1. Introduction
2. Model and methods
Figure 1. Schematic diagram of the network model. Promotion and state transition is represented by colored and black arrow-headed lines respectively. The promotion of degradation is indicated by cycle-headed lines. |
• | • The form of ATM switches between active and inactive, but their total amount is almost unchanged [26]. These facts make the kinetic terms of ATM’s generation and degradation are not included in the model, which is the main difference between ATM’s equations and other substances |
• | • Under resting conditions, p53 is rapidly degraded after production [4]. In other words, inactive p53 has poor stability. After DNA damage, p53 is activated into a stable phosphorylated form [24]. Therefore, the equation of p53 should consist of birth, death, activation, and deactivation dynamic terms The degradation rate of p53 should be greater than that of p53*, so we set ${\bar{d}}_{{\rm{p}}53}\gg {k}_{1}{\bar{d}}_{{\rm{p}}53}$ |
• | • Interestingly, Mdm2, the main negative regulator of p53, changes exactly the ‘opposite’ to that of p53 after DNA damage. The ‘opposite’ here refers to stability and enzyme activity, i.e. the phosphorylated Mdm2 induced by DNA damage is rapidly degraded on the one hand, and the effect of Mdm2* ubiquitin ligase is weakened on the other hand [25]. In the kinetic equations, the degradation rate of Mdm2 should be much smaller than that of Mdm2*, i.e. ${\bar{d}}_{\mathrm{mdm}2}\ll {k}_{2}{\bar{d}}_{\mathrm{pmdm}2}$. Besides, the kinetic terms in the Mdm2 equations are similar to those of p53. |
• | • Without involving complex situations, the equations of Wip1 and 14-3-3 Σ only contain a synthesis term and a degradation term |
Table 1. Simulation parameters. |
Parameter | Description | Value and unit |
---|---|---|
ATMt | The total level of ATM | 5 C |
kpatm | Maximum activation rate of ATM | 0.25 min−1 |
ρ0 | The proportional weight of ATM* autocatalysis | 0.96 – |
j0 | Half-saturated concentration of ATM* autocatalysis | 5 C |
kqatm | Maximum inactivation rate of ATM | 0.45 min−1 |
ρ1 | The proportional weight of ATM* inactivation catalyzed by Wip1 | 0.95 – |
j1 | Half-saturated concentration of Wip1 in the catalytic ATM* deactivation | 0.9 C |
sp53 | Basal production rate of p53 | 1.2 C · min−1 |
kqp53 | Maximum inactivation rate of p53 | 0.8 min−1 |
ρ2 | The proportional weight of p53* inactivation catalyzed by Wip1 | 0.91 – |
j2 | Half-saturated concentration of Wip1 in the catalytic p53* deactivation | 0.9 C |
kpp53 | Maximum activation rate of p53 | 1 min−1 |
ρ3 | The proportional weight of p53 activation catalyzed by ATM* | 0.96 – |
j3 | Half-saturated concentration of ATM* in the catalytic p53 activation | 5 C |
dp53 | Maximum degradation rate of p53 | 2.5 min−1 |
ρ4 | The proportional weight of p53 degradation catalyzed by Mdm2 | 0.97 – |
j4 | Half-saturated concentration of Mdm2 in the catalytic p53 degradation | 1.5 C |
smdm2 | Maximum generation rate of Mdm2 | 4.3 C·min−1 |
ρ5 | The proportional weight of Mdm2 generation catalyzed by p53* | 0.95 – |
j5 | Half-saturated concentration of p53* in the catalytic Mdm2 generation | 1.5 C |
kqmdm2 | Maximum dephosphorylation rate of Mdm2 | 3 min−1 |
ρ6 | The proportional weight of Mdm2* dephosphorylation catalyzed by Wip1 | 0.98 – |
j6 | Half-saturated concentration of Wip1 in the catalytic Mdm2* dephosphorylation | 0.9 C |
kpmdm2 | Maximum phosphorylation rate of Mdm2 | 2.5 min−1 |
ρ7 | The proportional weight of Mdm2 phosphorylation catalyzed by ATM* | 0.96 – |
j7 | Half-saturated concentration of ATM* in the catalytic Mdm2 phosphorylation | 5 C |
dmdm2 | Maximum degradation rate of Mdm2 | 1.03 min−1 |
ρ8 | The proportional weight of Mdm2 degradation catalyzed by 14-3-3 Σ | 0.93 – |
j8 | Half-saturated concentration of 14-3-3 Σ in the catalytic Mdm2 degradation | 0.7 C |
swip1 | Maximum synthesis rate of Wip1 | 0.16 C·min−1 |
ρ9 | The proportional weight of Wip1 generation catalyzed by p53* | 0.95 – |
j9 | Half-saturated concentration of p53* in the catalytic Wip1 generation | 0.35 C |
dwip1 | Basal degradation rate of Wip1 | 0.013 min−1 |
s14-3-3 Σ | Maximum synthesis rate of 14-3-3 Σ | 0.5 C·min−1 |
ρ10 | The proportional weight of 14-3-3 Σ generation catalyzed by p53* | 0.99 – |
j10 | Half-saturated concentration of p53* in the catalytic 14-3-3 Σ generation | 0.4 C |
d14-3-3 Σ | Basal degradation rate of 14-3-3 Σ | 0.175 min−1 |
k1 | The ratio of p53* to p53 degradation rate | 0.1 – |
k2 | The ratio of Mdm2* to Mdm2 degradation rate | 10 - |
Table 2. Initial condition. |
Protein | Initial | Protein | Initial | Protein | Initial | Protein | Initial | Protein | Initial |
---|---|---|---|---|---|---|---|---|---|
ATM* | 0 C | ATM | 5 C | p53* | 0.062 C | p53 | 0.809 C | Mdm2* | 0.097 C |
Mdm2 | 1.942 C | Wip1* | 0.627 C | 14-3-3 Σ | 0.03 C |
3. Results
3.1. Reproduce the dynamic behaviors of p53
Figure 2. Time evolution diagram of total p53 (blue line) and total Mdm2 (red line). (a) kpatm = 0.25; (b) kpatm = 0.5; (c) kpatm = 0.25 if $t\in [500,1400]\quad \min $, else, kpatm = 0. |
Figure 3. Time evolution diagrams of total p53 ([p53] + [p53*]) in some special situations. |
3.2. Robustness of oscillation to parameters variation
Figure 4. Codimensional 1 bifurcation diagram. HB represents the Hopf bifurcation point; the red line is the stable equilibrium state; the black line is the unstable equilibrium state. The green dots are the upper and lower bounds of the stable limit cycles; the blue cycles are the upper and lower bounds of the unstable limit cycles. |
Table 3. Robustness analysis of p53 oscillations. |
Parameter | Oscillation range | Dd% | Id% | S |
---|---|---|---|---|
ATMt | 4.332–5.060 | 13.36% | 1.20% | 7.75% |
kpatm | 0.214–0.253 | 14.40% | 1.20% | 8.35% |
ρ0 | 0.959–0.977 | 0.10% | 17.71% | 0.93% |
j0 | 4.883–6.614 | 2.34% | 32.28% | 15.06% |
kqatm | 0.444–0.524 | 1.33% | 16.44% | 8.26% |
ρ1 | 0.665–1.000 | 30.00% | 5.26% | 20.12% |
j1 | 0.534–0.985 | 40.67% | 9.44% | 29.69% |
sp53 | 0.934–1.217 | 22.17% | 1.42% | 13.16% |
kqp53 | 0.787–1.060 | 1.63% | 32.5% | 14.78% |
ρ2 | 0.301–1.000 | 66.92% | 9.89% | 53.73% |
j2 | 0.216–1.009 | 76.00% | 12.11% | 64.73% |
kpp53 | 0.774–1.015 | 22.60% | 1.50% | 13.47% |
ρ3 | 0.959–0.991 | 0.10% | 3.23% | 1.64% |
j3 | 4.863–7.715 | 2.74% | 54.30% | 22.67% |
dp53 | 2.466–3.135 | 1.36% | 25.40% | 11.94% |
ρ4 | 0.882–0.973 | 9.07% | 0.31% | 4.91% |
j4 | 1.079–1.529 | 28.07% | 1.93% | 17.25% |
smdm2 | 4.218–5.976 | 1.91% | 38.98% | 17.25% |
ρ5 | 0.930–0.951 | 2.11% | 0.11% | 1.12% |
j5 | 0.651–2.073 | 56.60% | 38.20% | 52.20% |
kqmdm2 | 2.231–12.266 | 15.63% | 308.87% | 69.22% |
ρ6 | 0.000–1.000 | 100% | 2.04% | 100.00% |
j6 | 0.000–3.095 | 100% | 243.89% | 100.00% |
kpmdm2 | 0.977–2.723 | 60.92% | 8.92% | 47.19% |
ρ7 | 0.951–1.000 | 0.94% | 4.17% | 2.51% |
j7 | 4.269–50.336 | 14.62% | 906.72% | 84.36% |
dmdm2 | 0.691–1.054 | 32.91% | 2.33% | 20.80% |
ρ8 | 0.906–0.995 | 2.58% | 6.99% | 4.68% |
j8 | 0.682–0.923 | 2.57% | 31.86% | 15.02% |
swip1 | 0.152–0.216 | 5.00% | 35.00% | 17.39% |
ρ9 | 0.903–0.985 | 4.95% | 3.68% | 4.34% |
j9 | 0.308–0.358 | 12.00% | 2.29% | 7.51% |
dwip1 | 0.009–0.014 | 30.77% | 7.69% | 21.74% |
s14-3-3 Σ | 0.379–0.513 | 24.20% | 2.60% | 15.02% |
ρ10 | 0.982–1.000 | 0.81% | 1.01% | 0.91% |
j10 | 0.397–0.434 | 0.75% | 8.50% | 4.45% |
d14-3-3 Σ | 0.171–0.235 | 2.29% | 34.29% | 15.76% |
k1 | 0.082–0.313 | 18.00% | 213.00% | 58.48% |
k2 | 2.554–13.447 | 74.46% | 34.47% | 68.08% |
3.3. Dynamics of p53-Mdm2 module regulated by 14-3-3 Σ and Wip1
Figure 5. (a) Codimension 2 bifurcation diagram. The green line represents the Hopf bifurcation, and the black line represents the Fold bifurcation. The blue area indicates the bistable state; the white area indicates the monostable state; the yellow area indicates the oscillation. (b)–(d) are pseudo-phase plane, black dots represent stable fixed points, and black closed orbit represents stable limit cycle. |
Figure 6. Pseudo-phase plane analysis. The blue line and red line represent the pseudo-collar slope lines. The gray line indicates orbit. Black dot represents unstable fixed point. |
Figure 7. Codimension 1 bifurcation diagram. sup- or sub- HB represents the supcritical or subcritical Hopf bifurcation point respectively; the red line is the stable equilibrium state; the black line is the unstable equilibrium state. The green dots are the upper and lower bounds of the stable limit cycles; the blue cycles are the upper and lower bounds of the unstable limit cycles. |
3.4. Impact of Wip1 and 14-3-3 Σ expression levels on the programmed cell death
Figure 8. The schematic diagram of ECP, the blue solid line is the time history of p53*, the blue dashed line is the threshold Th, and the gray area is the accumulation of p53 that is conducive to apoptosis. |
Figure 9. The functional relationship between ECP and kqatm (green inverted triangles), kqp53 (cyan upright triangles), kqmdm2 (black diamonds), swip1 (red squares), and s14-3-3Σ (pink five-pointed stars). |
Figure 10. ECP as a function of swip1 and s14-3-3Σ. The colors indicate the value of ECP, as the color bar on the right. The red arrow points to the direction of canceration. |