Topological Constraints on Noise Propagation in Gene Regulatory Networks

Gene expression, the production of protein from DNA and mRNA in the biological cell, is inherently stochastic. Cells with identical DNA exhibit fluctuations or 'noise' in gene expression. This noise propagates over gene regulatory networks (GRNs), which encode gene-gene interactions. The propagated 'extrinsic' noise interacts and combines with 'intrinsic' noise to affect biological decisions. Consequently, it is essential to understand how GRN topology affects total noise. Recently, uncertainty principles were established for noise propagation over GRN. In particular, in ring GRNs, exactly one node can have noise reduction below the intrinsic limit. We establish necessary and sufficient conditions for noise reduction in ring GRN. Specifically, for two- and three-node rings, an odd number of negative regulations is necessary for noise reduction. Further, sufficiency is ensured if sensitivities to input for feedforward and feedback regulations are bounded from below and above, respectively. These constraints are valid even if the ring GRN are regulated by an upstream gene. Finally, we use graph theory to decompose noise propagation in a general directed network over its strongly connected components. The combination of graph theory and stochastic processes may be a general framework for studying noise propagation.