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FIGURE 1: Experimentally measured bistable dose-response curves (a and b), their corresponding cell cycle regulatory networks (c and d), and their approximations using Eq. (5) in e and f, respectively. (a) Experimentally measured Cdk1 activity (a.u.) in function of cyclin B concentration (nM) (Pomerening et al., 2003). (e) Approximated response curve using (b,Cx,Cy) = (1.88, 55 nM, 0.5) as parameters. (b) Experimentally measured APC/C activity (a.u.) in function of active cyclin B-Cdk1 (nM) (Kamenz et al., 2021). (f) Approximated response curve using (b,Cx,Cy) = (1.95,20 nM, 0.5) as parameters.
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FIGURE 2: (a) Input (X)–output (Y) response curves given by Eq. (5) with Cx = 50 and Cy = 0.5 for increasing bistability by increasing the parameter b. (b) The same response curves as in a, but now the output Y is multiplied by the input X.
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FIGURE 3: (a) Simulation of the model for the cell cycle oscillator built on two interlinked bistable switches. (b) In time by was changed from 1.94 to 1.73 to 1.91. (c) The detected period for every consecutive cell cycle: in black for the simulated time series in a, and in red for the experimentally measured periods (mean period ± one SD) (Satoh, 1977; Anderson et al., 2017).
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FIGURE 4: Simulation of the model for the cell cycle oscillator built on cyclin B-Cdk1 bistability [see Eqs. (13)–(14)]. Time series (a and e) and phase space (b and f). Phase diagram in by and Cxy (c and g) showing the period of the oscillation in blue color map, the Hopf bifurcations (blue line), and the intersection of the cyclin B-nullcline with the folds of the cyclin B-Cdk1-nullcline (orange line). Bifurcation diagram in Cxy (d and h) for by =1.88 showing the value of active cyclin B-Cdk1 of the stationary state (in red) and the maximum of the oscillation (in blue). The Hopf bifurcations are indicated by H1,2 and the fold of cycles by FC.
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FIGURE 5: (a) Phase diagram of model (13)–(14) in the parameter space (ks,d2). Different bifurcations are indicated: Hopf bifurcations (H1, H2, H3-blue), folds of cycles (FC–green), and saddle-node bifurcations (SN1, SN2–red). TB1, TB2 are two codimension-two Takens–Bogdanov points. When stable oscillations exist, their period is also indicated in a blue color scale. (b, c) Bifurcation diagrams in function of cyclin synthesis ks for different values of the degradation rate d2. All other parameters are given in Table 1.
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FIGURE 6: (a) Simulation of model (15)–(16) for the cell cycle oscillator built on Cdk1-APC/C bistability: time series (a) and phase space (b). (c) (bz,Cxy)-phase diagram with Hopf bifurcations in blue. Parameters are given in Table 1.
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FIGURE 7: (a) Phase diagram of model (15)–(16) in the parameter space (ks,d2). Different bifurcations are indicated: Hopf bifurcations (H1–blue), fold of cycles (FC–green), and saddle-node bifurcations (SN1, SN2–red). TB1, TB2 and TB3 are three codimension two Takens–Bogdanov points. When stable oscillations exist, their period is also indicated in a blue color scale. (b) Bifurcation diagrams in function of cyclin synthesis ks for d2 = 0.1,1, and 5 min−1. All other parameters are given in Table 1.
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FIGURE 8: Simulation of the model for the cell cycle oscillator built on two interlinked bistable switches: time series (a) and phase space projections (b and c). Parameters are as in Table 1.
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FIGURE 9: (a) Phase diagram of model (6)–(8) in the parameter space (ks,d2). Different bifurcations are indicated: Hopf bifurcations (H–blue) and saddle-node bifurcations (SN–red). TB is a codimension-two Takens–Bogdanov point. When stable oscillations exist, their period is also indicated in a blue color scale. (b) Bifurcation diagrams in function of cyclin synthesis ks for d2 = 0.05,0.1,0.5,1,2 and 10 min−1. (c) Phase space projection of the cyclin B/APC/C and cyclin B/Cdk1 nullclines (in blue), cyclin B-Cdk1-nullcline (in orange), APC/C-nullcline (in green), and oscillation (in gray) for the indicated values of ks. All other parameters are given in Table 1.
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FIGURE 10: Phase diagrams of model (6)–(8) showing different oscillatory regimes. (a–c) (Cx,Cxy)-phase diagrams where the amplitude of the active Cdk1 oscillation is represented with the blue color map, the blue lines represent the Hopf bifurcations, the orange dashed lines correspond to the intersection of F1,2(Cdk1) with the cyclin B/APC/C-nullcline, and the green dashed lines represent the intersection of the F1,2(APC) with the cyclin B-Cdk1-nullcline. The parameters are given in Table 1 for panel a. The (cyclin B, active Cdk1) switch has been made ultrasensitive using by =1.7 in b. The (active Cdk1, APC/C) switch has been made ultrasensitive using bz = 1.7 in c. (d) The (by, bz)-phase diagram for parameters given in Table 1. The amplitude of the active Cdk1 oscillation is represented with the blue color map, the blue curves represent a Hopf bifurcation, and the dashed orange and green lines represent the transition of (cyclin B, active Cdk1) and (active Cdk1, APC/C) switches from S-shaped to ultrasensitive, respectively. (e–h) Phase space projection of the cyclin B/APC/C-nullcline (in blue), cyclin B-Cdk1-nullcline (in orange), APC/C-nullcline (in green), and oscillation (in gray). Parameter values are Cx = 55 nM, Cxy = 5 nM for e, Cx = 40 nM, Cxy = 40 nM for f, Cx = 30 nM, Cxy = 46 nM for g, and by = 1.7, bz = 1.95 for h. The other parameters are given in Table 1.
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