Arnold Tongues in Cell Dynamiccs

Oscillating genetic patterns have been observed in networks related to the transcription factors NFkB, p53 and Hes1 [1]. We have identified the central feed-back loops leading to oscillations. By applying an external periodic signal, it is possible to lock the internal oscillation to the external signal. For the NF-kB systems in single cells we have observed that the two signals lock when the ration between the two frequencies is close to basic rational numbers [2]. The resulting response of the cell can be mapped out as Arnold tongues. When the tongues start to overlap we may observe a chaotic dynamics of the concentration in NF-kB [2,3]. The group of Savas Tay (ETH, Zurich) has in single cell dynamics of the NF-kB system observed transitions from one tongue to the other when they overlap. We investigate this effect by Gillespie simulations and find interesting time correlation for the transitions probabilities when switching from one tongue to the other.

[1] B. Mengel, A. Hunziker, L. Pedersen, A. Trusina, M.H. Jensen and S. Krishna, “Modeling oscillatory control in NF-kB, p53 and Wnt signaling”, Current Opinion in Genetics and Development 20, 656-664 (2010).

[2] M.H. Jensen and S. Krishna, “Inducing phase-locking and chaos in cellular oscillators by modulating the driving stimuli”, FEBS Letters 586, 1664-1668 (2012).

[3] N. Mitarai, U. Alon and M.H. Jensen, “Entrainment of linear and non-linear systems under noise”, Chaos, Chaos 23, 023125 (2013)