Zhang W and Zou X (2012), Synchronization ability of coupled cell-cycle o...

XB-ART-46080

BMC Syst Biol 2012 Jan 01;6 Suppl 1Suppl 1:S13. doi: 10.1186/1752-0509-6-S1-S13.

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Synchronization ability of coupled cell-cycle oscillators in changing environments.

Zhang W , Zou X .

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BACKGROUND: The biochemical oscillator that controls periodic events during the Xenopus embryonic cell cycle is centered on the activity of CDKs, and the cell cycle is driven by a protein circuit that is centered on the cyclin-dependent protein kinase CDK1 and the anaphase-promoting complex (APC). Many studies have been conducted to confirm that the interactions in the cell cycle can produce oscillations and predict behaviors such as synchronization, but much less is known about how the various elaborations and collective behavior of the basic oscillators can affect the robustness of the system. Therefore, in this study, we investigate and model a multi-cell system of the Xenopus embryonic cell cycle oscillators that are coupled through a common complex protein, and then analyze their synchronization ability under four different external stimuli, including a constant input signal, a square-wave periodic signal, a sinusoidal signal and a noise signal. RESULTS: Through bifurcation analysis and numerical simulations, we obtain synchronization intervals of the sensitive parameters in the individual oscillator and the coupling parameters in the coupled oscillators. Then, we analyze the effects of these parameters on the synchronization period and amplitude, and find interesting phenomena, e.g., there are two synchronization intervals with activation coefficient in the Hill function of the activated CDK1 that activates the Plk1, and different synchronization intervals have distinct influences on the synchronization period and amplitude. To quantify the speediness and robustness of the synchronization, we use two quantities, the synchronization time and the robustness index, to evaluate the synchronization ability. More interestingly, we find that the coupled system has an optimal signal strength that maximizes the synchronization index under different external stimuli. Simulation results also show that the ability and robustness of the synchronization for the square-wave periodic signal of cyclin synthesis is strongest in comparison to the other three different signals. CONCLUSIONS: These results suggest that the reaction process in which the activated cyclin-CDK1 activates the Plk1 has a very important influence on the synchronization ability of the coupled system, and the square-wave periodic signal of cyclin synthesis is more conducive to the synchronization and robustness of the coupled cell-cycle oscillators. Our study provides insight into the internal mechanisms of the cell cycle system and helps to generate hypotheses for further research.

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Species referenced: Xenopus
Genes referenced: cdk1 plk1

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	Figure 1. Two-parameter bifurcation diagrams for four groups of parameters. (A) The bifurcation diagram for the coupling strength k and the synthetic rate α1. The whole region is divided into three regions I, II and III. I and III: stable regions. II: oscillation region. (B) The bifurcation diagram for the coupling strength k and the coefficient k0. I: oscillation region. II: stable region. (C) The bifurcation diagram for the degradation rate km and the coefficient k0. I: oscillation region. II: stable region. The behavior of the system in the region between two lines is unclear. (D) The bifurcation diagrams for the degradation rates β1 and β2. I, II and III: stable regions. Region IV: oscillation region.
	Figure 2. Bifurcation diagrams for the Hill coefficients. (A)The single parameter bifurcation diagram of n, n1, n2 and n3, with an increase of the parameter n the concentration of CDK1 decreases slightly but the concentration of CDK1 increases gradually with increases in n1, n2 and n3. The changing of n2 is very sensitivity to the concentration of CDK1. (B)(C) and (D) are two-parameter bifurcation diagrams for each pair among n, n1, n2 and n3, respectively. I: stable region, and II and III: oscillation regions.
	Figure 3. The coupled system switches from stable period oscillations to the stable steady state. (A) The coupled system switches from stable period oscillation when α2 = 1.6 to the stable steady state (C) when α2 = 1, the other parameters are set as Table 1. The coupled system switches from the stable period oscillation (B) when α3 = 1.6 to the stable steady state (D) when α3 = 1.2.
	Figure 4. The coupled system switch from a stable steady state to stable period oscillations when intrinsic noises are added and the parameter α2 is changed. The coupled system switches from a stable steady state to stable period oscillations if inner noises are introduced when K2 is set to be between two different synchronization intervals (The intensity of inner noise is 0.001 and the parameter α2 changes from 0.9 to 1.7)
	Figure 5. The coupled system switch from a stable steady state to stable period oscillations when intrinsic noises are added and the parameter α3 is changed. The coupled system switches from a stable steady state to stable period oscillations if inner noises are introduced when K2 is between two different synchronization intervals (The intensity of inner noise is 0.001 and the parameter α3 changes from 1.2 to 1.6)
	Figure 6. The effects of parameter K2 on the synchronization time in two different synchronization intervals. (A) The synchronization time of K2 in the first synchronization interval. It shows the synchronization time increases with an increase in K2. (B) The synchronization time of K2 in the second synchronization interval. It shows the synchronization time decreases with an increase of K2.
	Figure 7. The effects of the coupling strength on the synchronization time under different impulse signals and impulse strengths. (A) The effects of the coupling strength k on the synchronization time under a constant signal input with the impulse strengths 0.1, 0.15 and 0.2, respectively. (B) The effects of the coupling strength k on the synchronization time under a square wave signal input with the impulse strengths 0.4, 0.6, 0.8 and 1, respectively. (C) The effects of the coupling strength k on the synchronization time under a sine signal input with the impulse strengths 0.8, 1, 1.2 and 1.4, respectively. (C) The effects of the coupling strength k on the synchronization time under the Gauss noise input with the mean strengths 0.01, 0.03, 0.05 and 0.07, respectively and standard deviation 0.001. Where the constant signal: the cyclic synthesis α1 = aq. The square wave signal, the sine signal and the noise signal are corresponding to the following formula: α1t=aqift modt0<t10otherwise, α1t=aqsintifsint>00otherwise, α1t=aq+bq*rand. Where aq is the signal strength, t0 is set to 4 and t1 is set to 2 in the square wave signal, bq is set to 0.001 when aq is 0.01 and bq is set to 0.01 when aq is larger than 0.01 in the noise signal.
	Figure 8. The effects of the signal strength on the robustness in three different signal inputs under the variation of parameters at 10% and 20%. (A) The effects of the signal strength on the robustness index under a constant signal, a square wave signal and a sine signal at the variation of 10%. (B) The effects of the signal strength on the robustness index under a constant signal, a square wave signal and a sine signal at the variation of 20%.
	Figure 9. The effects of the signal strength on the robustness in noise signal input under the variation of parameters at 10% and 20%. The legend with circle and star lines represent the variation of 10% and 20%, respectively.
	Figure 10. The bifurcation diagram for the coupling strength k. (A) Bifurcation diagram for the coupling strength k. When the coupling strength is increased to the range between two saddle-node bifurcations, the coupled system can exhibit bistability and also exhibits some hysteresis, i.e., CDK1 converges to a low or a high state depending on the initial conditions. But when k is set to 1.6 and the synthetic rate α1 of CDK1 is changed into the square wave signal, the coupled system behaviors as (B): a pulse input drives CDK1 into the upper state and then oscillates, or oscillates all the time which depends on the initial state of the coupled system. (C) The bifurcation diagram for k when K2 = 0.35, α3 = 1.6 and other parameters are set as Table 1. (D) shows that the pulse input drives CDK1 into the upper state and cannot get back. SN: Saddle-Node point. HB: Hopf bifurcation point.
	Figure 11. The simplified diagrams of the Xenopus Embryonic cell cycle and global coupling style between oscillators. (A) The simplified diagrams of the Xenopus Embryonic cell cycle (redrawn from [1]). (B) Global coupling style between N oscillators.

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