DIRECTED GRAPH THEORY FOR THE ANALYSIS OF BIOLOGICAL REGULATORY NETWORKS

MARTHA TAKANE^1*SAUL BERNAL-GONZÁLEZ¹JESÚS MAURO-MORENO¹GUSTAVO GARCÍA-LÓPEZ¹Francisco F. De-Miguel^2*

• ¹Universidad Nacional Autonoma de Mexico Instituto de Matematicas, Mexico City, Mexico
• ²Instituto de Fisiologia Celular-Neurociencias, National Autonomous University of Mexico, México City, Mexico

ABSTRACT. 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 of the biological system." 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 the transition from a regulatory network to its logical digraph and vice versa. Furthermore, we demonstrate that the spectral properties of the so-called state matrix provide mathematical evidence explaining why the elements within the attractors and limit cycles carry information about the dynamics of the biological system. 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.