# Fibrosis Protein-Protein Interactions from Google Matrix Analysis of MetaCore Network
## 1 Introduction
For the analysis of PPI characterizing fibrosis, we added a group of 10 external proteins with indexes \(K_{x}=1,2,\ldots,10\). The choice of these 10 proteins is explained below in detail, but in short, these external proteins are those which affect, according to our network analysis, the internal proteins in the strongest manner. Thus, in total we have the PPI fibrosis network with 54 proteins (nodes). They are ordered by their global index \(K_{g}=1,2,\ldots,54\) in Table 1 (first 4 \(K_{t}\), then 20 \(K_{u}\), 20 \(K_{d}\) and 10 \(K_{x}\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \(K_{g}\) & \(K_{t,u,d,x}\) & \(K\) & \(K^{*}\) & \(K_{\mathbf{M}}\) & \(K^{*}_{\mathbf{M}}\) & **Protein** \\ \hline  
& RAPGEF4 \\ \hline \end{tabular}
\end{table} 
Table 1: Table of the subset of \(N_{r}=54\) selected fibrosis proteins (nodes). Here, \(K_{g}\) represents the global index of this group, \(K_{t,u,d,x}\) represent the index of the four subgroups of 4 TFG-\(\beta\) proteins, 20 up-proteins, 20 down-proteins and 10 additional X-proteins; \(K\) (\(K^{*}\)) represents the local PageRank (CheiRank) index obtained from the reduced Google matrix \(G_{\text{R}}\) (\(G_{\text{R}}^{*}\)) for this group of 54 proteins; \(K_{\text{M}}\) (\(K^{*}_{\mathbf{M}}\)) indexes represent the PageRank (CheiRank) index for the global MetaCore network of \(N=40\),079 nodes; the last column gives the associated protein names. To analyze the properties of this PPI fibrosis network, we use the developed commercial MetaCore network database of Clarivate [8]. This network database has been shown to be useful for analysis of various specific biological problems (see, e.g., [9; 10]). At present, the MetaCore network has \(N=40,\)079 nodes with \(N_{\ell}=292,\)191 links (without self-connections) with on average \(n_{\ell}=N_{\ell}/N\approx 7.3\) links per node [11]. The nodes are given mainly by proteins but there are also certain molecules and molecular clusters catalyzing the interactions with proteins. This MetaCore PPI network is directed and nonweighted. In addition, its network links mark the bifunctional nature of interactions leading to the activation or the inhibition of one protein by another one. For some nodes, link action is neutral or unknown. Thus, overall, the MetaCore network is a network with activation or inhibition directed links showing that a protein A acts on protein B. We note that this network is based on a detailed analysis of world literature describing experimental results of how one protein acts on another one. The construction of this network has been performed during several years and is now continued at Clarivate [8]. Scientific biological results obtained with this MetaCore network can, for example, be found at [9; 10]. This MetaCore network represents a commercial product actively used by the world's leading pharmaceutic companies [8]. We note that at present, new types of computational methods are actively being developed, e.g., using DeepMind methods [12], with new possibilities of predicting new structures and interactions between proteins. Such methods appear to be very promising. Indeed, they can add new interaction links between proteins in the MetaCore network. However, the creation of such a global PPI network as MetaCore with almost all proteins requires long work of gathering all available interactions between proteins and representing these interactions in a format of directed network which is very useful for scientific analysis of multiple PPI. We note that there are also other types of PPI networks developed by other companies and research groups (e.g., TRANSPATH [13], REACTOME [14]). Here, we present a universal mathematical analysis based on Google matrix methods which can be also applied to other PPI networks, such as [13; 14]. However, here, we present the analysis only for the MetaCore network available to us.