Analogue information processing in NF-κB gene regulatory system

Nuclear factor-κB (NF-κB) participates in various cellular processes to encompass cell fate through differential gene expression, but the underlying molecular mechanism behind this phenomenon is still elusive. Two factors in this process can control the gene expression for determining the cell's fate: (i) synthesized proteins may have a considerable lifetime and (ii) gene activation may be slow or delayed. To address the first factor, we argue that the NF- $\kappa$ B system experiences cellular variability, often considered the origin of environmental noise for protein production, which influences cellular decisions at the molecular level as they have a considerable lifetime. We employ unified coloured noise approximation to obtain analytical expressions for the protein mean number obtained from our theoretical model and stochastic simulation. We find that these fluctuations influence mean protein numbers and induce bimodality. However, for the second factor, we rely on experimental findings, where the time delay in gene activation plays an essential role in protein production. Our bifurcation analysis demonstrates that the system exhibits saddle-node bifurcations for the instantaneous case, but it experiences the Hopf bifurcation and oscillates between two states in the presence of the time delay. In a nutshell, as NF-κB dynamics influence downstream expression, this study may provide insight into how to adjust parameters to control gene expressions.