# Fibroblast-Specific Protein Interactions for Myocardial Fibrosis from MetaCore Network
## 3 Results  
### Numerical Results  
All results presented in this work were obtained with maximal iteration time \(\tau_{max}=100\) and \(R=100\),000 realizations for (potential) different initial conditions and random permutations for the Monte Carlo procedure. Also when we describe a particular initial condition it is implicitly understood that the permanently fixed 10 red and \(n_{b}\) blue nodes given above are indeed fixed to these values from the very beginning, i.e., if we say \"that there are \(n_{ib}\) initial blue values\" (with \(n_{ib}=0,1,\ldots\)) we mean that there are \(n_{ib}\) blue nodes _on the set of \(N_{v}\) variable (non-fixed) nodes_, other variable nodes are initialized to white values and the fixed nodes have still their 10 red and \(n_{b}\) blue values (in particular the value of \(n_{ib}\) does not include the number of \(n_{b}\) fixed blue nodes). In Figure 1, we show for the case of white or zero initial spins (for the \(N_{v}\) variable nodes; i.e., \(n_{ib}=0\)) and the three cases of \(n_{b}=6,10,14\) initial blue fixed nodes (and 10 fixed nodes as explained above) the probability distribution (normalized by an integral) of fraction of red outcomes \(f_{r}(i)\) for the nodes obtained by a histogram with bin width \(\Delta f_{r}=0.01\). These distributions are strongly peaked at values close to \(f_{r,peak}\approx 0.93\) (\(n_{b}=6\)), \(f_{r,peak}\approx 0.81\) (\(n_{b}=10\)), \(f_{r,peak}\approx 0.65\) (\(n_{b}=14\)) for roughly 82% of nodes. The secondary peak at \(f_{r}=0\) corresponds to the fraction (number) \(\approx 0.18\) (7230) of nodes (same value for the three cases of \(n_{b}\)) which stay white after 100 iterations for all \(R=100\),000 pathway realizations. This set of stable white nodes correspond to nodes not connected to the small number of fixed red and blue nodes. However, despite the small value of only 10 initial fixed red notes the majority of the other nodes has a red outcome, especially for \(n_{b}=6\) and with slightly reduced values for \(n_{b}=10,14\). Note that the bin width \(\Delta f_{r}=0.01\) in Figure 1 is still the rather large and histogram computations with \(\Delta f_{r}=0.001\) and \(\Delta f_{r}=0.0001\) show that the actual peaks of the distributions are much sharper as visible in Figure 1.  
Figure 1: Probability density \(p(f_{r})\) of \(f_{r}\) for \(n_{b}=6,10,14\) obtained from a histogram with bin width \(\Delta f_{r}=0.01\) using \(N_{v}=N-10-n_{b}\) data points \(f_{r}(i)\) corresponding to the number of variable nodes. For each node \(i\) the value \(f_{r}(i)\) is obtained as the fraction of red outcome (\(\sigma_{i}=+1\)) of \(R=100\),000 pathway realizations for this node using the initial condition with no initial (variable) blue nodes, i.e., \(n_{ib}=0\). The distributions are normalized by \(\int_{0}^{1}p(f_{r})\,df_{r}\) and there are strongly peaked at values close to \(f_{r,peak}\approx 0.93\) (\(n_{b}=6\)), \(f_{r,peak}\approx 0.81\) (\(n_{b}=10\)), \(f_{r,peak}\approx 0.65\) (\(n_{b}=14\)) for roughly 82% of nodes. The secondary peak at \(f_{r}=0\) corresponds to the fraction \(\approx 0.180\) of nodes (same value for the three values of \(n_{b}\)) which stay white after 100 iterations for all \(R=100\),000 pathway realizations. It seems that with the white initial condition (except for the fixed nodes) nearly all nodes (except those in the stable white set) have a large probability for red outcome. Therefore, we try to reduce this red outcome by choosing a certain number \(n_{ib}\) of initial blue (variable) nodes and other nodes with white initial values. The question is also where to place these blue nodes. In a first model we choose the \(n_{ib}\) blue nodes at different random positions (for each of the \(R=100\),\(000\) pathway realizations) on the full set of variable nodes giving a set of \(N_{r}=N_{v}\) of possible blue initial nodes with possible values \(n_{ib}=0,\ldots,N_{r}\) and a corresponding fraction \(f_{ib}=n_{ib}/N_{r}\).