Designing synthetic biological systems by multiple time scale networks

Rhythmic phenomena exist in living organisms at all levels with periods ranging from less than a second to years [1]. From both theoretical and experiment viewpoints, it is a great challenging problem in biological science to model, analyze and further predict the periodic behaviors of biosystems. This paper aims to develop a new methodology to analyze and design biological oscillating networks with time delays, by using multiple time-scale networks (MTN) which are composed of a cyclic feedback network (CFN) and multiple positive feedback networks (PFN). We show that a MTN with certain conditions has no stable equilibria but stable periodic oscillations, depending on the total time delay of the CFN, although it has a complicated network structure including both positive and negative feedback loops. As an implementation example, a biologically plausible two-gene synthetic model with genes lac and cI is designed to demonstrate the theoretical result. The detail theoretical and numerical results are described in [1].