Designing Synthetic Biological Systems by Multiple Time Scale Networks
Ruiqi Wang, Junya Nakai, Mitsuyasu Seo, and Luonan Chen
Osaka Sangyo University

This paper aims at developing a new methodology to model and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using multiple time-scale networks. Fast reactions constitute positive feedback-loop networks, while slow reactions consist of a cyclic feedback-loop network. Multiple time scale properties are exploited to simplify the model according to singular perturbation theory. We show that the proposed network has no stable equilibria but stable periodic orbits when certain conditions are satisfied. Such a property is ideal to design and construct a gene oscillator in a synthetic biological system. To demonstrate our theoretical results, we design a biologically plausible synthetic gene regulatory network by cI and Lac genes in an engineered circuit, which is assumed to be implemented in a eukaryotic cell and verified by the numerical simulation. In particular, we show both theoretically and numerically that time delay can be used as an important control parameter both quantitatively and qualitatively to change dynamics of the systems, in particular for the amplitudes and periods of periodic oscillations. Moreover, the effects of degradation rates of mRNA and protein on oscillatory and steady state regions also were examined, which indicates that degradation rates of mRNAs or proteins can be used as control parameters to change the global dynamics of the systems.