# Fibrosis Protein-Protein Interactions from Google Matrix Analysis of MetaCore Network
## 3 Results
### Bifunctionality of Fibrosis Network  
Here, we present in short certain results for the bifunctional MetaCore network. The doubled Ising MetaCore network has \(N_{I}=80,\!158\) nodes and \(N_{I,\ell}=939,\!808\) links. We compute the reduced Google matrix \(G_{\mathrm{R}}\) for the doubled number of nodes \(2\times 54=108\) (by attributing \((+)\) and \((-)\) labels to each node) for the fibrosis proteins of Table 1. Here, we present only some selected characteristics; all data for the Ising Google matrix are available at [27]. In Figure 7, we show the magnetization \(M(j)=(P_{+}(j)-P_{-}(j))/(P_{+}(j)+P_{-}(j))\) of proteins of Table 1 with their location on the PageRank-CheitRank plane \((K,K^{*})\). Remember that \(P_{\pm}(j)\) is the PageRank value of the node \(j\) with label \((\pm)\) and that the sum satisfies \(P(j)=P_{+}(j)+P_{-}(j)\) where \(P(j)\) is the PageRank value of the node \(j\) of the simple network. The magnetization is positive for nodes which are more likely to be activated, or in other words, which have on average more incoming activation links (and/or coming from other nodes with larger PageRank values) than inhibition links, while negative values correspond to nodes being more likely to be inhibited by other nodes. According to Figure 7, the majority of proteins have values of \(M\) being close to zero (neutral action on average coming from other nodes), but there are also some nodes \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Rank** & \(K_{g}(b)\) & \(K_{t,K}(b)\) & \(K\) &
\begin{tabular}{c} **Protein** \\ **(b)** \\ \end{tabular} & \(D_{s}^{\left(\mathrm{u/d}\right)}(b)\) & \(P_{\mathrm{r}}(b)\) \\ \hline
1 & 45 & \(K_{x}=1\) & 1 & \(\beta\)-catenin & 6.809993 & 0.175768 \\
2 & 46 & \(K_{x}=2\) & 2 & p53 & 6.789229 & 0.171249 \\
3 & 47 & \(K_{x}=3\) & 3 & ESR1 & 4.513399 & 0.113285 \\
4 & 48 & \(K_{x}=4\) & 4 & STAT3 & 4.109638 & 0.104088 \\
5 & 49 & \(K_{x}=5\) & 5 & Re1A & 3.343309 & 0.085443 \\
6 & 50 & \(K_{x}=6\) & 6 & PPAR-\(\gamma\) & 3.086237 & 0.070668 \\
7 & 51 & \(K_{x}=7\) & 7 & IKK-\(\beta\) & 1.696330 & 0.043249 \\
8 & 52 & \(K_{x}=8\) & 8 & SNAIL1 & 1.477019 & 0.034269 \\
9 & 53 & \(K_{x}=9\) & 10 & MMP-14 & 1.368302 & 0.029121 \\
10 & 2 & \(K_{t}=2\) & 9 & TGF-\(\beta\) 1 & 1.081828 & 0.029166 \\
11 & 54 & \(K_{x}=10\) & 12 & Flotillin-1 & 0.787569 & 0.016863 \\
12 & 3 & \(K_{t}=3\) & 13 & TGF-\(\beta\) 2 & 0.333981 & 0.012451 \\
13 & 4 & \(K_{t}=4\) & 20 & TGF-\(\beta\) 3 & 0.159633 & 0.004157 \\
14 & 1 & \(K_{t}=1\) & 30 & TGF-\(\beta\) 0 & 0.081261 & 0.002090 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of the sum \(D_{s}^{\left(\mathrm{u/d}\right)}(b)=\sum_{4=5}^{44}D_{ab}\) (i.e., the \(a\)-sum is over up- and down-proteins) for \(b\) belonging to the TGF-\(\beta\) or the X-proteins subgroups. The list is ordered with respect to decreasing \(D_{s}^{\left(\mathrm{u/d}\right)}(b)\) values with the first column giving the corresponding ranking index; the second and third columns giving the \(K_{g},K_{t,x}\) indexes; the fourth and fifth columns containing the local PageRank index \(K\) and the name of the protein \(b\); and the sixth and seventh columns giving the values of \(D_{s}^{\left(\mathrm{u/d}\right)}(b)\) and the local PageRank probability \(P_{\mathrm{r}}(b)\). Both \(K\) and \(P_{\mathrm{r}}(b)\) correspond to the group of 54 fibrosis proteins of Table 1.  
with with significant positive values such as RAPGEF4 (at \(K=18,K^{*}=17,K_{g}=20\), \(K_{u}=16,M=0.690937\)) corresponding to the only red box (maximum value of 1 in units of the color bar) and HMGCS2 (at \(K=23,K^{*}=27,K_{g}=35,K_{d}=11,M=0.550286\)) with an orange-brown box (value of 0.8 in units of the color bar). There are about a further nine proteins with various degrees of green color (\(M\) values between 0.2 and 0.4 corresponding to 0.3 to 0.6 in units of the color bar).