The Dynamics of Stress p53-Mdm2 Network Regulated by p300 and HDAC1

We construct a stress p53-Mdm2-p300-HDAC1 regulatory network that is activated and stabilised by two regulatory proteins, p300 and HDAC1. Different activation levels of observed due to these regulators during stress condition have been investigated using a deterministic as well as a stochastic approach to understand how the cell responds during stress conditions. We found that these regulators help in adjusting p53 to different conditions as identified by various oscillatory states, namely fixed point oscillations, damped oscillations and sustain oscillations. On assessing the impact of p300 on p53-Mdm2 network we identified three states: first stabilised or normal condition where the impact of p300 is negligible, second an interim region where p53 is activated due to interaction between p53 and p300, and finally the third regime where excess of p300 leads to cell stress condition. Similarly evaluation of HDAC1 on our model led to identification of the above three distinct states. Also we observe that noise in stochastic cellular system helps to reach each oscillatory state quicker than those in deterministic case. The constructed model validated different experimental findings qualitatively.


Introduction
The p53 is a 20-Kb tumor suppressor gene located on the small arm of human chromosome 17 that acts as a hub for a network of signalling pathways essential for cell growth regulation and apoptosis. It comprises of 393 amino acids and is divided into several structural and functional domains: the transactivation domain (TAD; residues 1-40), the proline-rich domain (PRD; residues 61-94), the DNA-binding domain (DBD; residues 100-300), the tetramerization domain (4D; residues 324-355) and the C-terminal regulatory domain (CTD; residues 360-393) [1]. Over the recent years many names have been accredited to p53 viz. Guardian of the Genome [2]; Death Star [3] and Cellular Gatekeper [4] and is regulated by a number of cellular proteins [5]. It is well established that p53 is accountable for preventing improper cell proliferation and maintaining genome integration following genotoxic stress. In normal proliferating cells, p53 is kept in low concentrations and exists mainly in an inactive latent form with a short half-life of 15-30 minutes [6]. This is due to interaction between p53 and Mdm2 the predominant negative regulator of p53. However, cellular insults activates p53 and its level increases rapidly. The activation of p53 is a result of several posttranslational modifications including phosphorylation, acetylation, sumoylation and neddylation [7]. Phosphorylation of Ser-15 and 37 at the amino terminus of p53 prevents Mdm2 binding, thus stabilizing p53. Also phosphorylation at Ser-15 increases p53 affinity for p300, thus promoting acetylation of p53 carboxy terminal by p300 [8]. Further the p53 in-turn activates the p53targeted genes including those involved in cell cycle arrest and DNA repair, as well as apoptosis and senescence related genes. The activation of the p53-targeted genes leads to cell cycle arrest that forces cell to choose either to repair the DNA damage to restore its normal function or cell death (apoptosis). Further, it has been observed that p53 acetylation is a reversible process and for it Mdm2 recruits HDAC1 (a histone deacetylase) to form a Mdm2-HDAC1 complex which deacetylates p53. Interestingly, it was also shown that p300 can form a complex with Mdm2 in vitro and in vivo [9,10] and this complex (Mdm2-p300) facilitate Mdm2 mediated p53 degradation. Moreover, it has also been reported that Mdm2-p53-p300 complex exists that is also thought to promote ubiquitylation and degradation of p53 [11]. Thus p300 plays dual role and exerts two opposite effects on p53 in cells i.e., it can either interact with Mdm2 promoting Mdm2-mediated ubiquitylation and degradation of p53 [9] or acetylate and stabilize p53. This remains puzzling.
There have been different mathematical techniques to study cellular and sub-cellular processes such as deterministic and stochastic models [12,13]. Stochastic model provide detail picture of molecular interaction in the microscopic systems (small systems with small number of molecules accomodated in each system) that leads the system dynamics as noise-driven process [13,14]. The model further highlights the important role of noise in the system dynamics, for example detection and amplification of weak noise, the phenomenon known as stochastic resonance [15,16], lifting of cellular expression at different distinct expression state [17] and noise in gene expression can drive stochastic switching among such states [18,19], noise induced stochastic phenotypic switching to different new level in living cells [20] etc. However, deterministic model provides qualitative picture of the cellular or sub-cellular processes.
The aim of the present study is (i) to understand some of the basic issues of p53 autoregulation induced by regulators p300 and HDAC1, (ii) to elucidate the functional relationship of p300 and HDAC1 in regulating p53 function, (iii) how do these regulators lifts the normal p53-Mdm2 network to different stress states and (iv) what could be the role of noise in such situations.

Materials and Methods
Stress p53{Mdm2 model regulated by p300 and HDAC1 In normal proliferating cells, p53 is usually maintained at low levels due to p53 and Mdm2 protein feedback mechanism [21]. In unstressed condition the p53 binds to the regulatory region of Mdm2 gene and stimulates its transcription into messenger RNA (mRNA) with a transcription rate constant k 3 , followed by translation into Mdm2 protein with a rate constant k 2 [22]. The degradation of Mdm2-mRNA, Mdm2 and genesis of p53 occurs with basal rate of k 4 , k 5 and k 6 respectively. The Mdm2 protein then interacts physically with p53 to form Mdm2-p53 complex with the rate of k 8 . Mdm2 functions as an E3 ubiquitin ligase and brings about ubiquitylation of multiple lysine residues (K370, K372, K373, K381, K382 and K386) [23] present in the Cterminal domain of p53 [11]. The ubiquitylation marks p53 for degradation via the 26S proteasome, with rate k 7 . The Mdm2-p53 complex can also dissociate to Mdm2 and p53 with rate constant k 9 . Mdm2 and p300 have been shown to interact with rate constant k 20 to form Mdm2-p300 complex, which facilitates p53 polyubiquitination and degradation at rate constant of k 1 [9,24]. Although there is no direct evidence reported to the best of author's knowledge in the literature for the degradation of Mdm2-p300 complex, however it has been shown that the p19ARFbinding domain of Mdm2 overlaps with its p300-binding domain suggesting that p19ARF could interfere with the Mdm2/p300 interaction [9]. Therefore, we can assume it is possible that Mdm2-p300 complex can be broken so as to interact with other proteins. Thus in normal unstressed cell, p53 is maintained at low level in an active state with short half-life of 15-30 minutes by Mdm2 and the cells are able to proliferate.
However, under stressed conditions the p53 is stabilized through various post translational modifications which lead to increase its level. Of the various mechanisms, phosphorylation of p53 is the most well studied and it is reported that multiple kinases phosphorylate various residues which increase the level of p53 protein. One of these protein kinases is ATM which upon activation by DNA damage phosphorylates p53 with a rate k 12 at serine 15 [25] which is critical for p53 activation and stabilization. Strikingly, the phosphorylation of serine 15 mediated by ATM acts as a nucleation event that promotes subsequent sequential modification of many residues. To achieve this, interconversion of inactivated and activated ATM takes place, with rate constants k 10 and k 11 respectively. The ATM-initiated phosphorylation reduces the affinity of p53 for Mdm2 while increases interactions with HATs like CBP=p300 [8,26]. Consequently, dephosphorylation of p53 with a rate k 13 also takes place to counter this phosphorylation. It has been demonstrated that p300 protein is a co-activator of p53 which potentiates its transcriptional activity as well as biological function in vivo [27]. However, it has also been shown that formation of p300 Mdm2 p53 ternary complex leads to suppressing p53 acetylation and activation [28]. The transcription activation domain (TAD) of p53 binds tightly to p300 with formation rate constant k 15 . The p53 p300 complex hence formed, causes p53 acetylation with rate constant k 16 at multiple lysine residues (K370, K372, K373, K381, K382) of its C-terminal regulatory domain [27,29]. The lysine residues (K370, K372, K373, K381, and K382) are the common sites for both acetylation and ubiquitination [30,31]. Thus their acetylation causes the inhibition of ubiquitination resulting p53 protein stability which is evident from the observation that acetylated p53 has half-life of greater than two hours [32]. Simultaneously, formation and degradation of p300 occurs with rate constants k 23 and k 14 respectively. Mdm2, p53 and p300 have also been demonstrated to exist in a ternary complex (k 19 ) which is incapable of acetylating p53 [28]. In the complex, TAD1 domain of p53 interacts with Mdm2 while TAD2 interacts with p300 [11]. As mentioned earlier, phosphorylation increases the affinity of p53 towards p300 while decreasing its affinity towards Mdm2. After phosphorylation, the ternary complex dissociates, with rate constant k 21 into Mdm2 and p53{p300 complex, in which both TAD1 and TAD2 of p53 interact with p300 [11]. p300 can then acetylate and stabilize p53. Stabilized p53 functions as a tumor suppressor and induces high levels of Mdm2, which in turn promotes p53 degradation by recruiting a p53 deacetylase, HDAC1 with rate constant k 24 . HDAC1 binds Mdm2 in a p53 dependent manner with binding rate constant k 18 and deacetylates p53 with rate constant k 17 at all known acetylated lysines in vivo [33]. Moreover, analysis has indicated the presence of MDM2, SMAR1 and HDAC1 complex under conditions of inhibited translation only 12 h post damage rescue while there is lack of complex formation 24 h post damage rescue, thereby suggesting degradation of the Mdm2-HDAC1complex [34]. HDAC1 is generated and degraded in cells with rate constants k 24 and k 22 respectively. The unmodified lysine residues can then serve as the substrates for Mdm2-mediated ubiquitylation resulting in p53 degradation and thus completing the feedback loop. The molecular species involved in the biochemical network are listed in Table 1 and the chemical reaction channels in the network are shown in Table 2. The schematic picture of the stress p53{Mdm2 autoregulatory biochemical reaction network model via p300 and HDAC1 based on the experimental evidences and reports mentioned above is shown in Fig. 1.

Stochastic description of biochemical reaction network
We now consider a configurational statẽ X X (t)~(X 1 ,X 2 , . . . ,X N ) T of the system of size V at any instant of time t defined by N molecular species undergoing M elementary reactions. The change in configurational state during any interval of time ½t,tzDt is due to random interaction of the participating molecules that leads to decay and creation of specific molecular species in state vectorX X (t) during the time interval [13,14,35]. Therefore the trajectory of this state vectorX X (t) as a function of time in the configurational space follows Markov process [13,14] and the dynamics of this vector becomes noiseinduced stochastic process [13]. If we define P(X X ,t) as the configurational probability of obtaining the stateX X at time t, then the time evolution of P(X X ,t) will obey Master equation [13,14,36]. Even though the Master equation for complex system could be very difficult to solve analytically, different algorithms have been devised to solve the system dynamics numerically depending on p53-Mdm2 Network Regulation by p300 and HDAC1 PLOS ONE | www.plosone.org the nature of the system. For example, stochastic simulation algorithm (Gillespie algorithm) for reaction system without considering time delay [13], stochastic simulation algorithm for time delay reaction system [37,38], t-leap algorithm which is approximated algorithm of stochastic simulation algorithm for very complex reaction network [39], hybrid algorithm for reaction networks consisting of both slow and fast reactions [40] etc.
The Master equation for the stochastic system can be approximated to simpler Chemical Langevin equations (CLE) based on two important realistic approximations applied on the the system [41]. This can be done by defining a function F (X X ,Dt) which is the number of a particular reaction fired during the time interval ½t,tzDt with DtT0 and applying the two approximations: first applying lim Dt?0 F (X X ,Dt) which leads to the prophensity functions (v) of the reactions fired remain constant during the time interval, and secondly applying lim Dt?? F (X X ,Dt) condition which gives rise vtTT1 [41]. These two conditions are true for large population size of each variables in state vectorX X which is valid for natural systems. These two conditions allow the function F to approximate to Poisson distribution function and then to Normal distribution function with same mean and standard deviation. If molecular concentration is defined by fx xg~1 V fX X g and linearize Normal distribution function, the Master equation leads to the following CLE of the vectorx x(t), where, G~X M i~1 n ij v i ½x(t) is the macroscopic contribution term and H~1ffiffiffiffi V p X M i~1 n ij v i fx(t)g ½ 1=2 is the stochastic contribution term to the dynamics. j i = lim dt?0 N i (0,1)= ffiffiffiffi dt p is uncorrelated, statistically independent random noise parameters which satisfy j i (t)j j (t 0 ) = d ij d(t{t 0 ). {n} is the stoichiometric matrix of the reactions in the network.
The classical deterministic equations can be obtained from the CLE equation (1) at thermodynamics limit [41] i.e. at V ??, N?? but N=V~constant. This leads to H?0 and the equation (1) becomes noise free deterministic equation, The same equation (2) can also be be retrieved from the biochemical reaction network by translating them into a set of differential equations based on standard principles of Mass-action law of biochemical reaction kinetics.
The stress p53{Mdm2{p300{HDAC1 model network we study is defined by N~14 (14 molecular species) and M~24 (24 reaction channels). The molecular species, possible reactions, kinetic laws and the rate constants in this model are listed in Table 1 and Table 2 respectively. The state vector at any instant of time t is given by,x x(t)~(x 1 , . . . ,x 14 ) T , where the variables in the vector are various proteins and their complexes which are listed in Table 1. The classical deterministic equations constructed from these reaction network are given by, Unbounded p300 protein x 8

14.
Mdm2 p300 Mdm2 and p300 complex where, fk i g and fx i g, i~1,2, . . . ,N(N~14) represent the sets of rate constants of the reactions listed in Table 2 and concentrations of the molecular populations listed in Table 1.
Following the same procedure as we have discussed above, we reach the following CLE for the network shown in Fig. 1, Table 1 and Table 2.  [11,23].
where, fj i g are random number which satisfy j i (t)j j (t 0 ) = d ij d(t{t 0 ), and V is the system's size.

Results and Discussion
Several researchers have studied the oscillations of p53{Mdm2 network in detail [22,[43][44][45][46], however to the best of our knowledge this study is the first one that uses systems biology approach for understanding the complex role of p300 and HDAC1 on p53. We numerically solved the set of deterministic differential equations (1)- (14), and stochastic CLE (15)-(29) by using standard algorithm of 4th order Runge-Kutta method of numerical integration [42]. We thus study the impact of p300 and HDAC1 on p53 activation and stabilization to understand the fate of the cell.

Impact of p300 on p53{Mdm2 activation
We first present the deterministic results on p53-Mdm2 regulatory network on exposure to different concentrations of p300 i.e. at different rate constants, k p300 (Fig. 2). For small values of k p300 ( = 0.04) (lower p300 concentration), p53 is first activated for some time (*30hours) and then resumes its normal condition indicated by its constant level (*12:4) which is the level of stabilization. The range of activation is increased as k p300 increases (increase of p300 concentration) as well as there is rise in the level of stabilization. However, when k p300~0 :06{0:08, p53 maintains sustain oscillations which leads to increasing level of activation as a consequence. With further increment of p300 concentration level, p53 dynamics that was at sustain oscillations switched to damped oscillations and subsequently p53 concentration is stabilized at a constant level. This activity suggests that the capping of the cterminal of p53 is higher and there is no decrement in the p53 levels as a result of which p53 is stabilized. The results obtained are consistent with the experimental observations which indicates that acetylation of p53 is responsible for its activation [27,31] and stabilization [29,32]. If we further increase the value of k p300 , p53 activation decreases maintaining p53 stability but at higher values ( §80). Hence we identify two conditions where p53 is stabilized, one at lower values (nearly normal cell condition) and the other at larger values (cell death condition) of k p300 and in between p53 is activated.
Similarly, Mdm2 dynamics as a function of time for different values of p300 concentration levels is shown (Fig. 3) that demonstrates counter behaviour as expected. The two dimensional recurrence plots of (p53{Mdm2), (p53{p300) and (Mdm2{p300) are presented in Fig. 4 which provides clear and qualitative picture of the above facts. The emergence of sustain/ limit-cycle oscillation (activated p53 level) from fix point oscillation (stabilized p53 level), and then from sustain oscillation to again fix point oscillation is observed as one increase the concentration of p300.

Impact of HDAC1 on p53{Mdm2 network
Several studies suggest that HDAC1 is involved in the deacetylation of p53 which has a potent impact on p53{Mdm2 regulatory dynamics [29,31,47,48]. It has been found that HDAC1 makes complex protein, HDAC1{Mdm2 which  deacetylates and then ubiquitinates the acetylated p53. Because of this process of interaction of HDAC1 with p53, both p53 and Mdm2 levels get stabilized. In our numerical simulation, we kept p300 concentration level fixed by keeping k p300~0 :08 throughout the simulation and allow HDAC1 concentration to vary by changing k HDAC1 value. The results are shown in Fig. 5 (a)-(f). In these plots we observe that at lower concentration of HDAC1 (k HDAC1~0 :0002), the p53 activation is large due to pre-existing p300, as indicated by the sustained oscillation ( Fig. 5 (f)). This activity suggests that there is regular decay and creation of p53, due to the presence of high levels of p300 and hence the impact of HDAC1 concentration level is not very significant. As the HDAC1 concentration increases (increasing k HDAC1 value), there is regular and competitive effect between p300 and HDAC1 for p53 that decreases p53 activation as indicated by decrease in p53 concentration level (Fig. 5 (c)-(e)). Further, if we increase the    Similarly, we present the simulation results of Mdm2 as a function of time for different HDAC1 levels ( Fig. 6 (a)-(f)). We observe similar behaviour of Mdm2 as p53 which shows a transition from sustain oscillation to fix point oscillation as one increase the HDAC1 concentration level. These results indicate that HDAC1 stabilizes p53 as well as Mdm2 concentration levels.
We also present the two dimensional recurrence plots of the (p53{Mdm2), (p53{HDAC1) and (Mdm2{HDAC1) for demonstrating these facts (Fig. 7). The clear indication of transition from sustain/limit cycle oscillation to fix point oscillation as k HDAC1 is increased, is shown in the plots indicating transition from activation of p53 and Mdm2 to stabilized state.

Stability analysis of p53 and Mdm2
We then checked how concentration level of p53 varies as a function of k p300 (!concentration levels of p300). This is done by defining a parameter called expose time (g) which can be stated as the amount of time the system is exposed to a particular concentration level of p300 or HDAC1. The calculation of p53 or Mdm2 concentration level induced by the exposition of the system to p300 or HDAC1 is done by obtaining its level just after the expose time (time slice calculation). Fig. 8 shows variation of p53 concentration levels as a function of k p300 for different expose times of 10-100 hours for a fixed value of k HDAC1~0 :04. The plots clearly show the activated and stabilized regimes. The activated regime is where the p53 levels fluctuate as a function of k p300 (induced by p300 levels). In the plots, p53 level starts activation from k p300 *0:27 because of the interaction among p53, Mdm2 and p300 with small level of HDAC1 giving rise to fluctuation in p53 level. This could be due to acetylation and deacetylation which leads to capping (which prohibits p53 to decay) and uncapping (which leads to p53 decay) of p53 due to p300. This p53 level fluctuation persists till k p300 *0:55 and then increases its level without fluctuation till k p300 *2:74 indicating only the capping of p300, then its level remain constant. Interestingly the range of activation of k p300 in p53 for all expose times remain the same in [0. 27-2.74].
The stabilized regimes are where p53 level is not affected by the variation in k p300 (p300 level variation). Initially, within the range of k p300 [0-0.27], the p53 level is not much affected indicating that the cell resumes its normal condition maintaining its minimum level (*13) which we call first stabilization regime. However, in   The activation and stabilization of Mdm2 induced by p300 is shown in Fig. 9. Since Mdm2 is counter part of p53 which is activated by p53, similar results are obtained as in the case of p53. The first stabilization regime is within [0-0.23] values of k p300 , followed by activation regime [S0.23-0.7] and finally second stabilization regime [S0.7-?]. The increased level of p53 in the second stabilization regime are capped p53 level which are prohibited from decay and taking part in any other reactions and therefore is not able to activate Mdm2 level. Hence its level reduces to minimum as soon as the second stabilization regime is reached.
Next we study the impact of HDAC1 on p53 stabilization in our system. This is done by keeping the value of k p300 fixed at 0.08 and simulating the level of p53 as a function of k HDAC1 for different exposure times 10-100 hours (Fig. 10). From the plots one can see the activation of p53 at low k HDAC1 values due to p300 impact but not due to HDAC1 contribution. As k HDAC1 value increases, the p53 level starts decreasing due the deacetylation of p53 which allow it to degrade and take part in reactions. The activation of p53 with fluctuation persists till (k HDAC1 ƒ0:1). After (k HDAC1 T0:1), p53 level remains constant for a short period of time and then its level starts increasing without fluctuation. This behaviour indicates that HDAC1 has suppressing impact on p53 activation. This pattern is same for all exposure times as is shown in the plots (Fig. 10). The same pattern is found for Mdm2 also which in fact is the counterpart of p53. The activated and stabilized regimes are shown in the Fig. 11.
We then present the results of amplitudes of p53, (A p53 ) and time period, (T p53 ) as a function of k p300 and k HDAC1 to understand the how p300 and HDAC1 influence the amplitude and time period of p53 oscillations (Fig. 12). The calculation of p53 amplitude is done as in the following. For sustain oscillation we took time range of [100-200] hours in our calculation and then calculated the average of it. Then we take 50 such time series for different initial conditions and determine the average of p53 amplitude again ( Fig. 12 and 13). The points in the plots are average points with error bars. For damped oscillations, we take the available number of oscillations and calculated the average of those oscillations which is found to be equivalent to the distance between x-axis and line which shows no oscillation (stable line) approximately. Similarly, for stabilized regime we determine  and finally its value remains constant. This in fact is the consequence of first stability (normal condition) where the impact of p300 is negligible, then activation of p53 due to interaction of p300 with p53 and other proteins and then stabilization of p53. These three regimes can also be seen in the case of T p300 versus k 300 plot.
However, in the case of A p53 and T p53 induced by HDAC1, the first stability condition is not observed because the cell is already activated with a constant p300 level i.e. at constant k p300~0 :08. In this case A p53 level decreases as k HDAC1 increases till k HDAC1~0 :03 and the remains constant. However, T p53 increases till k HDAC1~0 :03 and then stabilized.
Similarly we calculated A Mdm2 and T Mdm2 as a function of k 300 and k HDAC1 respectively and the results are shown in Fig. 12. For both the parameters similar behaviour was obtained as in the case of p53.
Deterministic steady state solutions: impact of HDAC1 and p300 on p53 The steady state solutions in deterministic case can be obtained by putting the conditions to the set of differential equations (3)-(16) and solving for various variables fx Ã i g. Following this procedure we first solve for x Ãd 1 (steady state solution of p53) as a function of x Ã 11 (steady state solution of HDAC1). The result is given by, where, C~k (p300 synthesis rate is larger than HDAC1 degradation rate), a will contribute positive to x Ãd 1 , otherwise it will give negative contribution.
Proceeding in the same way, the steady state solution of x Ãd 2 (Mdm2) can be obtained as a function of x Ã 11 . The result is given by, where, r~k should be larger than degradation rate of HDAC1 (k 22 ) provided the condition. This behaviours can be seen in Fig. 8. Next we solve for steady state solution x Ãd 2 of Mdm2 as a function of x Ã 8 (steady state solution of p300) to study the impact of p300 on Mdm2. The result is given by, Now we solve steady state solution of x Ãd 1 as a function of x Ã 8 to understand the impact of p300 on p53. The result is given by, where, s~k 4 k 20 k 2 k 3 , u~k x Ãd 2 (x Ã 8 ) is given by the equation (33). The equation (34) indicates that x Ãd 1 is increased by increase in x Ã 8 but decrease in x Ãd 2 . Further if k 24 , the sysnthesis rate of HDAC1 is increased then x Ãd 1 will also be increased. It can also be seen from s and (34) that x Ãd 1 ! 1 k 2 (synthesis rate of Mdm2).
The role of noise and stabilization on p53{Mdm2 regulation Now we present the role of noise on p53 and Mdm2 dynamics. This is done by solving the CLE equations (16)-(29) numerically. The results for different system size parameter, V (1-50) at constant values of k p300 and k HDAC1 , are shown in Fig. 14 (a)-(f). It has been observed that for V~1, no oscillation in p53 is seen. However, as V increases the oscillation starts emerging and when V~25 and 50 sustained oscillations are observed with increasing p53 level. After V~50 i.e. for V §50, the p53 level remains constant i.e. it exhibits sustained oscillatory behaviour. The p53 dynamics is noise induced stochastic process and the strength of noise decreases as V increases. The same behaviour is also seen in Mdm2 dynamics keeping all conditions the same (Fig. 14 (a)-(f)). Now we present the impact of p300 on p53 and Mdm2 in stochastic system by simulating p53 and Mdm2 levels as a function of k p300 for different V (Fig. 15). The result for V~10 shows similar pattern as we found in the deterministic case, but the two conditions of stabilization and activation is achieved earlier with respect to k p300 in stochastic case than that of the deterministic case as shown in the insets of the Fig. 15. Further, as one increases V , the values k p300 for getting the two conditions of stabilization and activation are increased.
The dynamics of p53 concentration remains constant with small fluctuations around the constant values of V (*1{15) even though there is a small damping behavior at initial few hours. We then define T V as the critical time below which the dynamics either shows damped or fixed point (stabilized) oscillations. The plot (T V {V ) in Fig. 16 shows the damped, stabilized and oscillatory regimes. To generate this plot we took 50 simulations for a certain fixed set of parameters and points in the curves show average values with error bars which are correct up to of the order of 5-10 percent in our calculation as shown in Fig. 16. The plots show how system size, which can be taken as noise parameter (as V increases noise strength decreases and vice versa), drives the system at different states, namely, damped, stabilized (no oscillation) and sustain oscillation regimes.
We also study the impact of exposure time (g) on p53 activation and stabilization for different values of V keeping the value of g constant. We can see from the two left panels with insets in Fig. 15 that as g increases the conditions of stabilization and activation are obtained faster.
The results showing the impact of p300 on p53 in stochastic system for different V s and gs are presented in Fig. 17. We also get the similar behaviour in the case as obtained in the case of Mdm2 as shown in Fig. 18.      ing terms, the synthesis rate of HDAC1, x Ãd 2 and x Ã 8 and their variation give significant contributions to the noise terms in equations (39) and (40). However noise contribution in equation (40) is negative contributor to the deterministic part.

Conclusion
The interaction of p300 with p53 allows p53 to be acetylated which prohibits it from decaying and allows it to participate in other reactions. This excess in p300 level eventually leads to increase in capped p53 whose population cannot be controlled and subjects the cell to stress condition. If the excess in p300 level is strong enough it may lead to cell death due to uncontrolled p53, similar to cancer. We observe this phenomena in our simulation results in qualitative sense via three different stages/conditions, namely, first stabilization or normal condition where impact of p300 is negligible, second activation of p53 due to significant interaction between p300 and p53, and third uncontrolled growth of capped p53 due to interaction with excess p300 leading to second stabilization level which may represent cell death condition. The same behaviour is seen in Mdm2 simulation results. The three conditions of stabilization and activation are obtained but the second stabilization level is obtained at lower level as compared to first stabilization level. This may be due to the fact that the increase of capped p53 cannot activate Mdm2 as is done normally, and goes to lower minimum level.
The interaction of HDAC1 with p53 will cause deacetylation of capped p53 which leads p53 to participate in other reactions and able to decay. This may help the already stressed cell to bring back to its normal condition. However excess of HDAC1 will cause excess deacetylation of p53 and will allow the cell to come back far beyond to its normal condition leading to stress. Our results supports these findings.
Noise has interesting but contrasting roles in stochastic system depending upon its strength. If its strength is strong then it has destructive impact on the signal processing in and outside the system etc. However if its strength is weak then it exhibit constructive role, for example weak signal detection, amplification and processing the signal etc. In our study, we found that if the system size is very small where the noise strength is very strong with respect to system size, the associated noise destroy the signal in the system which is in agreement with the theoretical claim. But if the system size is increased in our study where noise strength is comparatively weaker, the signal is resumed in normal with noise induced dynamics in each variable. Moreover, in stochastic system, the p53/Mdm2 is activated by small concentration level of p300/HDAC1 as compared to those in deterministic case and reach stabilization much much faster as compared to deterministic system. Further increase in system size reduces the noise fluctuation in the dynamics of each variable and when V ??, the noise strength is negligible and the system goes to classical deterministic system.
In the present study we determine only the impact of p300 and HDAC1 on p53{Mdm2 regulatory network. For developing any realistic model one needs to incorporate other proteins which influence p53 protein simultaneously and then study the impact collectively. Our study is just one step forward towards understanding p53 regulatory network.