  Directed protein networks with only a few thousand of nodes are rather
complex and do not allow to extract easily the effective influence of one
protein to another taking into account all indirect pathways via the global
network. Furthermore, the different types of activation and inhibition actions
between proteins provide a considerable challenge in the frame work of network
analysis. At the same time these protein interactions are of crucial importance
and at the heart of cellular functioning. We develop the Google matrix analysis
of the protein-protein network from the open public database SIGNOR. The
developed approach takes into account the bi-functional activation or
inhibition nature of interactions between each pair of proteins describing it
in the frame work of Ising-spin matrix transitions. We also apply a recently
developed linear response theory for the Google matrix which highlights a
pathway of proteins whose PageRank probabilities are most sensitive with
respect to two proteins selected for the analysis. This group of proteins is
analyzed by the reduced Google matrix algorithm which allows to determine the
effective interactions between them due to direct and indirect pathways in the
global network. We show that the dominating activation or inhibition function
of each protein can be characterized by its magnetization. The results of this
Google matrix analysis are presented for three examples of selected pairs of
proteins. The developed methods work rapidly and efficiently even for networks
with several million of nodes and can be applied to various biological
networks.
