Biochemical oscillations in delayed negative cyclic feedback: Existence and profiles

In this paper, we propose a theoretical framework to systematically analyze the existence and the profiles of chemical oscillations in gene regulatory networks with negative cyclic feedback. In particular, we analytically derive the existence conditions and the profiles of oscillations in terms of reaction kinetic parameters and reveal dimensionless quantities that essentially characterize the oscillatory dynamics. These discoveries then allow us to provide general biological insights that are useful for the design of synthetic gene circuits and wet-lab experiments. We point out that time delays due to splicing and transport play an important role for both of the existence and the profiles of oscillations. To this end, we first show that local instability leads to oscillations in cyclic gene regulatory networks, and we derive the existence conditions based on local instability analysis. Then, we analyze the period, phase and amplitude of oscillations using multivariable harmonic balance analysis. These results are demonstrated with two existing biochemical networks, the Pentilator and a self-repression network of a Hes protein.