Two‐dimensional polynomial type canonical relaxation oscillator model for p53 dynamics

Abstract

p53 network, which is responsible for DNA damage response of cells, exhibits three distinct qualitative behaviours; low state, oscillation and high state, which are associated with normal cell cycle progression, cell cycle arrest and apoptosis, respectively. The experimental studies demonstrate that these dynamics of p53 are due to the ATM and Wip1 interaction. This paper proposes a simple two‐dimensional canonical relaxation oscillator model based on the identified topological structure of ATM and Wip1 interaction underlying these qualitative behaviours of p53 network. The model includes only polynomial terms that have the interpretability of known ATM and Wip1 interaction. The introduced model is useful for understanding relaxation oscillations in gene regulatory networks. Through mathematical analysis, we investigate the roles of ATM and Wip1 in forming of these three essential behaviours, and show that ATM and Wip1 constitute the core mechanism of p53 dynamics. In agreement with biological findings, we show that Wip1 degradation term is a highly sensitive parameter, possibly related to mutations. By perturbing the corresponding parameters, our model characterizes some mutations such as ATM deficiency and Wip1 overexpression. Finally, we provide intervention strategies considering our observation that Wip1 seems to be an important target to conduct therapies for these mutations.

1 Introduction

p53 (tumour suppressor gene) network regulates the DNA damage response of the cell [1]. Studies show that p53 level is at low values under normal unstressed conditions [2]. However, upon the exposure of ionising radiation (IR), double‐strand breaks (DSBs) in DNA occur and p53 level starts to oscillate as pulses [3]. With the onset of p53 oscillation, cell cycle is arrested to avoid the proliferation of the damaged cell, and at the same time, DNA damage is being repaired by repair molecules [4]. If the DNA damage is not repaired in a certain period comprised a few pulses, then p53 level goes to the fixed high level and maintained at that level to trigger apoptosis [5–7]. If the damage is repaired, then p53 level goes back to the low level and normal cell cycle progression is continued [8]. This mechanism of DNA damage response gives cell some flexibility such that the decision of apoptosis is made only if attempts to repair fail [3, 6, 9, 10].

A 17‐dimensional mathematical model is proposed in [11] for describing the aforementioned flexibility of p53 dynamics showing three qualitative modes: low level (unstressed conditions), oscillations (cell cycle arrest), and a sustained high level (apoptosis). However, such high‐dimensional models suffer from pointing out critical structures underlying qualitative dynamics and from describing the overall mechanism in a concise way. Thus, it is inevitable to replace detailed explanations by abstractions, approximations, or idealisations for the sake of reaching mechanistic explanations.

In a work by the authors of this paper [12], to overcome these issues, a two‐dimensional reduced model is proposed by eliminating non‐essential elements, so enabling to interpret qualitative p53 dynamics from the reduced model perspective identifying the essential elements, such as ATM (ataxia‐telangiectasia mutated), Wip1 (wild‐type p53‐induced phosphatase 1), and P53DINP1 (P53‐dependent Damage‐Inducible Nuclear Protein 1). However, the reduced model developed in [12], although low dimensional, is highly non‐linear which does not allow to derive the analytical conditions regarding p53 dynamics.

In this paper, following a minimalist approach, we propose a two‐dimensional canonical relaxation oscillator model with only polynomial terms to describe the qualitative behaviour of p53 dynamics. The model is based on the interaction between ATM and Wip1 variables which have been observed as the most essential elements of the p53 network in a series of works [9, 11, 12]. Two‐dimensional canonical relaxation oscillator model could be more useful in comparison to the large‐scale models. Canonical models provide information [13] about the overall system behaviours due to the mechanistic explanations that they provide. A few to mention, Lotka–Volterra [14, 15], Goodwin [16], and Fitzhugh–Nagumo [17] canonical models have been found very useful in the contexts of ecology, biochemistry, and neurophysiology, respectively. The canonical approach reveals valid predictions by virtue of the modularity in biology, which allows for investigations both in molecular level and functional level.

The outline of the paper is constructed as follows. In Section 2, we identify the simplified topological structure of interactions between ATM and Wip1 proteins. In Section 3, we propose a two‐dimensional relaxation oscillator model with polynomial terms only based on the identified structure and we demonstrate that the model shows three main modes of p53 dynamics. In Section 4, we show how these different modes emerge from the model by mathematical analysis and give parametric conditions leading the three modes. In Section 4, we reveal the sensitive parameters that may be related to mutations and provide some intervention strategies.

2 Indispensability of ATM and Wip1 for three modes of p53 dynamics

In p53 network, DSBs upon the exposure of IR are detected by ATM [18, 19]. When there are DSBs in DNA, repair molecules form a complex with DSBs (DSBCs) [20, 21]. The produced DSBCs activate ATM by phosphorylation. Experimental studies show that even at low IR doses, causing a few DSBs, ATM is rapidly phosphorylated, so transforming to the active ATM (ATM*), suggesting that ATM is highly sensitive to DSBs [22]. Auto‐activation (positive feedback loop) property of ATM* allows for this rapid activation and gives the property of switch‐like behaviours of ATM. Mathematical models suggest that ATM has bistability property due to this auto‐activation [20, 23]. According to these suggestions, ATM* level stays at a low steady state in the absence of DSBCs and switches to a high steady state in the case of DSBCs. Activation of ATM is regarded as the first step of DNA damage response [22].

One of the important roles of ATM is signalling the DNA damage to p53 [24]. After IR exposure, p53 activity is elevated due to the phosphorylation by ATM. p53 activation increases the expression of Wip1, which is known to be an inhibitor of ATM*. Thus, this sequence of interactions form a negative feedback loop between ATM and Wip1 [25] (see Fig. 1a for an illustration).

Fig. 1.

Simplified diagram of ATM and Wip1 interaction (arrow‐headed lines represent enhancement, bar‐headed lines represent inhibition effect, and the arrow‐headed lines feeding back represent the auto‐activation)

(a) In normal unstressed conditions, ATM is in inactive form. Upon the exposure of gamma irradiation, DSBCs occur and ATM is activated. ATM phosphorylates p53, elevating active p53 (i.e. p53*) levels. p53* increases the activity of Wip1 which in turn deactivates ATM*, (b) By hiding p53*, ATM* can be considered as if it activates Wip1 in a direct way, (c) In apoptosis, Wip1 feedback loop is cut off such that ATM* cannot activate Wip1, thus Wip1 cannot deactivate ATM*. So, ATM* level is stuck at a high level indicating the initiation of apoptosis

Although other interactions exist, such as Wip1 inhibition of p53 and ATM inhibition of p53 inhibitor Mdm2 [3, 9], only the feedback loop interaction between ATM and Wip1 dynamics has been reported as the indispensable interaction for p53 oscillations [9]. In addition, there are experimental studies demonstrating that interplay between ATM and Wip1 is the key in regulation of cell cycle checkpoints and apoptosis [25–28]. The importance of ATM and Wip1 negative feedback loop in describing of p53 network dynamics has been demonstrated also in the mathematical models of [9, 11, 12]. In this paper, we focus on the construction of p53 dynamics by the topological structure of ATM and Wip1 interaction.

The ATM and Wip1 proteins separately or together have been included in the models that attempt to simulate p53 oscillations. For example, Mouri et al. [23] suggested a model which shows the activation of ATM by DSBs and reveal that ATM dynamics must possess the bistability property to account for the switch‐like behaviours. Ma et al. [20] proposed a model in which a bistable ATM sensor module switches p53‐Mdm2 oscillator on or off. However, this model does not include Wip1 feedback loop which is now known indispensable for oscillations [9]. By employing wet lab experiments, the work in [9] shows also that Mdm2 and p53 interaction is not even responsible for oscillations. On the other hand, Zhang et al. [11] proposed a comprehensive mathematical model, in which ATM switches p53 oscillations on or off. In the model of [11], feedback loop interaction between ATM and Wip1 is indispensable for oscillations. The in silico studies by Zhang et al. [11] describe the two‐phase dynamics of p53, in which p53 is driven into high levels only after ending the oscillation phase when the DNA damage is unrepairable. In the model of [11], the cut‐off of Wip1 feedback loop is the key to change of dynamics of p53 network, i.e. the transition from oscillatory to fixed high‐level state. The biological mechanism underlying the cut‐off of Wip1 feedback loop is the extinction of p53 proteins with ser15/20 phosphorylation (p53arrester), which carries the activation signal from ATM to Wip1 [11, 12]. With the extinction of p53 arrester protein in the medium (or dropping to a very low value), this activation signal is cut‐off.

There are several other models that try to replicate the oscillatory dynamics of p53 [29]; however, the models that keep ATM and Wip1 interaction on the forefront and study the high and low states as well as oscillations are rare [11, 12]. Focusing on the interaction between the bistable ATM characteristics and Wip1 feedback loop, in this paper, we propose a two‐dimensional canonical relaxation oscillator model that is able to replicate the three distinct qualitative behaviours of p53 network: (i) low equilibrium state, (ii) oscillations, and (iii) high equilibrium state. For the sake of obtaining lower dimensional models, p53 may simply be hidden, so ATM* is directly linked to Wip1 activity [9, 12] (see Fig. 1b for this elimination). In the proposed model, ATM has been chosen as the representative variable of p53 dynamics, due to the direct relation between p53 and ATM [9]. Three qualitative modes of p53 dynamics can be represented by the constructed canonical model as will be demonstrated in Section 3.

Low equilibrium state of p53* is observed when there is no DSBC activity. In this case, ATM* stays at low level, so is p53* indicating normal cell cycle progression [2]. The oscillation of p53* is observed when there is DSBC activity. In this case, ATM* level switches to a high equilibrium state rapidly. ATM* increases the activity of Wip1 which in turn deactivates ATM* even in the presence of DSBC. As ATM* level falls, Wip1 activity (level) decreases too. With the decrease in inhibitory Wip1 activity, ATM is again activated by DSBCs if DSBCs still persist. Thus, ATM repeatedly checks for the existence of DSBs [9]. If ATM is activated by the existent DSBCs, Wip1 level is re‐activated. As this sequence of activations/reactivations is repeated, ATM* and Wip1 oscillations are formed (Figs. 1a and b). Since p53 is on the feedback loop between ATM* and Wip1, we can further assume that p53* level oscillates too, indicating the presence of cell cycle arrest. Indeed, this is in agreement with biological findings where it has been shown that p53 oscillations result from the recurrent initiation of ATM activity [9]. High equilibrium state of p53* indicating apoptosis is observed when there are DSBCs and at the same time, Wip1 feedback loop is cut off. (See Fig. 1c for an abstraction of this cut‐off.) In this case, DSBCs activate ATM* and the increased ATM* level is stuck at a high level due to the absence of inhibitory Wip1 activity. With the increased ATM* level, p53* level also stays at a high level indicating the initiation of apoptosis.

The introduced two‐dimensional canonical relaxation oscillator model of this paper is developed based on the above explained abstract topological structure of p53 dynamics.

3 Two‐dimensional polynomial type canonical relaxation oscillator model for p53 dynamics

We propose a canonical relaxation oscillator model with polynomial terms only, for p53 network as defined by the ordinary differential equations (ODEs) of two state variables in (1) and (2) under the constraint given in (3), which is able to replicate the three qualitative behaviours of p53 network. When relevant parameters of the model are adjusted properly, the model is able to show: (i) low equilibrium state, (ii) oscillations, and (iii) high equilibrium state. Although the proposed model shows these three behaviours distinctively by means of the appropriate choices of the parameters, the model can be extended to higher dimensions with the introduction of new state variables and new ODEs, so manipulating those relevant parameters automatically. Herein, we focus on the emergence of the three behaviours from the topological structure between the bistable ATM and Wip1 feedback loop and we propose the following canonical model with polynomial terms, each has the interpretability of a known interaction between ATM and Wip1

$$\dot{x}=u=\frac{1}{{\tau }_1}\left(-x\left[r\left(x^2-\left(a+b\right)x\right)+ab+cy-rd\right]\right)$$ (1)

$$\dot{y}=v=\frac{1}{{\tau }_2}\left(z+mx-ny\right)$$ (2)

$$ab-d+\frac{cz}{n}<0$$ (3)

Above, x and y stand for ATM* and Wip1 levels, respectively, $\dot{x}=(\mathrm{d}x/\mathrm{d}t)$ and $\dot{y}=(\mathrm{d}y/\mathrm{d}t)$. The parameters $a,b,c,d,n,z,{\tau }_1,\text{and}{\tau }_2$ are positive constants while m is non‐negative. The parameter $r$ is an external signal modelled as a constant whose value may change between 0 and 1 indicating the severity of the DNA damage. The inequality in (3) ensures the existence of a trapping region when $r=1$, which leads to oscillations as detailed in Section 4.2.1. The model is specifically given in parametric form so that it can be used as a guide in developing extension models or producing models that are more complex in order to fit a particular experimental data. In addition, (3) draws the line between healthy functioning cells and potentially cancerous cells as will be detailed in Section 5.

The right side of (1) is set as a third‐order polynomial, since bistability dynamics can be obtained with having at least three equilibrium points, two of which are asymptotically stable and the third is a repeller [30, 31]. In this way, bistable dynamics of ATM is constructed by the minimalist approach followed in this paper. Negative feedback loop to be provided by Wip1 can be obtained with introducing the term $-x\left(cy\right)$ in (1) and the term mx in (2). The parameter m is used for modelling the cut‐off of Wip1 negative feedback loop as with setting it to zero (Fig. 1c ). The positive values of m are for modelling the link between ATM and Wip1. To possess the desired p53 dynamics, the values of the model parameters must be chosen such that the constraint in (3) is satisfied. Throughout the analyses in the paper, $b>a$ is assumed with no loss of generality.

The parameter r in (1) models the input $N_c/\left(1+N_c\right)$, having a saturation property, where $N_c$ is the number of DSBCs. As $N_c$ increases, the term r goes to 1 asymptotically. This assumption is fitted to the consideration of [11] in modelling the activation of ATM by DSBCs. If there is no DSBC activity ($N_c=0$), indicating no DNA damage, then the parameter r becomes zero.

In the introduced model, the variable x (ATM) is regulated negatively by y (Wip1), while y is regulated positively by x. In addition, x has a self‐activation property with the term ‘rdx’ and ‘$\left(a+b\right)x^2$’, which stands for the auto‐activation property of ATM. Wip1 is known to reset ATM activity even in the presence of DNA damage [25]. Thus, the term ‘$-xcy$’ is included in (1) for the strong inhibition property of Wip1 on ATM. When the terms are expanded, ‘$-r{x}^3$’ appears, which is needed for bistability property of ATM [20, 23] when $r=1$ and also stands for the self‐degradation of ATM together with the term ‘$-xab$’.

Equation (2) has a constant production rate of z. The term ‘mx’ stands for the activation of y by x and the term ‘$-ny$’ stands for the degradation of y. Since y feeds back negatively to x, this means x suppresses itself after a time delay by a feedback loop.

From systems theoretical perspective, the intuitive idea behind the model derives from the relaxation nature of oscillations in the reduced oscillator model presented in another paper by the authors [12]. Mimicking those oscillations in [12], the criticality lies in the interplay between the terms $\left(x^2-\left(a+b\right)x\right)+ab$, i.e. $\left(x-a\right)\left(x-b\right)$ and $cy-d$ when $r=1$ in (1), which can be seen via a dynamic route approach assuming a large time scale separation (for a similar analysis, see [12]). More detailed information about how the model is designed from a dynamic route approach perspective can be found in Supplementary material (see Figs. S1–S3 and Table S2).

Since the bistable ATM dynamics and Wip1 negative feedback loop constitute the backbone of p53 network dynamics, a biologically meaningful mathematical model must have these properties. Equation (1) provides the bistable property of ATM, while (2) provides the feedback loop property of Wip1. Owing to the direct relation between ATM* and p53 dynamics, x can be considered as the representative variable of p53 dynamics as well. Fig. 2 gives time evaluation of x level demonstrating the aforementioned three distinct modes of p53 dynamics for the constant parameters $P^{\ast }=\{a=5,b=10,c=15,d=70,n=0.8,$ $z=0.5,{\tau }_1=1,{\tau }_2=1\}$, and adjustable (mode design) parameters $m\in \left\{0,1.25\right\}$ and $r\in \left\{0,1\right\}$. Herein, the low equilibrium state mode is obtained when $r=0$ and $m=1.25$, the oscillation when $r=1$ and $m=1.25$, and high equilibrium state when $r=1$ and $m=0$. These selected values of the parameters will be used throughout the analyses, unless stated otherwise.

Fig. 2.

Numerical simulation of the ATM‐Wip1 oscillator defined by ( 1) and ( 2) replicating three qualitative behaviours of p53 dynamics. (a.u. stands for arbitrary unit.) ATM is at low state in the intervals [0–20] and [30–50], provided that no damage exists, i.e. $r=0$. ATM oscillates in the intervals [15, 30] and [50, 70] provided that DSBCs occur and Wip1 feedback loop is on, i.e. r = 1 and m = 1.25. ATM stops oscillating and goes to the high state, so initiating apoptosis after 70 a.u. provided that DSBCs still exist but Wip1 feedback loop is cut off, i.e. r = 1 and m = 0

Since ATM* (x) dynamics is known to be fast and Wip1 (y) feedback is relatively slow, the proposed model must also account for this time scale separation property. Indeed, the mechanism of the proposed model is suitable to be split into fast and slow dynamics for this timescale separation since the dynamics of $x$ naturally tends to be faster than the dynamics of y, due to the jumps in x dynamics as will be detailed in the following sections. We provide also time scaling parameters ${\tau }_1$ and ${\tau }_2$, for further tuning the periods of the oscillations.

4 Analysis of three modes of the model

To demonstrate the three modes of the model, we employ a parametric local stability analysis in this section. We show which parameters and which ranges of these parameters are critical for the existence of any of these three dynamical behaviours. The analysis of the model is tractable due to the polynomial nature of the non‐linearities.

In the analysis of a model, the location and the Lyapunov stability of equilibrium points are of great importance. The location of equilibriums of a two‐dimensional system given with ODEs can be found by the intersection of nullclines that are defined as $\dot{x}=0$ and $\dot{y}=0$ in $x-y$ space. Stability information of these equilibria can be obtained from the local stability analysis that involves computation of eigenvalues of Jacobian matrix evaluated at those equilibria. There are three nullclines of the system. Equation (1) has two nullclines each of which makes $\dot{x}$ zero. One of the nullclines of (1) is the vertical axis given in (4), and the other nullcline is the set of points defined by (5). Equation (2) has one nullcline defined by (6)

$$x_{\text{nullcline}1}:x=0$$ (4)

$$x_{\text{nullcline}2}:y_x=-\frac{r{x}^2-r\left(a+b\right)x+ab-rd}{c}$$ (5)

$$y_{\text{nullcline}}:y_y=\frac{mx+z}{n}$$ (6)

As stated, the equilibria of the system are the points where $\dot{x}$ and $\dot{y}$ are both zero. Therefore, the equilibria can be found from the intersection of (4) and (6), and the intersection of (5) and (6). One of the equilibria is located at the intersection of (4) and (6), which is $\left(0,(z/n)\right)$. At the intersection of (5) and (6), there can be one or two equilibria depending on the value of r. Clearly, if r is zero, then the right‐hand side of (5) reduces to a constant, namely $-(ab/c)$, so that the resulting horizontal line (5) intersects with (6) at only one point for $m\ne 0$. If r is non‐zero, then the resulting parabola defined by (5) and the line (6) intersect at two points. Thus, the system has two or three equilibrium points depending on the parameter r under the constraint in (3) and the condition $m\ne 0$ whenever $r=0$.

If r is zero, then there is one more equilibrium point of the system in addition to the point $\left(0,(z/n)\right)$. In this case, (5) reduces to a constant function, $y_x=-(ab/c)$, and the intersection point of (5) and (6) becomes $\left(-(n/m)\left((ab/c)+(z/n)\right),-(ab/c)\right)$, which is in the third quadrant.

For a non‐zero r, two more equilibrium points in addition to $\left(0,(z/n)\right)$ can be found by equating the right‐hand sides of (5) and (6) and then obtaining the quadratic (7) whose roots correspond to these two equilibrium points

$$r{x}_{\text{eq}}^2-\left(r\left(a+b\right)-\frac{cm}{n}\right)x_{\text{eq}}+\left(ab-rd+\frac{cz}{n}\right)=0$$ (7)

By using the root formula, two equilibrium points can be expressed as in Table 1. Stability of the equilibria can be determined from the Jacobian matrix evaluated at those points. Since the proposed model (1)–(2) is of two‐dimensional, the Jacobian matrix obtained as (8) in terms of model parameters is a 2 × 2 matrix, having two eigenvalues

$$\mathit{J}=\left[\begin{array}{cc}1/{\tau }_1& 0\\ 0& 1/{\tau }_2\end{array}\right]\left[\begin{array}{cc}\frac{\mathrm{d}u}{\mathrm{d}x}& -c{x}_{\text{eq}}\\ m& -n\end{array}\right]$$ (8)

where

$$\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}x}& =-\left(r{x}_{\text{eq}}^2-r\left(a+b\right)x_{\text{eq}}+ab+c{y}_{\text{eq}}-rd\right)\\ & -x_{\text{eq}}\left(2r{x}_{\text{eq}}-r\left(a+b\right)\right)\end{array}$$

Table 1.

Equilibrium points of the proposed canonical system model

Parameters$r\text{and}m$	Equilibrium point at the intersection of (4) and (6)	Equilibrium point(s) at the intersection of (5) and (6)
$0<r\le 1$	$\left(0,z/n\right)$	$x-$ components of the equilibrium points: $x_{1,2}^{\ast }={\displaystyle \frac{\left(\left(a+b\right)-(cm/rn)\right)\pm \sqrt{{\left(\left(a+b\right)-(cm/rn)\right)}^2-(4/r)\left(ab-rd+(cz/n)\right)}}{2}}$
		$y-$ components of the equilibrium points: $y_{1,2}^{\ast }={\displaystyle \frac{m{x}_{1,2}^{\ast }+z}{n}}$
$r=0,m\ne 0$	$\left(0,z/n\right)$	$\left(-{\displaystyle \frac{n}{m}\left(\frac{ab}{c}+\frac{z}{n}\right),-\frac{ab}{c}}\right)$

For the two‐dimensional system, the stability of equilibrium points can be easily determined from the determinant and the trace of the Jacobian matrix without actually finding the eigenvalues. If the determinant $\text{det}\left(\mathit{J}\right)$ of the Jacobian matrix is negative, then these two eigenvalues are opposite in sign, indicating that the equilibrium is a saddle point. If $\text{det}\left(\mathit{J}\right)>0$, then the signs of the eigenvalues are both negative or positive depending on the sign of the trace. If $\text{det}\left(\mathit{J}\right)>0$ and the trace is negative, then the eigenvalues are both negative, meaning that the equilibrium point is asymptotically stable, i.e. an attractor. If $\text{det}\left(\mathit{J}\right)>0$ and the trace is positive then the eigenvalues are both positive, meaning that the equilibrium point is unstable, i.e. a repeller. Following this discussion, the stability analyses of the equilibrium points given in Table 1 will be carried out in Sections 4.1–4.4.

4.1 Normal cell cycle progression: low equilibrium state

In Section 4.1, we show how the proposed two‐dimensional canonical model is capable of replicating the low equilibrium state. This will be done by determining the position of equilibria by means of nullclines and the stability of these equilibria by the eigenvalue analysis of the Jacobian at the equilibria.

Non‐existence of DSBC activity in DNA, the case of normal cell cycle progression is modelled by setting r to zero, so the equations in (1) and (2) become:

$$\dot{x}=\frac{1}{{\tau }_1}\left(-x\left(ab+cy\right)\right)$$ (9)

$$\dot{y}=\frac{1}{{\tau }_2}\left(z+mx-ny\right)$$ (10)

The nullclines in this case are obtained as:

$$x_{\text{nullcline}1}:x=0$$ (11)

$$x_{\text{nullcline}2}:y_x=-\frac{ab}{c}$$ (12)

$$y_{\text{nullcline}}:y_y=\frac{mx+z}{n}$$ (13)

The system defined by (9) and (10) with $m\ne 0$ has only two equilibrium points. One of them is located at the intersection of (11) and (13), while the other one is located at the intersection of (12) and (13). As given in the last row of Table 1, these equilibrium points are:

$$\begin{array}{rl}\left(x_{\text{eq}1},y_{\text{eq}1}\right)& =\left(0,\frac{z}{n}\right)\\ \left(x_{\text{eq}2},y_{\text{eq}2}\right)& =\left(-\frac{n}{m}\left(\frac{ab}{c}+\frac{z}{n}\right),-\frac{ab}{c}\right)\end{array}$$

The stability of the equilibrium points can be determined by using the parametric Jacobian matrix (8) to be evaluated at these equilibrium points. Let us start with the first equilibrium point. The Jacobian matrix evaluated at $\left(x_{\text{eq}1},y_{\text{eq}1}\right)=\left(0,(z/n)\right)$ is obtained as

$${\mathit{J}}_{\left(x_{\text{eq}1},y_{\text{eq}1}\right)}^{r=0}=\left[\begin{array}{cc}1/{\tau }_1& 0\\ 0& 1/{\tau }_2\end{array}\right]\left[\begin{array}{cc}-\left(ab+\frac{cz}{n}\right)& 0\\ m& -n\end{array}\right]$$ (14)

whose determinant is

$$\det \left({\mathit{J}}_{\left(x_{\text{eq}1},y_{\text{eq}1}\right)}^{r=0}\right)=n\left(ab+\frac{cz}{n}\right)\frac{1}{{\tau }_1{\tau }_2}$$ (15)

The determinant in (15) is always positive since all parameters are positive. Therefore, we have to check the sign of the trace to determine the asymptotical stability of $\left(x_{\text{eq}1},y_{\text{eq}1}\right)$. The trace of (14) which is given as

$$\text{trace}\left({\mathit{J}}_{(x_{\text{eq}1},y_{\text{eq}1})}^{r=0}\right)=-\left(ab+\frac{cz}{n}\right)\frac{1}{{\tau }_1}-\frac{n}{{\tau }_2}$$ (16)

which is always negative. So, it is concluded that the equilibrium point $\left(x_{\text{eq}1},y_{\text{eq}1}\right)$ always behaves as an attractor when $r=0$ (i.e. no DSBC activity, so the indicator of normal cell cycle progression).

The Jacobian matrix evaluated at the second equilibrium point $\left(x_{\text{eq},2},y_{\text{eq},2}\right)=\left(-(n/m)\left((ab/c)+(z/n)\right),-(ab/c)\right)$ is

$${\mathit{J}}_{\left(x_{\text{eq}2},y_{\text{eq}2}\right)}^{r=0}=\left[\begin{array}{cc}1/{\tau }_1& 0\\ 0& 1/{\tau }_2\end{array}\right]\left[\begin{array}{cc}0& c\frac{n}{m}\left(\frac{ab}{c}+\frac{z}{n}\right)\\ m& -n\end{array}\right]$$ (17)

whose determinant is obtained as

$$\det \left({\mathit{J}}_{\left(x_{\text{eq},2},y_{\text{eq},2}\right)}^{r=0}\right)=-n\left(ab+\frac{cz}{n}\right)\frac{1}{{\tau }_1{\tau }_2}$$

The determinant of the Jacobian matrix evaluated at $\left(x_{\text{eq},2},y_{\text{eq},2}\right)$ is always negative. No matter what the trace of the Jacobian is, it implies that $\left(x_{\text{eq}2},y_{\text{eq}2}\right)$ which is always located in the third quadrant is a saddle point.

In conclusion, in the phase space, there is one asymptotically stable equilibrium state which is located at $\left(0,z/n\right)$, and one saddle equilibrium state located at $\left(-(n/m)\left((ab/c)+(z/n)\right),-(ab/c)\right)$. By looking at the organisation of equilibrium points and vector field as sketched in Fig. 3, it is clear that all trajectories that start in the first quadrant (i.e. the only biologically meaningful region) goes to the attractor $\left(0,z/n\right)$. From biological point of view, when $r=0$ indicating no DSBC activity, the representative variable ATM* goes to zero. Therefore, the proposed model is capable of replicating the low state of p53 dynamics when there is no damage, as demonstrated by the trajectories of the system in Fig. 3.

Fig. 3.

Low equilibrium state of x is observed when there is no DSBC activity, i.e. r = 0. Trajectories originated in the first quadrant all go to the stable equilibrium (0,z/n). The big arrows show the direction of flow in the

$x=0$

nullcline

4.2 Cell cycle arrest: oscillations

In this subsection, we show how the proposed model is capable of replicating the known oscillations of p53 network. When there are DSBs in DNA, the number $N_c$ of DSBCs raises from zero and the parameter r (i.e. $N_c/\left(1+N_c\right)$) goes asymptotically to 1 as $N_c$ increases. In this case, the oscillatory behaviour of p53 is known to be observed. Replacing the parameter $r$ with its limit value of 1 for the sake of simplicity, the system of (1) and (2) become:

$$\dot{x}=\frac{1}{{\tau }_1}\left(-x\left(x^2-\left(a+b\right)x+ab+cy-d\right)\right)$$ (18)

$$\dot{y}=\frac{1}{{\tau }_2}\left(z+mx-ny\right)$$ (19)

4.2.1 Determining locations and stability types of equilibria by using nullclines and Jacobian

Now, we will determine all equilibrium points of (18)–(19) under the constraint in (3). We first investigate the general characteristics of the nullclines since their organisation in the phase space is crucial for the characteristics of the system dynamics such as the location of equilibrium points and the directions of the trajectories. When $r=1$, the nullclines become:

$$x_{\text{nullcline}1}:x=0$$ (20)

$$x_{\text{nullcline}2}:y_x=-\frac{x^2-\left(a+b\right)x+ab-d}{c}$$ (21)

$$y_{\text{nullcline}}:y_y=\frac{mx+z}{n}$$ (22)

Nullcline $y_x$ is a second‐order polynomial (a parabola) that opens down since the coefficient $-1/c$ of the quadratic term is negative. It has one maximum point at $x=\left(a+b\right)/2$. The nullcline $y_x$ cuts the y‐axis at $\left(d-ab\right)/c$, while $y_y$ cuts the y‐axis at $z/n$. As imposed by the constraint of (3), we restrict the choice of parameters such that $z/n$ is always smaller than $\left(d-ab\right)/c$ to allow just one non‐trivial equilibrium point, namely an equilibrium other than the origin, in the first quadrant (Fig. 4) for all choices of parameters. y_x cuts x‐axis at two points as can be found easily by the root formula which is shown in (23) and (24)

$$a^{\mathrm{\prime }}=\frac{\left(a+b\right)-\sqrt{{\left(a+b\right)}^2-4\left(ab-d\right)}}{2}$$ (23)

$$b^{\mathrm{\prime }}=\frac{\left(a+b\right)+\sqrt{{\left(a+b\right)}^2-4\left(ab-d\right)}}{2}$$ (24)

$b^{\mathrm{\prime }}$ (24) is always positive, while $a^{\mathrm{\prime }}$ (23) is always negative, since constraint (3) also implies that the term ‘d−ab’ must be positive. Fig. 4 illustrates a possible organisation of nullclines in x–y space.

Fig. 4.

Possible nullcline organisations in the phase space for $r=1$ under the constraint in ( 3 ). $y_x$ cuts y‐axis at $\left(d-ab\right)/c$, while $y_y$ cuts y‐axis at z/n. Depending on the choice of parameters, $y_x$ may have intersection with $y_y$ on the left of the maximum point (as in (a)) or on the right of the maximum point (as in (b)) in the first quadrant. When $r=1$, $\left(x_{\text{eq}1},y_{\text{eq}1}\right)$ is always a saddle and $\left(x_{\text{eq}3},y_{\text{eq}3}\right)$ is always an attractor

(a) If $x_{\text{eq}2}>(a+b)/2$, then $(x_{\text{eq}2},y_{\text{eq}2})$ is definitely an attractor. Choices of parameters are $P^{\ast },m=0.25\text{and}r=1$, (b) If $x_{\text{eq}2}<(a+b)/2$, then $(x_{\text{eq}2},y_{\text{eq}2})$ may be a repeller or an attractor depending on the values of the parameters ${\tau }_1$, ${\tau }_2$, and n. For our choices of parameters, i.e. $P^{\ast }\text{and}m=0.75\text{and}r=1,$ it is a repeller

It will be shown in further below that, under the constraint in (3), there is always a unique (non‐trivial) equilibrium point in the first quadrant such that it is either asymptotically stable, preventing oscillations, or unstable allowing oscillations in the first (biologically meaningful) quadrant.

There are three equilibrium points when r is 1. One of the equilibrium points, $\left(x_{\text{eq}1},y_{\text{eq}1}\right)$, is $\left(0,z/n\right)$ which is at the intersection of (20) and (22) while the other two real equilibrium points, $(x_{\text{eq}2},y_{\text{eq}2})$ and $(x_{\text{eq}3},y_{\text{eq}3})$, are located at the intersections of (21) and (22), and found as:

$$x_{\text{eq}2,3}=\frac{\left(\left(a+b\right)-(cm/n)\right)\pm \sqrt{{\left(\left(a+b\right)-(cm/n)\right)}^2-4\left(ab-d+(cz/n)\right)}}{2}$$ (25)

$$y_{\text{eq}2,3}=\frac{m{x}_{\text{eq}2,3}+z}{n}$$ (26)

Since we limit our model with the constraint in (3), i.e. $ab-d+(cz/n)<0$, there are two non‐trivial equilibrium points; one, $(x_{\text{eq}2},y_{\text{eq}2})$, corresponding to the unique intersection point of (21) and (22) in the first quadrant, and the other, $\left(x_{\text{eq}3},y_{\text{eq}3}\right)$, in the third quadrant (Fig. 4).

After finding the location of the equilibrium points, now we investigate the stability of the equilibrium points by computing the Jacobian matrix at those points. Since r is 1, the Jacobian matrix (8) evaluated at $\left(x_{\text{eq}1},y_{\text{eq}1}\right)=\left(0,(z/n)\right)$ is obtained as

$${\mathit{J}}_{\left(x_{\text{eq}1},y_{\text{eq}1}\right)}^{r=1}=\left[\begin{array}{cc}\frac{1}{{\tau }_1}& 0\\ 0& \frac{1}{{\tau }_2}\end{array}\right]\left[\begin{array}{cc}-\left(ab+\frac{cz}{n}-d\right)& 0\\ m& -n\end{array}\right]$$ (27)

whose determinant is

$$\det \left({\mathit{J}}_{\left(x_{\text{eq}1},y_{\text{eq}1}\right)}^{r=1}\right)=n\left(ab+\frac{cz}{n}-d\right)\frac{1}{{\tau }_1{\tau }_2}$$ (28)

The determinant at the equilibrium point $\left(x_{\text{eq}1},y_{\text{eq}1}\right)$ is always negative, due to the constraint $ab+(cz/n)-d<0$ in (3). Thus, we conclude that $\left(x_{\text{eq}1},y_{\text{eq}1}\right)$ is a saddle point.

To calculate the Jacobian matrix at $(x_{\text{eq}2},y_{\text{eq}2})$ and $\left(x_{\text{eq}3},y_{\text{eq}3}\right)$, whose $x-$components, i.e. $x_{\text{eq}2}$ and $x_{\text{eq}3}$, are non‐zero, the Jacobian matrix (8) can be rewritten as:

$$\begin{array}{rl}& {\mathit{J}}_{(x_{\text{eq}},y_{\text{eq}})=\left(x_{\text{eq}2,3},y_{\text{eq}2,3}\right)}^{r=1}\\ & =\left[\begin{array}{cc}\frac{1}{{\tau }_1}& 0\\ 0& \frac{1}{{\tau }_2}\end{array}\right]\left[\begin{array}{cc}\left(\frac{1}{x}{\left.\frac{\mathrm{d}x}{\mathrm{d}t}\right|}_{x=x_{\text{eq}}}\right)-x_{\text{eq}}\left(2x_{\text{eq}}-\left(a+b\right)\right)& -c{x}_{\text{eq}}\\ m& -n\end{array}\right]\end{array}$$ (29)

Since, by definition, ${\left.(\mathrm{d}x/\mathrm{d}t)\right|}_{x=x_{\text{eq}}}$ is zero and $x\ne 0$ at $(x_{\text{eq}2},y_{\text{eq}2})$ and $\left(x_{\text{eq}3},y_{\text{eq}3}\right)$, then the Jacobian matrix in (29) reduces to:

$$\begin{array}{rl}& {\mathit{J}}_{(x_{\text{eq}},y_{\text{eq}})=\left(x_{\text{eq}2,3},y_{\text{eq}2,3}\right)}^{r=1}\\ & =\left[\begin{array}{cc}1/{\tau }_1& 0\\ 0& 1/{\tau }_2\end{array}\right]\left[\begin{array}{cc}-x_{\text{eq}}\left(2x_{\text{eq}}-\left(a+b\right)\right)& -c{x}_{\text{eq}}\\ m& -n\end{array}\right]\end{array}$$ (30)

whose determinant is

$$\begin{array}{rl}& \det \left({\mathit{J}}_{(x_{\text{eq}},y_{\text{eq}})=\left(x_{\text{eq}2,3},y_{\text{eq}2,3}\right)}^{r=1}\right)\\ & =\frac{2n{x}_{\text{eq}}}{{\tau }_1{\tau }_2}\left(x_{\text{eq}}-\frac{\left(a+b-cm/n\right)}{2}\right)\end{array}$$ (31)

If we calculate the determinant (31) at $\left(x_{\text{eq}2},y_{\text{eq}2}\right)$, it is seen that the determinant is always positive since $x_{\text{eq}2}$ is positive and $x_{\text{eq}2}>((a+b-cm/n)/2)$ as can be deduced from (25). If we calculate determinant (31) at $\left(x_{\text{eq}3},y_{\text{eq}3}\right)$, again it is seen that the determinant is always positive, since $x_{\text{eq}3}<\left((a+b-cm/n)/2\right)$ and $x_{\text{eq}3}<0$. Thus, the determinants of the Jacobian matrix at $(x_{\text{eq}2},y_{\text{eq}2})$ and $\left(x_{\text{eq}3},y_{\text{eq}3}\right)$ are both positive and we need to check the trace of the Jacobian to determine the types of stabilities of these equilibrium points. The trace of (30) is

$$\begin{array}{rl}& \text{trace}\left({\mathit{J}}_{(x_{\text{eq}},y_{\text{eq}})=\left(x_{\text{eq}2,3},y_{\text{eq}2,3}\right)}^{r=1}\right)\\ & =-\frac{1}{{\tau }_1}2x_{\text{eq}}\left(x_{\text{eq}}-\frac{\left(a+b\right)}{2}\right)-\frac{1}{{\tau }_2}n\end{array}$$ (32)

Since $x_{\text{eq}3}$ is negative, the trace (32) is always negative. This result together with $\det \left({\mathit{J}}_{\left(x_{\text{eq}3},y_{\text{eq}3}\right)}^{r=1}\right)>0$ implies that $(x_{\text{eq}3},y_{\text{eq}3})$ is always an attractor (see Fig. 4). The case for $(x_{\text{eq}2},y_{\text{eq}2})$ is subject to some conditions. If $x_{\text{eq}2}$ is greater than $((a+b)/2)$ (as in Fig. 4a ), then the trace (32) is always negative. Therefore, we conclude that if the condition $x_{\text{eq}2}>((a+b)/2)$ holds, then $(x_{\text{eq}2},y_{\text{eq}2})$ is definitely asymptotically stable. However, the reverse is not true. If $x_{\text{eq}2}<((a+b)/2)$ (as depicted in Fig. 4b ), then the trace can be negative or positive, so $(x_{\text{eq}2},y_{\text{eq}2})$ can be stable or unstable according to the values of the parameters ${\tau }_1$, ${\tau }_2$ and the degradation term of $y$, i.e. n.

Now that we have investigated the stability of equilibria when $r=1$, we will show that there is a set of parameter values that guarantees the oscillation. For this, we will apply Poincare–Bendixson theorem, which states that if there is a trapping region, there is either an attractor or a periodic solution in that trapping region [32]. The trapping region $\mathcal{R}$ for our case is defined in (34), which contains $(x_{\text{eq}2},y_{\text{eq}2})$ inside, as illustrated in Fig. 5. The left corner of the trapping region is (0,z/n) and the right corner is $y_y\left(\overline{b}\right)$, where $\overline{b}$ is the positive value that satisfies $y_x\left(\overline{b}\right)=z/n$. Since $(x_{\text{eq}2},y_{\text{eq}2})$ is the only equilibrium in this trapping region, its type of stability will be the determinant for the existence of a stable steady state or a periodic solution. If $(x_{\text{eq}2},y_{\text{eq}2})$ is stable, then all trajectories that start in the positive quadrant will be attracted to it, if $(x_{\text{eq}2},y_{\text{eq}2})$ is unstable, then there will be a periodic solution according to the Poincare–Bendixson theorem. The reasoning that $\mathcal{R}$ is a trapping region which allows oscillation if $(x_{\text{eq}2},y_{\text{eq}2})$ is unstable and allows attractor if $(x_{\text{eq}2},y_{\text{eq}2})$ stable, is given in the Supplementary material of this paper

$$\mathcal{R}=\left\{\left.\left(\begin{array}{c}x\\ y\end{array}\right)\in R^2|0\le x\le \overline{b};\frac{z}{n}\le y\le \frac{z}{n}+\frac{m}{n}\overline{b}\right)\right\}$$ (33)

Fig. 5.

Illustration of trapping region

$\mathcal{R}$

We have had deduced from (32) that if $x_{\text{eq}2}<\left(a+b\right)/2$, then there would be a set of parameter values that $(x_{\text{eq}2},y_{\text{eq}2})$ may be unstable. We now aim to find this exact interval where $(x_{\text{eq}2},y_{\text{eq}2})$ is definitely unstable so that oscillations can occur. For this, we re‐write (32) as a quadratic polynomial:

$$\text{trace}\left({\mathit{J}}_{(x_{\text{eq}},y_{\text{eq}})=\left(x_{\text{eq}2},y_{\text{eq}2}\right)}^{r=1}\right)=-\frac{1}{{\tau }_1}2x_{\text{eq}}^2+\frac{1}{{\tau }_1}{x}_{\text{eq}}\left(a+b\right)-\frac{1}{{\tau }_2}n$$

The trace equation is a parabola that opens down and has a maximum point at $\left(a+b\right)/4$. The roots of the trace equation are:

$$r_{\text{tr}1,2}=\frac{\left(a+b\right)\pm \sqrt{{\left(a+b\right)}^2-({\tau }_1/{\tau }_2)8n}}{4}$$ (34)

For the real roots, the following inequality is always satisfied:

$$0<r_{\text{tr}1}<r_{\text{tr}2}<\frac{a+b}{2}$$

The trace is always positive for the $x_{\text{eq}}$ values that satisfy $r_{\text{tr}1}<x_{\text{eq}}<r_{\text{tr}2}$, since it is a parabola that opens down (Fig. 6c ). In other intervals (i.e. $r_{\text{tr}1}>x_{\text{eq}}$ and $x_{\text{eq}}>r_{\text{tr}2}$), the trace is negative. In addition, in the case of imaginary roots, the trace is always negative as well (Fig. 6d ). Thus the interval, $r_{\text{tr}1}<x_{\text{eq}2}<r_{\text{tr}2}$, where $r_{\text{tr}1}$ and $r_{\text{tr}2}$ are real and positive numbers, is the exact interval, in which oscillations occur (Figs. 6a and c).

Fig. 6.

Even if the condition

$(x_{eq2},y_{eq2})<(a+b)/2$

is satisfied, depending on the parameter settings, the oscillations may not exist

(a) The oscillation exists for parameter settings $a=5,b=10,c=15,d=70,m=0.75,n=0.8,z=0.5,{\tau }_1={\tau }_2=1$ which makes $(x_{\text{eq},2},y_{\text{eq},2})$ unstable. Red arrows show the jumps in trajectories, which correspond to fast part of dynamics, (b) The oscillation does not exist for parameter settings $a=5,b=10,c=15,d=70,m=75,n=80,z=50,{\tau }_1={\tau }_2=1$, which make $(x_{\text{eq}2},y_{\text{eq}2})$ stable. Note that we have specifically chosen parameters such that $y_y$ do not change in these two cases as can be seen in (c) and (d), (c) The oscillation exists if $x_{\text{eq}2}$ falls between the roots of the trace equation as in the case of (a). The trace equation is scaled with 0.1 to fit into the graph, (d) The oscillation does not exist if the trace equation has no real root as in the case of (b). The trace equation is scaled with 0.01 to fit into the graph

As the parameter n or time scale ratio ${\tau }_1/{\tau }_2$ gets bigger, the interval $\left[r_{\text{tr}1},r_{\text{tr}2}\right]$ gets smaller, for instance, the situation changes from Figs. 6a and c to Figs. 6b and d. This finding is crucial since it points out that a fine tuning of the parameter n (i.e. Wip1 degradation term), and the time scale separation between the x and y dynamics (i.e. ${\tau }_1/{\tau }_2$) are the key for a larger space of parametric uncertainties that allow oscillation. As ${\tau }_1/{\tau }_2$ increases, the distinction between fast and slow dynamics disappears, since as ${\tau }_1$ gets bigger, fast ATM dynamics become slower. Consequently, the time scale separation and Wip1 degradation term are two important factors for oscillations: a fine time scale separation is indispensable for oscillations and sufficiently large n will destroy the oscillations no matter what the other parameters are. The jumps in trajectory correspond to the fast parts (Fig. 6a ), while other portions correspond to slow parts in the periodic trajectory. The proposed model can be characterised by jump phenomenon and time scale separation in dynamics. In fact, we had shown by (34) that time scale separation is critical for the oscillations. Thus, the proposed model is an example of a relaxation oscillator.

4.2.2 Hypothesis about oscillatory and non‐oscillatory p53 dynamics observed experimentally

In the experiment by Geva‐Zatorsky et al. [33], it is observed some portion of irradiated cells did not show oscillations. Although ATM and Wip1 structure is capable of showing oscillations as we have shown, there may be some parametric variations that remove oscillations, e.g. a high Wip1 degradation rate. It would be intriguing to conduct experiments to examine if those non‐oscillating cells have a higher Wip1 degradation rate, n, than oscillating cells.

4.3 Apoptosis: high equilibrium state

The high steady state of x value is established when there are DSBCs in DNA and at the same time, Wip1 feedback loop is cut off. In this case, Wip1 is not influenced by ATM (i.e. x) anymore. Therefore, we take the parameter m as zero or a small number such that the situation in Fig. 4a , i.e. $x_{\text{eq}2}>((a+b)/2)$, is satisfied, where $(x_{\text{eq}2},y_{\text{eq}2})$ is definitely asymptotically stable. In this case, all the trajectories that start in the region $\mathcal{R}\mathrm{\setminus }\left\{\left\{\left(x,y\right)\in R^2|x=0\right\}\right\}$ go to $\left(x_{\text{eq}2},y_{\text{eq}2}\right)$, whose x‐component indicates the high level of representative x variable. We do not repeat the stability calculations due to the length concerns; however, we do the simulation for m = 0 as in Fig. 7. It should be noted that the case of $m=0$ (or a small value of m) makes a similar effect to the one performed by P53DINP1, which cuts off Wip1 feedback loop by accumulating in Zhang's model [11, 12].

Fig. 7.

In apoptosis, Wip1 feedback loop is cut off (m = 0) and there is still DSBC activity (r = 1). In this case, there is only one stable equilibrium point at a high level in positive quadrant. There are two stable nodes. However, only the one on the right is physiologically meaningful and the stable node on the left is not reachable from positive quadrant

5 Characterisation of mutations by the proposed canonical model and recovery by the parameter n

The proposed model (1)–(3) is based on ATM and Wip1 interaction and now we evaluate our model's predictive ability to characterise the known mutations of ATM and Wip1 from the literature. Wip1 overexpression and ATM deficiency are two mutations that cause cancer. Wip1 (product of PP1MD gene) overexpression is a type of cancer that is characterised by the high levels of Wip1 in the cell [34]. This situation can be embedded into our model by increasing the Wip1 production rate z in (2) such that it violates the constraint (3). Herein, constraint (3) can be seen as a border between normal and deficient p53 dynamics. In this case, the proposed model loses its ability to oscillate. Thus, the cell becomes defective in producing oscillations, so leading to cell cycle arrest (Fig. 8a ).

Fig. 8.

Characterisation of mutations and recovery from mutations

(a) Wip1 over‐expression. The parameters are r = 1, a = 5, b = 10, c = 15, d = 70, m = 1.5, n = 0.8, z = 2.1. The parameter z is increased from 0.5 to 2.1 to model the mutation Wip1 over‐expression. In this case, condition (3) is violated. Nullclines do not intersect in the positive quadrant and ability to oscillate is lost, (b) Recovering the ability to oscillate in case of Wip1 over‐expression in (a). By increasing the parameter n from 0.8 to 2.4, the oscillations are recovered, (c) ATM deficiency. The parameters are r = 0.75, a = 5, b = 10, c = 15, d = 70, m = 1.5, n = 0.8, z = 0.5. The parameter r is decreased from 1 to 0.75 to model the mutation ATM deficiency. In this case, again condition (3) is violated. Oscillations are lost, (d) Recovering the ability to oscillate in the case of ATM deficiency in (c). By increasing the parameter n to 3.2, the oscillations are recovered

Since oscillations are important for arresting cell cycle [6], and any defect in cell cycle arrest is the prerequisite of cancer [35–37], we speculate that Wip1 overexpression may cause cancer by removing cell's ability to arrest cell cycle. To recover the oscillations, the parameter n can be increased again so that constraint (3) can be satisfied (Fig. 8b ). This is in agreement with findings [38, 39] that show Wip1 overexpression can be recovered by Wip1 degradation.

ATM deficiency is a mutation and characterised by ATM that loses sensitiveness to the damage. This mutation can be studied by analysing the parameter r. In our model, parameter r $(r=N_c/(1+N_c))$ is a measurement of ATM's detection level of DSBCs. We replace r by $r/k$ where $k>1$ with the aim of decreasing the ATM's sensitiveness. This parametric change moves the $y_x$ nullcline downward (Fig. 8c ). Thus, the nullclines may not intersect at an oscillatory interval and the cell may be defective in oscillations. Indeed, some studies demonstrate that mutation in ATM causes defective cell cycle checkpoint activation [37, 40, 41].

Darlington et al. [27] showed in wet lab experiment that the absence of Wip1 rescues ATM deficiency phenotypes in mice. We model the absence of Wip1 by increasing the degradation parameter, n. With the increase in n, the slope of $y_y$ decreases and for a sufficiently large n, it may have an intersection at a high level indicating initiation of apoptosis. Thus, we hypothesise that this rescue may be due to the initiation of apoptosis. As, in ATM deficiency $y_x$ nullcline moves downward, then this could be compensated by moving $y_y$ nullcline downward too and making them cut through interval of oscillation again (Fig. 8d ). However, further wet lab experiments are needed to be conducted to validate this hypothesis.

6 Conclusion

In this paper, we proposed a two‐dimensional relaxation oscillator model with polynomial terms for p53 network. The oscillator is a new oscillator for biological systems. This simple model is rich in dynamics and valuable for intuitive understanding of p53 dynamics. The results obtained by the introduced model emphasise the importance of ATM and Wip1 in p53 dynamics. Owing to its simplicity, the model has the potential of being used in further analytical studies of p53 network in the context of non‐linear systems theory. We emphasise the importance of Wip1 degradation term and show mathematically that this parameter is critical for recovering the cell's ability to oscillate in the case of mutations ATM deficiency and Wip1 overexpression. The criticality of time scale separation is also shown. The analyses on the canonical model revealed that the production rate of Wip1 and sensitivity of ATM to DNA damage are critical parameters that are related to cancer. Wip1 feedback loop is important for different behaviours of p53 dynamics. Thus, Wip1 feedback loop may be a control element for adjusting P53 dynamics. This perspective is important since it hypothesises that cancer treatment strategies might use Wip1 feedback loop as a target to control mutated cells, encouraging experimentalists to go in this direction. We showed that the proposed canonical oscillator model is a relaxation oscillator. We give an analytical range on the parameters that result in oscillations. This analytical range also includes the time‐scale separation property. The findings emphasise the importance of time scale separation in the dynamics of relaxation oscillations.