The transcription factor NF-κB performs various cell functions, such as regulating proliferation and differentiation and blocking apoptosis, by inducing the expression of multiple genes. The shuttling of NF-κB between the cytoplasm and nucleus is involved in its transcriptional activity in the canonical NF-κB pathway. The transcription of the NF-κB target genes is regulated by the phosphorylation of both IκBα and the RelA subunit of NF-κB, suggesting that these phosphorylation events are crucial for the oscillation. In this study, we constructed a new mathematical model of NF-κB activation to explore the modulation of the oscillation by the phosphorylation of IκBα and RelA. Based on a stability analysis around the equilibrium point, we confirmed that IκBα phosphorylation added a structure with a stable periodic solution to the phosphorylation model. The stable periodic solution appeared to transiently respond to the attenuation of the concentration of active IKKβ. Because the NF-κB oscillation is caused by the periodic solution, the amplitude and period of the NF-κB oscillation in the phosphorylation model was constant regardless of the initial conditions; we defined this property as the reproducibility of the oscillation. On the other hand, the amplitude and period of the NF-κB oscillation depended on a parameter related to the RelA phosphorylation, suggesting that the oscillation period is regulated by RelA phosphorylation. In addition, the region of the periodic solution that is dependent on active IKKβ also depends on a parameter related to RelA phosphorylation. Therefore, we conclude that the phosphorylation of both IκBα and RelA regulates the robustness of the NF-κB signaling module oscillation. That is, by appropriately controlling the phosphorylation process, it becomes possible to control the NF-κB oscillation and appropriately induce the NFkB-dependent expression gene. We anticipate that this study will contribute to the future elucidation of the mechanism underlying the nuclear cytoplasmic (N-C) oscillation of NF-κB.
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