  Inflammatory responses in eucaryotic cells are often associated with
oscillations in the nuclear-cytoplasmic translocation of the transcription
factor NF-kB. In most laboratory realizations, the oscillations are triggered
by a cytokine stimulus, like the tumor necrosis factor alpha, applied as a step
change to a steady level. Here we use a mathematical model to show that an
oscillatory external stimulus can synchronize the NF-kB oscillations into
states where the ratios of the internal to external frequency are close to
rational numbers. We predict a specific response diagram of the TNF-driven
NF-kB system which exhibits bands of synchronization known as "Arnold tongues".
Our model also suggests that when the amplitude of the external stimulus
exceeds a certain threshold there is the possibility of coexistence of multiple
different synchronized states and eventually chaotic dynamics of the nuclear
NF-kB concentration. This could be used as a way of externally controlling
immune response, DNA repair and apoptotic pathways.
