Properties of Boolean dynamics by node classification using feedback loops in a network

Biological networks keep their functions robust against perturbations. Many previous studies through simulations or experiments have shown that feedback loop (FBL) structures play an important role in controlling the network robustness without fully explaining how they do it. Hence, there is a pressing need to more rigorously analyze the influence of FBL structures on network robustness.

In this paper, I propose a novel node classification notion based on the FBL structures involved. More specifically, I classify a node as a no-FBL-in-upstream (NFU) or no-FBL-in-downstream (NFD) node if no feedback loop is involved with any upstream or downstream path of the node, respectively. Based on those definitions, I first prove that every NFU node is eventually frozen in Boolean dynamics. Thus, NFU nodes converge to a fixed value determined by the upstream source nodes. Second, I prove that a network is robust against an arbitrary state perturbation subject to a non-source NFD node. This implies that a network state eventually sustains the attractor despite a perturbation subject to a non-source NFD node. Inspired by this result, I further propose a perturbation-sustainable probability that indicates how likely a perturbation effect is to be sustained through propagations. I show that genes with a high perturbation-sustainable probability are likely to be essential, disease, and drug-target genes in large human signaling networks.

Taken together, these results will promote understanding of the effects of FBL on network robustness in a more rigorous manner.