Oscillation Expression of NF- $\kappa$ B Driven by Transcription and Translation Time Delays

Oscillation expression of NF-κB signaling system can lead to nuclear NF-κB accumulation which involving various important cellular processes including immune response, inflammation and cancer development. Negative feedback circuits involved in NF-κB signaling system are generally believed to be responsible for the oscillation expression. In view of feedback loops involving time delays resulting from transcription, transcript splicing and processing, and protein synthesis, in this paper, a mathematical model with time delay to describe the core negative feedback circuits centered on NF-κB including IκBα and A20-like proteins is developed. Here, the time delay refer to the time demand of transcription and translation process of IκBα. First, the results suggest that time delay can drive oscillation dynamics of NF-κB signaling system. Concretely, there exists a critical value of time delay. When the time delay exceeds the critical value, the system exhibits oscillation dynamics, and with the increase of time delay, the amplitude and period increase accordingly. Second, analysis indicate that A20-like proteins are able to change the amplitude and period of the nuclear NF-κB oscillations. In addition, the effects of several important parameters on the nuclear NF-κB oscillations are also discussed. Finally, a prediction is formed, which is shortening time delay, repairing and maintaining the function of A20-like proteins, and increasing IκBα transcription rate may be new clues to related disease treatment.