Design principles of noise-induced oscillation in biochemical reaction networks: II. coupled positive and negative feedback loops

According to the chemical reaction network theory, the topology of a certain class of chemical reaction networks, regardless of the kinetic details, sets a limit on the dynamical properties that a particular network can potentially admit; the structure of a network predetermines the dynamic capacity of the network. We note that stochastic fluctuations can possibly confer a new dynamical capability to a network. Thus, it is of tremendous value to understand and be able to control the landscape of stochastic dynamical behaviors of a biochemical reaction network as a function of network architecture. Here we investigate such a case where stochastic fluctuations can give rise to the new capability of noise-induced oscillation in a subset of biochemical reaction networks, the networks with only three biochemical species whose reactions are governed by mass action kinetics and with the coupling of positive and negative feedback loops. We model the networks with master equations and approximate them by using linear noise approximation. For each network, we read the signal-to-noise ratio value, an indicator of amplified and coherent noise-induced oscillation, from the analytically derived power spectra. We classify the networks into three performance groups based on the average values of signal-to-noise ratio and robustness. We identify the common network architecture among the networks belonging to the same performance group, from which we learn that the coupling of negative and positive feedback loops generally enhance the noise-induced oscillation performance better than the negative feedback loops alone. The performance of networks also depends on the relative size of the positive and negative feedback loops; the networks with the bigger positive and smaller negative feedbacks are much worse oscillators than the networks with only negative feedback loops.