Stochastic focusing coupled with negative feedback enables robust regulation in biochemical reaction networks

Nature presents multiple intriguing examples of processes that proceed with high precision and regularity. This remarkable stability is frequently counter to modellers' experience with the inherent stochasticity of chemical reactions in the regime of low-copy numbers. Moreover, the effects of noise and nonlinearities can lead to 'counterintuitive' behaviour, as demonstrated for a basic enzymatic reaction scheme that can display stochastic focusing (SF). Under the assumption of rapid signal fluctuations, SF has been shown to convert a graded response into a threshold mechanism, thus attenuating the detrimental effects of signal noise. However, when the rapid fluctuation assumption is violated, this gain in sensitivity is generally obtained at the cost of very large product variance, and this unpredictable behaviour may be one possible explanation of why, more than a decade after its introduction, SF has still not been observed in real biochemical systems. In this work, we explore the noise properties of a simple enzymatic reaction mechanism with a small and fluctuating number of active enzymes that behaves as a high-gain, noisy amplifier due to SF caused by slow enzyme fluctuations. We then show that the inclusion of a plausible negative feedback mechanism turns the system from a noisy signal detector to a strong homeostatic mechanism by exchanging high gain with strong attenuation in output noise and robustness to parameter variations. Moreover, we observe that the discrepancy between deterministic and stochastic descriptions of stochastically focused systems in the evolution of the means almost completely disappears, despite very low molecule counts and the additional nonlinearity due to feedback. The reaction mechanism considered here can provide a possible resolution to the apparent conflict between intrinsic noise and high precision in critical intracellular processes.