDEG(s)) and E* ⊆ E. The function to obtain the final graph, G* = (Y*,E*), is described in the following pseudo-code:

The graph G* = (Y*,E*) is the fusion of all shortest paths found, where each node and each edge cannot be present more than once, the self-loops are not considered but the feed-backs and cycles were preserved. To ensure the identification of the initial models, the “block-recursive” criterion of Rigdon [25] and the “bow free” criterion of Brito and Pearl [26] were applied. The first affirms that reciprocal relationships, feedback loops, or covariances are segregated into groups, or blocks, with no more than two equations per block. The second affirms that a model is ensured if variables standing in direct causal relationships (directed edges) do not have correlated errors (bi-directed edges). So a new graph is attained in which the DEGs are connected by other DEGs, PIRSFs or NDEGs. In this way a model was created for each significant pathway found.

Successively, PIRSFs composite variables are defined considering not DEGs, present in shortest paths, as causal indicators of latent (hidden) constructs [27]. To generate the PIRSFs, a principal component analysis (PCA) was performed on genes belonging to a PIRSF and the principal component scores of the first principal component (PC1) were considered as the values that characterize the PIRSF. Only PIRSFs for which the PC1 represents 50% or more of the total variance are considered. At the end of process we have the initial SEM model.

The pathway graph conversion, the graph analysis, and the PC1 scores are obtained by graphite [28], igraph [24] and stats [29] R packages, respectively, while R functions for network analysis are implemented ad hoc, and are available Additional file 1.

SEM fitting

where θ=(β; ψ) is the list of the free parameters in the model of dimension t. The unknown parameters are estimated so that the implied covariance matrix Σ(θ) is close to the observed sample covariance matrix S.

The chi-square test is then χ² = -2logLRT = -2[logL(Σ(θ)) - logL(Σ₀)] with d = p(p + 1)/2-t degree of freedom (d.f.). logL( ) represents the log-likelihood of the model, p the number of genes, t the number of parameters of the fitted model. Not-significant P-values (P >0.05) indicate that the model provides a good fit to the data. The P-values are derived by using the χ²(d) distribution or a resampling bootstrap distribution [30].

P-values for RMSEA are set up from the non-central χ²(λ,d) distribution with non-centrality parameter, λ = (n-1) × d × 0.05² or from a resampling bootstrap distribution. The null hypothesis of close fit is not rejected if P > 0.05.

SRMR values <0.10 are assumed as an adequate fitting measure, whereas values <0.05 may be considered as a good fit [32].

Finally, the model refinement (step-d) is obtained adding new directed or bi-directed edges to the initial model. This modification was needed considering that the initial model is only a simplified representation of the whole pathway. The criteria used for the refinement are based on the combination of three elements. First, the modification indexes (MI), that is an estimate of the decrease in the χ²-score statistic that would result by freeing each fixed (=0) parameter in the model; second, z-tests (=parameter estimate/standard error) of the MLE; and finally, biological evidences obtained by STRING database [33] and by the existence of a direct path between the nodes that MI proposes to connect. The following heuristic stepwise strategy was used:

Heuristic stepwise procedure:

Input: list of the fixed (=0) parameters (paths and/or covariances) in the model.

Output: new free parameters (paths and/or covariances) in the model.

Multiple-group analysis

When data are observed from multiple subsamples, the representation of groups with “indicator variables”, considered as nodes, allows to recognize DEGs. Instead, “multiple-group analysis” allows to identify differentially regulated genes (DRGs) across groups.

In the “null” model (H₀), the mean or covariance estimates are constrained to be equal across groups; in the “alternative” model (H₁), they are allowed to differ across groups. The statistical significance is determined by comparison of LRT chi-square (χ²diff) values at a given degree of freedom (d.f. diff). If there is a significant difference (P < 0.05) in the chi-squared goodness-of-fit index, the groups differ significantly for one or more specific gene expression (nodes) and/or gene-relationships (edges). Finally, three path-coefficient differences are screened: 1) “up/down” expression (gene nodes), testing the “zero value” for the group indicator variable (C = experimental =1, and C = control = 0) path coefficients; 2) “up/down” regulation (gene edges), testing the “zero value” for the differences of path coefficients across groups; 3) “on/off” regulation (gene edges) with respect to a priori KEGG gene regulation target, testing the “zero value” of the edge coefficients across groups.

where SE( ) is the estimated standard error of the parameters. The statistic t_C can be used to test the conditional “up/down” expression level difference of one gene between groups, given the parents of the gene in the network. Similarly, t _D checks the conditional “up/down” expression regulatory differences of one gene on another between groups. Moreover, t₁ and t₂ check the “on/off” regulatory differences compared to a priori KEGG pathway. The P-values of these statistics (two-sided, for t_C and t_D and one-sided, for t₁ and t₂) are derived either asymptotically from the N(0,1)-distribution or empirically from the nonparametric-based or using model-based bootstrap distribution with B bootstrap samples (usually, B = 100, or 1000).

Note that the marginal bivariate test of DEGs with SAM approach can be regarded as the special case of the conditional test with t_C, when the pathway graph is G = (Y, ∅), so pa(y) = ∅ for all genes ∈Y.

We use the implementation provided by the lavaan [34] R package for estimation, evaluation, and modification of SEM data analysis, and R codes is available in Additional file 1.

FTLD-U analysis

The most of the dysregulated pathways, as the MPAK signalling pathway, the calcium signalling pathway, the gap junction and the ECM-receptor interaction, confirm the analysis of [35]. The dysregulated pathways with a significant p-PERT were the glutamatergic synapse and GABAergic synapse. The role of the glutamate in the acute and neurodegenerative processes were well described in literature [35–38]. Meldrum [36] illustrated three different pathological mechanisms of action of the glutamate in the neurodegeneration. Glutamate can be neurotoxic through an agonist effect on the N-methyl-D-aspartate (NMDA), α-amino-hydroxy-5-methyl-4-isoaxaleproprionicacid (AMPA), kainate or Group I metabotropic receptors. The relative contribution of these different classes of receptors vary according to the neurons involved and a variety of other circumstances. Selective neuronal death subsequent to the epileptics status appears to be highly dependent on NMDA receptor activation. Acute neuronal degeneration after transient global or focal cerebral ischemia seems to be dependent on both NMDA and AMPA receptors. Regarding the GABAergic pathway, a loss of glutamatergic pyramidal cells and calbindin-D28k-immunoreactive GABAergic neurons in the frontal and temporal cortices of patients with FTLD [39] and FTLD with motor neuron disease [40] was reported.

SEM analysis of glutamatergic synapse KEGG pathway

Four genes and one PIRSF of the glutamatergic model resulted influenced by the group: genes 1742 or PSD-95, 5532 or PPP3CB, 2785 or GNG3, 5534 or Ppp3r1 and the PIRSF of the glutamate receptor. An important role could be played by PSD-95 gene that is believed to be involved in the synapse maturation, in the induction of a network of neurotransmitter receptors, scaffolding proteins and ionotropic glutamate receptors [42]. The gene PPP3CB and the gene GNG3 are associated to the Wnt signalling correlated to the dysregulation in the case of progranulin deficiency [43]. To note that four of the nodes influenced by the group were involved in the perturbed edges described in the Table 1. Giving a look to the significant edges found, the relationships 22941 <-1742 and 22941 <-PIRSF “glutamate receptors” are well note in the literature. In fact, the gene 229141 or SHANK2 plays a critical role both in the integration of the various postsynaptic membrane proteins, cell-adhesion molecules, signal components, scaffolding proteins, and actin-based cytoskeleton, part of the PSD protein network (activated by the PSD-95) [44], and in the organization of the glutamate receptors [45]. The edges 3708 <-9456 that involves the gene Itpr1 and the gene HOMER1 were very interesting. The relationships are involved in the spinocerebellar ataxia in human as described by [46]. Also other links that include the PIRSF adenylate cyclase, GTP-binding regulatory protein and the PLC-beta could be useful to interpret the role of mutation in the progranulin gene group.

The comparison with KEGG edges revealed that 17 edges were not active (off) in the “mutant” group and 21 in the “control”. In contrast the edges activated (on) was 9 and 5 in mutant and control, respectively (cf. Table 3). The Figure 2 illustrates the model and the links found statistically significant between the yes/no mutant groups.

Multiple sclerosis

The first, B cell receptor (BCR) signaling pathway, is an important component of adaptive immunity. B cells produce and secrete millions of different antibody molecules, each of which recognizes a different antigen. This signalling ultimately results in the expression of immediate early genes that further activate the expression of other genes involved in B cell proliferation, differentiation and immunoglobulin (Ig) production as well as other processes. The role of B cells is well known in MS [47, 48]. The second one, Fc gamma R-mediated phagocytosis pathway, includes specialized cell types as macrophages, neutrophils, and monocytes that take part in host-defence mechanisms. Expression of the inhibitory Fc gamma receptor IIB (FcγRIIB) plays an important role during peripheral B cell development, which prevents memory B cells with low affinity or self-reactive receptors from entering the germinal center and becoming IgG positive plasma cells [49]. Furthermore, the decreased expression of Fc gamma RIIB or non-functional Fc gamma RIIB variants are consistently associated with the development of autoimmune tissue inflammation [49]–[51]. Considering the connection between the Fc gamma R-mediated phagocytosis and B cells, and the association of this pathway with autoimmune inflammation, we could conclude that also this pathway could be implicated in the MS phenotype.

Salmonella infection, the third pathway, may appear less interesting, nevertheless it is connected with response to infections that, as showed for the two previous pathways, plays an important role in MS.

SEM analysis of Fc gamma R-mediated phagocytosis KEGG

As an example, we analyzed the Fc gamma R-mediated phagocytosis pathway. The model was obtained starting from the KEGG pathway and finding the shortest paths between DEGs (no PIRSF reduction was performed). The graph reduction of the model specification was from KEGG pathway (mean degree = 10.404, number nodes(edges) = 94 (489)) to DEGs shortest paths model (mean degree = 3.647, number nodes(edges) = 17(31)). The initial model had a poor fit (χ² (df) = 339.410 (105), P < 0.001, RMSEA (P-close) =0.288 (<0.001), SRMR = 0.321) and the final model was adequate considering the SRMR index (0.098) and RMSEA (P-close) 0.111 (0.059). To reach the final model, twenty edges were added: fifteen using as driving criteria the presence of a directed path between the nodes in the original pathway, and five using STRING database information.

Two-group analysis of the final pathway-model revealed a significant global mean (χ² diff(df) = 40.3 (16), P < 0.001 of H₀: μ₁ = μ₂ subject to Σ₁ = Σ₂) and covariance differences (χ² diff (df) = 124.4 (48), P < 0.001 of H₀: Σ₁ = Σ₂ subject to μ₁≠ μ₂).

The comparison with KEGG edges showed that in the model pathway, 22 edges were not active (off) in the MS patients and 18 in controls. In contrast, the edges activated (on) were 6 and 10 in MS patients and control subjects respectively (cf. Table 6). The Figure 3 illustrates the pathway model and the links found statistically significant between the MS/control groups.