Stability of delay induced oscillations in gene expression of Hes1 protein model

Mathematical analysis of the Hes1 genetic oscillator system is presented. The considered model is a modification of the model proposed by Monk [N.A. Monk, Oscillatory expression of Hes1, p53, and NF- B driven by transcriptional time delays, Curr. Biol. 13 (2003) 1409–1413]. In this paper we model separately time delays connected with transcription and translation processes. Global existence, uniqueness and non-negativity of solutions are proved. The existence of a unique positive steady state is shown. The stability of the steady state is studied and it is shown that it depends on the sum of time delays. Conditions guaranteeing the occurrence of the Hopf bifurcation as well as its stability are proved. Presented analysis partially confirms the results obtained numerically by Jensen et al. [M. Jensen, K. Sneppen, G. Tiana, Sustained oscillations and time delays in gene expression of protein Hes1, FEBS Lett. 541 (2003) 176–177].