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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.


Bifurcation diagrams for Model 2 and Model 3.Left column displays bifurcation diagrams for Model 2. Right column displays bifurcation diagrams for Model 3. (A) Period as a function of the transcription rate of mRNA v. (B) Period as a function of v and k2, representing the transcription and translation rates, respectively. (C) Period as a function of k1 and Ka, representing the degradation rate and the threshold of mRNA, respectively. All other parameter values are taken from Table 1. Numbers in (B) and (C) indicate the period of oscillations along each curve.
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f2: Bifurcation diagrams for Model 2 and Model 3.Left column displays bifurcation diagrams for Model 2. Right column displays bifurcation diagrams for Model 3. (A) Period as a function of the transcription rate of mRNA v. (B) Period as a function of v and k2, representing the transcription and translation rates, respectively. (C) Period as a function of k1 and Ka, representing the degradation rate and the threshold of mRNA, respectively. All other parameter values are taken from Table 1. Numbers in (B) and (C) indicate the period of oscillations along each curve.

Mentions: First, the patterns of periodic solutions are different between Models 2 and 3 (Table 1). Model 2 shows sinusoidal shapes of both M and P, and Model 3 shows distinct patterns of M and P, where M undergoes sharp rise followed by an exponential decrease due to the prolonged increase of P exerting negative feedback on the synthesis of M. Second, we observe drastically different behaviors of the systems with changes in parameter space. The top row of Fig. 2 displays bifurcation diagrams showing the period of oscillations as a function of v, which is the synthesis rate of mRNA in both models. As v increases, the period of Model 2 evolves with a small increase initially and then decreases monotonically (left panel in Fig. 2(A)), whereas the period of Model 3 increases at a relatively faster rate and then decreases at a slower rate (right panel in Fig. 2(A)). The behavior of Model 2 results from the autocatalytic effect on the protein. As v increases, increased mRNA results in reaching the threshold for autocatalysis faster, which leads to faster inhibition of mRNA synthesis resulting in decrease of period. In Model 3, the period initially increases as a function of v, because faster increase of mRNA leads to prolonged high level of protein due to its inhibitory role on its own degradation, which then leads to extended repression of mRNA synthesis, as indicated by the long exponential decay pattern of mRNA decrease. However, once it passes the peak period at v = 148, the period decreases with increasing v, because the trough of the protein level is above the threshold for the positive feedback to result in the observed sharp protein increase as shown in Table 1. Therefore, prolonged protein abundance actually decreases when v > 148, resulting in a shorter duration of negative feedback. See Supplementary Information for numerical solutions at different values of v (Supplementary Fig. S1).


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

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

Bifurcation diagrams for Model 2 and Model 3.Left column displays bifurcation diagrams for Model 2. Right column displays bifurcation diagrams for Model 3. (A) Period as a function of the transcription rate of mRNA v. (B) Period as a function of v and k2, representing the transcription and translation rates, respectively. (C) Period as a function of k1 and Ka, representing the degradation rate and the threshold of mRNA, respectively. All other parameter values are taken from Table 1. Numbers in (B) and (C) indicate the period of oscillations along each curve.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f2: Bifurcation diagrams for Model 2 and Model 3.Left column displays bifurcation diagrams for Model 2. Right column displays bifurcation diagrams for Model 3. (A) Period as a function of the transcription rate of mRNA v. (B) Period as a function of v and k2, representing the transcription and translation rates, respectively. (C) Period as a function of k1 and Ka, representing the degradation rate and the threshold of mRNA, respectively. All other parameter values are taken from Table 1. Numbers in (B) and (C) indicate the period of oscillations along each curve.
Mentions: First, the patterns of periodic solutions are different between Models 2 and 3 (Table 1). Model 2 shows sinusoidal shapes of both M and P, and Model 3 shows distinct patterns of M and P, where M undergoes sharp rise followed by an exponential decrease due to the prolonged increase of P exerting negative feedback on the synthesis of M. Second, we observe drastically different behaviors of the systems with changes in parameter space. The top row of Fig. 2 displays bifurcation diagrams showing the period of oscillations as a function of v, which is the synthesis rate of mRNA in both models. As v increases, the period of Model 2 evolves with a small increase initially and then decreases monotonically (left panel in Fig. 2(A)), whereas the period of Model 3 increases at a relatively faster rate and then decreases at a slower rate (right panel in Fig. 2(A)). The behavior of Model 2 results from the autocatalytic effect on the protein. As v increases, increased mRNA results in reaching the threshold for autocatalysis faster, which leads to faster inhibition of mRNA synthesis resulting in decrease of period. In Model 3, the period initially increases as a function of v, because faster increase of mRNA leads to prolonged high level of protein due to its inhibitory role on its own degradation, which then leads to extended repression of mRNA synthesis, as indicated by the long exponential decay pattern of mRNA decrease. However, once it passes the peak period at v = 148, the period decreases with increasing v, because the trough of the protein level is above the threshold for the positive feedback to result in the observed sharp protein increase as shown in Table 1. Therefore, prolonged protein abundance actually decreases when v > 148, resulting in a shorter duration of negative feedback. See Supplementary Information for numerical solutions at different values of v (Supplementary Fig. S1).

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.