Dynamics of p53 and Wnt cross talk
Md.Zubbair Malik, Shahnawaz Ali, Md. Jahoor Alam, Romana Ishrat, R.K. Brojen Singh
aa r X i v : . [ q - b i o . S C ] M a r Dynamics of p53 and Wnt cross talk.
Md.Zubbair Malik , , Shahnawaz Ali , ,Md. Jahoor Alam , ,Romana Ishrat , R.K. Brojen Singh ∗ Centre for Interdisciplinary Research in Basic Sciences, Jamia Millia Islamia, New Delhi-110025, India. College of Applied Medical Sciences, University of Hail, P.O. Box 2440, Hail, Kingdom of Saudi Arabia. School of Computational and Integrative Sciences, Jawaharlal Nehru University, New Delhi-110067,India ∗ Corresponding author, E-mail: R.K. Brojen Singh - [email protected],
Abstract
We present the mechanism of interaction of Wnt network module, which is responsible for periodicsometogenesis, with p
53 regulatory network, which is one of the main regulators of various cellularfunctions, and switching of various oscillating states by investigating p − Wnt model. The variationin Nutlin concentration in p
53 regulating network drives the Wnt network module to different states,stabilized, damped and sustain oscillation states, and even to cycle arrest. Similarly, the change in Axinconcentration in Wnt could able to modulate the p
53 dynamics at these states. We then solve the setof coupled ordinary differential equations of the model using quasi steady state approximation. We,further, demonstrate the change of p
53 and
Gsk
Keywords: p
53 activation, Fixed point oscillations, Nutlin,Wnt, Sustain oscillations.
Introduction p
53, one of the largest hub in cellular network [1, 2], is considered to be one of the most importantkey regulators of cellular metabolic pathways [3, 4], regulates a number of cellular functions [5] and itsdynamics control even the fate of the cell when stress is given to the cell [6]. Further, p
53 suppression isobserved in various types of cancer either due to mutations or by ambiguous expression of control systemslike
M DM
HDM
M DM E p
53 [9,10], facilitates the diffusionof p
53 in the nucleus [11], which in turn lowers the level of p
53 within the cell [12]. In stressed cells, the p
53 triggers the cell to cell cycle arrest forcing it to choose its fate, either to repair or apoptosis [13–15]. p
53 dynamics is regulated by various signaling molecules as evident from various experimental andtheoretical reports [16]. On of the most important inhibitor of p − M DM
M DM p
53 pathway [18, 19],and inhibiting p
53 binding to
M DM
N utilin − a and 3 b are the two most active enantiomorphsthat are found to up-regulate the p
53 in p
53 dependent manner [21]. Further, it is also demonstratedthat nutlin -3 induces anti-angiogenic activities in endothelial cells probably via three mechanisms; firstby inhibiting endothelial cell migration; second by inducing cell cycle arrest; and third by increasingapoptotic tendency in endothelial cells [22]. Further, it was also shown that nutlin -3 treatment in thesecells leads to accumulation of p
53 [23], indicating the important impacts of nutlin -3 which interferencesthe p M DM
N utilin -3 stabilizes the p
53 dynamics and causes the activation of survival pathways [23].Somitogenesis in vertebrates is periodic formation of somites (vertabrae precursors) in the anteriorpresomitic mesoderm tissue (PSM) controlled by complex gene network known as segmentation clock [24],where, Notch, Wnt pathways [25, 26] and fibroblast growth factor (FGF) are the main components [27].Further, it has been shown that these three main pathways generate rhythms of specific oscillation rela-tionships [27], and cross-talk among them [28] as a basis of spatio-temporal self-organization of patternsduring sometogenesis [29]. This physiological oscillations is responsible for periodic spacing of somites,dynamic structures and regular segmented development in vertebrates [29]. In some studies in normal(stress) and cancerous cells, it has been reported that p
53 and Wnt modules are being coupled andcross-talk between them via different intermediate proteins or genes or signaling molecules [30–32]. Thestudies also reported that p
53 network suppresses the Wnt signaling cascade [31]. The Wnt signalingplays a pivotal role in determination of cell fate, which may probably through p
53 interaction [31, 32],but still it is not clear yet. Further, several other works reported that the intercellular signaling of Wntdepends on the β -catenin and glycogen synthase kinase -3 ( GSK -3) [33]. Moreover, p
53 can regulate thedynamics of
Axin
GSK -3 [34]. Recently, it is also reported that the miR −
34 family, which isdirectly transcribed by the p
53, links p
53 with Wnt signaling [32], in this process, and a set of
GSK -3-related Wnt genes, are directly targeted by p
53 and miR -34. These observations clearly suggest the closeconnection between the p
53 functionalities and Wnt in stress and cancer cells [30, 32]. However, the wayhow p
53 interacts with Wnt during sometogenesis and its impact on somite organization/reogranizationat molecular level is still an open question.
GSK -3, which is an important coupling molecule between p
53 and Wnt pathways, is widely expressedin broad range of cellular processes and being a Ser/Thr kinase it is a multifunctional protein [35, 36]. Itis active under resting condition and act as a core regulator in various disease pathways like cancer [35].Being a kinase, it confers selectivity and substrate specificity [35]. It also largely acts as a phosphorylatingand an important signaling agent for the degradation of large number of proteins, such as β -catenin inWnt pathway [37]. Further, it is a central player in Wnt signaling pathway recruited with the Axin
GSK -3 shows itsactivity in the protein destruction complex with
Axin
2, APC (adenomatous polyposis coli) and otherpartners mediating the destruction and phosphorylation of β -catenin [39]. GSK -3 also mediates p p GSK − p p
53 dynamics, and vice versa, are needed to be investigated in order to understandhow segmentation clock works.We study the coupling of Wnt oscillator which is one of the three important clocks in sometogenesis invertabrates and p
53 regulatory network to understand how does these two oscillators regulates each otherand process information during somite formation and pattern organization. Since p
53 interferes variousother cellular functions, it will be interesting to understand its regulating impacts on segmentation clockand vice versa. Presently we study these phenomena by modeling this p Materials and Methods
A. Coupling p53-Wnt Oscillators Model
The Wnt signaling pathway is authorized by β -catenin regulation via various intermediate interactionsteps (Fig. 1) [40,41]. In this signal processing, Wnt promotes the synthesis of Dishevelled (Dsh) protein,and then inhibits GSK
GSK
3, then interacts with
Axin (with rates k and k forforward and backward reactions as in Fig.1) in the presence of β -catenin to form transient complex which Wnt k P53 OscillatorWnt Oscillator
Dsh p53_MDM2p53_Gsk3 k k k k k k k k k Gsk3Axin βcatp
Axin_Gsk3 p53
MDM2
N t MDM2Nutilin βcat k k k k k k k k k k k k βcatp Nut_MDM2 βcat k k k k k k k Axin_mRNA βcatn k k k k MDM2_mRNA
Figure 1.
The schematic diagram of biochemical network of p
53 and Wnt modules and their cross talkvia GSK3.in turn phosphorylates β -catenin into β -catenin-P (with rate rate k ) [27, 41]. The β -catenin-P degradedquickly with rate k , on the other hand it is dephosphorylated with rate k to form freed β -catenin andthen degraded with a rate k [42]. The free β -catenin is then transported to nucleus ( β -catenin-n) withrate k , on the other hand β -catenin-n is transpoted from nucleus to cytoplasm with another rate k .Then β -catenin-n promotes the transcription of set of target genes including mAxin (mRNA of Axin,specifically homolog Axin2 in PSM tissue) at the rate k , with rate k , and mAxin also degrades with arate k [43]. On the other hand mAxin translates to Axin with a rate Axin protein which also degradeswith a rate k . This mechanism of Axin is just like a negative feedback loop which drives the oscillationsin the network components. Thus Axin β -catenin by forming a negative feedback loop [44].The Wnt pathway communicates with the p
53 Oscillator [45–47] through
GSK p − GSK p
53 at the rate k . Then this complex p − GSK k to the free p
53 and
GSK p
53 and
M DM
M DM k to form a complex N ut M DM
2, and then this complex dissociates to releaseback free Nutlin and
M DM
2. Then Nutilin indirectly interacts with p
53 via
M DM p
53 and
M DM
2. The Nutilin then degrades with rate constant k . Thusthe activated p
53 can now go to the nucleus and initiate the transcription of a number of target genesincluding
M DM M DM p
53 through a positive feedbackloop mechanism by making complex [49, 50].
Mathematical model of the network: Quasi steady state approximation
The stress p − W nt regulating network we study is defined by N = 13 molecular species (Table 1)corresponding to the reaction network description provided in Table 2. The state of the system at anyinstant of time t is given by the state vector, ~x ( t ) = ( x , x , . . . , x N ) T , where, N = 13 and T is thetranspose of the vector. By considering feedback mechanism of in p
53 and Wnt oscillators and couplingreaction channels of the two oscillators, we could able to reach the following coupled ordinary differentialequations (ODE), ddt ~x ( t ) = F F ... F (1)where, the functions in the equation (1) { F i ( x , x , . . . , x N ) } , i = 1 , , . . . ,
13 are given by, F = k (cid:20) k x − k x k + x − k x x + k ( k − x ) (cid:21) (2) F = k " k + k ( x ) k ( k ) k + ( x ) k − k (cid:18) x k + x (cid:19) (3) F = k (cid:20) k − k (cid:18) k k + k (cid:19) (cid:18) x k + x (cid:19) (cid:18) k − x k (cid:19) + k (cid:18) x k + x (cid:19)(cid:21) − k [ k x − k x + k x ] (4) F = k (cid:20) k (cid:18) k k + k (cid:19) (cid:18) x k + x (cid:19) (cid:18) k − x k (cid:19) − k (cid:18) x k + x (cid:19) − k x (cid:21) (5) F = k [ k x − k x ] (6) F = k ( − k x x + k k − k x ) − ( k x x + k x ) (7) F = k − k x x + k x − k x x + k x (8) F = k x − k x + k x − k x x + k x − k x x (9) F = k x − k x + k x (10) F = − k x + k x x − k x (11) F = k − k x x + k x − k x (12) F = k x x − k x (13) F = k x x − k x (14)The coupled ODEs (equation (1)) are very difficult to solve analytically. However, one can get ap-proximate analytical solution of these equations by using quasi-steady state approximation [51, 52]. Ingeneral, any biochemical reactions network involves two basic types of reaction, namely slow and fastreactions [51]. Therefore, the N = 13 variables in the system can be divided into sets of slow and fastvariables respectively. If ~x s ( t ) = ( x , x , . . . , x l ) T and ~x f ( t ) = ( x l +1 , x l +2 , . . . , x N ) T are slow and fastvariable vectors, then ~x ( t ) = (cid:0) ~x s , ~x f (cid:1) T . Then from equation (1)-(14) along with Table I, we have, ~x s ( t ) = x x x x x x ; ~x f ( t ) = x x x x x x x (15)Since the rate of complex formation is fast, and after this fast complex formation, the fast variablesimmediately retain steady state (equilibrium). Then using Henri-Michaelis-Menten-Briggs-Haldane ap-proximation [53], one can take quite fair assumption that the ODEs of variables of complex molecularspecies reach steady state equilibrium quite fast compared with the time evolution of slow variables [51].Then one can straight forward put, d~x f dt ≈ ~x ( t ) dt = ~x s ( t ) dt = ddt x x x x x x (16)and the ~x f become the following steady state values, ~x s → x ∗ x ∗ x ∗ x ∗ x ∗ x ∗ x ∗ → constant (17)The thirteen ordinary differential equations are now reduced to six ordinary differential equations. Thisallows to simplify the complex mathematical model to get approximate solutions of the system numericallysaving computational cost or analytically fron the reduced system if possible.We used standard Runge-Kutta method (order 4) of numerical integration to simulate equation (re-fode) to find the solution of the variables listed in Table 1 for the parameter values given in Table 2. Wethen analysed the constructed mathematical model to get possible approximate analytical solutions ofthe variables (slow variables) using quasi-steady state approximation. Results and Discussion
The cross talk between p
53 and Wnt via Gsk3 subjected to different stress conditions induced by nutlin,as well as Axin concentrations available in the system. In order to understand how they regulate eachother, we numerically simulated the coupled ordinary differential equations (1) with the parameters listed
Figure 2.
The dynamics of Axin and β -catenin for different values of k i.e. 0.06, 0.18, 0.24, 0.6 and1.2 for fixed vale of k = 0 .
02. The right hand panels are the two dimensional plots the Axin and β -catenin with nutlin for the same set of parameter values showing different state behaviors.in Table 2. The simulation results for the variable in the model are presented and compared to understandthe switching of different oscillating states induced by different stress conditions. The reason is that thesestates of the oscillations in Wnt network are responsible for different behavior of regular somite formation,and many other periodic phenomena during sometogenesis. The roles of Axin Gsk β -catenin andNutilin on the p
53 network dynamics is discussed in the context of the model we presented. Further,the signal processing between the two coupled network modules via the
Gsk p
53 generating effect due to
N utilin -3 a . On the other hand, we also studied theregulation of p
53 by Wnt due to available Axin concentration in the system.
Driving Wnt oscillating states by p The p
53 in p
53 regulatory network can be induced stress by the available concentration of stress inducermolecular species,
N utlin − a . The concentration level of N utlin − a is proportional to the creation rateconstant of N utlin − a , k (Fig. 1). Therefore, computationally we can able to monitor the amount ofstress induced to p
53 by changing the value of parameter k and supervising the dynamical behavior of p
53 (Fig. 2 and 3).The dynamics of p
53 (Fig. 2 upper left panels) for small values of k ( k h .
07) show single spike dueto sudden stress, and due to small k values, the dynamics become stabilized indicating normal natureof the system. Then, as the value of k increases, p
53 dynamics show damped oscillation indicatingthe inducing oscillation for certain interval of time ( t ≈ [0 , k = 0 . k and increases as k . This indicates that if the stress given to the system is small, the system first will goto the stress or excited state (indicated by oscillatory behavior), repair back the changes in the system Figure 3.
The similar plots for p
53 and GSK3 for the same set of values as in Figure 2.and come back to the normal condition. For sufficiently large values of k (0 . ≤ k ≤ .
55) the p t → ∞ . Further increase in the value of k ( k > .
55) force the sustain oscillation of p
53 to dampedoscillation whose time interval of damped oscillation decreases as k increases. This indicates that excessstress in p
53 due to Nutlin may become toxic to the system reflected in p
53 dynamics. If we increase thevalue of k ( k > .
3) the p
53 dynamics become stabilized. This may probably the case of apoptosisof the system. The results obtained are in consistent with the experimental observations which indicatesthat acetylation of p
53 is responsible for its activation and stabilization [54]. If we further increase thevalue of k , p
53 activation decreases maintaining p
53 stability, but at higher values ( ≤ p
53 and Nutlin (Fig. 2 upper right panels) for different valuesof k show different behaviors of the system, namely stable fixed point, damped oscillation towards afixed point, sustain oscillation, then damped and stabilized fixed point. This behaviors show the samebehaviors as shown in p
53 dynamics for different values of k .Similarly, we study the dynamics of Gsk k (Fig. 2 lower left panels) for corre-sponding values which we took for the case of p
53. In this case also we found the similar behaviors aswe got in p
53 case (Fig. 2), namely, stabilized, damped, sustain, then damped and stabilized states forthe corresponding values of k taken in p
53 case. The two dimensional plots of
Gsk p
53 via
Gsk p
53 is processed by Axin via
Gsk p
53. The Axin and β − Catenin dynamicsas a function of k , as well as two dimensional plots of these two molecules with Nutlin are similar Figure 4.
Plots of p
53, Axin and β -catenin as a function of time for k values 0.0002, 0.0003, 0.01,0.025 and 0.08, for fixed values of k . The upper right panels shows the two dimensional plots of p β -catenin with Axin for same set of k and k parameter values.qualitatively with the dynamics of stress p
53 and
Gsk k the stress p
53 can able to generate and arrest the Axin and β − Catenin cycles. Thus thedynamics of Axin and β − Catenin can be regulated by p
53 and probably can control the Wnt oscillator.This means that p
53 regulation has strong impact on Wnt oscillation, and regulate various mechanismsduring sometogenesis.
Dynamics of p regulated by Wnt The activation of Axin occurs after the
Gsk initiation from the signal received at the membrane. TheNutilin synthesis rate ( k ) is kept constant in this case ( k = 0 . p
53, we allow to change Axin synthesis rate k and see the dynamical behavior of p
53 induced by Axin, which is one of the most important molecularspecies in Wnt network module. The dynamics of the Axin and p
53 for different values of k (Fig. 4).For small values of k ( k < . k allows the Axindynamics to switch from stabilized state to damped state for certain range of time and then becomestabilized. The co-existence of these two states (damped and stabilized states) takes place for a certainrange on k (0 . < k < . k , the duration of damped oscillationalso increases. The switching of damped state to sustain oscillation state takes place for certain range of k , 0 . < k < . k force the sustain oscillation state to damped oscillationstate again, where the time interval of damped oscillation becomes smaller as k increases. If we increase k value, then Axin dynamics landed to stabilized state. Figure 5.
The amplitudes of p
53 ( x A ), β -catenin ( x A ), Gsk3 ( x A ) and M dm x A ) as a function of k for various k values 0.030, 0.035, 0.040, 0.045 and 0.050 (upper four panels). The lower four panels arethe maxima of p
53 ( x M ), β -catenin ( x M ), Gsk3 ( x M ) and p
53 ( x M ) for the set of parameter values.We then show the dynamics of β − Catenin as a function of k (Fig. 4) which indicates the switchingof the three different states driven by k . In this case we could able to find sustain oscillation very quicklyindicating that the β − Catenin can be quickly switch to stress condition and easily. Similarly, β − Catenincan reach second stabilized state (probably cycle arrest) quickly and easily as compared to other molecularspecies. The dynamical behavior of β − Catenin is supported by two dimensional plots of β − Catenin andAxin (Fig. 3 middle right panels) which exhibit fixed point oscillations (corresponding to two stabilizedstates), attractor towards the stable fixed points (corresponding to two damped oscillations), limit cycle(corresponding to sustain oscillation).Now the dynamics of p
53 as a function of k (Fig. 4 upper left panel) show that the patterns inthe dynamics of molecular species in Wnt oscillator are well captured in the p
53 dynamics. The p β − Catenin as a function of k with similar patterns and dynamics.The important dynamical behaviors those have been captured in p
53 dynamics due to change ofstates in molecular species in Wnt oscillator are synchronous in states and time periods of dynamics. Ifwe look at the dynamics of Axin and β − Catenin as a function of k , we can easily observe two importantchanges, (1) change in time period of oscillations of the molecular species, and (2) change in the patternof states. The increase in k force to decrease the time period of the Axin and β − Catenin dynamics. Atthe same time the switching of the various states takes place. These changes in the Axin and β − Cateninare systematically and synchronously well captured in qualitative sense in the p
53 dynamics (Fig. 4)during the communication of the two p
53 and Wnt oscillators via
Gsk
Figure 6.
Similar plots of p
53 ( x ), β -catenin ( x ), Gsk3 ( x ) and p
53 ( x ) as a function of k for thesame k values as in Figure 5. Amplitude death driven by
Gsk and p interaction We now study the impact of interaction of
Gsk p
53 ( k ) on Wnt and p
53 cross talk driven byNutlin concentration level in the system (Fig. 5). The rate of interaction of p
53 and
Gsk
Wnt cycle arrest induced by Nutlin
We calculated amplitudes of four molecular species Axin, β − Catenin,
Gsk
M dm k for five different values of k (Fig. 5 upper two rows). The amplitudes of the molecular speciesfor every values of k are calculated by averaging the amplitudes within the interval of time [100 , k : first amplitude death(AD) regime (for small k ); second finite amplitude or oscillation regime (OR) (for moderate k ); andthird amplitude death regime again (for large k ). The first AD regime may correspond to normal state(stable state) of the system, and second AD regime may correspond to apoptosis (cycle arrest) of thesystem. Further, we calculated maximum values of molecular species Axin, β − Catenin,
Gsk p k , then reaches amaximum amplitude, after which amplitude decreases slowly and then decreases monotonically. This isthe regime where the system is in activated or stress condition. It is also to be noted that as k increasesthe amplitude also increases and the range of activated regime also increases. This means that increasing1 k force the system to get activated quickly and protects the system from next stabilized regime (cyclearrest or apoptosis regime). Therefore, even though increase in Nutlin concentration drives the system toapoptosis regime quickly, one can modulate the interaction of p
53 and
Gsk p cycle arrest induced by Axin Now we study the impact of p
53 and
Gsk p
53 and Wnt oscillators. We calculated the amplitudes of these molecular species as a functionof Axin concentration level in the system ( k ) (Fig. 6 first two rows) for the same parameter values aswe had taken in the case of calculation of amplitudes of these molecular species with the variation ofNutlin concentration in the system. For the same parameter values we also calculated the maxima ofthese molecular species (Fig. 6 lower two rows).The change in k drives the amplitudes of all the molecular species to different oscillation states(AD, OR, AD) as we have obtain in the case of Wnt cycle arrest by k variation. The increase inAxin concentration level in the system forces amplitudes of the molecular species variables to pass thetrajectories of first AD (normal condition), then OR (stress condition) and then second AD (cycle arrestcondition). However, in this case the amplitudes of the variables do not change much as a function k ,but significant change in the range of second AD as a function of k . The results indicate the Axininduced p
53 cycle arrest, however, interaction of p
53 and
Gsk k so that the system can repair back to its normal condition. Steady state solution of the system
Since the direct and exact analytical solution of the set of coupled nonlinear differential equations (1),we used quasi steady state approximation [51, 52] to obtain approximation solution of the system. Usingsteady state values of the fast variables using (15), putting back to the ODE of x ( t ) in equation (16)and taking x k + x ∼ k x ∼ (cid:16) − k x (cid:17) ∼ x ) we get the following approximate solutions of x , x ( t ) ≈ u u + C e − u t (18)where, C is a constant, and u , u are also given by, u = k (cid:18) k x ∗ + k k − u k k k − k (cid:19) u = k k k x ∗ k k ( k x ∗ − k x ∗ ) (19)The equation (18) shows that x becomes constant (steady state) as t → ∞ . However, x depends onvarious other steady state variables as shown in equation (18) and (19).We then solve ODE of x ( t ) in equation (16) for x , and the result is given by, x ( t ) ≈ k − k x ∗ k + C e − k k t (20)where, C is a constant. Here, in this solution, we can get that x stabilizes to a constant value atsufficiently large time limit ( t → ∞ ).2Now, the approximate solution of x can be obtained by solving ODE of x in equation (16), usingequation (18) and (17), as given below, x ( t ) ≈ exp (cid:0) at − be − u t (cid:1) (cid:20) C − (cid:18) u u (cid:19) b a/u Γ (cid:18) − au , b (cid:19)(cid:21) (21)where, C is a constant. Similarly, u = k k k , a = k k u u and b = k k C u are also constants.The molecular species x can be solved directly from equation (16) and using equation (17). Theresult is given by, x ( t ) ≈ C e − k t + k k (22)where, C is a constant. Further, using the differential equation in x in equation (16) and equation (22)along with equation (17), we solve for x given by, x ( t ) ≈ u k R y u /k t y − C dy + C h C + k k e k t i ≈ πi u k C u k + C h C + k k e k t i (23)where, u = k + k x ∗ , u = k ( k k x ∗ ) and C is a constant. Similarly, the variable x can besolved using equations (16), (22) and (17) as follows, x ( t ) ≈ C − k x ∗ k R y − (cid:16) u k (cid:17) e − u k y dye − u t + u k e − k t ≈ C − k x ∗ k E u k (cid:16) u k (cid:17) e − u t + u k e − k t (24)where, u = k + k k k , u = k C and C are constants. E is the error function. Conclusion
The cross talk between p
53 and Wnt oscillators can influence each other such that the dynamics oneoscillator can regulate the dynamics of the other and vice versa. The concentration level of Nutlin candrive the p
53 network module to different oscillating states corresponding to different stress states of thesystem. This changes in the oscillating states can be well communicated to Wnt network module via
Gsk p
53 network module via
Gsk p
53 and Wnt network modules can cross talk between them and regulate each other due to changes ineach oscillator by processing information between them. However, excess changes in each network modulecan act as toxic to each module and arrest the cycle of the other, and vice versa.We then demonstrate that any other changes in the regulation of the network module, may probablyby catalytic reaction or external stimuli which alters the rate of the reaction of a particular reaction, can3probably try to protect the system from apoptosis upto certain range of system’s condition within whichthe system can repair back the changes to come back to normal condition. We attempt to numericallydemonstrate this phenomena by allowing the change in the rate constant of p
53 and
Gsk p
53 and Wnt network modules upto certain range of excess concentrationlevels of Nutlin and Axin to protect themselves from cycle arrest, namely apoptosis. This means that oneimportant role of catalytic reaction and external stimuli (which do not affect the reactions and networkbut rate of reactions) is to protect the system from apoptosis or system’s failure by giving a chance ofrepair the failure or defect by itself within the system automatically. Therefore, we need to find outvarious such reactions and stimuli in cellular network in order to control and prevent the cell from celldeath, due to cellular breakdown by diseases or other external mechanisms, by regulating these importantreactions and stimuli.
References
1. L Collavin, A Lunardi, G Del Sal. 2010 p53-family proteins and their regulators: hubs and spokesin tumor suppression. Cell Death and Differentiation 17, 901911.(doi:10.1038/cdd.2010.35)2. Oren, M., Damalas, A., Gottlieb, T., Michael, D., Taplick, J., Leal, J.F., Maya, R., Moas, M., Seger,R., Taya, Y., et al. 2002 Regulation of p53: intricate loops and delicate balances. Biochem Pharmacol64, 865-871. (doi:S0006295202011498).3. Itahana, K., Dimri, G.Campisi, J. 2001 Regulation of cellular senescence by p53. Eur J Biochem268, 2784-2791.(doi: 10.1046/j.1432-1327.2001.02228.x).4. Vogelstein, B., Lane, D. & Levine, A.J. 2000 Surfing the p53 network. Nature 408, 307-310.(doi:10.1038/35042675)5. Agarwal, M.L., Taylor, W.R., Chernov, M.V., Chernova, O.B., Stark, G.R. 1998 The p53 network.J Biol Chem 273, 1-4.(doi: 10.1074/jbc.273.1.1)6. Purvis, J.E., Karhohs, K.W., Mock, C., Batchelor, E., Loewer, A., Lahav, G. p53 dynamics controlcell fate. Science 336, 1440-1444. (doi:336/6087/1440)7. Lowe SW, Cepero E, Evan G. 2004 Intrinsic tumour suppression. Nature 432, 307-315.19.(doi:10.1038/nature03098).8. Momand, J., Wu, H.H., Dasgupta, G. 2000 MDM2–master regulator of the p53 tumor suppressorprotein. Gene 242, 15-29. (doi:S0378-1119(99)00487-4)9. Toledo F,Wahl GM. 2006 Regulating the p53 pathway: in vitro hypotheses, in vivo veritas. Nat RevCancer 6, 909-923.7. (doi:10.1038/nrc2012)10. Honda R, Tanaka H,Yasuda H. 1997 Oncoprotein MDM2 is a ubiquitin ligase E3 for tumor sup-pressor p53. FEBS Lett 420, 25-27.(doi:S0014-5793(97)01480-4).11. Kobet E, Zeng X, Zhu Y, Keller D,Lu H. 2000 MDM2 inhibits p300-mediated p53 acetylationand activation by forming a ternary complex with the two proteins. Proc Natl Acad Sci U S A 97,12547-12552.(doi:10.1073/pnas.97.23.12547 97/23/12547).12. Oliner JD, Kinzler KW, Meltzer PS, George DL, Vogelstein B. 1992 Amplification of a gene en-coding a p53-associated protein in human sarcomas. Nature 358, 80-83.29.(doi:10.1038/358080a0).13. Chen, J., Wu, X., Lin, J., Levine, A.J. 1996 mdm-2 inhibits the G1 arrest and apoptosis functionsof the p53 tumor suppressor protein. Mol Cell Biol 16, 2445-2452.414. Haupt, Y., Barak, Y., Oren, M. 1996 Cell type-specific inhibition of p53-mediated apoptosis bymdm2. EMBO J 15, 1596-1606.15. Arora, A., Gera, S., Maheshwari, T., Raghav, D., Alam, M.J., Singh, R.K. Agarwal, S.M.The dynamics of stress p53-Mdm2 network regulated by p300 and HDAC1. PLoS One 8, e52736.(doi:10.1371/journal.pone.0052736PONE-D-12-06209).16. Oren, M. 2003 Decision making by p53: life, death and cancer. Cell Death Differ 10, 431-442.(doi:10.1038/sj.cdd.44011834401183).17. Shangary, S., Wang, S. 2009 Small-molecule inhibitors of the MDM2-p53 protein-protein interactionto reactivate p53 function: a novel approach for cancer therapy. Annu Rev Pharmacol Toxicol 49,223-241. (doi:10.1146/annurev.pharmtox.48.113006.094723).18. Momand, J., Zambetti, G.P., Olson, D.C., George, D., Levine, A.J. 1992 The mdm-2 oncogeneproduct forms a complex with the p53 protein and inhibits p53-mediated transactivation. Cell 69,1237-1245. (doi:0092-8674(92)90644-R).19. Vassilev LT, Vu BT, Graves B, Carvajal D, Podlaski F, Filipovic Z, Kong N, Kammlott U, LukacsC, Klein C et al. 2004 In vivo activation of the p53 pathway by small-molecule antagonists of MDM2.Science 303, 844-848.(doi:10.1126/science.1092472 1092472).20. Logan IR, McNeill HV, Cook S, Lu X, Lunec J, Robson CN. 2007 Analysis of the MDM2 antagonistnutlin-3 in human prostate cancer cells. Prostate 67, 900-906.(doi:10.1002/pros.20568).21. Villalonga-Planells R, Coll-Mulet L, Martinez-Soler F, Castano E, Acebes JJ, Gimenez-Bonafe P,Gil J, Tortosa A. 2011 Activation of p53 by nutlin-3a induces apoptosis and cellular senescence inhuman glioblastoma multiforme. PLoS One 6, e18588.(doi:10.1371/journal.pone.0018588).22. Binder, B.R. 2007 A novel application for murine double minute 2 antagonists: thep53 tumor suppressor network also controls angiogenesis. Circ Res 100, 13-14. (doi:10.1161/01.RES.0000255897.84337.38).23. Lee SY, Shin SJ,Kim HS. 2013 ERK1/2 activation mediated by the nutlin3induced mitochondrialtranslocation of p53. Int J Oncol 42, 1027-1035.(doi:10.3892/ijo.2013.1764).24. Aulehla, A., AgarwalAgarwalPourquie, O. 2008 Oscillating signaling pathways during embryonicdevelopment. Curr Opin Cell Biol 20, 632-637. (doi:10.1016/j.ceb.2008.09.002).25. Gibb, S., Zagorska, A., Melton, K., Tenin, G., Vacca, I., Trainor, P., Maroto, M., Dale, J.K.2009 Interfering with Wnt signalling alters the periodicity of the segmentation clock. Dev Biol 330,21-31.(doi:10.1016/j.ydbio.2009.02.035)26. Gibb, S., Maroto, M. Dale, JK. 2010 The segmentation clock mechanism moves up a notch. TrendsCell Biol 20, 593600.(doi: 10.1016/j.tcb.2010.07.001)27. Goldbeter, A., Pourquie, O. 2008 Modeling the segmentation clock as a network of cou-pled oscillations in the Notch, Wnt and FGF signaling pathways. J Theor Biol 252, 574-585.(doi:10.1016/j.jtbi.2008.01.006).28. Dequeant, M.L., Glynn, E., Gaudenz, K., Wahl, M., Chen, J., Mushegian, A., Pourquie, O. 2006 Acomplex oscillating network of signaling genes underlies the mouse segmentation clock. Science 314,1595-1598. (doi:10.1126/science.1133141).529. Pourquie, O. 2003 The segmentation clock: converting embryonic time into spatial pattern. Science301, 328-330.(DOI: 10.1126/science.1085887).30. Peng, X., Yang, L., Chang, H., Dai, G., Wang, F., Duan, X., Guo, L., Zhang, Y., Chen, G.Wnt/beta-catenin signaling regulates the proliferation and differentiation of mesenchymal progenitorcells through the p53 pathway. PLoS One 9, e97283. (doi:10.1371/journal.pone.0097283).31. Kim NH, Kim HS, Kim NG, Lee I, Choi HS, Li XY, Kang SE, Cha SY, Ryu JK, Na JMet al. 2011 p53 and microRNA-34 are suppressors of canonical Wnt signaling. Sci Signal 4,ra71.(doi:10.1126/scisignal.2001744).32. Cha YH, Kim NH, Park C, Lee I, Kim HS, Yook JI. 2012 MiRNA-34 intrinsically links p53 tumorsuppressor and Wnt signaling. Cell Cycle 11, 1273-1281.(doi:10.4161/cc.19618).33. Tauriello DV, Maurice MM. 2010 The various roles of ubiquitin in Wnt pathway regulation. CellCycle 9, 3700-3709.(doi:10.4161/cc.9.18.13204).34. Kim, N.H., Cha, Y.H., Kang, S.E., Lee, Y., Lee, I., Cha, S.Y., Ryu, J.K., Na, J.M., Park, C., Yoon,H.G., et al. 2013 p53 regulates nuclear GSK-3 levels through miR-34-mediated Axin2 suppression incolorectal cancer cells. Cell Cycle 12, 1578-1587. (doi:10.4161/cc.24739).35. Doble BW, Woodgett JR. 2003 GSK-3: tricks of the trade for a multi-tasking kinase. J Cell Sci116, 1175-1186.(doi: 10.1242/jcs.00384)36. Woodgett JR. 2001 Judging a protein by more than its name: GSK-3. Sci STKE 2001, re12.18.(doi:10.1126/stke.2001.100.re12 2001/100/re12).37. Aberle H, Bauer A, Stappert J, Kispert A, Kemler R. 1997 β -catenin is a target for the ubiquitin-proteasome pathway. EMBO J 16, 3797-3804.(doi:10.1093/emboj/16.13.3797).38. Nakamura, T., Hamada, F., Ishidate, T., Anai, K., Kawahara, K., Toyoshima, K., Akiyama, T. 1998Axin, an inhibitor of the Wnt signalling pathway, interacts with beta-catenin, GSK-3beta and APCand reduces the beta-catenin level. Genes Cells 3, 395-403.( DOI: 10.1046/j.1365-2443.1998.00198.x)39. Taelman VF, Dobrowolski R, Plouhinec JL, Fuentealba LC, Vorwald PP, Gumper I, Sabatini DD,De Robertis EM. 2010 Wnt signaling requires sequestration of glycogen synthase kinase 3 inside mul-tivesicular endosomes. Cell 143, 1136-1148.(doi:S0092-8674(10)01356-5. 10.1016/j.cell.2010.11.034).40. Logan IR, McNeill HV, Cook S, Lu X, Lunec J, Robson CN. 2007 Analysis of the MDM2 antagonistnutlin-3 in human prostate cancer cells. Prostate 67, 900-906.(doi:10.1002/pros.20568).41. Jensen, P.B., Pedersen, L., Krishna, S., Jensen, M.H. A Wnt oscillator model for somitogenesis.Biophys J 98, 943-950. (doi:10.1016/j.bpj.2009.11.039).42. Clevers H. 2006 Wnt/ β -catenin signaling in development and disease. Cell 127, 469-480.(doi:10.1016/j.cell.2006.10.018).43. Huelsken J, Behrens J. 2002 The Wnt signalling pathway. J Cell Sci 115, 3977-3978.(doi:10.1242/jcs.00089)44. Wang HY, Huang YX, Qi YF, Zhang Y, Bao YL, Sun LG, Zheng LH, Zhang YW, Ma ZQ, LiYX. 2013 Mathematical models for the Notch and Wnt signaling pathways and the crosstalk betweenthem during somitogenesis. Theor Biol Med Model 10, 27. (doi:10.1186/1742-4682-10-27).45. Proctor CJ, Gray DA. 2008 Explaining oscillations and variability in the p53-Mdm2 system. BMCSyst Biol 2, 75.(doi:10.1186/1752-0509-2-75).646. Devi, G.R., Alam, M.J., Singh, R.K. Synchronization in stress p53 network. Math Med Biol.(doi:10.1093/imammb/dqv002).47. Alam, M.J., Devi, G.R., Ravins, Ishrat, R., Agarwal, S.M., Singh, R.K. Switching p53 states by cal-cium: dynamics and interaction of stress systems. Mol Biosyst 9, 508-521. (doi:10.1039/c3mb25277a).48. Proctor CJ, Gray DA. 2010 GSK3 and p53 - is there a link in Alzheimer’s disease? Mol Neurode-gener 5,7. (doi:10.1186/1750-1326-5-7).49. Chickarmane V, Nadim A, Ray A, Sauro HM. 2005 A p53 Oscillator Model of DNA Break RepairControl. arXiv preprint q-bio/051002.50. Lahav G, Rosenfeld N, Sigal A, Geva-Zatorsky N, Levine AJ, Elowitz MB, Alon U. 2004 Dynam-ics of the p53-Mdm2 feedback loop in individual cells. Nat Genet 36, 147-150.(doi:10.1038/ng1293ng1293).51. J.D. Murray. 2003 Mathematical Biology I and II. Springer-Verlag 3rd Edition.52. M. Schauer and R. Heinrich. 1983 Quasi-steady state approximation in the Mathematical modelingof Biochemical reaction networks. Math. Biosc. 65, 155-170.53. MG Pedersen, AM Bersani and E Bersani. Quasi steady-state approximations in complex intracel-lular signal transduction networks-a word of caution. J. Math. Chem. 43, 1318 (2007).54. Gu W,Roeder RG. 1997 Activation of p53 sequence-specific DNA binding by acetylation of the p53C-terminal domain. Cell 90, 595-606.(doi:10.1016/S0092-8674(00)80521-8)7 Table 1-List of molecular species and their intial concentrationS.No. Species Name Description Notation Intial Concen-tration(nM) Axin
Axin x . Axin
Axin x . β -catenin UnPhosphorylated β -catenin x . β -catenin p Phosphorylated β -catenin x . β -catenin n Nuclear β -catenin x . Gsk x . p
53 unbound p
53 protein x . M DM
M DM x . M DM
M DM x . p M DM M DM p
53 complex x . x . M DM
M DM x . p Gsk p
53 and
Gsk x . Table 2 List of parameterS.No. Notation Description Values Reference K Rate constant for binding of
Gsk
Axin . nM min − [27]2. K Rate constant for dissociation of
Gsk Axin . min − [27]3. K Rate of β -catenin synthesis 0 . nM min − [27]4. K Rate of β -catenin entry into the nucleus 0 . min − [27]5. K Rate of β -catenin exit from the nucleus 1 . min − [27]6. K Rate of phosphorylation of β -catenin by the Gsk . nM min − [27]7. K Concentration of Dishevelled(Dsh)protein 2 . nM [27]8. K Total
Gsk . nM [27]9. K Rate of inhibition by Dsh for β -catenin p by the Axin Gsk . nM [27]10. K Rate constant for β -catenin p by the Axin Gsk . nM [27]11. K Rate constant of dephosphorylation of β -catenin 1 . nM min − K Maximun rate constant for β -catenin phosphora-tion 0 . nM [27]13. K Rate constant for degradation of unphosphory-lated β -catenin 0 [27]14. K Rate constant for degradation of phosphorylated β -catenin 7 . min − K Rate of transcription of the
Axin . nM min − [27]16. K Rate of transcription of the
Axin β -catenin 1 . nM min − [27]17. K Rate for induction by nuclear β -catenin of Axin . nM [27]18. K Maximum Rate of degradation of
Axin . nM min − [27]19. K Rate constant for degradation of
Axin . nM [27]20. K Rate of transcrioption of
Axin . nM min − [27]21. K rate for induction by transcription factor of Axin . nM [27]22. K Rate of synthesis of
Axin . min − [27]23. K Maximum rate of degradtion of
Axin . nM min − [27]24. K Michaelis rate for degradtion of
Axin . nM [27]25. K Hill coefficients 2 . k Scaling factor for Wnt oscillator 1 . k M DM . min − [45]28. k M DM . min − [45]29. k M DM m RN A degradation 0 . min − [45]30. k M DM . min − [45]31. k p
53 synthesis 4 . min − [45]32. k p
53 degradation 0 . min − [45]33. k p M DM . min − [45]9 Table 2 Continue...S.No. Notation Description Values Reference k p M DM . min − [45]35. k Nutilin formation 0 . min − Estimated36. k Nutilin
M DM . min − Estimated37. k Nutilin
M DM . min − Estimated38. k Nutilin degaradation 0 . min − Estimated39. k p Gsk . min − [48]40. k p Gsk . min − [48]41. k M DM mRN A synthesis 0 . min −1