Stochastic Focusing and Defocusing in Biological Reaction Networks: Lessons Learned from Accurate Chemical Master Equation (ACME) Solutions

Biological reaction networks are stochastic due to random thermal fluctuations. Stochasticity plays important roles in regulation of biochemical reaction networks when the copy numbers of molecular species are small and often results in unexpected outcomes. For example, it has been shown that a basic enzymatic reaction system can display stochastic focusing (SF) by increasing the sensitivity of the network as a result of the increasing signal noise [1]. Although stochastic simulation algorithm has been widely used to study such systems, it is ineffective in examining rare events and this becomes a significant issue when the tails of probability distributions are relevant as is the case of SF. Here we use the ACME method for the exact solution of the discrete Chemical Master Equation and study the probability landscape of product molecules in the basic enzymatic reaction system used in the original SF study [1]. Examinations of the effects of signal molecules under different stochastic processes show that SF is at play as stochastic changes enhance the system sensitivity. However, we also observed that the noise in signaling under certain stochastic processes in the same reaction network lead to a decrease in the system sensitivities, thus the network experiences stochastic defocusing. We further show that signal molecules following certain stochastic processes in the same reaction network can give rise to noise-induced bistability in the distribution of product molecules. These results highlight the fundamental role of stochasticity in biological reaction networks and the need for exact computation of probability landscape of the molecules in the system. It also points to possible importance of positive and negative feedback loops in such networks for control of the intrinsic noise.[1] Paulsson et. al, 2000, PNAS.