Complex Dynamics and Periodic Oscillation Mechanism in Two Novel Gene Expression Models with State-Dependent Delays

The reaction time delay in the transcription process depends on the concentration of the protein because the transportation of mRNA from the nucleus to the cytoplasm becomes saturated. Thus the gene regulatory network is a state-dependent delayed model. This study aims to provide some mathematical explanations for the dynamics of the system, such as the linear stability and periodic oscillation, using mathematical techniques, such as formal linearization, linear stability analysis, the method of multiple scale (MMS), and the normal form. First, Hopf bifurcation of the state-dependent delayed gene regulatory networks model in the gene expression is analyzed by the method of multiple scales (MMS). Mechanism of periodic oscillations is obtained by Hopf bifurcation. The findings show that when degradation effects of the mRNA and protein are very strong, the oscillatory gene expression disappears. Then, a more realistic version of the aforementioned model with both constant and state-dependent time delays is established due to the existence of the constant time delay in the protein degradation process. Its nonresonant double Hopf bifurcation is found and analyzed using MMS. Interesting complex dynamic phenomena, such as periodic, quasi-periodic, and global period-[Formula: see text] solutions, are also discovered. These observations indicate that both state-dependent delay and constant delay could induce richer dynamics of the system, and the modified model may potentially describe the real dynamical mechanism (both the transcription process and the degradation process) more accurately in the gene expression. The findings may provide important guidance or hints to understand the real dynamic mechanism of the gene expression process.