Feedback-mediated circulation and persistence of stochastic fluctuations in gene regulatory circuits

Feedback plays a significant role in biochemical networks that govern a multitude of cellular functions, including development, adaptation, and homeostasis. Yet, how feedback topology controls stochastic fluctuations remains incompletely understood. Here, we develop a theoretical framework for two-node feedback motifs composed of activating and repressive regulatory interactions between two transcription factors. Under the linear noise approximation, we identify a feedback-driven contribution to node-wise fluctuations, termed cyclic noise, that arises specifically from loop closure. Cyclic noise is the component of fluctuations that circulates through the regulatory circuit. Its sign and magnitude distinguish whether feedback amplifies or attenuates node-wise fluctuations. We further show that feedback-mediated noise circulation leaves a temporal signature in the decay of steady-state autocorrelation, revealing how loop closure modifies the persistence of fluctuations. We thus provide a minimal framework for understanding how feedback architecture regulates both the magnitude and the temporal persistence of noise in gene regulatory circuits.