A Minimal Spatio-Temporal Model of the NF-$\kappa$B Signalling Pathway Exhibits a Range of Behaviours

Abstract

In animal cells, the transcription factor NF-$\kappa$B regulates many stressful, inflammatory, and innate immune responses. Experiments have revealed that, in response to cell stimulation, NF-$\kappa$B can exhibit oscillatory dynamics where the nature of these dynamics can influence the pattern of NF-$\kappa$B-dependent gene expression. Oscillations in NF-$\kappa$B are believed to depend on a negative feedback loop linking NF-$\kappa$B and one of its downstream products, namely $\hbox {I}\kappa \hbox {B}\alpha$. This negative feedback loop is enhanced by cell stimulation. However, it also exists in the absence of cell stimulation. Here we propose a minimal spatio-temporal model of the NF-$\kappa$B signalling pathway, composed of partial differential equations. Through numerical simulations, we find various combinations of behaviours before and during cell stimulation: equilibrium dynamics (rapid convergence to a solution that is everywhere constant) before cell stimulation, followed by oscillatory dynamics during cell stimulation; oscillatory dynamics before and during cell stimulation; oscillatory dynamics before cell stimulation, followed by equilibrium dynamics during cell stimulation; and equilibrium dynamics before and during cell stimulation. In each case, when cell stimulation ceases, the model quickly returns to its pre-stimulation behaviour. All of these different combinations of behaviours occur for similar sets of parameter values. Therefore, our results may help to explain why, in experiments on the NF-$\kappa$B pathway involving populations of cells, only a certain fraction of the cells exhibit oscillatory dynamics.