A theoretical exploration of birhythmicity in the p53-Mdm2 network

Experimental observations performed in the p53-Mdm2 network, one of the key protein modules involved in the control of proliferation of abnormal cells in mammals, revealed the existence of two frequencies of oscillations of p53 and Mdm2 in irradiated cells depending on the irradiation dose. These observations raised the question of the existence of birhythmicity, i.e. the coexistence of two oscillatory regimes for the same external conditions, in the p53-Mdm2 network which would be at the origin of these two distinct frequencies. A theoretical answer has been recently suggested by Ouattara, Abou-Jaoudé and Kaufman who proposed a 3-dimensional differential model showing birhythmicity to reproduce the two frequencies experimentally observed. The aim of this work is to analyze the mechanisms at the origin of the birhythmic behavior through a theoretical analysis of this differential model. To do so, we reduced this model, in a first step, into a 3-dimensional piecewise linear differential model where the Hill functions have been approximated by step functions, and, in a second step, into a 2-dimensional piecewise linear differential model by setting one autonomous variable as a constant in each domain of the phase space. We find that two features related to the phase space structure of the system are at the origin of the birhythmic behavior: the existence of two embedded cycles in the transition graph of the reduced models; the presence of a bypass in the orbit of the large amplitude oscillatory regime of low frequency. Based on this analysis, an experimental strategy is proposed to test the existence of birhythmicity in the p53-Mdm2 network. From a methodological point of view, this approach greatly facilitates the computational analysis of complex oscillatory behavior and could represent a valuable tool to explore mathematical models of biological rhythms showing sufficiently steep nonlinearities.     

Analysis of a minimal model for p53 oscillations

Oscillatory behaviours in genetic networks are important examples for studying the principles underlying the dynamics of cellular regulation. Recently the team of Alon has reported a surprisingly rich oscillatory response of the p53 tumor suppressor to irradiation stress et al. [Lahav, G., Rosenfeld, N., Sigal, A., Geva-Zatorsky, N., Levine, A.J., Elowitz, M.B., Alon, U., 2004. Dynamics of the p53-Mdm2 feedback loop in individual cells. Nat. Genet. 36 (2), 147-150; Geva-Zatorsky, N., Rosenfeld, N., Itzkovitz, S., Milo, R., Sigal, A., Dekel, E., Yarnitzky, T., Liron, Y., Polak, P., Lahav, G., Alon, U., 2006. Oscillations and variability in the p53 system. Mol. Syst. Biol. 2, 2006.0033]. Several models for this system have been proposed by different groups, based essentially on negative feedback loops. In this paper we investigate in detail oscillations and stability in a deterministic time delayed differential model of the core circuit for p53 expression. This model is representative of a class of modelling approaches of this system, based on a "minimal" set of well-established biomolecular regulations. Depending on the protein degradation rates we show the existence of bifurcations between a stable steady state and oscillations both in presence and absence of stress.     

p53-Mdm2 loop controlled by a balance of its feedback strength and effective dampening using ATM and delayed feedback

When the genomic integrity of a cell is challenged, its fate is determined in part by signals conveyed by the p53 tumour suppressor protein. It was observed recently that such signals are not simple gradations of p53 concentration, but rather a counter-intuitive limit-cycle behaviour. Based on a careful mathematical interpretation of the experimental body of knowledge, we propose a model for the p53 signalling network and characterise the p53 stability and oscillatory dynamics. In our model, ATM, a protein that senses DNA damage, activates p53 by phosphorylation. In its active state, p53 has a decreased degradation rate and an enhanced transactivation of Mdm2, a gene whose protein product Mdm2 tags p53 for degradation. Thus the p53-Mdm2 system forms a negative feedback loop. However, the feedback in this loop is delayed, as the pool of Mdm2 molecules being induced by p53 at a given time will mark for degradation the pool of p53 molecules at some later time, after the Mdm2 molecules have been transcribed, exported out of the nucleus, translated and transported back into the nucleus. The analysis of our model demonstrates how this time lag combines with the ATM-controlled feedback strength and effective dampening of the negative feedback loop to produce limit-cycle oscillations. The picture that emerges is that ATM, once activated by DNA damage, makes the p53-Mdm2 oscillator undergo a supercritical Hopf bifurcation. This approach yields an improved understanding of the global dynamics and bifurcation structure of our time-delayed, negative feedback model and allows for predictions of the behaviour of the p53 system under different perturbations.     

The p53 and Mdm2 families in cancer.

Cells within an organism are occasionally exposed to either intracellular or environmental stress. Such stress often has genotoxic potential that enhances the probability of cancer. Two gene families, the p53 family (p53, p63 and p73) and the Mdm2 family (Mdm2 and MdmX), serve as major integrators of the signals generated by genotoxic and oncogenic stress. Their co-ordinated modulation ensures an optimal response to stress and decreases the likelihood of cancer. Work over the past year has provided better understanding of the p53-Mdm2 module that lies in the heart of this regulatory network, and of the intricate interplay between the various members of the network.     

MDM2-HDAC1-mediated deacetylation of p53 is required for its degradation

The tumor suppressor p53 is stabilized and activated in response to cellular stress through post-translational modifications including acetylation. p300/CBP-mediated acetylation of p53 is negatively regulated by MDM2. Here we show that MDM2 can promote p53 deacetylation by recruiting a complex containing HDAC1. The HDAC1 complex binds MDM2 in a p53-independent manner and deacetylates p53 at all known acetylated lysines in vivo. Ectopic expression of a dominant-negative HDAC1 mutant restores p53 acetylation in the presence of MDM2, whereas wild-type HDAC1 and MDM2 deacetylate p53 synergistically. Fibroblasts overexpressing a dominant negative HDAC1 mutant display enhanced DNA damage-induced p53 acetylation, increased levels of p53 and a more pronounced induction of p21 and MDM2. These results indicate that acetylation promotes p53 stability and function. As the acetylated p53 lysine residues overlap with those that are ubiquitylated, our results suggest that one major function of p53 acetylation is to promote p53 stability by preventing MDM2-dependent ubiquitylation, while recruitment of HDAC1 by MDM2 promotes p53 degradation by removing these acetyl groups.     

The enhancement of stability of p53 in MTBP induced p53–MDM2 regulatory network

We have modeled an MTBP-MDM2–p53 regulatory network by integrating p53–MDM2 autoregulatory model (Proctor and Gray, 2008) with the effect of a cellular protein MTBP (MDM2 binding protein) which is allowed to bind with MDM2 (Brady et al., 2005). We study this model to investigate the activation of p53 and MDM2 steady state levels induced by MTBP protein under different stress conditions. Our simulation results in three approaches namely deterministic, Chemical Langevin equation and stochastic simulation of Master equation show a clear transition from damped limit cycle oscillation to fixed point oscillation during a certain time period with constant stress condition in the cell. This transition is the signature of transition of p53 and MDM2 levels from activated state to stabilized steady state levels. We present various phase diagrams to show the transition between unstable and stable states of p53 and MDM2 concentration levels and also their possible relations among critical value of the parameters at which the respective protein level reach stable steady states. In the stochastic approach, the dynamics of the proteins become noise induced process depending on the system size. We found that this noise enhances the stability of the p53 steady state level.     

Steady States and Oscillations in the p53/Mdm2 Network

p53 is activated in response to events compromising the genetic integrity of a cell. Recent data show that p53 activity does not increase steadily with genetic damage but rather fluctuates in an oscillatory fashion (Lahav et al., Nature Genetics, 36, 147-150, 2004). Theoretical studies suggest that oscillations can arise from a combination of positive and negative feedbacks or from a long negative feedback loop alone. Both negative and positive feedbacks are present in the p53/Mdm2 network, but it is not known what roles they play in the oscillatory response to DNA damage. We developed a mathematical model of p53 oscillations based on positive and negative feedbacks in the p53/Mdm2 network. According to the model, the system reacts to DNA damage by moving from a stable steady state into a region of stable limit cycles. Oscillations in the model are born with large amplitude, which guarantees an all-or-none response to damage. As p53 oscillates, damage is repaired and the system moves back to a stable steady state with low p53 activity. The model reproduces experimental data in quantitative detail. We suggest new experiments for dissecting the contributions of negative and positive feedbacks to the generation of oscillations.     

Cyclin G: a regulator of the p53-Mdm2 network

A recent study published in the April issue of Molecular Cell has shown that cyclin G, a p53 target, is a regulatory component of the active PP2A holoenzyme, which activates Mdm2 through dephosphorylation. These findings suggest that cyclin G is a key regulator of the p53-Mdm2 network.     

Reconstitution of Mdm2-dependent post-translational modifications of p53 in yeast

p53 mediates cell cycle arrest or apoptosis in response to DNA damage. Its activity is subject to a tight regulation involving a multitude of post-translational modifications. The plethora of functional protein interactions of p53 at present precludes a clear understanding of regulatory principles in the p53 signaling network. To circumvent this complexity, we studied here the minimal requirements for functionally relevant p53 post-translational modifications by expressing human p53 together with its best characterized modifier Mdm2 in budding yeast. We find that expression of the human p53-Mdm2 module in yeast is sufficient to faithfully recapitulate key aspects of p53 regulation in higher eukaryotes, such as Mdm2-dependent targeting of p53 for degradation, sumoylation at lysine 386 and further regulation of this process by p14(ARF). Interestingly, sumoylation is necessary for the recruitment of p53-Mdm2 complexes to yeast nuclear bodies morphologically akin to human PML bodies. These results suggest a novel role for Mdm2 as well as for p53 sumoylation in the recruitment of p53 to nuclear bodies. The reductionist yeast model that was established and validated in this study will now allow to incrementally study simplified parts of the intricate p53 network, thus helping elucidate the core mechanisms of p53 regulation as well as test novel strategies to counteract p53 malfunctions.     

Novel recurrent neural network for modelling biological networks: Oscillatory p53 interaction dynamics

Understanding the control of cellular networks consisting of gene and protein interactions and their emergent properties is a central activity of Systems Biology research. For this, continuous, discrete, hybrid, and stochastic methods have been proposed. Currently, the most common approach to modelling accurate temporal dynamics of networks is ordinary differential equations (ODE). However, critical limitations of ODE models are difficulty in kinetic parameter estimation and numerical solution of a large number of equations, making them more suited to smaller systems. In this article, we introduce a novel recurrent artificial neural network (RNN) that addresses above limitations and produces a continuous model that easily estimates parameters from data, can handle a large number of molecular interactions and quantifies temporal dynamics and emergent systems properties. This RNN is based on a system of ODEs representing molecular interactions in a signalling network. Each neuron represents concentration change of one molecule represented by an ODE. Weights of the RNN correspond to kinetic parameters in the system and can be adjusted incrementally during network training. The method is applied to the p53-Mdm2 oscillation system – a crucial component of the DNA damage response pathways activated by a damage signal. Simulation results indicate that the proposed RNN can successfully represent the behaviour of the p53-Mdm2 oscillation system and solve the parameter estimation problem with high accuracy. Furthermore, we presented a modified form of the RNN that estimates parameters and captures systems dynamics from sparse data collected over relatively large time steps. We also investigate the robustness of the p53-Mdm2 system using the trained RNN under various levels of parameter perturbation to gain a greater understanding of the control of the p53-Mdm2 system. Its outcomes on robustness are consistent with the current biological knowledge of this system. As more quantitative data become available on individual proteins, the RNN would be able to refine parameter estimation and mapping of temporal dynamics of individual signalling molecules as well as signalling networks as a system. Moreover, RNN can be used to modularise large signalling networks.     

Harmonic Oscillations in Homeostatic Controllers: Dynamics of the p53 Regulatory System

Homeostatic mechanisms are essential for the protection and adaptation of organisms in a changing and challenging environment. Previously, we have described molecular mechanisms that lead to robust homeostasis/adaptation under inflow or outflow perturbations. Here we report that harmonic oscillations occur in models of such homeostatic controllers and that a close relationship exists between the control of the p53/Mdm2 system and that of a homeostatic inflow controller. This homeostatic control model of the p53 system provides an explanation why large fluctuations in the amplitude of p53/Mdm2 oscillations may arise as part of the homeostatic regulation of p53 by Mdm2 under DNA-damaging conditions. In the presence of DNA damage p53 is upregulated, but is subject to a tight control by Mdm2 and other factors to avoid a premature apoptotic response of the cell at low DNA damage levels. One of the regulatory steps is the Mdm2-mediated degradation of p53 by the proteasome. Oscillations in the p53/Mdm2 system are considered to be part of a mechanism by which a cell decides between cell cycle arrest/DNA repair and apoptosis. In the homeostatic inflow control model, harmonic oscillations in p53/Mdm2 levels arise when the binding strength of p53 to degradation complexes increases. Due to the harmonic character of the oscillations rapid fluctuating noise can lead, as experimentally observed, to large variations in the amplitude of the oscillation but not in their period, a behavior which has been difficult to simulate by deterministic limit-cycle models. In conclusion, the oscillatory response of homeostatic controllers may provide new insights into the origin and role of oscillations observed in homeostatically controlled molecular networks.     

Direct p53 transcriptional repression: in vivo analysis of CCAAT-containing G2/M promoters

In response to DNA damage, p53 activates G(1)/S blocking and apoptotic genes through sequence-specific binding. p53 also represses genes with no target site, such as those for Cdc2 and cyclin B, key regulators of the G(2)/M transition. Like most G(2)/M promoters, they rely on multiple CCAAT boxes activated by NF-Y, whose binding to DNA is temporally regulated during the cell cycle. NF-Y associates with p53 in vitro and in vivo through the alphaC helix of NF-YC (a subunit of NF-Y) and a region close to the tetramerization domain of p53. Chromatin immunoprecipitation experiments indicated that p53 is associated with cyclin B2, CDC25C, and Cdc2 promoters in vivo before and after DNA damage, requiring DNA-bound NF-Y. Following DNA damage, p53 is rapidly acetylated at K320 and K373 to K382, histones are deacetylated, and the release of PCAF and p300 correlates with the recruitment of histone deacetylases (HDACs)-HDAC1 before HDAC4 and HDAC5-and promoter repression. HDAC recruitment requires intact NF-Y binding sites. In transfection assays, PCAF represses cyclin B2, and a nonacetylated p53 mutant shows a complete loss of repression potential, despite its abilities to bind NF-Y and to be recruited on G(2)/M promoters. These data (i) detail a strategy of direct p53 repression through associations with multiple NF-Y trimers that is independent of sequence-specific binding of p53 and that requires C-terminal acetylation, (ii) suggest that p53 is a DNA damage sentinel of the G(2)/M transition, and (iii) delineate a new role for PCAF in cell cycle control.