  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 the master
equations and approximate them, using the linear noise approximation. For each
network, we read the signal-to-noise ratio value, an indicator of amplified and
coherent noise-induced oscillation, off from the analytically derived power
spectra. We classify the networks into three performance groups based on the
average values of the signal-to-noise ratio and the 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.
