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A theoretical exploration of birhythmicity in the p53-Mdm2 network.

Abou-Jaoudé W, Chaves M, Gouzé JL - PLoS ONE (2011)

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: COMORE Project-team, INRIA Sophia Antipolis, Sophia Antipolis, France. wabou@sophia.inria.fr

ABSTRACT
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.

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Subdivision of the phase space and transition graph for Model 3.(A) Subdivision of the phase space for Model 3 in 8 domains delimited by the thresholds KP, KMn, KMc and the additional threshold K (in red). (B) Transition graph of Model 3. The graph contains a branching point in domain D23 and two embedded cycles. From this domain, the system can either go to domain D13 or domain D22 depending on the initial conditions.
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pone-0017075-g007: Subdivision of the phase space and transition graph for Model 3.(A) Subdivision of the phase space for Model 3 in 8 domains delimited by the thresholds KP, KMn, KMc and the additional threshold K (in red). (B) Transition graph of Model 3. The graph contains a branching point in domain D23 and two embedded cycles. From this domain, the system can either go to domain D13 or domain D22 depending on the initial conditions.

Mentions: In order to reproduce the two folds forming the outward excursion which characterizes the large amplitude oscillatory regime in Model 1 and Model 2, we added another threshold K (K<KMn) for p53 level. As the evolution of Mn in each domain is now monotone, the introduction of this new threshold allows changing the sign of the derivative of Mn as the system crosses threshold K and thus recovering in particular the fold observed in domain D21 in Model 2 (Figure 6). The space of variables can thus be decomposed into 8 domains (D11, D21, D12, D22, D13, D23, D14, D24), defined by the thresholds KMc, KMn plus the additional threshold K for p53, and KP for nuclear Mdm2 (Figure 7). The equations of evolution in each domain of the phase space are detailed in Table S3 (Model 3).


A theoretical exploration of birhythmicity in the p53-Mdm2 network.

Abou-Jaoudé W, Chaves M, Gouzé JL - PLoS ONE (2011)

Subdivision of the phase space and transition graph for Model 3.(A) Subdivision of the phase space for Model 3 in 8 domains delimited by the thresholds KP, KMn, KMc and the additional threshold K (in red). (B) Transition graph of Model 3. The graph contains a branching point in domain D23 and two embedded cycles. From this domain, the system can either go to domain D13 or domain D22 depending on the initial conditions.
© Copyright Policy
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3038873&req=5

pone-0017075-g007: Subdivision of the phase space and transition graph for Model 3.(A) Subdivision of the phase space for Model 3 in 8 domains delimited by the thresholds KP, KMn, KMc and the additional threshold K (in red). (B) Transition graph of Model 3. The graph contains a branching point in domain D23 and two embedded cycles. From this domain, the system can either go to domain D13 or domain D22 depending on the initial conditions.
Mentions: In order to reproduce the two folds forming the outward excursion which characterizes the large amplitude oscillatory regime in Model 1 and Model 2, we added another threshold K (K<KMn) for p53 level. As the evolution of Mn in each domain is now monotone, the introduction of this new threshold allows changing the sign of the derivative of Mn as the system crosses threshold K and thus recovering in particular the fold observed in domain D21 in Model 2 (Figure 6). The space of variables can thus be decomposed into 8 domains (D11, D21, D12, D22, D13, D23, D14, D24), defined by the thresholds KMc, KMn plus the additional threshold K for p53, and KP for nuclear Mdm2 (Figure 7). The equations of evolution in each domain of the phase space are detailed in Table S3 (Model 3).

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: COMORE Project-team, INRIA Sophia Antipolis, Sophia Antipolis, France. wabou@sophia.inria.fr

ABSTRACT
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.

Show MeSH