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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 graph of transitions for Model 2.(A) Subdivision of the phase space for Model 2 in 6 domains delimited by the thresholds KP, KMn and KMc. (B) Graph of the transitions followed by the two oscillatory regimes composing birhythmicity shown in Figure 6. The small amplitude oscillatory regime of short period crosses domains D22, D12, D13 and D23 successively (in red). The large amplitude oscillatory regime of long period crosses domains: D22, D21, D11, D12, D13 and D23 successively (in green).
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pone-0017075-g005: Subdivision of the phase space and graph of transitions for Model 2.(A) Subdivision of the phase space for Model 2 in 6 domains delimited by the thresholds KP, KMn and KMc. (B) Graph of the transitions followed by the two oscillatory regimes composing birhythmicity shown in Figure 6. The small amplitude oscillatory regime of short period crosses domains D22, D12, D13 and D23 successively (in red). The large amplitude oscillatory regime of long period crosses domains: D22, D21, D11, D12, D13 and D23 successively (in green).

Mentions: The space of variables can thus be decomposed into 6 domains (D11, D21, D12, D22, D13, D23) delimited by the threshold values of the step functions: KP, KMn and KMc (Figure 5A). The equations of evolution in each domain of the phase space are detailed in Table S1. In each domain, the equations are affine and stable and one can calculate a so-called target equilibrium point of the domain towards which the system will tend (Table S2). The target equilibrium point is an analogous of the focal point defined in a class of piecewise linear diagonal models [34] (see next section). If a target equilibrium point of a domain Dij does not belong to its domain, then the system starting from Dij will leave Dij at some time as it will reach sooner or later a boundary of the domain. If a target equilibrium point of a domain Dij belongs to Dij, it corresponds to a stable equilibrium point of the system. Importantly, the equation of Mn depends on Mc. It follows that the sign of the derivative of Mn can change in each domain of the phase space according to Mc. The transition graph, which is the graph defining the possible transitions between the different domains [35], can thus not be directly derived from the position of the target equilibrium points (see next section).


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 graph of transitions for Model 2.(A) Subdivision of the phase space for Model 2 in 6 domains delimited by the thresholds KP, KMn and KMc. (B) Graph of the transitions followed by the two oscillatory regimes composing birhythmicity shown in Figure 6. The small amplitude oscillatory regime of short period crosses domains D22, D12, D13 and D23 successively (in red). The large amplitude oscillatory regime of long period crosses domains: D22, D21, D11, D12, D13 and D23 successively (in green).
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3038873&req=5

pone-0017075-g005: Subdivision of the phase space and graph of transitions for Model 2.(A) Subdivision of the phase space for Model 2 in 6 domains delimited by the thresholds KP, KMn and KMc. (B) Graph of the transitions followed by the two oscillatory regimes composing birhythmicity shown in Figure 6. The small amplitude oscillatory regime of short period crosses domains D22, D12, D13 and D23 successively (in red). The large amplitude oscillatory regime of long period crosses domains: D22, D21, D11, D12, D13 and D23 successively (in green).
Mentions: The space of variables can thus be decomposed into 6 domains (D11, D21, D12, D22, D13, D23) delimited by the threshold values of the step functions: KP, KMn and KMc (Figure 5A). The equations of evolution in each domain of the phase space are detailed in Table S1. In each domain, the equations are affine and stable and one can calculate a so-called target equilibrium point of the domain towards which the system will tend (Table S2). The target equilibrium point is an analogous of the focal point defined in a class of piecewise linear diagonal models [34] (see next section). If a target equilibrium point of a domain Dij does not belong to its domain, then the system starting from Dij will leave Dij at some time as it will reach sooner or later a boundary of the domain. If a target equilibrium point of a domain Dij belongs to Dij, it corresponds to a stable equilibrium point of the system. Importantly, the equation of Mn depends on Mc. It follows that the sign of the derivative of Mn can change in each domain of the phase space according to Mc. The transition graph, which is the graph defining the possible transitions between the different domains [35], can thus not be directly derived from the position of the target equilibrium points (see next section).

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