# Fibrosis Protein-Protein Interactions from Google Matrix Analysis of MetaCore Network
## 3 Results
### Sensitivity of Fibrosis Proteins
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(K_{s}\) & \(K_{g}(a)\) & \(K_{u,d}(a)\) & \(\mathbf{Protein}(a)\) & \(K_{g}(b)\) & \(K_{tx}(b)\) & \(\mathbf{Protein}(b)\) & \(D_{ab}\) \\ \hline
1 & 25 & \(K_{d}=1\) & CLEC3B & 53 & \(K_{x}=9\) & MMP-14 & 0.263109 \\
2 & 22 & \(K_{u}=18\) & GALNT3 & 46 & \(K_{x}=2\) & p53 & 0.259298 \\
3 & 13 & \(K_{u}=9\) & C1 QTNF3 & 50 & \(K_{x}=6\) & PPAR-\(\gamma\) & 0.225877 \\
4 & 26 & \(K_{d}=2\) & SCAR5 & 52 & \(K_{x}=8\) & SNALL1 & 0.219938 \\
5 & 27 & \(K_{d}=3\) & SLC10A6 & 50 & \(K_{x}=6\) & PPAR-\(\gamma\) & 0.214345 \\
6 & 29 & \(K_{d}=5\) & MYOC & 54 & \(K_{x}=10\) & Flotillin- & 0.200157 \\
7 & 19 & \(K_{u}=15\) & LAMA1 & 46 & \(K_{x}=2\) & p53 & 0.199892 \\
8 & 43 & \(K_{d}=19\) & FMO2 & 47 & \(K_{x}=3\) & ESR1 & 0.196550 \\
9 & 29 & \(K_{d}=5\) & MYOC & 45 & \(K_{x}=1\) & \(\beta\)-catenin & 0.196394 \\
10 & 39 & \(K_{d}=15\) & GFRA1 & 45 & \(K_{x}=1\) & \(\beta\)-catenin & 0.184019 \\
11 & 6 & \(K_{u}=2\) & FGF21 & 46 & \(K_{x}=2\) & p53 & 0.182339 \\
12 & 20 & \(K_{u}=16\) & RAPGEF4 & 45 & \(K_{x}=1\) & \(\beta\)-catenin & 0.182303 \\
13 & 28 & \(K_{d}=4\) & CXCL5 & 46 & \(K_{x}=2\) & p53 & 0.181444 \\  
}=1\) & ADAMTS16 & 46 & \(K_{x}=2\) & p53 & 0.173822 \\
40 & 14 & \(K_{u}=10\) & ANO4 & 46 & \(K_{x}=2\) & p53 & 0.173770 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of 40 top protein pairs \((a,b)\) with strongest sensitivity matrix element \(D_{ab}\), with \(a\) belonging to the subgroups of up- or down-proteins and \(b\) belonging to the subgroups of TGF-\(\beta\) and X-proteins. The first column gives the ranking index \(K_{s}\) of \(D_{ab}\) matrix elements ordered by a decreasing value, the second to fourth columns provide the \(K_{g},K_{u,d}\) indexes and the name of the protein \((a)\), the fifth to seventh columns provide the \(K_{g},K_{t,x}\) indexes and the name of the protein \((b)\), and the eighth column shows the value of \(D_{ab}\). See also Figure 5, which shows a color density plot for all matrix elements \(D_{ab}\), and Table 1 for the list of considered proteins. An ordered list of all 560 values of sensitivity influence values \(D_{ab}\) of TGF-\(\beta\) or X-proteins (for “\(b\)”) on up-/down proteins (for “\(a\)”) is available at [27]. We mention that the appearance of MMP-14 (\(K_{x}=9\)) at the top position of Table 2 is the reason why we selected this protein as one of the five top nodes in the net diagrams discussed in the last subsection. For the net diagrams shown in Figure 4, the other four top nodes were simply chosen as the first two up- (\(K_{u}=1,2\)) and down-proteins (\(K_{d}=1,2\)). However, for the net diagrams shown in Appendix A Figure A2, the two top up- and down-nodes were also chosen by the criterion of top positions in Table 2 resulting in \(K_{u}=9,18\) and \(K_{d}=1,2\). We also computed the effective TGF-\(\beta\) sensitivity on up- or down-proteins (noted \(a\)) defined by the sum \(D_{s}^{\rm(TGF-\beta)}(a)=\sum_{b=1}^{4}D_{ab}\). Ordering these values in decreasing order, we obtain the ranking index \(K_{s}^{\rm(TGF-\beta)}=1,\ldots,40\) whose dependence on \(K_{u}\) and \(K_{d}\) is visible in Appendix A Figure A4. We see that for the up-proteins we have 14 ranking values located at \(K_{s}^{\rm(TGF-\beta)}\leq 20\) and for the down-proteins only 6 values at \(K_{s}^{\rm(TGF-\beta)}\leq 20\) (with 3 values at \(K_{s}^{\rm(TGF-\beta)}=18,19,20\)). This shows that the overall influence of TGF-\(\beta\) proteins is somewhat stronger on the up-proteins, compared to the down-proteins. However, we mention that the different values of \(D_{s}^{\rm(TGF-\beta)}(a)\) used to determine this ranking have only modest size variations in the interval 0.0250 to 0.0465 with most values between 0.040 and 0.043. Furthermore, overall, the external X-proteins have a much higher influence (on up- and down-proteins) than the TGF-\(\beta\) proteins.