Synchronization ability of coupled cell-cycle oscillators in changing environments
© Zhang and Zou; licensee BioMed Central Ltd. 2012
Published: 16 July 2012
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.
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.
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.
Oscillations play a vital role in many dynamic cellular processes, and two typical examples of genetic oscillators are the cell cycle oscillators [1, 2] and circadian clocks . Understanding the molecular mechanisms that are responsible for oscillations and their collective behaviors is important for clarifying the dynamics of cellular life and for designing efficient drug doses. Synchronization is a type of typical collective behavior and is a basic motion in nature that can explain many natural phenomena [4, 5]. Recent studies have shown that cellular communication is accomplished by synchronization, and a number of simulations and fundamental experimental studies have also confirmed synchronization mechanisms in some interacting or independent biological systems [6–9]. The revealed synchronization mechanisms and the dynamics of control in multi-cellular systems are essential for understanding inherent mechanisms of living organisms at both the molecular and cellular levels [10–12].
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 that confirm that the interactions in the cell cycle can produce oscillations and predict behaviors such as synchronization [13–16], but much less is known about how the various elaborations and collective behavior of the basic oscillators can affect the robustness of the system and how cells use the information to control the cell cycle.
The experiments indicated that the cyclin-dependent kinases (CDKs) are not solely responsible for establishing the global cell-cycle transcription program, although they have a function in the regulation of cell cycle transcription, and the precise cell cycle could be controlled by coupling a transcription factor network oscillator with the cyclin-CDK oscillator . To elucidate various synchronization mechanisms (from the viewpoint of the dynamics) by investigating the effects of various biologically plausible couplings and external stimuli, in this paper we use the three-order ordinary differential equation (ODE) model of the Xenopus embryonic cell cycle that was presented in the literature  as a basic model for one oscillator and study the synchronization for a network of N oscillators in which all of the units were indirectly coupled by interacting with a common environment. We present the coupled model of cell cycle oscillators and the synchronization feature of the coupled system, and we determine the synchronization intervals of system parameters and analyze the effects of parameters on the period and amplitude when synchronization is achieved.
Furthermore, the recent biological experiments found that cell cycle oscillations in Xenopus early embryonic extracts might not be driven by constant cyclin B synthesis ( and ). Therefore, we consider the cyclin synthesis rate as four possible impulse signals, including a constant input signal, square-wave periodic signal, a sinusoidal signal and a noise signal, and investigate the synchronization ability under different external stimuli by defining two measures, including the synchronization time and the robustness index. These studies are viewed as an important step toward the comprehensive understanding of mechanisms of the Xenopus embryonic cell cycle.
Synchronization of a population of N-cell cycle oscillators
The parameter settings of the coupled system
Parameter sensitivity analysis of the coupled system
The range of the parameter distributions is set to be a random number between [0, 1], and we obtain an average over 100 runs. All of the results are normalized, and the effects of the parameter changes on the amounts of the three variables and the complex protein R in equation (2) (Additional file 2). From Additional file 2 we can see that the most sensitive parameter is K1, followed by α1, Ka, K2, K3, β2, β3, α3 and km.
Synchronization intervals for the selected parameters
The bifurcation diagram for the parameters of the variations in the complex protein CDK1 (C1) of the first oscillator in the coupled system (Additional files 3, 4, 5). From Additional file 3, we find an interesting phenomenon, which is that there are two stable states for parameters K2 (Additional file 3 (B)) and β2 (Additional file 4 (A)), when K2 varies in [0, 0.8] and when β2 varies in [0, 2], respectively.
The synchronization intervals for the sensitive parameters
From Table 2 we can see that there are two synchronization intervals for K2, and the other parameters have only one synchronization interval. Although there are two stable states for the degradation rate β2, there is only one synchronization interval. We can also observe that the more sensitive parameters have smaller synchronization intervals.
The effects of sensitive parameters on the synchronization period and amplitude
(A)The effects of the activation coefficients K1, K2, and K3 in the Hill functions
From Additional file 6 we can observe that the activation coefficients K1 and K3 have the same influence on the period and amplitude, which is that the oscillation period and amplitude are almost linearly decreased with increases in K1 and K3.
However, the activation coefficient K2 has distinct influences on the period and amplitude of the synchronization system in different synchronization intervals (Additional file 7). In the first interval [0.185, 0.22], the period increases and the amplitude is almost the same, but in the second interval, the period and amplitudes decrease.
(B)The effects of α1 and α3 on the period and amplitude when synchronization is achieved
The simulated course of the period and amplitude with changes in α1 and α3 are depicted (Additional file 8). From Additional file 8 we can see that the oscillation period and amplitude decreased with an increase in α1 and increase with an increase in α3, but the change of the period for both α1 and α3 is obvious and the change in the amplitude is slight. This observation further demonstrates that the activation rates can adjust the oscillation period in the coupled system, which is the same as in the single oscillator of interlinked positive and negative feedback .
(C) The effects of coupling parameters on the period and amplitude when synchronization is achieved
The effects of the coupling strength k, the ratio coefficient k0 and the activation coefficients KL and Ka on the period and amplitude are shown in Additional files 9 and 10. With an increase in these parameters, the oscillation periods for parameters KL, Ka and K increase, but the oscillation period for parameter K0 decreases. The trend of the oscillation amplitudes is similar to the periods except for the coupling strength k. However, the influence of the coupling parameters on the period is greater than the influence on the amplitude, especially for the activation coefficient Ka of the Hill function of Ci.
Comparisons of synchronization abilities based on the synchronization time and robustness index
In this study, we investigated the synchronization feature of one coupling system of N cell-cycle oscillators that were coupled through a common complex protein. The work of Mclsaac. R et al.  analyzed the spatial synchronization oscillation of Xenopus embryos that was triggered by the fertilization-initiated calcium wave; this investigation may offer insights into determining the components of the complex protein R.
There are also some limitations to our approach. In our proposed coupled model, we chose three components, which composed a negative feedback loop as the basic model; this configuration captured the main features of the cell cycle but may have limitations for interpreting the details of the mechanism of the cell cycle, for example, adding the positive feedback of the Wee1 as well as Cdc25 on the cyclin CDK1 may contribute a more widely tunable period and amplitude of the oscillation .
Although we have mainly examined effects of the most sensitive parameters and coupled parameters on the cellular dynamics, there are also other important factors that may play important roles in biological processes and should be further investigated from theoretical viewpoints.
In this paper, a new dynamical global coupled model for cell cycle oscillators is presented. Through bifurcation analysis and numerical simulations, we determined synchronization intervals of the coupled system. Our simulation results show that the more sensitive parameters have smaller synchronization intervals. Furthermore, we find that there are two synchronization intervals of the activation coefficient in the Hill function of the activated CDK1 that activate the Plk1, and different synchronization intervals have distinct influences on the period and amplitude of the synchronization system. Afterwards, when this parameter shifts from two different synchronization intervals, the coupled system can switch from stable period oscillations to a stable steady state. Computational results through the two metrics, the synchronization time and the robustness index, indicate that a larger coupling strength has a shorter synchronization time for the three signals, and the robustness index for the square-wave periodic signal of cyclin synthesis is strongest in comparison to the other signals. These results suggest that the reaction process in which the activated cyclin-CDK1 activates the Plk1 has a very important influence on the synchronization features of the coupled system. The square-wave periodic signal of cyclin synthesis is more beneficial to the synchronization and robustness of the coupled cell-cycle oscillators.
Our work not only can be viewed as an important step toward the comprehensive understanding of the mechanisms of the Xenopus embryonic cell cycle but also can provide a guide for further biological experiments.
Models and methods
Model of coupled cell cycle regulatory oscillators
where the parameters αi, βi (i = 1, 2 and 3), K1, K2 and K3 are set to be the same as those in the Literature , except for the Hill coefficients n1, n2 and n3, which are set to be 4.
To reveal the internal mechanism of the Xenopus embryonic cell cycle, we assume that all of the cells are coupled indirectly through the common extracellular medium, in other words, they are coupled through a complex protein (R) that excites the protein of Cyclin-CDK1 in the core cell cycle regulatory pathway. The diagram for global coupling of the cell cycle oscillators is shown as in Figure 11(B).
Synchronization of a population of N-cell cycle oscillators
The coupled system is defined to achieve synchronization when E reaches zero in a limited amount of time. In our simulation, we assume that the system achieves synchronization when the synchronization error E is smaller than 1e-5.
Parameter sensitivity analysis of the coupled system
The range of the parameter distributions is set to be a random number between [0, 1] and we obtain an average over 100 runs; all of the results are normalized.
Identification of the synchronization intervals for the selected parameters
To analyze the effects on the synchronization when the parameters change, we perform a bifurcation analysis for the sensitive parameters and the coupling parameters by varying the chosen parameter and fixing the other parameters.
Calculation of the synchronization time and robustness index
where M is the number of equally divided regions according to the distribution of the oscillation period and bk is the number of the distribution of periods of the kth region; N is the total number of the distribution of periods that are obtained through using the Latin sampling method  by a variation of the parameters 10% or 20%. (In our study, N = 1000). Obviously, 0 ≤ r ≤ 1, where r = 1 corresponds to perfect synchronization and perfect robustness (M = 1 and b1 = N), and r = 0 corresponds to no synchronization and poor robustness (M = N and bk = 1).
This work was supported by the Chinese National Natural Science Foundation under Grant 61173060.
This article has been published as part of BMC Systems Biology Volume 6 Supplement 1, 2012: Selected articles from The 5th IEEE International Conference on Systems Biology (ISB 2011). The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/6/S1.
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