Bistability and chaos in the discrete two-gene Andrecut-Kauffman model

Figure 1.  Schematic representation of DNA transcription

Figure 2.  Schematic representation of the RNA translation

Figure 3.  The two coordinates ($ x $ and $ y $) of selected trajectories observed for $ \varepsilon = 0.8 $, $ \beta = 0.1 $, $ n = 4 $, and different values of $ \alpha_1 $ and $ \alpha_2 $

Figure 4.  Bifurcation diagrams of variables $ x $ and $ y $ for model (2) with fixed parameters $ \varepsilon = 0.7 $, $ \beta = 0.2 $, $ n = 4 $ for $ \alpha_1 = \alpha_2 $ (a), (c) and $ \alpha_2 = 200 $ (b), (d)

Figure 5.  Two attractors observed in model (2) with the parameters $ \varepsilon = 0.7 $, $ \beta = 0.2 $, $ n = 4 $, and $ \alpha_1 = \alpha_2 = 200 $

Figure 6.  Distribution of the computed Lyapunov exponent for $ 1024 \times 1024 $ starting points taken from $ (1, 99) \times (1, 99) $. Parameters used in this simulation are $ \alpha_1 = 50 $, $ \alpha_2 = 85 $, $ \beta = 0.2 $, $ \varepsilon = 0.7 $, and $ n = 3 $

Figure 7.  Pairs of parameters $ \beta $ and $ \varepsilon $ for models (1) and (2) for which there exists a parameter $ \alpha\in [0,100] $, or a pair of parameters $ (\alpha_1,\alpha_2)\in[0,100]\times[0,100] $, respectively, with the positive maximum Lapunov exponent. Pairs with the positive maximum Lyapunov exponent observed in both models are marked in blue, while those with this observation made only for model (2) are marked in orange

Figure 8.  The maximum Lyapunov exponent computed for model (2) for $ \alpha_1, \alpha_2 \in (0,250) $ with $ \beta = 0.2 $, $ \varepsilon = 0.6 $, and $ n = 3 $

Figure 9.  Bifurcation diagram of the $ x $ variable computed along the segment in the parameter space plotted in red in the previous figure (Figure 8), parametrized as $ (\alpha_1, \alpha_2) = (20+130t, 10+215t) $ for $ t \in [0,1] $

Figure 10.  Asymmetry of the model assessed for $ \alpha_1, \alpha_2 \in (0,250) $, $ \beta = 0.2 $, and $ n = 3 $, measured by the two quantities introduced in the text: $ S' $ shown as the black line and $ p^{\pm} $ indicated by the shaded area

Figure 11.  An attracting cycle (blue) and a chaotic attractor (orange) observed in model (2), with the parameters $ \alpha_1 = 43.6 $, $ \alpha_2 = 75.7 $, $ \varepsilon = 0.9 $, $ \beta = 0.2 $, and $ n = 3 $ mentioned in Table 1. Approximations of attraction basins of the attractors are indicated by the corresponding bright colors