Transition and identification of pathological states in p53 dynamics for therapeutic intervention

Abstract

We study a minimal model of the stress-driven p53 regulatory network that includes competition between active and mutant forms of the tumor-suppressor gene p53. Depending on the nature and level of the external stress signal, four distinct dynamical states of p53 are observed. These states can be distinguished by different dynamical properties which associate to active, apoptotic, pre-malignant and cancer states. Transitions between any two states, active, apoptotic, and cancer, are found to be unidirectional and irreversible if the stress signal is either oscillatory or constant. When the signal decays exponentially, the apoptotic state vanishes, and for low stress the pre-malignant state is bounded by two critical points, allowing the system to transition reversibly from the active to the pre-malignant state. For significantly large stress, the range of the pre-malignant state expands, and the system moves to irreversible cancerous state, which is a stable attractor. This suggests that identification of the pre-malignant state may be important both for therapeutic intervention as well as for drug delivery.

Figure 1

Interaction network for $\mathrm{p}{53}_A$-$\mathrm{p}{53}_M$-MDM2-ARF-Stress. Dashed arrow shows movement from nucleus to cytoplasm or vice versa, while solid arrow, and bars corresponds to activation, and inhibition on respective node. (Modified network from).

Figure 2

The left column show three different form of stress discussed about. In the top row (A₁) (normal), (A₂) (apoptosis), and (A₃) (cancer) display the time course of ${\text{p53}}_{\mathrm{A}}$ (green), and ${\text{p53}}_{\mathrm{M}}$ (red) for constant stress of magnitude 1.00, 1.75, and 2.50 respectively ($K_3$=1000.0). (B₁) (normal), (B₂) (apoptosis), and (B₃) (cancer) display the time series for averaged oscillatory stress of magnitude 1.00, 1.50, and 2.50 respectively ($K_3$=1000.0). And (C₁) (normal), (C₂) (recovery from initial cancer stage), and (C₃) (cancer) display the time series for decaying stress of magnitude 1.00, 3.5, and 4.50 respectively ($K_3$=500.0 and $\lambda$=0.05 h${}^{-1}$).

Figure 3

Dynamics on the phase plane for the time series results in the Fig. 2. Green color indicates active state, blue color apoptotic, and red color cancer state. The dot shows the attractor (end point of the trajectory).

Figure 4

The left column show three different form of stress discussed about. ${\mathrm{A}}_1$, ${\mathrm{A}}_2$, and ${\mathrm{A}}_3$ display the steady state behaviour against magnitude of stress for different K${}_3$ values 1000.0, 500.0, and 100.0 respectively driven with constant stress. ${\mathrm{B}}_1$, ${\mathrm{B}}_2$, and ${\mathrm{B}}_3$ display the steady state behaviour against magnitude of stress for different K${}_3$ values 1000.0, 500.0, and 100.0 respectively driven with oscillatory stress. ${\mathrm{C}}_1$, ${\mathrm{C}}_2$, and ${\mathrm{C}}_3$ display the steady state behaviour against amplitude for different K${}_3$ values 1000.0, 500.0, and 100.0 respectively driven with decaying stress. Yellow region, cyan region, and grey region correspond to active, apoptotic, premalignant, and cancer state respectively. In panel ${\mathrm{C}}_1$, and ${\mathrm{C}}_2$ (wheat region) black line (upper line), and blue line (lower line) show maximum of $\mathrm{p}{53}_M$, and maximum of $\mathrm{p}{53}_A$ in T${}_{\mathit{ps}}$ (see the text) time region, which corresponds to the initial cancer condition.

Figure 5

Hysteresis plot for three different type of stress as discussed in main text. (A) Increasing the magnitude of stress, apoptotic state moves towards cancer phase while decreasing stress, the cancer phase does not come back to apototic phase. (B) and (C) show similar pattern for oscillatory and decaying stress respectively. Solid and dashed arrow correspond to increasing and decreasing stress magnitude respectively. For solid arrow the initial value of variable corresponds to apoptotic phase while dashed arrow correspond to cancer phase. Parameters used in the figure correspond to the parameters used in Fig. 4${\mathrm{B}}_2$, ${\mathrm{C}}_2$, and ${\mathrm{C}}_2$ respectively.

Figure 6

The left column show three different form of stress discussed about. Second column (A–C) show two parameter steady state behavior of the system for three different form of driven stress (a–c) respectively. ${\mathrm{A}}_1$, ${\mathrm{A}}_2$, ${\mathrm{A}}_3$, and ${\mathrm{A}}_4$ correspond to the time course of ${\text{p53}}_{\mathrm{A}}$ (green), and ${\text{p53}}_{\mathrm{M}}$ (red) for the parameter set (0.5,500.0), (1.6, 500.0), (1.27, 100.0), and (2.0, 200.0) respectively on the heat map A. ${\mathrm{B}}_1$, ${\mathrm{B}}_2$, ${\mathrm{B}}_3$, and ${\mathrm{B}}_4$ correspond to the time course for the parameter set (0.5,500.0), (1.3, 500.0), (0.96, 100.0), and (2.0, 200.0) respectively on the heat map B. And ${\mathrm{C}}_1$, ${\mathrm{C}}_2$, ${\mathrm{C}}_3$, and ${\mathrm{C}}_4$ for the parameter set (2.0,500.0), (4.0, 500.0), (2.0, 100.0), and (4.0, 100.0) respectively on the heat map C. First, and second term in the parameter set correspond to magnitude of stress (I), and $K_3$ respectively. Green, yellow, and red region indicate active, apoptotic, and cancer phase respectively on the heap map (A–C).

Figure 7

The left column show decaying stress. Second column, (A–C), show two parameter cancer recovery behavior for the parameter set (magnitude of stress, $K_3$), (magnitude of stress, $K_4$), and (magnitude of stress, $\lambda$) respectively driven with same decaying stress. ${\mathrm{A}}_1$, ${\mathrm{A}}_2$, ${\mathrm{A}}_3$, and ${\mathrm{A}}_4$ correspond to the time course for the parameter set (3.25,700.0), (3.8, 700.0), (3.25, 500.0), and (3.8, 500.0) respectively on the heat map A. ${\mathrm{B}}_1$, ${\mathrm{B}}_2$, ${\mathrm{B}}_3$, and ${\mathrm{B}}_4$ correspond to the time course for the parameter set (3.25,300.0), (3.8, 300.0), (3.25, 1.0), and (3.8, 1.0) respectively on the heat map B ($K_3=500.0$). ${\mathrm{C}}_1$, ${\mathrm{C}}_2$, ${\mathrm{C}}_3$, and ${\mathrm{C}}_4$ correspond to the time course for the parameter set (3.0,0.07), (6.0,0.07), (3.0,0.03), and (6.0,0.03) respectively on the heat map C ($K_3=1000.0$, $K_4=10.0$). On the heat map green color shoes lowest recovery time, while red shows highest recovery time or no recovery (in case of cancer).