Fig. 1 | Nature Communications

Fig. 1

From: On chaotic dynamics in transcription factors and the associated effects in differential gene regulation

Fig. 1

Dynamics from coupled oscillators and emergence of chaos. a Schematic picture of a simplified NF-κ B network with a single negative feedback loop which can generate oscillations. b Schematic picture of oscillations in the external TNF concentration, represented by the changing shade of blue. c Dynamics that emerges when the NF-kB system is driven by a periodic TNF signal. The left panel shows schematically that there are Arnold tongues, triangular regions of the TNF amplitude-period parameter space where NF-kB oscillations can be synchronised to the TNF signal. Outside the Arnold tongues, e.g. point 0, there is no synchronisation. As TNF amplitude increase the Arnold tongues start overlapping and the behaviour becomes more complex. Keeping the TNF period fixed (here we used T = 50 min), as we increase the the TNF amplitude we enter three distinct states: Point 1: A single Arnold tongue, only allowing one oscillation state. Point 2: Overlap of Arnold tongues, allowing two stable oscillation states. The presence of noise can cause transitions (mode-hopping). Point 3: Chaotic dynamics, with apparently unpredictable trajectories. The trajectories corresponding to these points are shown in in the middle panels. Red and blue trajectories correspond to two different initial conditions in a deterministic simulation. The rightmost panels show the dynamics of NF-κ B vs. time in stochastic simulations where intrinsic noise is implemented using the Gillespie algorithm. d Schematic figure of the polymerase binding for genes that have NF-κB (green spheres) as a transcription factor. e Profile of the Hill function in Eq. (2) for different values of affinity and cooperativity. Red: h = 2 and K = 1.0 (HAG). Purple: h = 4 and K = 4.5 (LAG). Green: Example of intermediate values with h = 3 and K = 2.0 (MAG). Vertically, light blue: a representative NF-κB oscillation

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