Bifurcation Analysis on p53 Dynamics and Its Physiological Functions

Cell physiological behaviour controlled by p53 can be analysed using bifurcation theory that views cell behaviour as a dynamical system (Tyson et al., 2001) as discussed in Chapter 3. The bifurcation analysis was performed using a MATLAB package called DDE-BIFTOOL (Engelborghs et al., 2002), a tool for bifurcation analysis of steady state solutions and periodic solutions of delay differential equations with constant delays. A saddle-node bifurcation and a supercritical Hopf bifurcation were obtained from the MATLAB scripts that use DDE-BIFTOOL and are shown in Figure 4-9. As mentioned previously (in Chapter 3), a general description of supercritical Hopf bifurcation from Strogatz’s textbook on nonlinear dynamics (Strogatz, 1994) is that “a supercritical Hopf bifurcation occurs when a stable spiral changes into an unstable spiral surrounded by a small, nearly elliptical [stable] limit cycle.” The meaning of a limit cycle is defined as “any simple oriented closed curve trajectory that does not contain singular points” (singular points are steady states at which the phase flow is

stagnant) (Edelstein-Keshet, 1988). Figure 4-9 shows three distinct responses of p53 depending on the bifurcation parameter ATM auto-activation kauto (damage signal amplification rate) (we used the

same bifurcation parameter as Sun et al. (2011)). These are saddle-node bifurcations SN1 and SN2 and supercritical Hopf bifurcations HB1 and HB2. Time course simulations for p53 in the regions of SN1, HB1 and HB2 are presented in Figure 4-10. Figure 4-10 (a) shows that for low kauto values (kauto ≤

0.04), the steady state value of the total p53 levels are very low, and starts to pulse (with only few pulses) after the saddle-node SN1 at the threshold value kauto=0.04 with characteristics of damped

oscillations that reach a higher steady state value at kauto=0.041. Figure 4-10 (b) shows that p53

pulses are excitable once it crosses the activation threshold. After supercritical Hopf bifurcation HB1 at kauto=0.0492, p53 starts to oscillate (repeated pulses) with a growing stable limit cycle with a

maximum and minimum amplitude shown in blue colour in Figure 4-9 and Figure 4-10 (c). It shows for the default model parameter value kauto=0.07 it generates oscillatory behaviour. After a second

Hopf bifurcation HB2 point at kauto=0.3434 (Figure 4-9), a stable steady state of the total p53 levels at

a much higher level is attained (Figure 4-10 (d)). These results lead us to conclude that p53 dynamics are both pulsatile (few pulses) and oscillatory depending on the strength of the stress signal.

(Although a series of repeated pulses with fixed amplitude and duration is the same as limit cycle oscillations, in literature, p53 dynamics with repeated pulses are known interchangeably as pulses (Batchelor et al., 2008; Lahav et al., 2004) and oscillations (Geva-Zatorsky et al., 2006). But, theoretical and computational biologists refer to it as sustained oscillations (Goldbeter, 2002)).

Figure 4-9 Bifurcation diagram of the system for total p53 (in µM) against the parameter kauto, the

ATM auto-activation rate. The dotted and dashed lines represent the stable and

unstable steady states, respectively. The saddle-node bifurcation occurs at the kauto=0.04

and kauto=−0.036. The supercritical Hopf bifurcation occurs at the kauto=0.0492 and

kauto=0.3434. Saddle-node bifurcation was not found in Sun et al. (2011) model.

We hypothesise that the above bifurcation analysis gives insight into four modes of p53 behaviour that link to the ways p53 decides cell physiology. Figure 4-10 (a)-(d) show the behaviour of total p53 protein concentration (in µM) in response to four different values of the parameter kauto. First, for

low value of kauto = 0.04, p53 shows a low steady state value of 0.0116 µM that corresponds to

homeostasis. Second, for kauto = 0.041, p53 levels pulse (few pulses with damped oscillations) that

activates DNA damage repair genes. Third, for kauto =0.07, p53 levels oscillate with a stable limit cycle

that activates cell cycle arrest and DNA damage repair genes. Finally, for high value of kauto = 0.4, p53

levels show a much higher steady state value of 0.259 µM that leads to apoptosis. As proposed by Tyson (2006), exposure of cells to radiation induced DNA damage can first cause ATM auto-activation kauto to increase and then decrease, as damage is repaired. Theoretically, this increase of parameter

value in kauto crosses the first saddle-node bifurcation point SN1 giving rise to a small number of p53

pulses (with damped oscillations). The existence of saddle-node bifurcation was a novel result that was not found in Sun et al. (2011) model and could be due to the positive feedback loop of p53 auto- regulation introduced in our model (positive feedback loop in general contributes to saddle-node bifurcation or bistability). Then as explained by Tyson (2004, 2006), its increase crosses the first Hopf

bifurcation point HB1 giving rise to robust limit cycle oscillations. We assumed that kauto above the

second bifurcation point HB2 leads to a very high steady state of p53 protein levels corresponding to p53 activation of apoptosis in response to severe damage. This analysis is supported by experimental results (Lai et al., 2007) that suggest that high p53 levels induce apoptosis.

Figure 4-10 The time course simulations for: (a) kauto =0.04; (b) kauto =0.041; (c) kauto =0.07; and (d) kauto

=0.4.

The illustration of the growing stable limit cycle after the first Hopf bifurcation point at kauto=0.0492 is

shown in Figure 4-11. It is a typical behaviour observed in dynamical systems where a stable steady state is lost after the supercritical Hopf bifurcation point and a stable limit cycle arises with growing amplitude (Goldbeter, 2002) for increasing kauto value as shown in Figure 4-11 (a) and (b) where kauto

Figure 4-11 Phase plane diagrams showing the growing stable limit cycle after the first Hopf bifurcation point at kauto=0.0492. These diagrams data are generated from XPP and

redrawn using MATLAB.

The oscillations emanating from the supercritical Hopf bifurcation of stable limit cycle with growing amplitude generated from DDE-BIFTOOL had period ranging from 4 to 6 hours as shown in Figure 4- 12 (a), which is consistent with the experimental findings (Geva-Zatorsky et al., 2006; Loewer et al., 2010). Further analysis of the frequency of the oscillations in Figure 4-12 (b) shows that the limit cycle is born at the bifurcation with a non-zero frequency and this feature classifies p53 excitable dynamics as Type II excitability (Rue & Garcia-Ojalvo, 2011). Type I excitability typically happens when the limit cycle is born at the bifurcation with zero frequency (Rue & Garcia-Ojalvo, 2011). The classification of Type I and Type II excitability has been discussed in Chapter 2 (please refer to Section 2.5.1 and Figure 2-19). These classifications are based on the study of neuronal excitability. Our results suggest that the information processing in the p53 network’s DNA damage response conforms to Type II excitability.

Figure 4-12 The period of the oscillations for different parameter values of kauto, ATM auto-activation

rate after the first Hopf bifurcation point HB1 at kauto =0.0492 (the thin vertical line

indicates this HB1 point) and the standard model parameter value of kauto =0.07 (the red

dotted vertical line). (b) The frequency of the corresponding oscillations in (a). The frequency of the limit cycle is born from non-zero frequency and this feature characterises p53 excitable dynamics as Type II excitability.