Stochastic fluctuations in gene expression play a decisive role in shaping cellular phenotypes. Understanding how biochemical noise propagates across different layers of cellular regulation is crucial for identifying how cells control their phenotypic variability.

In this work we quantify how biochemical noise propagates from the expression of a catalytic enzyme to its metabolic product. We use a stochastic model of a catalytic reaction under transcriptional repression from the product to enzyme expression. The model accounts for product catalysis and consumption, together with the expression and degradation of enzymatic molecules. Stochastic simulations suggest that feedback repression can both amplify or attenuate the stationary fluctuations in the product molecules as compared to the case of constitutive enzyme expression. To further understand this phenomenon we derive analytic estimates for the stationary distribution of the product. From these estimates we prove that noise amplification/attenuation depends non-monotonically on the promoter and repression strength. We show that a weak promoter or weak repression tends to attenuate noise, whereas a strong promoter or strong repression enhances the size of stochastic fluctuations.

We derive an analytic condition on the model parameters that lead to noise amplification/attenuation, revealing that highly sensitive promoters expand the parameter space for noise attenuation. In the case of switch-like promoters, attenuation is subject to a trade-off between promoter and repression strength, where noise attenuation can be achieved either by strong promoters that are weakly repressed or, conversely, by weak promoters under strong repression.

The propagation of noise between enzyme expression and metabolism has typically been neglected on the basis that metabolism operates under constant enzyme abundance. Environmental shocks, however, trigger dynamic changes in enzyme expression that adapt the metabolic state of the cell to the new environment. Our analysis reveals how transcriptional control can modulate stochastic fluctuations in metabolic products, opening new questions on how microbes control the population-level variability of their metabolic phenotypes in changing environments.

biochemical noise; metabolic reactions; transcriptional control; feedback regulation