Noise propagation in a class of metabolic networks

A metabolic pathway made of a cascade of biochemical reactions is considered, with a substrate which is eventually transformed into the final product by means of a sequence of reactions, each catalyzed by the same enzyme. The amount of the enzyme varies according to discrete noisy processes of production and elimination. A feedback acts on the final product clearance rate, exerted by the final product accumulation itself: higher final product levels lead to a faster dynamics. The aim of this note is to investigate how the noise scales with the length of the cascade and how the feedback impacts on the noise propagation. To this end, a Stochastic Hybrid System (SHS) formulation is exploited, with the enzyme production/clearance processes constituting the noise source. The noise propagation is measured in terms of the square of the coefficient of variation of the final product, and computations are carried out by means of the equations of moments, which are estimated in closed form after linearizing the SHS. Analytical solutions allow to infer information and to relate the noise propagation to the model parameters. Similarly to recent results occurring in other types of enzymatic reactions, the results highlight the influential role of feedback in noise reduction.