# Fibrosis Protein-Protein Interactions from Google Matrix Analysis of MetaCore Network
## 3 Results
### Sensitivity of Fibrosis Proteins  
In addition to the matrix components \(G_{\rm R},G_{\rm pr},G_{\rm rr},G_{\rm qr}\) and the network diagrams (of \(G_{\rm R}\) and \(G_{\rm rr}+G_{\rm qr}^{\rm(nd)}\)), it is also important to analyze the sensitivity matrix \(D_{ab}\) defined previously in Equation (7). This matrix \(D_{ab}\) gives the sensitivity of a protein \(a\) with respect to a small variation of the transition matrix element of \(G_{\rm R}\) from protein \(b\) to \(a\) on the basis of logarithmic derivative of the PageRank probability (see Section 2.5 and also Appendix A for more technical details on this). As described previously (see Section 2.6), we first compute the sensitivity matrix \(D_{ab}^{(166)}\) associated with \(G_{\rm R}^{(166)}\) being the reduced Google matrix for a larger intermediary subset containing the 44 TGF-\(\beta\), up- and down-proteins and further 122 external proteins having direct links (of the full MetaCore network) to the first five up- (\(K_{u}=1,\ldots,5\)) and the first five down-proteins (\(K_{d}=1,\ldots,5\)). This matrix is shown in Appendix A Figure A3. Then, from the set of 122 external proteins, we select the 10 proteins \(b\) with the largest effective sensitivity given by the sum \(D_{s}^{(5+5)}(b)=\sum_{a=5}^{9}D_{ab}^{(166)}+\sum_{a=25}^{29}D_{ab}^{(166)}\) (see Section 2.6) which form the group of 10 X-proteins. The 44 TGF-\(\beta\), up- and down-proteins, together with these 10 X-proteins, form our main group of 54 proteins given Table 1 and for which we present results of the reduced Google matrix in the last subsections. The sensitivity matrix \(D_{ab}\) of size \(54\times 54\) for this main group is shown in Figure 5 with zoomed parts visible in Figure 6.  
Figure 5: Color density plot of the sensitivity matrix \(D_{ab}\) of fibrosis proteins of Table 1; the axes and colors are defined as in Figure 2 (without saturation); the strongest top 40 sensitivity values are given in Table 2.  
Figure 6: Zoomed parts of sensitivity matrix \(D_{ab}\) of Figure 5. Both panels show a selected subregion of Figure 5 with the index \(a\) (vertical axis from top to down) belonging to the set of up-nodes (\(a=5,\ldots,24\) in panel (**a**)) or down-nodes (\(a=25,\ldots,44\) in panel (**b**)) and the index \(b\) (horizontal axis from left to right) corresponds to both panels to the four nodes of the TGF-\(\beta\) subgroup (\(b=K_{t}=1,\ldots 4\) for four left columns in each panel) and the 10 nodes of the \(X\)-proteins (\(b=45,\ldots 54\) or \(K_{x}=1,\ldots,10\) for 10 right columns in each panel). The list of all 560 sensitivity matrix values \(D_{ab}\) with \(a\) belonging to the subgroups of up- or down-proteins and \(b\) belonging to the subgroups of TGF-\(\beta\) and X-proteins is available at [27]. The strongest 40 \(D_{ab}\) values of this list are shown in Table 2. Among the top three pairs, we find that the protein MMP-14 gives the top sensitivity (influence) on the protein CLEC3B (\(D_{ab}=0.263109\)), next is the protein p53 giving the sensitivity (\(D_{ab}=0.259298\)) on the protein GALNT3, and the third place is for the sensitivity of C1 QTNF3 from PPAR-\(\gamma\) (\(D_{ab}=0.225877\)).