On enzymatic reactions: the role of a feedback from the substrate

This note investigates a basic enzymatic scheme, with a substrate transforming into a product by means of the catalytic action of an enzyme. The focus is in the role of a feedback regulating the enzyme production. The novelty of the paper is in the choice of the feedback, acting from substrate accumulation differently from previous cases already studied in the literature, where the feedback acts from the product or from the enzyme. The feedback scheme is studied according to both a deterministic and stochastic approach: the former providing the existence of a unique meaningful asymptotically stable equilibrium; the latter investigating how noise propagates with or without the feedback. Regards to the stochastic approach, the metabolic noise is evaluated in terms of the coefficient of variation of the product of the enzymatic reaction, aiming at measuring its fluctuations around the average steady-state. Numerical results are carried out according to Chemical Master Equations, showing a clear improvement, in terms of noise reduction, when the negative feedback is applied. Linear Noise Approximation has been as well exploited with the aim of finding analytical solutions for the metabolic noise, relating it to the model parameters