# Fibroblast-Specific Protein Interactions for Myocardial Fibrosis from MetaCore Network
## 1 Introduction
We note that the Google matrix algorithms [17; 18; 19] find a variety of useful applications in modern complex networks [20] including World Wide Web, Wikipedia, world trade etc. In our opinion an important advantage of the MetaCore network is that it presents a global network structure of PPI with about 40,000 proteins and important molecules. This allows to perform a mathematical modeling of fibrosis progression over the whole network. Thus in this work we present such a modeling which assumes that at an initial stage there is a certain number of proteins which activate a fibrosis progression of other proteins via their network links to other proteins. These initial proteins are always marked as red network nodes and assumed to be always red permanently generating fibrosis progression (thus being always red or with fixed Ising spin up with \(\sigma_{i}=1\)). Among the other proteins, we assume that there is an initial group of proteins which inhibit fibrosis progression being of fixed blue color (thus being always blue or with Ising spin down with \(\sigma_{i}=-1\)). The remaining other proteins (or network nodes) are assumed to be in neutral state of white color or spin being zero (\(\sigma_{i}=0\)). The fibrosis progression is modeled by the asynchronous Monte Carlo process when at each step a spin of a given node (which is not fixed) is determined as the total spin sign of other nodes linked to it with certain weights of Markov chain transitions. In a certain sense this rule corresponds to a situation when a person takes an average opinion of his network friends linked to him. After many of such Monte Carlo steps the system converges to a steady-state when all network spins obtain a fixed polarization being up (red or fibrosis activated) or down (blue or healthy protected) and some of the nodes may remain at zero neutral spin state (white) being practically disconnected from the red or blue network nodes. We note that the above Monte Carlo steps describe a process of opinion formation and the global vote of nodes (society members) between red and blue options. For complex social networks [20] there are numerous studies of opinion formation, reviewed in [21]. Various interesting features of voter models and opinion formation on networks had been obtained and described in [21; 22; 23; 24; 25; 26; 27]. Recently it was argued that the opinion formation on the world trade network can be linked with country preference to trade in one or another currency (e.g., US dollar or hypothetical BRICS currency) [28]. The important new element appeared in [28] is that opinion of certain network nodes (countries) is considered to be fixed since it is assumed that they prefer to trade always with fixed currency of USD or BRICS. A further development of this approach with nodes of fixed opinion is done in [29] for Wikipedia networks when initially there are two groups of nodes of fixed red and blue opinions but all other nodes have neutral state (white opinion or spin zero) and their fixed opinion emerges in the result of asynchronous Monte Carlo process. This process is viewed as the Ising Network Opinion Formation (INOF) model. In fact this INOF model is directly suitable for the modeling of fibrosis progression in the PPI MetaCore network and we are using it here with certain slight modifications described in the next Section. We call this modified system as the Ising Network Fibrosis (INFI) model. It should be pointed out that a somewhat similar asynchronous Monte Carlo process is used in models of associative memory (see e.g., [30; 31]). In this work, using the results reported in [9; 16], we take in the INFI model an initial state of 10 fixed red nodes that produce positive activation response of fibrosis on growth factor-\(\beta\) (TGF-\(\beta\)) considering them as fixed 10 red protein nodes. We also consider 6 (or 10, 14) fixed blue protein nodes with negative response (see details in next Section).