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Robustness and period sensitivity analysis of minimal models for biochemical oscillators.

Caicedo-Casso A, Kang HW, Lim S, Hong CI - Sci Rep (2015)

Bottom Line: Furthermore, our parameter sensitivity analysis and stochastic simulations reveal different rankings of hierarchy of period robustness that are determined by the number of sensitive parameters or network topology.In addition, systems with autocatalytic positive feedback loop are shown to be more robust than those with positive feedback via inhibitory degradation regardless of noise type.We demonstrate that robustness has to be comprehensively assessed with both parameter sensitivity analysis and stochastic simulations.

View Article: PubMed Central - PubMed

Affiliation: 1] Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA [2] Departamento de Matemáticas, Universidad del Valle, Cali, Valle, COL.

ABSTRACT
Biological systems exhibit numerous oscillatory behaviors from calcium oscillations to circadian rhythms that recur daily. These autonomous oscillators contain complex feedbacks with nonlinear dynamics that enable spontaneous oscillations. The detailed nonlinear dynamics of such systems remains largely unknown. In this paper, we investigate robustness and dynamical differences of five minimal systems that may underlie fundamental molecular processes in biological oscillatory systems. Bifurcation analyses of these five models demonstrate an increase of oscillatory domains with a positive feedback mechanism that incorporates a reversible reaction, and dramatic changes in dynamics with small modifications in the wiring. Furthermore, our parameter sensitivity analysis and stochastic simulations reveal different rankings of hierarchy of period robustness that are determined by the number of sensitive parameters or network topology. In addition, systems with autocatalytic positive feedback loop are shown to be more robust than those with positive feedback via inhibitory degradation regardless of noise type. We demonstrate that robustness has to be comprehensively assessed with both parameter sensitivity analysis and stochastic simulations.

No MeSH data available.


The effect of a reversible reaction on oscillatory behavior in Model 1.In (A), the period change is shown as a function of k7, the activation rate of autocatalysis, with or without a reversible reaction. The solid curve indicates the system with a reversible reaction when k6 = 0.21, and the dashed curve indicates the system without a reversible reaction when k6 = 0. The parameter value for k4 is drawn in dashed line in (B). In (B) a bifurcation diagram is shown for two parameters, k4 and k6. A region of oscillations is enclosed by the Hopf-bifurcation boundary. Each curve inside the boundary indicates a set of parameter values that produces a fixed period indicated on the curve. In the bottom panels, two bifurcation diagrams are displayed with a reversible reaction (C) and without a reversible reaction (D). The period of oscillations is given as a function of k4 and k7. The rest of the parameter values are taken from Table 1.
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f1: The effect of a reversible reaction on oscillatory behavior in Model 1.In (A), the period change is shown as a function of k7, the activation rate of autocatalysis, with or without a reversible reaction. The solid curve indicates the system with a reversible reaction when k6 = 0.21, and the dashed curve indicates the system without a reversible reaction when k6 = 0. The parameter value for k4 is drawn in dashed line in (B). In (B) a bifurcation diagram is shown for two parameters, k4 and k6. A region of oscillations is enclosed by the Hopf-bifurcation boundary. Each curve inside the boundary indicates a set of parameter values that produces a fixed period indicated on the curve. In the bottom panels, two bifurcation diagrams are displayed with a reversible reaction (C) and without a reversible reaction (D). The period of oscillations is given as a function of k4 and k7. The rest of the parameter values are taken from Table 1.

Mentions: The key difference of Model 1 from a substrate-depletion oscillator is the inclusion of a reversible reaction from Pp into P (Table 1). To investigate the effect of this additional reaction on the system, we explore the behavior of the period of oscillations as each parameter varies. This reversible reaction in the system can be eliminated by simply setting k6 = 0. Figure 1 highlights the effect of a reversible reaction on the oscillatory behavior of the system by using three parameters, k4, k6, and k7. All the other parameter values are kept constant. Figure 1(A) shows the period change as a function of k7 with or without a reversible reaction, which demonstrates that the system with a reversible reaction almost doubles the region of oscillations. Figure 1(B) indicates the relationship between two parameters k4 and k6 in terms of the period, and shows that a larger region of oscillations can be achieved by choosing an appropriate value of k6. Each curve inside the Hopf-bifurcation boundary indicates a set of parameter values that produce periodic solutions with fixed period. Figure 1(C,D) compare two-parameter bifurcation diagrams in the presence (k6 = 0.21) or absence (k6 = 0) of a reversible reaction, which confirms that a region of oscillation can be enlarged when the reversible reaction comes into play in the dynamics. We wondered if we would continue to observe enlarged oscillatory domain with the reversible reaction using a different network topology. We extended Model 1 to incorporate a negative feedback loop (Model 1′) and demonstrate that the aforementioned reversible reaction enlarges oscillatory domain in most parameter space (Supplementary Text S3). Molecular mechanisms of cell division cycles involve an autocatalytic activation of the CycB/CDC2 complex, where CycB/CDC2 activates its own activator, CDC25. This activation of CycB/CDC2 is antagonized by WEE1 creating a meticulously controlled reversible activation and inactivation of the CycB/CDC2 complex5. Based on our results, we hypothesize that this reversible reaction not only controls the activation of CycB/CDC2, but it may also enlarge the region of oscillations.


Robustness and period sensitivity analysis of minimal models for biochemical oscillators.

Caicedo-Casso A, Kang HW, Lim S, Hong CI - Sci Rep (2015)

The effect of a reversible reaction on oscillatory behavior in Model 1.In (A), the period change is shown as a function of k7, the activation rate of autocatalysis, with or without a reversible reaction. The solid curve indicates the system with a reversible reaction when k6 = 0.21, and the dashed curve indicates the system without a reversible reaction when k6 = 0. The parameter value for k4 is drawn in dashed line in (B). In (B) a bifurcation diagram is shown for two parameters, k4 and k6. A region of oscillations is enclosed by the Hopf-bifurcation boundary. Each curve inside the boundary indicates a set of parameter values that produces a fixed period indicated on the curve. In the bottom panels, two bifurcation diagrams are displayed with a reversible reaction (C) and without a reversible reaction (D). The period of oscillations is given as a function of k4 and k7. The rest of the parameter values are taken from Table 1.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4542697&req=5

f1: The effect of a reversible reaction on oscillatory behavior in Model 1.In (A), the period change is shown as a function of k7, the activation rate of autocatalysis, with or without a reversible reaction. The solid curve indicates the system with a reversible reaction when k6 = 0.21, and the dashed curve indicates the system without a reversible reaction when k6 = 0. The parameter value for k4 is drawn in dashed line in (B). In (B) a bifurcation diagram is shown for two parameters, k4 and k6. A region of oscillations is enclosed by the Hopf-bifurcation boundary. Each curve inside the boundary indicates a set of parameter values that produces a fixed period indicated on the curve. In the bottom panels, two bifurcation diagrams are displayed with a reversible reaction (C) and without a reversible reaction (D). The period of oscillations is given as a function of k4 and k7. The rest of the parameter values are taken from Table 1.
Mentions: The key difference of Model 1 from a substrate-depletion oscillator is the inclusion of a reversible reaction from Pp into P (Table 1). To investigate the effect of this additional reaction on the system, we explore the behavior of the period of oscillations as each parameter varies. This reversible reaction in the system can be eliminated by simply setting k6 = 0. Figure 1 highlights the effect of a reversible reaction on the oscillatory behavior of the system by using three parameters, k4, k6, and k7. All the other parameter values are kept constant. Figure 1(A) shows the period change as a function of k7 with or without a reversible reaction, which demonstrates that the system with a reversible reaction almost doubles the region of oscillations. Figure 1(B) indicates the relationship between two parameters k4 and k6 in terms of the period, and shows that a larger region of oscillations can be achieved by choosing an appropriate value of k6. Each curve inside the Hopf-bifurcation boundary indicates a set of parameter values that produce periodic solutions with fixed period. Figure 1(C,D) compare two-parameter bifurcation diagrams in the presence (k6 = 0.21) or absence (k6 = 0) of a reversible reaction, which confirms that a region of oscillation can be enlarged when the reversible reaction comes into play in the dynamics. We wondered if we would continue to observe enlarged oscillatory domain with the reversible reaction using a different network topology. We extended Model 1 to incorporate a negative feedback loop (Model 1′) and demonstrate that the aforementioned reversible reaction enlarges oscillatory domain in most parameter space (Supplementary Text S3). Molecular mechanisms of cell division cycles involve an autocatalytic activation of the CycB/CDC2 complex, where CycB/CDC2 activates its own activator, CDC25. This activation of CycB/CDC2 is antagonized by WEE1 creating a meticulously controlled reversible activation and inactivation of the CycB/CDC2 complex5. Based on our results, we hypothesize that this reversible reaction not only controls the activation of CycB/CDC2, but it may also enlarge the region of oscillations.

Bottom Line: Furthermore, our parameter sensitivity analysis and stochastic simulations reveal different rankings of hierarchy of period robustness that are determined by the number of sensitive parameters or network topology.In addition, systems with autocatalytic positive feedback loop are shown to be more robust than those with positive feedback via inhibitory degradation regardless of noise type.We demonstrate that robustness has to be comprehensively assessed with both parameter sensitivity analysis and stochastic simulations.

View Article: PubMed Central - PubMed

Affiliation: 1] Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA [2] Departamento de Matemáticas, Universidad del Valle, Cali, Valle, COL.

ABSTRACT
Biological systems exhibit numerous oscillatory behaviors from calcium oscillations to circadian rhythms that recur daily. These autonomous oscillators contain complex feedbacks with nonlinear dynamics that enable spontaneous oscillations. The detailed nonlinear dynamics of such systems remains largely unknown. In this paper, we investigate robustness and dynamical differences of five minimal systems that may underlie fundamental molecular processes in biological oscillatory systems. Bifurcation analyses of these five models demonstrate an increase of oscillatory domains with a positive feedback mechanism that incorporates a reversible reaction, and dramatic changes in dynamics with small modifications in the wiring. Furthermore, our parameter sensitivity analysis and stochastic simulations reveal different rankings of hierarchy of period robustness that are determined by the number of sensitive parameters or network topology. In addition, systems with autocatalytic positive feedback loop are shown to be more robust than those with positive feedback via inhibitory degradation regardless of noise type. We demonstrate that robustness has to be comprehensively assessed with both parameter sensitivity analysis and stochastic simulations.

No MeSH data available.