# Fibroblast-Specific Protein Interactions for Myocardial Fibrosis from MetaCore Network
## 2 Data Sets and INFI Model Description
### Ising Spin Network, Monte Carlo Process for INFI Model  
In this work, we consider an Ising type of model using the symmetrized matrix \(\tilde{S}+\tilde{S}^{T}\) to model the effective spin interactions. We denote by \(\sigma_{i}\) the spin associated to the node (protein) \(i\) but here we allow for three different values being \(\sigma_{i}=+1\) (\"red state\", \"fibrosis state\"), \(\sigma_{i}=-1\) (\"blue state\", \"healthy state\") and also \(\sigma=0\) (\"white state\", \"undetermined or neutral state\"). We study this model using a specific Monte Carlo process similar to [29]. The white state will only be used in the initial condition and is supposed to be only temporary. First, we fix \(10\) specific nodes permanently to the red state, those with \(K_{g}=1,\ldots,K_{g}=10\) in the set of Table 1 corresponding to the four TGF proteins and the first \(6\)\(K_{u}\)-proteins with strongest positive response on the TGF stimulation [16]. We also fix \(n_{b}\) nodes permanently to the blue state being the nodes with \(K_{g}=25,\ldots,K_{g}=24+n_{b}\) corresponding to the first \(n_{b}\)\(K_{d}\)-proteins with strongest negative TGF-response. Here we choose mostly \(n_{b}=6\) (same number as the 6 \(K_{u}\) proteins with fixed red state) but we also present some data for the cases \(n_{g}=10\) (same total number of fixed red proteins) and \(n_{g}=14\). Therefore \(10+n_{b}\) nodes have already a permanently fixed spin value \(\sigma_{i}=\pm 1\). We implement a Monte Carlo procedure that modifies the other \(N_{v}=N-10-n_{b}\) remaining nodes with \"variable\" spin values (non fixed). For this at given iteration time \(\tau\), we update the \(N_{v}\) variable nodes by computing the sum over all nodes \(j\) of the network (fixed and variable):  
\[Z_{i}=\sum_{j(\
eq i)}(\tilde{S}_{ij}+\tilde{S}_{ji})\sigma_{j} \tag{2}\]  
and choosing \(\sigma_{i}(\tau+1)=+1\) if \(Z_{i}>0\), \(\sigma_{i}(\tau+1)=-1\) if \(Z_{i}<0\) and \(\sigma_{i}(\tau+1)=\sigma_{i}(\tau)\) (unchanged) if \(Z_{i}=0\). Without going into technical details, we mention that the numerical implementation of (2) exploits of course the sparse structure of the matrix \(\tilde{S}_{ij}\). This update is done for every node \(i\) in the set of \(N_{v}\) variable nodes in a random order using a random permutation (thus no repetitions) which is chosen at the beginning of the iteration process. Once a spin value for a node \(i\) has been updated, the new value \(\sigma_{i}(\tau+1)\) will be used for the update of the subsequent nodes \(\tilde{i}\). We note that somewhat similar asynchronous Monte Carlo process what used for bio-networks in [32; 33] but there the matrix elements \(\tilde{S}_{ij}+\tilde{S}_{ji}\) had values \(\pm 1\) and network sizes were relatively small (about 100). This full update procedure for one time step and the full set of \(N_{v}\) variable nodes is repeated up to the iteration time \(\tau_{\max}=100\) (with different random permutations for the update order at each time step). We also verify if at any given value of \(\tau\) there is ideal convergence where the spins no longer change. (This happens typically at \(\tau_{last}\approx 10\); see below for details). The full iteration procedure with \(\tau_{\max}=100\) time steps is repeated \(R=100\),\(000\) times with different random permutations and eventually also different random positions of initial blue or white nodes (depending on the precise type of initial condition). During this procedure, we compute the fraction of red outcome \(f_{r}(i)\) for each node as the number of times the node \(i\) has red state at \(\tau_{\max}\) (at different realizations) divided over \(R\) and also the overall average \(f_{r}=\sum_{i}f_{r}(i)/N_{v}\) over the nearly full network of variable nodes. To test the convergence, we also compute the overall average at intermediate values of \(\tau\). In order to test the procedure and also in the direct influence of the permanently fixed nodes, we first choose an initial condition where all \(N_{v}\) variable nodes are initialized to the white state.