Delay Hill dynamics in regulatory biological systems.

We explore one of the best-studied protein circuits in human cells, the negative feedback loop between the tumor suppressor p53 and the oncogene Mdm2 following nuclear irradiation. Using stochastic delay differential equations and the Gillespie algorithm, we illustrate the distinct oscillatory dynamics at the single-cell and population-cell levels which were found in the recent experiments. The oscillatory dynamics of p53-Mdm2 interaction appears as coherent resonance with delay and noise in individual cells. Dephasing mechanisms provide the origin of damped oscillation at the population level out of the sustained one at the single-cell level. The non-Gaussian nature of distributions of protein populations results from the interplay between time delay and nonlinearity of reaction processes. Our findings may lead to new insights related to the effects of noise and cancer therapy.