A Critical Quantity for Noise Attenuation in Feedback Systems

Feedback modules, which appear ubiquitously in biological regulations, are often subject to disturbances from the input, leading to fluctuations in the output. Thus, the question becomes how a feedback system can produce a faithful response with a noisy input. We employed multiple time scale analysis, Fluctuation Dissipation Theorem, linear stability, and numerical simulations to investigate a module with one positive feedback loop driven by an external stimulus, and we obtained a critical quantity in noise attenuation, termed as “signed activation time”. We then studied the signed activation time for a system of two positive feedback loops, a system of one positive feedback loop and one negative feedback loop, and six other existing biological models consisting of multiple components along with positive and negative feedback loops. An inverse relationship is found between the noise amplification rate and the signed activation time, defined as the difference between the deactivation and activation time scales of the noise-free system, normalized by the frequency of noises presented in the input. Thus, the combination of fast activation and slow deactivation provides the best noise attenuation, and it can be attained in a single positive feedback loop system. An additional positive feedback loop often leads to a marked decrease in activation time, decrease or slight increase of deactivation time and allows larger kinetic rate variations for slow deactivation and fast activation. On the other hand, a negative feedback loop may increase the activation and deactivation times. The negative relationship between the noise amplification rate and the signed activation time also holds for the six other biological models with multiple components and feedback loops. This principle may be applicable to other feedback systems.Author Summary: Many biological systems use feedback loops to regulate dynamic interactions among different genes and proteins. Here, we ask how interlinked feedback loops control the timing of signal transductions and responses and, consequently, attenuate noise. Drawing on simple modeling along with both analytical insights and computational assessments, we have identified a key quantity, termed as the “signed activation time”, that dictates a system's ability of attenuating noise. This quantity combining the speed of deactivation and activation in signal responses, relative to the input noise frequency, is determined by the property of feedback systems when noises are absent. In general, such quantity could be measured experimentally through the output response time of a signaling system driven by pulse stimulus. This principle for noise attenuation in feedback loops may also be applicable to other biological systems involving more complex regulations.