DIRECTED GRAPH THEORY FOR THE ANALYSIS OF BIOLOGICAL REGULATORY NETWORKS

Abstract

ABSTRACTRegulated biological networks are commonly represented as logical diagrams, in which the exact interactions between the elements remain out of sight. Here we propose a new type of excitation-inhibition graph based on Boolean logic, which we name “logical directed graph or simply, logical digraph of the biological system”. Such logical digraph allows the representation of every possible regulatory interaction among elements, based on Boolean interactions. The logical digraph contains information about the connectivity, dynamics, limit cycles, and attractors of the network. As proof of the application, the logical digraph was applied to analyze the functioning of the well-known neural network that produces oscillatory swimming in the mollusk Tritonia. Our method permits to transit from a regulatory network to its logical digraph and vice versa. In addition, we show that the spectral properties of the so-called state matrix provide mathematical evidence about why the elements in the attractors and limit cycles contain information about the dynamics of the biological system. Open software routines are provided for the calculations of the components of the network and the attractors and limit cycles. This approach offers new possibilities to visualize and analyze regulatory networks in biology.