Oscillations in the NF-kB signalling pathway.

NF-κB oscillations were suggested by Hoffmann et al from electro-mobility shift assays (EMSA) in population studies of IκBα-/- embryonic fibroblasts and simulated in a computational model. NF-κB oscillations were also observed by Nelson et al at the single cell level. The Hoffmann model gave a fairly good pre- diction of Nelson et al oscillatory experimental data using fluorescent proteins. A common comment on the source of oscillations is the existence of negative feedback loops. Just from the point of mathematics, we can set up a simple system containing a negative feedback loop that possesses oscillating behaviour resembling the ones observed in the experiments like Fonslet et al did. However, different models with similar structures can have dramatically different dynam- ical behaviour. In order to understand biological mechanisms, it is necessary to work on those real models that are based on experimental data even though models may be very large. In this paper, we are able to analyze the dynamical properties of Hoffmann’s large computational model (containing 24 variables and 64 parameters) by using computational and analytical methods and give an explanation of the source of oscillations. We find that the computational model can be treated as a fast-slow system where the level of total IκB Kinase (IKK) is treated as a slow variable. If we consider the limit in which the level of total IKK does not change at all, then we can take the level of total IKK as a parameter. Since the total NF-κB is conserved in the model, we can also view the total NF-κB as a parameter. If the actual variation of IKK is sufficiently slow, then orbits in the true system trace attractors in the reduced model. We find that for some range of the level of NF-κB, the reduced system experiences Hopf bifurcation twice while varying the level of total IKK. The damped oscillations observed in the computational system come from the existence of stable limit cycles and stable spirals in the reduced system family.