The complex dynamics of gene expression in living cells can be well-approximated using Boolean networks. The average sensitivity is a natural measure of stability in these systems: values below one indicate typically stable dynamics associated with an ordered phase, whereas values above one indicate chaotic dynamics. This yields a theoretically motivated adaptive advantage to being near the critical value of one, at the boundary between order and chaos. Here, we measure average sensitivity for 66 publicly available Boolean network models describing the function of gene regulatory circuits across diverse living processes. We find the average sensitivity values for these networks are clustered around unity, indicating they are near critical. In many types of random networks, mean connectivity <K> and the average activity bias of the logic functions <p> have been found to be the most important network properties in determining average sensitivity, and by extension a network’s criticality. Surprisingly, many of these gene regulatory networks achieve the near-critical state with <K> and <p> far from that predicted for critical systems: randomized networks sharing the local causal structure and local logic of biological networks better reproduce their critical behavior than controlling for macroscale properties such as <K> and <p> alone. This suggests the local properties of genes interacting within regulatory networks are selected to collectively be near-critical, and this non-local property of gene regulatory network dynamics cannot be predicted using the density of interactions alone.
Logic and connectivity jointly determine criticality in biological gene regulatory networks
Bryan C. Daniels, Hyunju Kim, Douglas Moore, Siyu Zhou, Harrison Smith, Bradley Karas, Stuart A. Kauffman, Sara I. Walker