Dynamical Hierarchy and Modularity in Gene Regulatory Networks

Modularity and hierarchy are two essential traits of biological organization. They pervade the logic of cellular computations, adaptive responses to changing environments and evolvability. However, no general agreement exists on how to properly measure them. Here, we provide a well grounded theoretical definition of dynamical hierarchy and modularity. This is possible through the identification of the dynamical backbone (DB), the minimal subgraph that contains all the dynamically essential components of any gene regulatory network. Our methodology is based on the most elementary trait behind any dynamic behavior: the principle of causality. In gene regulatory networks this principle is captured by the regulatory control of transcription factors on their target genes. As case studies, we analyzed the structure of the DB in both yeast and E. coli gene transcriptional regulatory networks. Although these webs display similar global topological patterns, their DBs exhibit dramatically different architectures. A marked top-down hierarchy is present in the E. coli net, whereas the yeast network displays a bow-tie structure. Several modules are identified in both systems, although their number and position within the DB is markedly different, suggesting two different forms of logic organization. Our method allows to unambiguously define the core dynamical modules and their hierarchical organization without the use of any tunable parameter and it can be applied to any arbitrary directed graph of causal dependencies.