Network design principle for robust oscillatory behaviors with respect to biological noise

Oscillatory behaviors, which are ubiquitous in transcriptional regulatory networks, are often subject to inevitable biological noise. Thus a natural question is how transcriptional regulatory networks can robustly achieve accurate oscillation in the presence of biological noise. Here, we search all two- and three-node transcriptional regulatory network topologies for those robustly capable of accurate oscillation against the parameter variability (extrinsic noise) or stochasticity of chemical reactions (intrinsic noise). We find that, no matter what source of the noise is applied, the topologies containing the repressilator with positive auto-regulation show higher robustness of accurate oscillation than those containing the activator-inhibitor oscillator, and additional positive auto-regulation enhances the robustness against noise. Nevertheless, the attenuation of different sources of noise is governed by distinct mechanisms: the parameter variability is buffered by the long period, while the stochasticity of chemical reactions is filtered by the high amplitude. Furthermore, we analyze the noise of a synthetic human nuclear factor κB (NF-κB) signaling network by varying three different topologies, and verify that the addition of a repressilator to the activator-inhibitor oscillator, which leads to the emergence of high-robustness motif—the repressilator with positive auto-regulation, improves the oscillation accuracy in comparison to the topology with only an activator-inhibitor oscillator. These design principles may be applicable to other oscillatory circuits.