Hierarchical organization in sparse gene regulatory networks shapes structural coherence and emergent regulatory coordination

How large-scale regulatory coordination in biological systems emerges from local signed and directed interactions in sparse gene regulatory networks (GRNs) remains an unanswered fundamental question. We introduce the coherence matrix, a topology-based framework that captures the consistency of regulatory influence between gene pairs by integrating information across all direct and indirect paths. Analysis of synthetic networks reveals that structural coherence - a metric derived from the coherence matrix - dictates global coordination: while highly coherent motifs maintain aligned regulatory coordination across widely varying network sparsity values, motifs with low coherence allow such coordination only at biologically unrealistic sparsity values. Our investigation of six whole-organism GRNs and further analysis of synthetic networks highlighted that hierarchical organization in GRNs a dense middle layer enriched in feedback loops that mediates coordination between input and output layers - serves as a structural buffer to allow regulatory coordination even for sparse networks. Finally, comparison with Escherichia coli transcriptomic modules further shows that the coherence matrix accurately predicts the sign of coordinated gene contribution, emphasizing its biological application, while also serving as a unifying descriptor integrating local interactions and global network architecture to explain the emergent regulatory coordination.