Detection of spreader nodes in human-SARS-CoV protein-protein interaction network

1. Protein–protein interaction network (PPIN)

When one protein interacts with another protein, it forms a network-like structure known as PPIN. Generally, it is portrayed as a graph where proteins are represented as nodes, and their corresponding connecting edges represent their interactions. Mathematically, PPIN can be highlighted as a graph G_nv, which consists of a set of vertices v(nodes) connected by edges e (links). Thus, G_nv = (v, e) (Saha et al., 2014; Saha et al., 2019a).

2. Level-1 and Level-2 proteins

In a PPIN, level-1 proteins of a node are those proteins that are in direct connection with that node, i.e., its immediate neighbors, whereas level-2 proteins are those proteins that are indirectly connected with level-1 proteins of that node, i.e., its indirect neighbors (Saha et al., 2014; Saha et al., 2019a).

3. Graph centrality

Graph centrality is one of the essential aspects for the identification of significant nodes in a PPIN. The centrality of a node defines how relevant the node is in a PPIN or how much a node is centrally located in a PPIN.

4. Betweenness centrality (BC)

BC (Anthonisse, 1971) is one of the ways of measuring a node’s impact on the transmission of information between every pair of nodes in a graph, considering that this transmission is always executed over the shortest path between them. Mathematically, it is defined as: ${\displaystyle C_B^{\left(u\right)}=\sum _{s\ne u\ne t}\frac{\rho \left(s,u,t\right)}{\rho \left(s,t\right)}}$ where ρ(s, t) is the total number of shortest paths from node s to node t,  and ρ(s, u, t) is the number of those paths that pass through u.

5. Closeness centrality (CC)

CC (Sabidussi, 1966) is a procedure for detecting nodes that transmit information within a network efficiently. Nodes with high closeness centrality values are considered to have the shortest distance to all available nodes in the network. It can be mathematically expressed as: ${\displaystyle C_C^{\left(u\right)}=\frac{\left|N_u\right|-1}{\sum _{v\in V}dist\left(u,v\right)}}$ where $\left|N_u\right|$ denotes the number of neighbors of node u and dist(u, v) is the distance of the shortest path from node u to node v.

6. Degree centrality (DC)

DC (Jeong et al., 2001) is considered the simplest among the available centrality measures that only count the degree of a node, i.e., the number of directly connected neighbors. Nodes having a high degree are said to be the highly connected module of the network. It is defined as: ${\displaystyle C_D^{\left(u\right)}=\left|N_u\right|}$ where $\left|N_u\right|$ denotes the number of neighbors of node u.

7. Local average centrality (LAC)

LAC (Li et al., 2011) of a node represents how close its neighborhood proteins are. It is defined to be the local metric to compute the essentiality of the node for transmission ability by considering its modular nature, the mathematical model of which is highlighted as: ${\displaystyle LAC\left(u\right)=\frac{\sum _{w\in N_u}de{g}_{C_u}^w}{\left|N_u\right|}}$ where C_u is the subgraph induced by N_u (i.e., the number of neighbors of node u) and $de{g}_{c_u}^w$ isthe total number of nodes that are directly connected in C_u.

8. Ego network

Ego network of node i (S_i) (Samadi & Bouyer, 2019) is defined as the grouping of node i itself along with its corresponding level-1 neighbors and interconnections. N (S_i) (Samadi & Bouyer, 2019) consists of the set of nodes which belong to the ego network, S_i i.e., {i} ∪ Γ(i).

9. Edge ratio

The edge ratio of node i (Samadi & Bouyer, 2019) is defined by the following equation: ${\displaystyle Edgeratio\left(i\right)=\frac{\left(\sum _{j\in \left(i\right)}|\Gamma \left(j\right)-N\left(S_i\right)|\right)+1}{\left(\frac{1}{2}\sum _{j\in \Gamma \left(j\right)}\left|{\Gamma }^{S_i}\left(J\right)\right|\right)+1}=\frac{E_{out}^{S_i}+1}{E_{in}^{S_i}+1}}$

where $E_{out}^{S_i}$ is the total number of interactions between the ego network S_i and the proteins outside it. $E_{in}^{S_i}$ is the total number of interactions among node i’s neighbors. Γ (i) denotes the level-1 neighbors of node i.S_i is considered to be Ego network. Γ^S_i(j) denotes node j’s neighbors which belongs S_i. In the edge ratio, $E_{out}^{S_i}$ is positively related to the non-peripheral location of node i. A large number of interactions resulting from the ego network denotes that the node has a high level of interconnectivity between its neighbors. On the other hand, $E_{in}^{S_i}$ is negatively related to the inter-module location of node i. It represents the fact that the interconnectivity between neighbors is usually connected to the number of structural holes available around the node. Thus, when the neighbor’s interconnectivity is low, the root or the central node i gains more control of transmission flow among the neighbors.

10. Jaccard dissimilarity

The similarity between two nodes is determined by Jaccard dissimilarity (Jaccard, 1912) based on their common neighbors. Jaccard dissimilarity of node i and j (dissimilarity(i, j)) is defined as: ${\displaystyle dissimilarity\left(i,j\right)=1-sim\left(i,j\right)=\frac{|\Gamma \left(i\right)\cap \Gamma \left(j\right)|}{|\Gamma \left(i\right)\cup \Gamma \left(j\right)|}}$ where $|\Gamma \left(\mathrm{i}\right)\cap \Gamma \left(\mathrm{j}\right)|$ refers to the number of common neighbors of i and $j.|\Gamma \left(\mathrm{i}\right)\cup \Gamma \left(\mathrm{j}\right)|$ is the total number of neighbors of i and j. The similarity degree between i and j is considered more when they have more common neighbors. Whereas, when dissimilarity between the neighbors of a node is high, it guarantees that the only common node among the neighbors is the central node, which is termed a structural hole situation (Samadi & Bouyer, 2019).

11. Neighborhood diversity

The neighborhood diversity (Samadi & Bouyer, 2019) is a significant parameter of a graph that is based on Jaccard dissimilarity. When the dissimilarity of the neighbors of a node is high, it assures that the central node is the only neighbor common among the neighbors of that node, i.e., it represents the structural hole situation. On the other hand, when a node’s neighborhood diversity reaches its greatest value, it reveals that the neighbors have no other closer path. Hence, the neighbors should transmit or communicate through this node. Mathematically, it is defined as: ${\displaystyle neighborhood\text{\_}diversity\left(i\right)=\sum _{j,k\in \left(i\right)}dissimilarity\left(j,k\right).}$

12. Node weight

Node weight (Wang & Wu, 2013) is a graph parameter used to assign weightage to a node in a graph. Node weight w_v of node v ∈ V in PPIN is interpreted as the average degree of all nodes in G_V′, a sub-graph of a graph G_V. It is considered as another measure to determine the strength of connectivity of a node in a network. Mathematically, it is represented by ${\displaystyle w_v=\frac{\sum _{u\in V^{{}^{\prime \prime }}}deg\left(u\right)}{\left|V^{{}^{\prime \prime }}\right|}}$ where V^′′ is the set of nodes in G_V′. ∣V^′′| is the number of nodes in G_V′. And deg(u) is the degree of a node u ∈ V^′′.

1. Identification of spreader nodes by spreadability index

The spreadability index of node i is defined as the ability of node i to mediate a viral infection in a PPIN. Mathematically it can be defined as: ${\displaystyle Spreadability\text{\_}index\left(i\right)=\left(Edgeratio\left(i\right)\times neighborhood\text{\_}diversity\left(i\right)\right)+Nodeweight\left(w_i\right).}$ Nodes having a high spreadability index are termed as spreader nodes, i.e., if the viral proteins establish interactions with these nodes, then the viral infection can be mediated to a more significant number of nodes in a much short amount of time compared to the other nodes in PPIN.

Figure 1 represents a sample PPIN where each protein is denoted as a node while edges mark its interactions with other proteins. The PPIN consists of 33 nodes and 53 edges. The PPIN data and the protein names and interactions are given as input to the Cytoscape, which generates the network view as highlighted in Fig. 1. Cytoscape is open-source software that is used for PPIN generation and visualization (Shannon et al., 2003). The spreadability index is computed on the synthetic PPIN, shown in Fig. 1, using essential PPIN characteristics in this PPIN, as stated earlier. The same is compared to DC, BC, CC, and LAC, highlighted in Tables 1 to 5.

In Fig. 1, it can be observed that nodes 1 and 24 are the essential spreaders. Node 1 connects the four densely connected modules of the PPIN, making this node the topper with the highest spreadability index. This node has been correctly ranked by all the methods except LAC and DC. Node 24, though, has a moderate edge ratio and node weight but is one of the most densely connected modules itself despite getting isolated from the main PPIN module of node 1. Moreover, node 24 has the highest neighborhood density. It establishes that the only path of transmission of information for nodes 26, 27, 25, 28, 29, 30, 31, 32, and 33 is node 24. Thus, if viral proteins of SARS-CoV establishes interaction with node 24, then all the connected nodes will be indirectly coming under the interaction of viral proteins as the connected nodes have no interactions with other central nodes except node 24. So, node 24 holds the second position for the spreadability index in our proposed methodology. Node 24 is not correctly identified as the second most influential spreader node by the other methods. Further assessment of the remaining nodes highlights the fact that the performance of the new attribute spreadability index in our proposed methodology is relatively better in comparison to the others.

2. Validation of spreader nodes by SIS model

To design the mathematical model for this infectious disease, the SIS Epidemic Model (Bailey, 1975) is used in this proposed methodology by classifying the proteins in SARS-CoV-human PPIN based on their interactivity status (for more details, please see “Studied Models in epidemiology” section of the supplementary document). SIS refers to Susceptible, Infected and Susceptible states, which are generally considered the three probable protein states in a PPIN. (1) S - The susceptible states are the states of those human proteins with which viral proteins have not yet interacted, but they are at risk of getting interacted. In general, every protein in PPIN is initially in a susceptible state. (2) I –These infected states are the states of those human proteins with which viral proteins have interacted, and the viral infection gets mediated. (3) S –The susceptible states are the states of those human proteins that have lost their interaction with the viral proteins (due to antiviral therapies or change in interface residues (Brito & Pinney, 2017)) and again become susceptible. The interaction rate of the viral proteins with human proteins, the loss rate of interactivity of the human protein with the viral proteins (general assumption is that any protein after coming out of the infected state gets into a susceptible state again in one day), and the total number of proteins are usually provided as input to SIS model. If a protein gets into an infected state and has many neighbors, any neighbor can mediate viral infection. So, the final result is generated after 50 iterations for each protein in the infected state. The total number of proteins in the susceptible state after 50 iterations in the neighborhood of each protein in an infected state divided by the total number of proteins in the PPIN gives the interaction capability of the protein in an infected state. Thus, the spreader nodes identified by the spreadability index are validated by the interaction rate as generated by the SIS model for them. It can be observed from Tables 1 to 5 that the proposed methodology has the highest SIS interaction rate of 2.46 with viral proteins (see Table 1) in comparison to others for their corresponding top 10 spreader nodes in the synthetic PPIN, as shown in Fig. 1.

3. Ranking of Spreader edges

To show the ranking of interacting spreader edges, two synthetic PPINs: PPIN-1 and PPIN-2, have been considered in Fig. 2. Node D, E, and F are the selected top spreader nodes in PPIN-1 by spreadability index, similarly explained with a synthetic PPIN in Fig. 1. To avoid the complexity in the diagram, the top 5 nodes in PPIN-2 (see Table 1) are selected as spreader nodes. Red-colored edges are the interconnectivity within PPIN-1, while black-colored edges show the interconnectivity within PPIN-2. Green-colored spreader edges (i.e., edges connected with spreader nodes) show the interconnectivity between PPIN-1 and PPIN-2. Ranking of a spreader edge measures the interaction ability of a spreader edge with the viral proteins, i.e., how many nodes get interacted with the viral proteins through that edge, and the viral infection gets mediated. Thus, all the spreading edges are ranked based on the average spreadability index of its connected spreader nodes. The ranked spreader edges in Fig. 2 are highlighted in Table 6.