All the biopsies analysed in the study were extracted from non-inflamed mucosa of the sigmoid colon. Paraffinised samples of colectomy were selected among three groups of patients: a first group consisting of four patients operated on UC in presence of deep ulcerations (ASC), what is a constituent marker of severity; a second sample made of three healthy subject with normal colonic mucosa and a last sample of four patients with refractory UC, i.e., superficial inflammation without deep ulcers (Figure 1(a)). The extracted RNA derives from low inflammatory areas of the colon. MicroRNA (small non-coding RNA containing between 22 - 25 nucleotides) expression was measured by specific chip of microarray Affymetrix.

Differential miRNA expression was performed using limma [11] by fitting a log-normal (LN) generalized linear model (GLM) that accounts for expression (mucosal) as well as group (UC/ASC).

In this stage a cohort of eleven miRNA microarrays was used with the aim of co-integrating the differential miRNA expression profiles not present in the intersection in pairwise of UC patients (i.e., UC, ASC and Control) and known human Protein-Protein Interaction (PPI, defined as miRNA-gene target product) from Genemania database [12] . Our approach is based on the assumption that genes with similar gene expression levels are translated into proteins that are more likely to interact. Recent works on gene expression and protein interaction data at genome-wide level expose such a conjecture: “Protein pairs encoded by co-expressed genes are much more likely to interact mutually than with any other type of proteins [13] [14] . Specifically, the rationale to transform the miRNA expression of a patient into a network is like this: We may want to represent a node in the graph for every protein encoded by a miRNA target gene provided its expression level was measured on this patient’s microarray. We create an edge between two given proteins of this type if these proteins are reported as interacting by Genemania, and genes are up or down-regulated at the same time with respect to a provided measure tag (see previous subsection). Herein, no distinction is made between coding gene and protein.

Briefly, we initially explore the geometric and algebraic behaviour of each co-integrated omic graph (UC, ASC and Control) by means of some few key spectral properties, namely: their spectra; i.e., the eigenvalue for a given factor measures the variance in all the variables which is accounted for by that factor. Their algebraic connectivity calculated in the Laplacian matrix as its second smallest eigenvalue. Eigencentrality; i.e., to weight the relative importance of a given i^th node in linking motifs within the graphs and defined as the i^th com- ponent of the eigenvector corresponding to the greatest eigenvalue; and their modularity by calculating the Fiedler’s vector; i.e., the vector corresponding to its algebraic connectivity [15] [16] . All the calculations were performed using MATLAB R2011a (maci64 architecture on a machine with a single 2.8 GHz processor and 8GB RAM distributed in two cores).

Definition 1 Lets M ∈ ℝ r × c and M ′ ∈ ℝ p × q be two real matrices, the Kronecker product M ⊗ M ′ ∈ ℝ r p × c q and column-stacking operator v e c ( M ) ∈ ℝ r c are defined as

M ⊗ M ′ : = [ M 11 M ′ M 12 M ′ ⋯ M 1 c M ′ M 21 M ′ M 22 M ′ ⋯ M 2 c M ′ ⋮ ⋮ ⋮ M r 1 B M r 2 M ′ ⋯ M r c M ′ ] ,

v e c ( M ) : = [ M * 1 ⋮ M * c ] ,

where M * k amounts the k^th column of M .

The Kronecker product and v e c operator meet the following relationship (e.g., [17] , Proposition 7.1.9):

v e c ( M N P ) = ( P T ⊗ M ) v e c ( N ) . (1)

Another standard condition of the Kronecker product exploited in this work is ( [17] , Proposition 7.1.6):

( M ⊗ M ′ ) ( N ⊗ N ′ ) = M N ⊗ M ′ N ′ . (2)

All these ideas are extendable to Reproducing Kernel Hilbert Spaces (RKHS). Let H be such a space, hence it is defined by a p.s.d. kernel κ : Y × Y → ℝ , where Y is a set of labels including the singular label ξ . H generates a feature map Φ : Y → H satisfying κ ( y , y ′ ) = 〈 Φ ( y ) , Φ ( y ′ ) 〉 H and mapping in H ξ to its zero element. We finally denote by Φ ( Y ) the matrix of a graph G associated to the feature map that enables lifting tensor algebra from Y to H [18] .

¹In some others context this matrix might be differently defined, e.g., spectral graph theory.

Graphs G s were constructed for each group of UC patients individually. These networks consisted of a set of n vertices V = { v 1 , v 2 , ⋯ , v n } endowed with order and edges E ⊂ V × V . The nature of the measured microRNA expression leads us to work on undirected graphs, i.e., if ( v p , v q ) ∈ E ⇔ ( v q , v p ) ∈ E . Aditionally, ( v p , v p ) ∉ E for any p. Now, we define paths on those UC graphs as a sequence of indices p 0 , ⋯ , p l ( l amounts path length) such that v p s − 1 ~ v p s , for all 1 ≤ s ≤ l . Our graphs are robustly connected since a path can be traced in each direction between each pair of vertices of the graph. We also associate a weight w p q > 0 to each edge ( v p , v q ) to capture the “strength” of an edge ( v p , v q ) . Then ( v p ≁ v q ) implies w p q = 0 whereas for undirected weighted graphs we have w p q = w q p . Now, let A : = w q p D − 1 be the adjacency matrix¹ of our weighted graphs with D a diagonal matrix measuring the node degrees, that is,

D p p = ∑ q   w q p . Thus it may be used as transition matrix in a stochastic process

since the sum of each of its columns is one. We transform a path on G s into random by applying P ( p s + 1 | p 1 , ⋯ , p s ) = A p s + 1 , p s what generates sequences of vertices v p 1 , v p 2 , v p 3 , ⋯ proportionally linked to their weights in pairwise following the above probability. Hence, the probability of transition between any pair of vertex v q and v p through a path of length p can be induced by the expression ( A p ) p q . Finally, we say that two graphs G = ( V , E ) and G ′ = ( V ′ , E ′ ) are isomorphic ( G ≅ G ′ ) if ( v p , v q ) ∈ E iff ( g ( v p ) , g ( v q ) ) ∈ E ′ , where g : V → V ′ is a bijection.

Henceforth, we note that all the definitions are generalised to the normalised case, whereas the edges are taken on a set with finite number of labels { 1,2, ⋯ , d } . In particular, we can take the induced RKHS H = ℝ d endowed with the usual inner product.

Intuitive definition: Random walk graph kernel has been extensively reported in literature to classify and measure similarities of graphs [18] [19] . The rationale of this algorithm is as follows: The random walk kernel on graph counts the number of walks shared by a couple of graphs. Two walks are said to be shared if their lengths and label sequences are the same. Subsequently, the calculated number of shared walks enables to measure the similarity of the two graphs. To infer a formal definition of random walk graph kernel, we might want to present some basic concepts in direct product of graphs. The direct product of two graphs G = { V , E } and G ′ = { V ′ , E ′ } is other graph, denoted by G × = { V × , E × } , where the node set

V × = { ( v p , v ′ s ) | v p ∈ V , v ′ s ∈ V ′ } , andthe edge set E × = { ( ( v p , v ′ s ) , ( v q , v ′ t ) ) | ( v p , v q ) ∈ E , ( v ′ s , v ′ t ) ∈ E ′ } . In particular, G × can be

associated to a weight matrix W × = A ⊗ A ′ (Definition 1) with non-zero entries provided the analogous edge is defined in the graph produced by the direct product. A random walk on the direct product graph G × amounts the trace of random walks on G and G ′ at once. Let c ( d ) and c ′ ( d ′ ) be the starting (stopping) probabilities of the random walks on G and G ′ , respectively. Then, the number of shared walks of length l on the direct product graph G × is calculated by ( d ⊗ d ′ ) ( A T ⊗ A ′ T ) l ( c ⊗ c ′ ) , where A and A ′ are the normalised adjacency matrices of G and G ′ , respectively [20] . This definition enables the review of all the shared walks per each unique lengths. However, this sum might not be convergent. Thus, we introduce a non-negative coefficient of decay μ ( l ) to get rid of the longer walks.

Kernel definition Formally, the expression for the random walk kernel on graph is as follows:

k ( G , G ′ ) : = ∑ l = 0 ∞     μ ( l ) d × T W × l c × . (3)

Hence, c × : = c ⊗ c ′ ( d × : = d ⊗ d ′ ) is the starting (stopping) probability distribution associated to the graph produced by the direct product. Therefore, if the coefficients μ ( l ) assure the convergence of (3), then (3) is a valid p.s.d. kernel ( [18] , Theorem 3).

We selected the conjugate gradient method for calculating the random walk kernel on our graphs since other methods such as the Sylvester or the spectral decomposition are not applicable for kernels on graphs in general [20] .

The computation of a random walk kernel on graph with μ ( l ) = λ l stands for inverting ( I − λ W × ) , an n 2 × n 2 matrix if each graph G and G ′ have n vertices. Lets M and v be a matrix and a vector respectively, conjugate gradient (CG) method is used to solve systems as M x = v efficiently [20] . More general, since these methods are thought of symmetric p.s.d. matrices, CGs solve as well other linear systems efficiently. CG solvers improve their performances as the matrix has a small number of different eigenvalues, or is rank deficient. Remarkably, in cases where the matrix M is sparse the computation speed of matrix-vector products can be increased significantly [21] .

The computation of the graph kernel (3) using CG maybe firstly described as the solution of the following linear system:

( I − λ W × ) x = c × , (4)

for x, then we compute d × T x . Next, it ought to contemplate proficient ways to solve (4) with the CG solver. We already know that W is a square matrix of size n 2 × n 2 . The application of the CG method to a direct approach needs O ( n 4 ) iterations to multiply W by a vector y. However, if we exploit the above extended vec-MNP formula (1) into RKHS ( [18] , Lemma 12) with some new matrix Y ∈ ℝ n × n with y = v e c ( Y ) and taking into account that in particular W × = A ⊗ A ′ (A and A ′ the normalised adjacency matrix for the graphs G and G ′ respectively), by ( [18] , Lemma 12) we can write

W × y = ( A ⊗ A ′ ) v e c ( Y ) = v e c ( A ′ Y A T ) . (5)

If A ~ Φ ( ⋅ ) ∈ ℝ d then we can compute the above multiplication of a matrix by a vector in time order of O ( d n 3 ) . Furthermore, even more efficient computation of A ′ Y A T is feasible provided that the matrices A and A ′ are sparse: Assuming that A and A ′ have O ( n ) non-x entries, then computing (5) takes only O ( n 2 ) time.

Finally, note that the nearest Kronecker product [22] is not appropriate to approximate W × since the number d of distinct labels in our labeled graph is not large enough.

As validation of our results, we also propose to compare the random walk kernel on graph and the family of Weisfeiler-Lehman kernels. The later consists of proficient kernels to be used on graphs presenting discrete node labels. Such family is built on the Weisfeiler-Lehman test of isomorphism between graphs [23] and its valid 1-dimensional variant [24] . It captures topological and label information iteratively mapping the graph of reference onto a sequence of graphs with nodes displaying characteristic attributes. This catenation of graphs originating from the Weisfeiler-Lehman test can establish a family of kernels, including an adequate kernel to compare patterns taking subtree shape. Notice how the edges and length of such a sequence produce a final complexity in linear terms.

Definition 2 Given the Weisfeiler-Lehman (WL) graph G a = ( V , E , l a ) of height a , its sequence is denoted by:

V = { G 0 , G 1 , ⋯ , G h } = { ( V , E , l 0 ) , ( V , E , l 1 ) , ⋯ , ( V , E , l h ) } , (6)

where h counts iterations, and { G 0 , ⋯ , G h } and { G ′ 0 , ⋯ , G ′ h } are respectively the sequences of G and G ′ associated to WL graphs.

Definition 3 Provided the so-called base kernel κ is fixed, then the definition of Weisfeiler-Lehman kernel for κ is

κ W L ( h ) ( G , G ′ ) = κ ( G 0 , G ′ 0 ) + κ ( G 1 , G ′ 1 ) + ⋯ + κ ( G h , G ′ h ) , (7)

where G 0 = G and l 0 = l , the WL sequence up to height a of G.

Finally, κ ( h ) is positive semidefinite if the base kernel κ is positive semidefinite [25] , Theorem 3.

Definition 4 Let Γ k ⊆ Γ be the set of node labels matching at least once in graphs G or G ′ at the end of the k-th iteration of the WL algorithm. We also fix Γ 0 as the set of original node labels of G and G ′ while Γ k are pairwise disjoint. Then, we presume every Γ k = { σ k 1 , ⋯ , σ k | Γ k | } is ordered. Define a map p k : { G , G ′ } × Γ k → ℕ such that p k ( G , σ k l ) amounts the count of the letter σ k l in a graph G . The Weisfeiler-Lehman subtree kernel on two graphs G and G ′ is as follows:

κ W L s u b t r e e ( h ) ( G , G ′ ) = 〈 ϕ W L s u b t r e e ( h ) ( G ) , ϕ W L s u b t r e e ( h ) ( G ′ ) 〉 , (8)

where for G (resp. G ′ )

ϕ W L s u b t r e e ( h ) ( G ) = ( p 0 ( G , σ 01 ) , ⋯ , p 0 ( G , σ 0 | Γ 0 | ) , ⋯ , p h ( G , σ h 1 ) , ⋯ , p h ( G , σ h | Γ h | ) ) .

This algorithm basically seeks matching of vertex identifiers assuming that the corresponding subgraphs match.

Definition 5 Provided a function w weighting the edges exits, we can described the corresponding base kernel κ E by ∑ e ∈ E ∑ e ′ ∈ E ′     δ ( α , α ′ ) δ ( β , β ′ ) κ w ( w ( e ) , w ( e ′ ) ) , where δ amounts Dirac kernel and κ w is the similarity captured by a kernel between weights. Hence by 6, the Weisfeiler-Lehman edge kernel turns into

κ W L e d g e ( h ) = κ E ( G 0 , G ′ 0 ) + κ E ( G 1 , G ′ 1 ) + ⋯ + κ E ( G h , G ′ h ) ,

where κ E = 〈 ϕ E ( G ) , ϕ E ( G ′ ) 〉 and ϕ E ( G ) is a vector of matching pairs ( α , β ) , α , β ∈ ∑ , which amounts sorted final vertices of an edge in G .

Definition 6 We also calculate the shortest path version of the Weisfeiler- Lehman kernel. Similarly, it is defined as

κ W L s h o r t e s t p a t h ( h ) = κ S P ( G 0 , G ′ 0 ) + κ S P ( G 1 , G ′ 1 ) + ⋯ + κ S P ( G h , G ′ h ) ,

where κ S P ( G , G ′ ) = 〈 ϕ S P ( G ) , ϕ S P ( G ′ ) 〉 and ϕ S P ( G ) denotes a vector composed by the counts of matches for triplets ( α , β , s p l ) in G / G ′ , where α , β ∈ ∑ are sorted final vertices of a shortest path and s p l ∈ ℕ 0 is the shortest path length.

So far, we described how to compare UC/ASC graphs, enabling the trace of the underlying similarity between them and their corresponding control samples by gene targets expression profiles from data. Now, we are interesting in performing inference on our topological model to characterise the genetic mechanisms of miRNA perturbations of gene graph in detail. In section 4, we discuss how inference schemes can be used on our estimated model to learn about downstream effects of miRNAs perturbations. We note that all of these inference schemes are based on enrichment analysis in functional annotations (calculation of Fisher’s test [26] is performed to quantitatively capture the functional enrichment of genes according to their annotation terms) using the gene ontology database (GO) [27] .

We analyse our sequence of graphs individually by comparing some algebraic characteristics.

As describe in section 3.3, we found that 2390 proteins (Figure S1) from Genemania [12] were reported by the gene expression levels of our miRNA microarrays (Figure 1(b)). The largest amount of those proteins (1071 for 330 miRNAs differentially expressed (see section 3.2) was identified in the ASC sample, whereas the UC patients sample matched in 804 (in 237 miRNAs differentially expressed); the remaining 515 (in 92 miRNAs differentially expressed) corresponded to the sample of control. These amounts seem to be consistent with the medical expectation of discovering, at a larger-scale, perturbed expression profiles involved in the pathways leading to deep ulcerations (ASC). Strikingly, the comparison of their spectra showed dissimilar conclusions; while the eigenvectors of ASC and Control patients exhibit similar patterns regarding UC patients (Figure 2(a)), the eigenvalue distributions of the three group of patients display the same Gaussian mixture models (Figure 2(b)). However, the algebraic connectivity in ASC and UC resembled each other with associated values of 21 and 18 what means almost twofold greater than the control group with a value of 10. No significant difference was detected among the remaining spectral parameters, i.e., eigencentrality or simple modularity Figure S1. Although we enhance important algebraic and geometric characteristics of our graphs, it seems that no conclusions might be made regarding their similarities per group.

To measure topological similarity among our three groups of graphs, i.e., UC, ASC and Control with a biological significant, we established a comparison

between interacting and co-regulated groups of target genes per sample of patient. To this task a random walk kernel on graph is the appropriate selection, as for this graph a random walk amounts a set of target genes in which continuous genes by the walk side are co-expressed and interact. To efficiently compute the random walk, we made use of the CG methods using the parameter λ = 0.001 with convergence threshold set to 10⁻⁶. In Figure 3(a) we contrast the scores of similarity measured by graph kernel computation of the conjugate gradient algorithm referred to UC patients modelled as labeled graphs with that of the direct sparse method. Our approach demonstrates how the group of ASC

patients is topologically more similar to control patients (0.35 as normalised score ∈ [ 0,1 ] ) than UC are (0.18). Here, closer to 1 means more similar graphs. We recall that the random walk kernel on graph measures the amount of walks shared by the couple of graphs involved in G × (section 3.7). This topological relationship between ASC patients and their group of control is, although relatively unexpected, entirely plausible from a biological and thus medical point of view. Validation of similarities using the Weisfeiler-Lehman Graph Kernels The reliability of our results is also validated by comparing the performances of the random walk and Weisfeiler-Lehman graph kernels. The latter consists of a triplet of robust methods (see methods) in capturing topological and label information on graphs. These algorithms confirmed the same scheme described in our results, i.e., ASC group is closer than UC patients to control group. Whereas a graphical visualisation of these data may be displayed in Figure 3(b), the specific normalised (by all the possible paths on the graph) values of the pairwise comparison between patients’ graphs are shown in the following Table 1.

We can perform inference on our topological model combining the similarity scores and functional enrichment analysis. Since ASC patients are topologically more similar than UC to Control (Figure 3 and Figure 5(a)), one natural idea is to explore the lack of or alternatively the low expression levels of miRNA-gene targets involved in enriched pathways from both ASC and control data with respect to UC patients. In the view of the enrichment analysis using GO (Table SI, Table SII and Table SIII) and the above inference constraint, the only enriched functional module fitting our topological model in the colon was that linked to lymphoid nodules (GO:0048541 with p-value = 2.45 e − 4 and q-value = 5.45 e − 2 associated to the Fisher Exact test). Such structures are the equivalent

to the Peyer’s patches (PPs) in the Ileum. This fact derives from the no detection of such enrichment in the production of immunoglobulins, i.e., sIgA and slgE, in ASC patients as compared to UC patients (Figure 4). Such a lack of production in sIgA is a consequence of the relative poorly enriched scores associated to the coding genes ID2 and STAT5, which control the intestinal immune network for sIgA production via negative regulation of class IgA/E class switching [28] and“on-off” recombination of immunoglobulin gene in developing pro-B cells [29] (Figure 5(d)―starts highlighted in red). Now, we are equipped with enough information to infer the following hypothesis: There exists a very low production of immune globulin A (sIgA) within ASC patients occasioned by malfunction of lymphoid nodules. Indeed, there is no immune system’s activation, whereas in UC patients we have over-expression of lymphoid nodules related pathways (Figure 5(d)).

The sIgA is an antibody―Y-shaped protein―that plays a critical role in immune function in the mucous membranes. This scenario matches the topological constraint yielded by our model between control and ASC patients. Furthermore, it is been already described how sIgA likely contains other propitious outcomes in overall immunity by means of a diminished inflammation in the digestive tract [30] . There is also evidence that sIgA (low/coding genes under-expression) secretion into body cavities in combination with malfunction of immune cells in PPs [31] [32] [33] is involved in allergic diseases (type 1 diabetes, Ulcerative Colitis/Crohn disease, hay fever or asthma). Thus, the formation of deep ulcers in some UC patients may be caused by the low production of sIgA as a consequence of lymphoid structures malfunction.