Directed graph theory for the analysis of biological regulatory networks

Synchronous regulated biological networks are often represented as logical diagrams, where the precise interactions between elements remain obscured. Here, we introduce a novel type of excitation-inhibition graph based on Boolean logic, which we term “logical directed graph” or simply, “logical digraph.” Such a logical digraph facilitates the representation of every conceivable regulatory interaction among elements, grounded in Boolean interactions. The logical digraph includes information about connectivity, dynamics, limit cycles, and attractors of the network. As proof of application, we utilized the logical digraph to analyze the operations of the well-known neural network that produces oscillatory swimming in the mollusk Tritonia. Our method enables a seamless transition between a regulatory network and its corresponding logical digraph, and vice versa. Additionally, we demonstrate that the spectral properties of the so-called state matrix provide mathematical evidence explaining why the elements within attractors and limit cycles contain information about the dynamics of the biological system. More specifically, the non-zero entries of the Perron-Frobenius eigenvector of the state matrix indicate the attractors and limit cycles of the network. We demonstrate that each connected component of the regulatory network has exactly one attractor or limit cycle. Open software routines are available for calculating the components of the network, as well as the attractors and limit cycles. This approach opens new possibilities for visualizing and analyzing regulatory networks in biology.