Quantifying effects of stochastic focusing and defocusing on noise propagation in biochemical reaction networks

Biochemical reaction networks are intrinsically stochastic. Such stochastic effects are pronounced when molecules have low copy numbers. In an enzymatic cascade reaction network, the formation of product molecules is dictated by the enzymatic signal. Here the noise level in the signaling molecules can have counterintuitive effects. Specifically, increased noise level can reduce the uncertainty in amplified responses of product formation. This phenomenon has been termed Stochastic Focusing. Of particular interest is the sensitivity of product formation to changes in the distribution of signaling enzyme molecules. However, this sensitivity between stochastic noise in signaling molecules and the product formation is yet to be quantified. In this study, we use the Accurate Chemical Master Equation (ACME) algorithm to (1) characterize the stochastic behavior of the enzymatic cascade reaction network and (2) define the system sensitivity of production formation to perturbation in distribution of the signaling molecule. Our approach is based on solving the discrete Chemical Master Equation (dCME) to obtain the exact probability landscape of this network. By perturbing the signal molecule distribution after reaching steady state, we define noise change as the difference in signal distribution widths before and after the shift. We find that the sensitivity of the network correlates exponentially with the level of signaling noise in both the stochastic focusing (increase in system sensitivity) and defocusing (decrease in system sensitivity) regimes. We further studied noise propagation in this enzymatic reaction network by examining the effects of different number of intermediate species required for forming the product molecule. Our findings provide a mechanistic understanding on how signaling noise controls product formation in biological reaction networks, enabling the design of better control mechanisms in cellular systems.