key: cord-0939735-krliloi7 authors: Qi, J.-P.; Shao, S.-H.; Li, D.-D.; Zhou, G.-P. title: A dynamic model for the p53 stress response networks under ion radiation date: 2006-10-31 journal: Amino Acids DOI: 10.1007/s00726-006-0454-3 sha: 6d34ce06351e718ea4651f1ea0ce8a2c1a42ef5c doc_id: 939735 cord_uid: krliloi7 P53 controls the cell cycle arrest and cell apoptosis through interaction with the downstream genes and their signal pathways. To stimulate the investigation into the complicated responses of p53 under the circumstance of ion radiation (IR) in the cellular level, a dynamic model for the p53 stress response networks is proposed. The model can be successfully used to simulate the dynamic processes of generating the double-strand breaks (DSBs) and their repairing, ataxia telangiectasia mutated (ATM) activation, as well as the oscillations occurring in the p53-MDM2 feedback loop. As a vital anticancer gene, p53 can regulate its downstream genes through their signal pathways, and further implement cell cycle arrest and cell apoptosis (Chou et al., 1999a (Chou et al., , 2000 Perez and Purdy, 1998; Vogelstein et al., 2000) . These functions are important to repair DNA damage, and eliminate the abnormal cells with DNA damage or deregulated proliferation, especially to modulate the activity of anticancer agents (Chou et al., 1997; Pauklin et al., 2005; Ritter et al., 2002) . Recently, several models have been proposed to explain the damped oscillations of p53 in cell populations (Lev Bar-Or et al., 2000; Ma et al., 2005) . However, the dynamic mechanism of the single-cell responses is not completely clear yet, and the complicated regulations among genes and their signal pathways need to be further addressed. Based on the previous models (Lev Bar-Or et al., 2000; Ma et al., 2005) and stimulated by the impact of the bioinformatical approaches on the biomedicine (Chou, 2004 ), a dynamic model for the p53 stress response networks under IR was proposed, along with the kinetics of DSB generation and repair, ATM activation, as well as the p53-MDM2 feedback loop module. Furthermore, analyses were given on the threshold time for the switch-like ATM activation, as well as on the number and amplitude of oscillations between p53 and MDM2 in response to IR. Using differential equations and graphic approaches to study various dynamical and kinetic processes of biological systems can provide useful insights, as indicated by many previous studies on a series of important biological topics, such as enzyme-catalyzed reactions (Chou, 1989a; Chou and Forsen, 1980; Chou et al., 1979; King and Altman, 1956; Kuzmic et al., 1992; Lin and Neet, 1990; Zhou and Deng, 1984) , internal motions of biomacromolecules (Chou, 1984 (Chou, , 1987 (Chou, , 1988 (Chou, , 1989b Chou and Chen, 1977; Chou et al., 1989; Chou and Mao, 1988; Han, 1992; Han and Wang, 1992; Martel, 1992; Sobell et al., 1983) , diffusion-controlled reactions in enzyme systems (Chou and Jiang, 1974; Chou and Zhou, 1982; Cotes and Sceats, 1988; Zhou et al., 1983; Zhou and Zhong, 1982) , protein folding kinetics (Chou, 1990) , inhibition kinetics of processive nucleic acid polymerases and nucleases (Althaus et al., 1993a, b; Chou et al., 1994a) , soliton transport in protein and DNA (Chou et al., 1994b; Sinkala, 2006; Zhou, 1989) , analysis of codon usage (Chou and Zhang, 1992; Zhang and Chou, 1994) , base frequencies in the anti-sense strands (Zhang and Chou, 1996) , hepatitis B viral infections (Xiao et al., 2006) , HBV virus gene missense mutation (Xiao et al., 2005) , and visual analysis of SARS-CoV (Gao et al., 2006; Wang et al., 2005) . In this study, we used differential equations and graphic methods to investigate the dynamic and kinetic processes of p53 stress response networks under the effects of IR. Under the genome stresses, numerous co-factors are involved in enhancing p53-mediated transcription (Magne et al., 2006) . The interactions among these co-factors make the model more complicated. Therefore, some simplification procedures are needed in order to allow the model to incorporate more biochemical information (Tyson, 1999; Tyson and Novak, 2001) . To realize this, let us take the following criteria or assumptions: (1) only the vital components and interactions are taken into account in the model; (2) all the localization issues are ignored; (3) the simple linear relations are used to describe the interactions among the components concerned; and (4) there are enough substances to keep the system ''workable'' (Tyson, 1999) . The scheme of the integrated model is given in Fig. 1 . In the DSBs generation and repair module, the acute IR induces DSBs stochastically and forms DSB-protein complexes (DSBCs) at each of the damage sites after interacting with the DNA repair proteins. As a sensor of genome stress, ATM is activated by the DSBCs signal transferred from DSBs. ATM activation switches on or off the p53-Mdm2 feedback loop, further regulating the downstream genes to control the cell cycle arrest and the cell apoptosis in response to genome stresses (Weller, 1998) . Under the continuous effect of acute IR dose, DSBs occur and trigger two major repair mechanisms in eukaryotic cells: homologous recombination (HR) and nonhomologous end joining (NHEJ) (Rapp and Greulich, 2004; Rothkamm et al., 2003) . DSB repair is a first-order process if the break ends associated with the same DSB are rejoined, or a second-order process if the break ends associated with two different DSBs are rejoined (Daboussi et al., 2002) . Meanwhile, about 60-80% of DSBs are rejoined quickly, whereas the remaining 20-40% of DSBs are rejoined more slowly (Budman and Chu, 2005; Daboussi et al., 2002) . In the first part of our model, we implement the module of DSBs generation and repair process. As shown in Fig. 2 , the module contains both the fast and slow kinetics, each is composed of a reversible binding of repair proteins and DSB lesions into DSBCs, and an irreversible process from the DSBCs to the fixed DSBs (Daboussi et al., 2002; Rapp and Greulich, 2004) . Some experimental data suggest that the quantity of the resulting DSBs within different IR dose domains obey a Poisson distribution (Ma et al., 2005) . In accordance with the experiments, we assume that the stochastic number of the resulting DSBs per time scale is proportional to the number generated by a Poisson random function during the period of acute radiation (Ma et al., 2005) . The DSBs generation process is formulated as follows: where [D T ] is the concentration of the total resulting DSBs induced by continuous IR in both fast and slow repair process, k T the parameter to set the number of DSBs per time scale, and a ir the parameter to set the number of DSBs per IR dose. Moreover, we assume that adequate repair proteins are available around DSBs sites, and 70% of the initial DSBs are fixed by the fast repair process. Each DSB can be in one of the following three states: intact DSB, DSBC, and fixed DSB. Thus, we have the following differential equations: where [D] , [C] , and [F d ] represent the respective concentrations of DSB, DSBC, and fixed DSB; k dc , k cd , k fd are the transition rates among the above three states; k dc , and k cross represent the repair rates of the first-order and the second-order process (Daboussi et al., 2002) , and the subscripts '1' and '2' refer to the fast and the slow kinetics. As a DNA damage detector, ATM exists as a dimer in unstressed cells. After IR is applied, intermolecular autophosphorylation occurs, causing the dimer to dissociate rapidly into the active monomers. The active ATM monomer (ATM à ) can prompt the p53 expression further (Budman and Chu, 2005; Daboussi et al., 2002) . Like (Chou et al., 1999b) , here we use the superscript à to represent the activate state. Based on the existing model of ATM switch (Ma et al., 2005) , we present an ATM activation module under IR. Figure 3 shows the module scheme, which includes three components: ATM dimer, inactive ATM monomer, and ATM à . Here, we assume that DSBCs is the main signal transduction from DSBs to p53-MDM2 feedback loop through ATM activation, and the rate of ATM activation is a function of the amount of DSBCs and the self-feedback of ATM à . Furthermore, the total concentration of ATM is a constant, including ATM dimer, ATM monomer and ATM à (Ma et al., 2005) . As a detector of DNA damage, ATM activation plays an important role in triggering the regulatory mechanisms of p53 stress response networks (Kohn and Pommier, 2005; Oren, 2003) . After the acute IR is applied, phosphorylation of inactive ATM monomers is promoted first by DSBCs and then rapidly by means of the positive feedback from ATM à , accounting for the intermolecular autophosphorylation (Ma et al., 2005) . The main formulations are as follows: where [ATM d ], [ATM] and [ATM à ] represent the concentrations of ATM dimer, ATM monomer, and active ATM monomer respectively; k undim is the rate of ATM undimerization, and k dim the rate of ATM dimerization; k ar is the rate of ATM monomer inactivation, and k af the rate of ATM monomer activation. In addition, f is the function of ATM activation, the term a 1 C implies the fact that DSBs somehow activate ATM molecules at a distance, a 2 [ATM à ] indicates the mechanism of autophosphorylation of ATM, and a 3 C[ATM à ] represents the interaction between the DSBCs and ATM à (Bakkenist and Kastan, 2003; Ma et al., 2005) . As shown in Fig. 4 , p53 and its principal antagonist, Mdm2 is transactivated by p53, form a p53-MDM2 feedback loop. ATM à can elevates the transcriptional activity of p53 by prompting phosphorylation of p53 and degradation of MDM2 protein (Lev Bar-Or et al., 2000) . This negative feedback loop can produce oscillations in response to the sufficiently strong IR dose (Ma et al., 2005) . To account for a decreased binding affinity between inactive p53 and p53 à , we assume that MDM2-induced degradation of inactive p53 is faster than that of p53 à . The main differential equations used in this module are as follows: To ensure the accuracy of the simulation results, let us consider the fact that the valid parameter sets should obey the following rules (Kohn and Pommier, 2005; Lev Bar-Or et al., 2000; Ma et al., 2005) : (1) the model must contain oscillations. This is important as there has been experimental evidence that oscillations occur between p53 and MDM2 after cell stress; (2) the mechanism used to mathematically describe the degradation of p53 by MDM2 is accurate only for low concentrations of p53; (3) the concentration of p53 à is much higher than that of inactive p53 after the system reaching an equilibrium. Based on these three rules and the existing parameter sets used in (Ma et al., 2005) , we obtained the kinetics of p53 stress response networks under acute IR dose through simulation platform in MATLAB7.0. The detailed parameter sets used in our model can be found in Tables 1-3 . During the simulation process, we applied 8 Gy IR to generate DSBs fraction. In order to agree with the experimental results that the measured 30-40 DSBs per Gy occurred in the single cell (Ma et al., 2005) , the stochastic number of resulting DSBs were generated by using a Poisson random function with a mean of 35x as continuous IR dose of x Gy was applied. Figure 5a displays a stochastic trace of the resulting DSBs versus the constant radiation time. Compared with the Monte Carlo methods (Ma et al., 2005) , our simulation method is more suitable for the Figure 5b shows the dynamic traces of resulting DSBs, DSBCs, and intact DSBs in response to 8 Gy IR in both fast and slow repair kinetics. The ATM activation module was established to describe the switch-like dynamics of the ATM activation in response to DSBCs increasing, and the regulation mechanisms during the process of the ATM transferring DNA damage signals to the p53-MDM2 feedback loop. Under the cooperative function of DSBCs and the positive selffeedback of ATM à , the ATM would reach the equilibrium state within minutes because of the fast phosphor-ylation (Ma et al., 2005) . As shown in Fig. 6a , the concentration of ATM à increases fast against the decreasing of ATM dimer and ATM monomer, both of which reach the cross point behind the IR by about 9 minutes. In Fig. 6b delineates the kinetic process of ATM à versus IR dose increasing, and the process of ATM reaching the saturation after IR overpass the threshold value by about 0.5 Gy. The immunoblot studies (Bakkenist and Kastan, 2003) show a rather abrupt onset of activated ATM that starts at 0.1-0.2 Gy and reaches saturation at about 0.4 Gy. The step-like traces in Fig. 6a suggest that the ATM module can produce an on-off switching signal to the p53-MDM2 feedback loop; the simulation shown in Fig. 6b qualitatively resembles the results in (Bakkenist and Kastan, 2003; Ma et al., 2005) and the actual data reported in (Bakkenist and Kastan, 2003) except for the saturation at about 0.5 Gy. The p53-MDM2 feedback loop is a vital part in controlling the downstream genes and regulation pathways to fight against the genome stresses (Lev Bar-Or et al., 2000; Ritter et al., 2002; Weller, 1998) . In response to the input signal of ATM à , the p53-Mdm2 module generates one or more oscillations. The response traces of p53 and Mdm2 protein under continuous application of 8-Gy IR from time 0 are shown in Fig. 7a . Upon the activation by ATM à and decreased degradation by Mdm2, the total amount of p53 proteins increases quickly. Due to the p53dependent induction of Mdm2 transcription, the increase of Mdm2 proteins is sufficiently large to lower the p53 level, which in turn reduces the amount of the Mdm2 proteins. The oscillation pulses shown in Fig. 7a have a period of 400 min, and the phase difference between p53 and Mdm2 is about 100 min. Moreover, the first pulse is slightly higher than the second, quite similar with the experimental observations (Ma et al., 2005) and the simulation results (Lahav et al., 2004) . By comparing the results in Fig. 7b and c, we can see that two or more pulses of p53 and MDM2 are generated after increasing the IR doses. Our simulations are fully in accordance with the observations that, in average, the greater the number of p53 pulses, the more severe damage that can be triggered (Lev Bar-Or et al., 2000; Ritter et al., 2002; Weller, 1998) . These results show that our model of p53 stress response network yields roughly equal-sized pulses with a mean number that increases with the IR strength and the continuous IR time. A set of differential equations, combined with the Poisson random function, were proposed to model the DSBs generation and repair process. It is demonstrated according to our model that ATM exhibits a strong sensitivity and switch-like behaviour in response to the number of DSBs, fully consistent with the necessary outcome to transfer the stress signal to the p53-MDM2 feedback loop and arrest the cell cycle in response to the acute IR (Ma et al., 2005) . Once the IR dose is sufficiently large, the p53-MDM2 feedback loop will produce oscillations. Especially, the number and amplitude of the oscillations are different according to the cell types and the IR dose domains (Ma et al., 2005) . 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