  Oscillation has an important role in bio-dynamical systems such as circadian
rhythms and eukaryotic cell cycle. John Tyson et. al. in Nature Review Mol Cell
Biol 2008 examined a limited number of network topologies consisting of three
nodes and four or fewer edges and identified the network design principles of
biochemical oscillations. Tsai et. al. in Science 2008 studied three different
network motifs, namely a negative feedback loop, coupled negative feedback
loops, and coupled positive and negative feedback loops, and found that the
interconnected positive and negative feedback loops are capable of generating
frequency-tunable oscillations. We enumerate 249 topologically unique network
architectures consisting of three nodes and at least three cyclic inhibitory
edges, and identify network architectural commonalities among three functional
groups: (1) most frequency-tunable yet less robust oscillators, (2) least
frequency-tunable and least robust oscillators, and (3) less frequency-tunable
yet most robust oscillators. We find that Frequency-tunable networks cannot
simultaneously express high robustness, indicating a tradeoff between frequency
tunability and robustness.
