A Novel Pathway Network Analytics Method Based on Graph Theory

Abstract

A biological pathway is an ordered set of interactions between intracellular molecules having collective activity that impacts cellular function, for example, by controlling metabolite synthesis or by regulating the expression of sets of genes. They play a key role in advanced studies of genomics. However, existing pathway analytics methods are inadequate to extract meaningful biological structure underneath the network of pathways. They also lack automation. Given these circumstances, we have come up with a novel graph theoretic method to analyze disease-related genes through weighted network of biological pathways. The method automatically extracts biological structures, such as clusters of pathways and their relevance, significance of each pathway and gene, and so forth hidden in the complex network. We have demonstrated the effectiveness of the proposed method on a set of genes associated with coronavirus disease 2019.

1. Introduction

Identifying the functional correlations among molecular components is very crucial to accurately deciphering the structure–function interdependencies. Usually, most of the biological activities are not stemming from a single molecule but a set of molecules interacting in a concerted way (for instance, polygenic disorders). Consequently, deciphering biology under the context of networks is very crucial and promising. From the perspective of graph theory, a biological network consists of a set of nodes representing specific biological entities. Two nodes are connected by an edge depicting an affiliation between them. Based on the characteristic of a network, an edge can be directed or undirected. The weight of an edge defines similarity or dissimilarity between the two participating nodes, such as semantic similarity and Pearson's coefficient. For instance, in protein–protein interaction (PPI) networks, nodes and edges represent proteins and physical interactions, respectively (Rual et al., 2005; Stelzl et al., 2005). For another instance, in metabolic networks, nodes serve as metabolites and edges are links for these metabolites engaging in the same biochemical reactions (Jeong et al., 2000; Duarte et al., 2007). In this context, accurately identifying functional modules (i.e., clusters) in a biological network is very critical because it helps us to figure out the underneath structure, interactions, and dynamics of cell functions (Bu et al., 2003; Spirin and Mirny, 2003).

Applying biological network analytics to a set of genes related to a specific disease has been done before by several researchers. For example, Hu et al. (2017) have taken a pathway-based approach and applied a network analysis method to understand the molecular features of Alzheimer's disease. Other researchers have proposed mining algorithms targeting PPI networks for Homo sapiens. Theses algorithms are designed to discover functional modules (clusters) of protein complexes. This is because such densely connected subgraphs usually lead to substantial biological knowledge at the molecular level. As an example, Sriwastava et al. (2017) have proposed a quasi-clique mining method for detecting these dense regions. Melo et al. (2016) have used a machine learning approach to detect such hot spots. In their study they used 27 algorithms with different cost functions and reported the best algorithm.

In this article, we have proposed a novel methodology for analyzing complex network of biological pathways. It is scale free, that is, there is no hard thresholding to discard edges based on weight. It runs in four stages. At first, it picks a set of enriched biological pathways for a given set of disease-related genes. Later, it constructs a weighted network where each pathway acts as a node. Two nodes are connected by an edge if they have some common biological entities and the weight of this edge refers to that similarity. In this study, our similarity score is well defined and its values are ranging from 0 to 1. At the third stage, it clusters the network into a set of nonoverlapping groups having highest modularity. Finally, it ranks the pathways based on their significance.

The rest of the article is organized as follows. Our proposed methodology is described in Section 2. Results and some relevant discussions are portrayed in Section 3 and Section 4 concludes the article.

2. Methods

Our proposed algorithm consists of four basic steps as stated earlier. At the beginning, we find a set of statistically significant biological pathways with respect to a curated list of disease-related genes. We then form a network of pathways by employing an innovative weighted network construction method. At the third step, we detect a set of subnetworks by clustering the entire network. Finally, we analyze each of the subnetworks based on closeness centrality. Pseudo code of our proposed method can be found in Algorithm 1. Next, we illustrate our proposed method in detail.

2.1. Identification of significantly enriched pathways

At first, we find a set of biological pathways from the database of pathways (Reactome, KEGG, etc.) with respect to a given set of disease-related genes (i.e., a set of genes known to be responsible for a specific disease, such as Alzheimer's, Parkinson's, or COVID-19). Later, we employ hypergeometric test that uses the hypergeometric distribution (https://en.wikipedia.org/wiki/Hypergeometric_distribution) to calculate the statistical significance of a biological pathway with respect to the given set of genes. Specifically, we computed a hypergeometric p-value for each of the biological pathways to assess whether a pathway is over-represented with those genes. Finally, we choose a set of enriched pathways having Bonferroni corrected p-value <0.05.

2.2. Construction of a weighted network

We build an undirected weighted graph to investigate interlinks and interactions among the enriched biological pathways. Let $G (V, E, w)$ be our undirected weighted graph where each enriched pathway $v \in V$ acts as a vertex. Two vertices v_i and v_j will be connected by an edge $e \in E$ iff (1) v_i and v_j share at least x common genes and (2) the similarity between v_i and v_j is greater than a threshold, t. In our experiment, x and t were set to 2 and 0, respectively (i.e., there is no hard thresholding). We define similarity between any two vertices in a way so that it mimics a specific biological theme between them, if any. The similarity between any two genes can roughly be defined by their common gene ontology (GO) terms. The “biological process” subontology of GO (GO-BP) is widely used to evaluate sets of relationships between genes. It is due to the fact that genes annotated with the same (or related) GO-BP terms are functionally homogeneous. Consequently, two pathways will be functionally similar if they contain a set of functionally related genes between them. By considering this observation, we compute the pair-wise Jaccard index between any two pathways. Suppose, v_i and v_j pathways consist of G_i and G_j sets of genes of size m and n, respectively. Let us assume, $G_{i} = {g_{1}^{i}, g_{2}^{i}, \dots, g_{m}^{i}}$ and $G_{j} = {g_{1}^{j}, g_{2}^{j}, \dots, g_{n}^{j}}$ . We then extract the number of common GO-BP terms and divide it by the total number of unique terms of each pair of genes gⁱ and g^j to get the Jaccard index. We add all such indices and normalize the final value by dividing it by the number of such pairs to get the similarity score. As a result, the minimum and maximum values of such a score will be 0 and 1, respectively. Intuitively, the higher the score, the more will be two pathways functionally similar. Now, the score constitutes the weight of the edge between v_i and v_j. Assume $g_{p}^{i}$ and $g_{q}^{j}$ ( $1 \leq p \leq m$ and $1 \leq q \leq n$ , respectively) contain the sets of $b_{p}^{i}$ and $b_{q}^{j}$ GO-BP terms, respectively. The similarity score between any two vertices can then be mathematically formulated as: $S (v_{i}, v_{j}) = \frac{\sum_{m} \sum_{n} \frac{| b_{m}^{i} \cap b_{n}^{j} |}{| b_{m}^{i} \cup b_{n}^{j} |}}{m \times n} .$ (1)

Algorithm 1 Pathway Network Analytics Method
Input: A set of disease-related genes G, shared genes s, and common genes c.
Output: A set of functional modules, pathway influences.
1: Let D be a database of biological pathways (Reactome, KEGG, etc.);
2: Pick a set of pathways $P_{i n t e r i m} \in D$ with respect to G where each pathway $p \in P$ contains at least c number of common genes in G;
3: Retain statistically significant pathways $P \in P_{i n t e r i m}$ by employing hypergeometric over-representation test;
4: Initialize a weighted network N where each pathway $p \in P$ acts as a node;
5: for each distinct and ordered pair $(p_{i}, p_{j}) \in P$ do
6: $t_{i} \leftarrow p_{i} \cap G$
7: $t_{j} \leftarrow p_{j} \cap G$
8: if $\| t_{i} \cap t_{j} \| \geq s$ then
9: Compute the similarity score $l_{p_{i} p_{j}}$ between p_i and p_j using Equation 1;
10: if $l_{p_{i} p_{j}} > 0$ then
11: Add an edge $e_{p_{i} p_{j}}$ in network N;
12: Weight of edge $e_{p_{i} p_{j}}$ , $w_{p_{i} p_{j}} \leftarrow l_{p_{i} p_{j}}$ ;
13: end if
14: end if
15: end if
16: Calculate the influence score of each pathway $p \in P$ using closeness centrality;
17: Cluster the network N by a suitable graph clustering algorithm;
18: for each cluster $q \in Q$ do
19: Calculate the influence score of each pathway $p \in q$ using closeness centrality;
20: end for
21: Return the clusters, influence scores, etc.

2.3. Identification of subnetworks

Clustering is one of the most widely used techniques for exploratory data analysis. The goal here is to divide the biological pathways into several groups such that each group of pathways represents a specific and distinct biological event/theme. As the network of biological pathways often constitutes a small number of nodes ( $\leq 50$ ), we employed an optimal community structure prediction algorithm (Brandes et al., 2007). It calculates the optimal community structure of a graph, by maximizing the modularity measure over all possible partitions. Note that modularity optimization is an NP-complete problem (NP = nondeterministic polynomial time). Consequently, all known algorithms have exponential time complexity in worst case. So, it is impossible to run exact algorithms on a graph with large number of nodes having dense connections. Louvain method is an attractive alternative in this context (Blondel et al., 2008). The method is a greedy optimization method having time complexity , where n is the number of nodes in the network.

2.4. Identification of important pathways

In graph theory and network analysis, centrality is a very crucial notion in identifying influential nodes in a graph. It is used to measure the importance of distinct nodes in a graph. Applications include but are not limited to identifying the most influential person(s) in a social network, key infrastructure nodes in the Internet or urban networks, super-spreaders of disease, and so forth. Depending on the definition of centrality, it comes in contrasting essences. Simplest one is the degree centrality, which is defined as the number of edges incident on a particular node. A natural extension of degree centrality is closeness centrality. In a connected graph, closeness centrality of a node is a measure of centrality in a network, calculated as the reciprocal of the sum of the length of the shortest paths between the node and all other nodes in the graph (Bavelas, 1950). Consequently, the more “central” a node is, the more closer it is to all other nodes in the graph. Mathematically it is defined as $C (x) = \frac{1}{\sum_{y} d (y, x)}$ , where $d (y, x)$ is the distance between node x and node y. We have used closeness centrality as a metric to identify influential pathways in both the entire network and corresponding subnetworks. We normalize the score by multiplying it by $n - 1$ where n is the total number of nodes in the network. It is to be noted that we have replaced the weight of each edge with $1 - w e i g h t$ before computing the closeness centrality.

3. Results and Discussions

3.1. Data set employed

To demonstrate the effectiveness of our proposed methodology, we have performed rigorous experimental evaluations by considering a set of human protein-coding genes linked to SARS-CoV-2 infection and COVID-19 disease. These genes are curated from GENCODE (https://www.gencodegenes.org/human/covid19.html). The list consists of 560 genes extrapolated from recent publications and in collaboration with other projects, such as recently published drug repurposing studies by Zhou et al. (2020) and Gordon and Jang (2020). Throughout the article, we dubbed this set of genes as covid-genes.

3.2. Outcomes and relevant discussions

As stated in Section 2, our network analytics method runs in four stages. Next, we illustrate the experimental evaluations based on those stages in detail.

3.2.1. Pathway enrichment analysis

At first, we chose a set of enriched biological pathways from the database of pathways. In this study, Reactome (Croft et al., 2010) pathways were utilized to decipher the biological theme from the set of 560 covid-genes. Here is a brief review of Reactome database. Reaction is the “nucleus” of the Reactome data model. A set of entities, such as nucleic acids, proteins, complexes, and small molecules, engages in reactions to form a network of biological interactions that are ultimately assembled into pathways. Some notable examples of biological pathways in Reactome database consist of signaling, innate and acquired immune function, transcriptional regulation, translation, apoptosis, and classical intermediary metabolism.

After employing hypergeometric test, we detect 30 Bonferroni corrected (adjusted $p < 0.05$ ) biological pathways. Table 1 contains these 30 significant pathways. It is to be noted that we only retain those enriched pathways each having at least five genes in common with covid-genes to discard potentially spurious and smaller pathways. Next we discuss about some of the enriched pathways. At first, consider interferon signaling pathways. Interferons, also known as type I, type II, and type III interferons in humans, are proteins produced by cells in response to infection. Insufficient or inappropriately timed activation of interferon signaling may contribute to severe cases of COVID-19 caused by the SARS-CoV-2 (Blanco-Melo et al., 2020; Hadjadj et al., 2020; Park and Iwasaki, 2020). Now, consider basigin interactions pathway. Basigin (BSG, in short) also known as extracellular matrix metalloproteinase inducer (EMMPRIN) or cluster of differentiation 147 (CD147) is a protein encoding gene (Kasinrerk et al., 1992). According to Wang et al. (2020), host cell-expressed basigin (CD147) may bind spike protein of SARS-CoV-2 and possibly be involved in host cell invasion.

Table 1.

Bonferroni Corrected Enriched Reactome Pathways

Pathway	Reactome ID	p	G_p	G_c
Interferon alpha/beta signaling	R-HSA-909733	5.93E-12	70	22
Interferon signaling	R-HSA-913531	4.57E-09	158	29
Influenza infection	R-HSA-168254	2.06E-08	63	17
Basigin interactions	R-HSA-210991	3.63E-08	26	11
MHC class II antigen presentation	R-HSA-2132295	4.98E-08	59	16
Disease	R-HSA-1643685	9.80E-07	508	54
Metabolism of angiotensinogen to angiotensins	R-HSA-2022377	1.05E-06	17	8
Interactions of Rev with host cellular proteins	R-HSA-177243	2.33E-06	37	11
Peptide hormone metabolism	R-HSA-2980736	3.14E-06	53	13
Influenza life cycle	R-HSA-168255	3.14E-06	53	13
Host interactions of HIV factors	R-HSA-162909	4.17E-06	89	17
Mitochondrial protein import	R-HSA-1268020	6.85E-06	65	14
Nuclear import of Rev protein	R-HSA-180746	7.54E-06	34	10
Transport of ribonucleoproteins into the host nucleus	R-HSA-168271	1.82E-05	30	9
NEP/NS2 interacts with the cellular export machinery	R-HSA-168333	1.82E-05	30	9
ISG15 antiviral mechanism	R-HSA-1169408	1.82E-05	30	9
Export of viral ribonucleoproteins from nucleus	R-HSA-168274	2.44E-05	31	9
Activation of matrix metalloproteinases	R-HSA-1592389	2.44E-05	31	9
Metabolism of proteins	R-HSA-392499	2.59E-05	2000	146
Trafficking and processing of endosomal TLR	R-HSA-1679131	2.91E-05	13	6
TRAF3-dependent IRF activation pathway	R-HSA-918233	2.91E-05	13	6
Rev-mediated nuclear export of HIV RNA	R-HSA-165054	4.24E-05	33	9
Nuclear pore complex disassembly	R-HSA-3301854	5.49E-05	34	9
Neutrophil degranulation	R-HSA-6798695	5.66E-05	485	47
Cellular response to heat stress	R-HSA-3371556	6.93E-05	99	16
SUMOylation of DNA replication proteins	R-HSA-4615885	1.13E-04	37	9
Regulation of glucokinase by GRP	R-HSA-170822	1.35E-04	30	8
Degradation of the extracellular matrix	R-HSA-1474228	1.42E-04	105	16
Negative regulators of DDX58/IFIH1 signaling	R-HSA-936440	1.44E-04	23	7
TRAF6-mediated IRF7 activation	R-HSA-933541	1.75E-04	17	6

G_p refers to the number of genes a pathway contains. G_c refers to the common genes between a pathway and covid-genes.

DDX58, refers to the DExD/H-box helicase 58 gene; GRP, gastrin releasing peptide; IFIH1, interferon induced with helicase C domain 1; IRF, interferon regulatory factors; MHC, major histocompatibility complex; NEP/NS2, nuclear export protein; TLR, toll-like receptor; TRAF3, TNF receptor associated factor.

3.2.2. Entire network analysis

After finding the statistically significant pathways, we build a network as stated in Section 2. We found 243 covid-genes (out of 560) in the enriched 30 Reactome pathways. Enrichment analyses based on GO-BP and disease ontology (DO) terms have been performed with respect to those 243 covid-genes.

3.2.2.1. GO-BP enrichment analysis

One of the main uses of the GO terms is to perform enrichment analysis on a given set of genes. For instance, an enrichment analysis will find which GO terms are over-represented (or under-represented) using annotations for that set of genes. We have performed enrichment analysis on 243 covid-genes based on GO-BP terms and retained 163 GO-BP Bonferroni corrected (adjusted $p < 0.05$ ) terms. Top 10 enriched GO-BP terms are given in Table 2. Most of the terms are associated with COVID-19.

Table 2.

Bonferroni Corrected Top 10 Enriched GO-BP Terms

ID	Description	Gene ratio	p	Adjusted p
GO:0019058	Viral life cycle	39/243	3.61E-28	1.35E-24
GO:0043312	Neutrophil degranulation	47/243	2.17E-27	8.14E-24
GO:0002283	Neutrophil activation involved in immune response	47/243	2.85E-27	1.07E-23
GO:0042119	Neutrophil activation	47/243	6.97E-27	2.61E-23
GO:0002446	Neutrophil mediated immunity	47/243	7.62E-27	2.85E-23
GO:0043903	Regulation of symbiosis encompassing mutualism through parasitism	32/243	2.23E-25	8.34E-22
GO:1903900	Regulation of viral life cycle	28/243	2.66E-25	9.98E-22
GO:0050792	Regulation of viral process	30/243	8.35E-25	3.13E-21
GO:0051607	Defense response to virus	32/243	9.31E-24	3.49E-20
GO:0009615	Response to virus	35/243	1.88E-22	7.04E-19

GO-BP, “biological process” subontology of gene ontology.

3.2.2.2. DO enrichment analysis

Like GO, the DO is a formal ontology of human disease. We have performed enrichment analysis on the set of top 243 covid-genes as already noted based on DO terms and retained 7 DO Bonferroni corrected (adjusted $p < 0.05$ ) terms (Table 3). Almost all of the retained enriched DO terms are associated with COVID-19 disease. For instance, Aldhaleei et al. (2020) reported the first case of hepatitis B virus reactivation caused by COVID-19 in a young adult with altered mental status and severe transaminitis.

Table 3.

Bonferroni Corrected Enriched Disease Ontology Terms

ID	Description	Gene ratio	p	Adjusted p
DOID:2237	Hepatitis	31/150	5.23E-11	2.93E-08
DOID:8469	Influenza	16/150	2.4E-10	1.35E-07
DOID:2043	Hepatitis B	19/150	3.72E-09	2.08E-06
DOID:3459	Breast carcinoma	20/150	2.81E-05	0.0157
DOID:1883	Hepatitis C	15/150	2.86E-05	0.016
DOID:184	Bone cancer	15/150	4.66E-05	0.0261
DOID:3347	Osteosarcoma	14/150	4.72E-05	0.0264

3.2.2.3. Pathway significance

As noted earlier, after constructing the network, we compute the influence of each pathway based on closeness centrality. The corresponding network is shown in Figure 1. The influence of each pathway with respect to the entire network is proportional to the diameter of its representative circle. “Metabolism of proteins” and “Trafficking and processing of endosomal TLR” pathways possess the highest (0.91) and lowest (0.36) centrality scores, respectively.

FIG. 1.

Entire network built from 30 statistically significant pathways.

3.2.3. Subnetwork analysis

After constructing the weighted network, we cluster the network to find functional modules. Since the weight of an edge corresponds to the functional similarity between two pathways, see Section 2, each of the subnetworks consisting of highly interconnected pathways should mimic a specific biological theme or functionality. Note that our network is scale free, that is, there is no thresholding on weights. Also we do not need to provide the number of clusters a priori. The clustering algorithm automatically dismantles the entire weighted network into three groups, namely C1, C2, and C3. Figure 1 shows the entire network along with cluster annotations. We have also computed pathway centrality scores for each of the subnetworks as shown in Figure 2.

FIG. 2.

Closeness centrality values for pathways in clusters C1, C2, and C3.

At first, consider C1 cluster. It consists of six pathways and all of them are related to immune systems in humans. Therefore, our proposed method accurately classifies a set of interrelated and analogous pathways into a group. According to the centrality measure, “ISG15 antiviral mechanism” is the most influential pathway in this cluster. It is a potential regulator of the immune response from viral infection. As reported by Swaim et al. (2020), viral de-ISGylases, including SARS-CoV-2 PL $^{p r o}$ , positively modulate ISG15 secretion. Now, see Table 4 for the top six most occurring covid-genes in cluster C1. IRF3 appears in all the six pathways. It plays a critical role in the innate immune system's response to viral infection (Collins et al., 2004).

Table 4.

Top Six Most Occurring Covid-Genes in Each Cluster

C1			C2			C3
Gene	N_P	%O	Gene	N_P	%O	Gene	N_P	%O
IRF3	6	100	CTSG	6	67	RAE1	14	100
DDX58	5	83	CTSD	5	56	NUP88	14	100
TRIM25	5	83	MME	4	44	NUP58	14	100
IRF7	4	67	CTSB	4	44	NUP98	14	100
ISG15	4	67	CTSK	4	44	NUP54	14	100
MX1	3	50	CTSS	4	44	NUP210	14	100

N_P refers to the number of pathways a specific gene is found.

$% O$ represents the fraction of such pathways.

C2 subnetwork contains nine pathways and is very interesting. It is a mixture of protein metabolism and immune system-related pathways. Several studies (such as Odegaard and Chawla, 2013) demonstrated the strong link between immune cell function and protein metabolism. Table 4 contains top six most occurring covid-genes in cluster C2. The Cathepsin G gene (CTSG) has been found in six (out of nine) pathways. According to Akgun et al. (2020), it was significantly altered in naso-oropharyngeal samples of SARS-CoV-2 patients.

Finally, C3 consists of 14 pathways. Almost all of them are related to some specific viral infections. See Table 4 for the top six most occurring covid-genes in cluster C3. All the top genes have been found in all the 14 pathways. As reported by Addetia et al. (2020), SARS-CoV-2 ORF6 disrupts nucleocytoplasmic transport through interactions with RAE1 and NUP98.

4. Conclusions

In this article, we have proposed a formal framework to decipher complex structure among the interacting biological pathways. To begin with, a set of enriched biological pathways is identified with respect to a set of disease-related genes. An innovative weighted network is then constructed. It is scale free, that is, there is no hard thresholding to discard edges based on weights. The weighted network is then disassembled to find a set of nonoverlapping and functionally different clusters. We have demonstrated its effectiveness by employing a set of genes potentially associated with COVID-19.

Footnotes

Author Disclosure Statement

The authors declare they have no competing financial interests.

Funding Information

This research has been supported in part by the National Science Foundation (NSF) grants 1743418 and 1843025.

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