School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Abstract

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.

Background

Oscillations play a vital role in many dynamic cellular processes, and two typical examples of genetic oscillators are the cell cycle oscillators

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

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

Furthermore, the recent biological experiments found that cell cycle oscillations in Xenopus early embryonic extracts might not be driven by constant cyclin B synthesis (

Results

Synchronization of a population of N-cell cycle oscillators

For simplicity, we analyze the case of ten identical oscillators (N = 10), and the same results can be obtained when N is set to be greater than 10. By the numerical simulation, all of the parameters of the coupled system that can reach synchronization are obtained (Table

The parameter settings of the coupled system

**
α**

**
α**

**
α**

**
β**

**
β**

**
β**

**
K**

**
K
_{2}
**

**
K**

0.1

3

3

3

1

1

0.5

0.5

0.5

**
n**

**
n**

**
n**

**
n
**

**
k**

**
K**

**
k**

**
k
**

**
K**

4

4

4

3

2

0.5

1.5

1

0.5

**The synchronization behavior of the coupled oscillators**. The coupled system achieved synchronization when the parameters were set as in Table

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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 _{1}, followed by α_{1}, K_{a}, K_{2}, K_{3}, β_{2}, β_{3}, α_{3 }and k_{m}.

**The sensitivity of the coupled system to the perturbation of parameters**. (A) Sensitivity of CDK1 to the perturbation of parameters. (B) Sensitivity of PLK1 to the perturbation of parameters. (C) Sensitivity of APC to the perturbation of parameters. (D) Sensitivity of R to the perturbation of parameters.

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Synchronization intervals for the selected parameters

The bifurcation diagram for the parameters of the variations in the complex protein CDK1 (C_{1}) of the first oscillator in the coupled system (Additional files _{2 }(Additional file _{2 }(Additional file _{2 }varies in [0, 0.8] and when β_{2 }varies in [0, 2], respectively.

**The bifurcation diagrams for K _{1}, K_{2}, α_{1 }and α_{3}**. (A) The bifurcation diagrams of the activation coefficients K

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**The bifurcation diagrams for the degradation rates**. (A) The bifurcation diagrams of degradation rates β_{2}. (B) The bifurcation diagrams of degradation rates β_{3}. (C) The bifurcation diagrams of the degradation rate of complex protein R. (D) The coupled system achieved an asymptotically steady state when k_{m }= 1.25.

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**The bifurcation diagrams for the coupling parameters**. (A) The bifurcation diagrams for the activation coefficients K_{L }in the Hill function. (B) The bifurcation diagrams for the activation coefficients K_{a }in the Hill function. (C) The bifurcation diagrams for the coupling strength k. (D) The bifurcation diagram for the activation constant k_{0}.

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Furthermore, we searched for the synchronization intervals of these parameters through numerical simulations. We assume that the system achieves synchronization when the synchronization error is smaller than 1e-5. The synchronization intervals obtained for the parameters are shown in Table

The synchronization intervals for the sensitive parameters

**
K**

**
K**

**
K**

**
α**

**
α**

**
β**

[0.48, 0.57]

[0.185, 0.22]

[0.48, 0.57]

[0.48, 0.57]

[0.09, 0.21]

[2.2, 3.5]

[0.9, 1.3]

**
β**

**
k**

**
K**

**
K**

**
k
**

**
k**

[0.89, 1.3]

[1.3, 1.6]

[0.44, 0.55]

[0.46, 0.52]

[0.92, 1.3]

[1.85, 2.3]

From Table _{2}, and the other parameters have only one synchronization interval. Although there are two stable states for the degradation rate β_{2}, there is only one synchronization interval. We can also observe that the more sensitive parameters have smaller synchronization intervals.

To further investigate the dynamical features of the system in the synchronization intervals, we provide some two-parameter bifurcation diagrams with the XPPAUT software _{1}, n_{2 }and n_{3 }are greater than 3 when the coupled Hill coefficient n is set to be no smaller than 3, indicating the rationality of the parameter settings for the Hill coefficients in Table

Two-parameter bifurcation diagrams for four groups of parameters

**Two-parameter bifurcation diagrams for four groups of parameters**. (A) The bifurcation diagram for the coupling strength k and the synthetic rate _{1}. 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 k_{0}. I: oscillation region. II: stable region. (C) The bifurcation diagram for the degradation rate k_{m }and the coefficient k_{0}. 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 _{1 }and _{2}. I, II and III: stable regions. Region IV: oscillation region.

Bifurcation diagrams for the Hill coefficients

**Bifurcation diagrams for the Hill coefficients**. (A)The single parameter bifurcation diagram of n, n_{1}, n_{2 }and n_{3}, with an increase of the parameter n the concentration of CDK1 decreases slightly but the concentration of CDK1 increases gradually with increases in n_{1}, n_{2 }and n_{3}. The changing of n_{2 }is very sensitivity to the concentration of CDK1. (B)(C) and (D) are two-parameter bifurcation diagrams for each pair among n, n_{1}, n_{2 }and n_{3}, respectively. I: stable region, and II and III: oscillation regions.

The effects of sensitive parameters on the synchronization period and amplitude

(A)The effects of the activation coefficients K_{1}, K_{2}, and K_{3 }in the Hill functions

From Additional file _{1 }and K_{3 }have the same influence on the period and amplitude, which is that the oscillation period and amplitude are almost linearly decreased with increases in K_{1 }and K_{3}.

**The effects of K _{1 }and K_{3 }on the period and amplitude**. The above two diagrams show the effects of K

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However, the activation coefficient K_{2 }has distinct influences on the period and amplitude of the synchronization system in different synchronization intervals (Additional file

**The effects of K _{2 }on the period and amplitude**. The above two diagrams show the effects of K

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When K_{2 }varies in the interval [0.35, 0.42], and the parameter α_{2 }changes from 1.6 to 1.0 (Figure _{3 }changes from 1.6 to 1.2 (Figure

The coupled system switches from stable period oscillations to the stable steady state

**The coupled system switches from stable period oscillations to the stable steady state**. (A) The coupled system switches from stable period oscillation when α_{2 }= 1.6 to the stable steady state (C) when α_{2 }= 1, the other parameters are set as Table 1. The coupled system switches from the stable period oscillation (B) when α_{3 }= 1.6 to the stable steady state (D) when α_{3 }= 1.2.

To consider the influence of noise on the features of the system, we introduce the inner noise in the system (2). Figure _{2 }changes from 0.9 to 1.7 or the parameter α_{3 }changes from 1.2 to 1.6, respectively, indicating that the coupled system can switch between a stable state and a stable periodic orbit regardless of whether there is noise.

The coupled system switch from a stable steady state to stable period oscillations when intrinsic noises are added and the parameter α_{2 }is changed

**The coupled system switch from a stable steady state to stable period oscillations when intrinsic noises are added and the parameter α _{2 }is changed**. The coupled system switches from a stable steady state to stable period oscillations if inner noises are introduced when K

The coupled system switch from a stable steady state to stable period oscillations when intrinsic noises are added and the parameter α_{3 }is changed

**The coupled system switch from a stable steady state to stable period oscillations when intrinsic noises are added and the parameter α _{3 }is changed**. The coupled system switches from a stable steady state to stable period oscillations if inner noises are introduced when K

(B)The effects of α_{1 }and α_{3 }on the period and amplitude when synchronization is achieved

The simulated course of the period and amplitude with changes in α_{1 }and α_{3 }are depicted (Additional file _{1 }and increase with an increase in α_{3}, but the change of the period for both α_{1 }and α_{3 }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

**The effects of α _{1 }and α_{3 }on the period and amplitude when synchronization is achieved**. The left two diagrams show the effects of α

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(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 k_{0 }and the activation coefficients K_{L }and K_{a }on the period and amplitude are shown in Additional files _{L}, K_{a }and K increase, but the oscillation period for parameter K_{0 }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 K_{a }of the Hill function of C_{i}.

**The effects of parameters K _{L}, K_{a}, k and k_{0 }on the period when synchronization is achieved**. With an increase in these parameters in their synchronization intervals, the oscillation periods for parameters K

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**The effects of parameters K _{L}, K_{a}, k and k_{0 }on the amplitude when achieved synchronization**. With an increase in these parameters in their synchronization intervals, the oscillation amplitudes for parameters K

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Comparisons of synchronization abilities based on the synchronization time and robustness index

To evaluate the synchronization ability of a coupled system, we simulated two metrics, the synchronization time and the robustness index (see Methods). First, we analyzed the effect of K_{2 }on the synchronization time (Figure _{2 }in the first interval and the synchronization time decreased with an increase of K_{2 }in the second interval. We also observed that the synchronization time in the first interval was much shorter than the synchronization time in the second interval, but the synchronization was very sensitive to changes in the initial values.

The effects of parameter K_{2 }on the synchronization time in two different synchronization intervals

**The effects of parameter K _{2 }on the synchronization time in two different synchronization intervals**. (A) The synchronization time of K

Figure

The effects of the coupling strength on the synchronization time under different impulse signals and impulse strengths

**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 α_{1 }= aq. The square wave signal, the sine signal and the noise signal are corresponding to the following formula: _{0 }is set to 4 and t_{1 }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

The effects of the signal strength on the robustness in three different signal inputs under the variation of parameters at 10% and 20%

**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%.

The effects of the signal strength on the robustness in noise signal input under the variation of parameters at 10% and 20%

**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.

Discussion

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.

The cell division cycle of the Xenopus embryo was demonstrated to consist of two phases: interphase and metaphase

The bifurcation diagram for the coupling strength k

**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 _{1 }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 K_{2 }= 0.35, _{3 }= 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.

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.

Conclusions

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

The simplified reaction diagram of the Embryonic cell cycle is depicted in Figure

The simplified diagrams of the Xenopus Embryonic cell cycle and global coupling style between oscillators

**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

For cell _{1}), and the inactivation rate is proportional to the concentration of CDK1* (C_{i}) times a Hill function of APC*(A_{i}). The activation of Plk1 (P_{i}) by CDK1* is proportional to the concentration of inactive Plk1 (we also assume the total concentration of active and inactive Plk1 to be constant, specifically 1-P_{i}) times a Hill function of CDK1*(C_{i}), and the inactivation is proportional to Plk1*(P_{i}). The activation of APC (A_{i}) by Plk1 is proportional to the concentration of inactive APC (1- Ai) times a Hill function of Plk1 (Pi), and the rate of inactivation of APC is described by simple mass action kinetics. The resulting three-ODE model is the following:

where the parameters α_{i}, β_{i }(i = 1, 2 and 3), K_{1}, K_{2 }and K_{3 }are set to be the same as those in the Literature _{1}, n_{2 }and n_{3}, 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

The ODE equations for N cell oscillators (denoted by i = 1, 2,.., N) are written as follows:

Synchronization of a population of N-cell cycle oscillators

In order to quantify the level of synchronization of the coupled system, we introduce the synchronization error proposed in

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

To investigate the effects of parameter changes on the amount of all of the variables in the coupled system, we make the sensitivity analysis of parameters with an approach proposed in _{0}, we have the following:

The solution can be approximated by expanding the Taylor series about the nominal solution _{0}).

The sensitivity function _{0}, the sensitivity function suffices to approximate the solution. Then, we can calculate the sensitivity of the system parameters by solving the following sensitivity equation (See

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

To quantify the speediness and robustness of the synchronization, we use two quantities, the synchronization time and robustness index, to evaluate the synchronization ability under different conditions. The synchronization time is calculated according to the time when the synchronization error of the coupled system is smaller than 1e-5. The robustness index (r) is computed with the following formula, which is similar to the formulas in

where M is the number of equally divided regions according to the distribution of the oscillation period and b_{k }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 _{1 }= N), and r = 0 corresponds to no synchronization and poor robustness (M = N and b_{k }= 1).

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

XFZ designed the research and wrote the manuscript. WZ performed the methods and conducted the numerical experiments. All the authors read, edited and approved the final manuscript.

Acknowledgements

This work was supported by the Chinese National Natural Science Foundation under Grant 61173060.

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