Modeling the Control of Meiotic Cell Divisions: Entry, Progression, and Exit

Abstract

Upon nitrogen starvation, Schizosaccharomyces pombe exit the mitotic cell cycle and become irreversibly committed to the completion of meiosis program. Meiotic cell divisions are coordinated with sporulation events to produce haploid spores. In the last few decades, experiments on fission yeast have revealed different molecular players involved in two meiotic cell divisions, meiosis I (MI) and meiosis II (MII). How the MI entry, MI-to-MII transition, and MII exit occur because of the dynamics of the regulatory network is not well understood. In this work, we developed a comprehensive mathematical model of the network that describes the temporal dynamics of meiotic progression. The model accounts for the phenotypes of several experimental data (single and multiple mutations). We demonstrate the control strategy involving multiple feedback loops to yield two successive division cycles. The differential regulation of anaphase-promoting complex/cyclosome (APC/C) coactivators and its inhibitors is crucial for the dynamics of both MI-to-MII transition and MII exit. This model generates mechanistic insights that help in further experiments and modeling.

Significance

Meiosis is a specialized cell cycle program in eukaryotes to generate haploid gametes. The canonical cell cycle network is modified by meiosis-specific proteins to undergo one round of DNA replication followed by two divisions. The fission yeast Schizosaccharomyces pombe is an interesting model system to understand the interplay between cell cycle and meiosis-specific proteins at the systems level. In this work, we developed a comprehensive mathematical model of meiosis regulatory network in fission yeast for the first time, to our knowledge. This model captures the dynamics of meiosis under different genetic backgrounds and generates mechanistic insights that help in further experiments and modeling. We also relate our findings in fission yeast to the meiosis regulation in other organisms.

Introduction

Meiosis is a specialized cell cycle program in eukaryotes to generate haploid gametes (germ cells) and spores (yeasts) in response to developmental and environmental signals. During meiosis, diploid cells undergo DNA replication followed by two consecutive cell divisions, meiosis I (MI) and meiosis II (MII). In the MI, dyad chromosomes segregate, whereas in the MII, sister chromatids segregate as in mitosis. These two divisions are tightly controlled by the regulation of cyclin-dependent kinase (Cdk) and anaphase-promoting complex/cyclosome (APC/C). How the cell cycle network is modified to control the meiotic cell divisions is not well understood. In the last few decades, experiments in the fission yeast Schizosaccharomyces pombe have revealed the roles of different proteins in meiotic cell divisions and sporulation. This provides an opportunity to study how the meiotic progression and exit is controlled at the systems level by assembling the components into a regulatory network.

Under nitrogen starvation, the fission yeast enters the meiotic program and becomes irreversibly committed to the completion of meiosis with the activation of Mei2 (1). We have shown previously that multiple feedback loops between Pat1 kinase and transcriptional activator Ste11 ensure irreversible commitment by making the activation of Mei2 bistable (2). Mei2 is an RNA-binding protein required for entry into S phase and meiotic cell divisions. Mei2 binds to long noncoding RNA meiRNA to sequester the YTH-family, RNA-binding protein Mmi1, an inhibitor of meiosis. Mmi1 eliminates the meiosis-specific transcripts by binding to it, and its inactivation leads to the accumulation of proteins involved in the meiotic progression (3). One of the targets of Mmi1 is a forkhead-type transcription factor, Mei4, which controls the expression of middle-phase genes involved in meiotic cell divisions (3). The middle-phase genes include B-type cyclins (Cdc13, Cig2, and Rem1) that promote the activation of Cdk1 in MI and MII (4, 5, 6).

The inactivation of Cdk1 depends on APC/C, which targets cyclins to degradation by ubiquitination (7). In the fission yeast, the APC/C activity is regulated by mitotic coactivators Slp1 and Ste9 and by three coactivators Fzr1/Mfr1, Fzr2, and Fzr3 that are expressed, specifically in meiosis. Meiosis-specific APC/C inhibitor Mes1 distinctly regulates coactivators Slp1 and Fzr1 (8). Interestingly, Mes1 is not merely an inhibitor but is also a substrate of APC/C. The ubiquitination of Mes1 by APC/Slp1 is required for the metaphase-to-anaphase transition in MI (9). Further, the deletion of other APC/C coactivators Ste9, Fzr2, and Fzr3 leads to the delay in Mes1 destruction at MII (8). Therefore, different APC/C coactivators can regulate Mes1 destruction at different stages of meiosis. However, Ste9, Fzr2, and Fzr3 have a redundant function because the deletion of these APC/C coactivators has no apparent effect on the meiosis progression (8). On the other hand, Fzr1 deletion leads to the entry into the third nuclear (“meiosis-III-like”) division, and spore formation is affected (10,11). Its expression is boosted by a meiosis-specific transcriptional activator Cuf2 (11). Therefore, both Cuf2 and Fzr1 are required for the timely termination of MII and spore formation.

In this study, we propose a consensus picture of meiosis regulation in the fission yeast (Fig. 1 a). A comprehensive mathematical model of this network captures the dynamics of wild-type and several mutants. The model provides a systems-level interpretation of experimental data, deduces the molecular logic of meiosis, and generates insights that help in further experiments and modeling. We show that the mutual antagonism between APC/C coactivators and its inhibitors controls the dynamics of both the MI-to-MII transition and MII exit. We propose a control strategy involving multiple feedback loops to explain the meiosis program.

Figure 1.

(a) The regulatory network of meiotic progression and exit in fission yeast. (b) The dynamics of cyclin-B-Cdk1, Mes1, and different coactivators of APC/C is shown for the wild-type (control). To see this figure in color, go online.

Methods

Fig. 1 a shows the regulatory network involved in the progression of meiosis with Mei4 as an input. This network was assembled from several experimental observations in the literature for meiotic cell divisions and exit. The core of the regulatory network is the regulation of Cdk1 and APC/C by meiosis-specific factors. Cdk1 activity depends on the synthesis and degradation of cyclins and inhibitory Tyr-15 phosphorylation/dephosphorylation of Cdk1. Although there are different cyclins in fission yeast, a single cyclin-Cdk complex is sufficient to drive the meiotic cell cycle (12). Therefore, we considered only one cyclin (referred to as cyclin B) to simplify the model. Cyclin B synthesis is activated by Mei4, whereas its degradation is mediated by APC/C in complex with coactivators Slp1, Ste9, Fzr1, and Fzr2. We ignored Fzr3 because it has a redundant function as Fzr2 (8). Tyr-15 phosphorylation (inactivation) and dephosphorylation (activation) of Cdk1 depends on Wee1 kinase and Cdc25 phosphatase, respectively. Mei4 is shown to promote the synthesis of Cdc25 and inhibit the synthesis of Wee1 (13,14). CyclinB-Cdk1, in turn, activates Cdc25 and inactivates Wee1 by phosphorylation.

Cyclin-B-Cdk1 promotes the activation of APC/C-Slp1 (see below), whereas it inhibits APC/C-Ste9 by phosphorylation (15). APC/C-Fzr1 is less active in the mitotic extract compared with interphase extracts, which suggests that it can be regulated by Cdk1-dependent phosphorylation (8). However, APC-Fzr2 activation is independent of cyclin-B-Cdk1 because it is active in both interphase and mitotic extracts (8). Mei4 indirectly activates the synthesis of Fzr1 by activating the synthesis of transcriptional activator Cuf2 (Fig. 1 a; (16)). The expression profile of Fzr2 shows that it accumulates in MII (8). Because the mechanism controlling Fzr2 synthesis is unknown, we assumed that it is also a Cuf2 target, and its accumulation is slower compared with Fzr1. Mes1 inhibits all APC/C coactivators by forming a stoichiometric inhibitory complex. Mes1 transcription is boosted in a window between late MI and MII (17). The levels of Mes1 depends on a messenger RNA splicing factor, which is synthesized in the Mei4-dependent manner (18). APC/C coactivators also promote the degradation of Mes1.

The dynamics of individual components were captured by translating the network into a set of ordinary differential equations and algebraic equations (Table S1). The synthesis and degradation of cyclin B, Cuf2, splicing factor, Mes1, and Cdc25 were modeled by the law of mass action. The synthesis rate of Wee1 was reduced in the presence of Mei4 to account for the transcriptional repression. Cuf2-dependent synthesis of APC/C coactivators Fzr1 and Fzr2 was described by Hill equation. The Hill equation was used to capture the delay in Fzr2 accumulation and to eliminate the unknown intermediate step in its transcription. Cyclin-B-Cdk1-dependent phosphorylation of APC/C coactivator Fzr1 was described by Goldbeter-Koshland (GK) kinetics. This kinetics was previously used to model the APC/C coactivator regulation by multisite phosphorylation (19). The meiosis-specific model was integrated with an existing fission yeast cell cycle model (20). The cell cycle model describes the Cdk1-dependent inhibition of Wee1 and Ste9 and activation of Cdc25 by GK kinetics. Further, Cdk1-dependent activation of APC-Slp1 is indirect through an intermediate enzyme, which provides a time delay. Both intermediate enzyme and APC-Slp1 regulation are described by GK kinetics. The total level of Slp1 and Ste9 is modeled as a fixed parameter. Further, we considered that Mes1 forms a stoichiometric inhibitory complex with both phosphorylated and unphosphorylated Slp1, Ste9, and Fzr1. The regulation of Fzr1 by Cuf2, cyclin-B-Cdk1, and Mes1 are shown in Fig. S1.

The set of ordinary differential equations were solved numerically with XPPAUT (http://www.math.pitt.edu/∼bard/xpp/xpp.html) to obtain the temporal profiles. The dynamic variables represent relative protein concentrations and are dimensionless. Rate constants (k) have a dimension of min⁻¹. Michaelis constants (J), half-saturation constants (k_h), and other parameters are also dimensionless. We simulated the model starting with the parameter values of an existing fission yeast cell cycle model and with degradation rates of Mes1 and cyclin B deduced based on the experimental observations that Mes1 ubiquitination by APC/C-Slp1 and APC/C-Ste9 and Cdc13 ubiquitination by APC/C-Fzr1 and APC/C-Ste9 are higher compared with other coactivators (8). The complex formation is assumed to be rapid, with high equilibrium binding constant compared with synthesis, degradation, and phosphorylation rate constants. The parameter values were refined by simulating the model to capture the temporal dynamics of two meiotic cell divisions in the wild-type (5,21); time of appearance of Mes1, Fzr1, and Fzr2; and qualitative behavior of various mutants (single and multiple deletions) (8,21). Such a modeling approach was used previously to constrain the parameters of the budding yeast cell cycle and meiotic commitment (2,22, 23, 24). The model equations and parameter values of the wild-type are provided as part of the Supporting Material along with the XPPAUT code (Tables S1–S3). The parameter values used for the simulation of mutants are provided in Table S4. The gene deletion experiment was simulated by setting the rate of synthesis of protein to zero. The total level of a protein was set to zero if it is a fixed parameter. For temperature-sensitive mutants, the rate of activation was halved or set to zero depending on partial or complete inactivation, respectively. Cdk1 mutant was simulated by reducing the rate of cyclin synthesis. The ubiquitination rates of Mes1 by different APC/C coactivators were set to zero to mimic the nonubiquitinable form of Mes1.

We also studied the sensitivity of the model (system dynamics) by varying the parameter values of the wild-type (basal) over a range (twofold) in both directions. We checked manually for each parametric variation whether the model satisfies a rough criterion of two peaks of cyclin-B-Cdk1 activation. The parametric changes that yielded one or more than two divisions were identified. Further, one- and two-parameter bifurcation analyses were also performed using XPPAUT to study the effect of different parameter values and to understand the role of different feedback loops.

Results

The model describes the two peaks of cyclin-B-Cdk1 activation in the fission yeast

Numerical simulations of the equations were performed with the initial condition corresponding to G2 arrest obtained by deleting Mei4. Cyclin B is initially high, but Cdk1 activity is low because of Tyr-15 phosphorylation. The level of Cdc13 is shown to be high in Mei4-deleted cells (5). We simulated the model to capture the two peaks of cyclin-B-Cdk1 activation representing MI and MII cell divisions in the presence of Mei4. Fig. 1 b shows how the concentration of cyclin-B-Cdk1, Mes1, and different coactivators of APC/C vary during meiosis. The time required to complete the two peaks of cyclin-B-Cdk1 activity agrees reasonably well with the time required to complete two meiotic divisions in the fission yeast (21). The activation of cyclin-B-Cdk1 occurs rapidly in the MI compared with the MII. The APC-Slp1 activity reaches a peak at 30 min in MI and 90 min in MII. In the model, Mei4 forms an incoherent feed-forward loop (IC-FFL) involving cyclin-B-Cdk1 (positive arm) and Mes1 (negative arm) to control the activation of APC-Slp1 (see Fig. 1 a). Cyclin-B-Cdk1 promotes the activation of APC-Slp1, but the Mei4-dependent synthesis of Mes1 in MI can keep it inactive. Because APC-Slp1 ubiquitinates Mes1, it can overcome the Mes1 inhibition and promote the timely destruction of cyclin B at the end of MI. The degradation of cyclin B results in the inactivation of APC-Slp1, which, in turn, helps in the reaccumulation of cyclin B in MII.

The accumulation of APC/C coactivator Fzr1 starts from the end of MI, and it increases during the MI-to-MII transition with the activation of Cuf2 synthesis by Mei4 (8,16). However, APC-Fzr1 and APC-Ste9 remain inactive at the MI-to-MII transition. Further, the accumulation of Fzr2 is delayed compared with Fzr1, and it increases from the end of MII (8). The exit from MII is marked by complete degradation of cyclin B, which coincides with the activation of APC-Fzr1, APC-Fzr2, and APC-Ste9. The degradation of Mes1 is incomplete because Mes1 is both an inhibitor and substrate of APC/C (Fig. 1 b). Therefore, the complete disappearance of Mes1 at MII may also require the downregulation of its transcription.

The model captures the phenotypes of several mutants

In addition to the simulation of the wild-type, we also performed the simulation of different mutant strains to test the model. It is important that the model is consistent with known facts about the fission yeast meiosis. We compared the simulations with observed phenotypes of mutants. The equations and parameter values corresponding to wild-type condition were used with only parameters related to the mutation changed as described in the Methods (Table S4).

Mutations affecting the MI-to-MII transition

We first simulated mutations affecting Cdk1 activation. Temperature-sensitive mutations of Cdk1 or Cdc25 have a striking phenotype with cells undergoing only one division (4). Further, the experiment with the cdc13-cdc2 fusion protein shows that overexpression of it is required for the MII entry (12). The simulation of the model in the absence of the Mei4-dependent synthesis of cyclins (k_scycbm) shows the entry into MI and one peak of Cdk1 activity (Document S1. Figs. S1–S7 and Tables S1–S5, Document S2. Article plus Supporting Material a). Further, a decrease in the Cdc25 activation rate (V_ac25) also results in one peak of Cdk1 activity (Fig. 2 a). This suggests that both cyclin B synthesis and Cdc25 activity are required for the MII entry, and complete inactivation of cyclin-B-Cdk1 at MI will lead to the exit from meiosis. The model predicts that Cdk1-dependent regulation of APC/C coactivators is required for the MI-to-MII transition because a nonphosphorylatable Fzr1 mutant situation results in one division with complete degradation of cyclin B (Fig. S2 b). The model also captures the phenotype of mes1Δ strain, which shows a premature exit from MI (Fig. 2 b; (8)). One peak of cyclin-B-Cdk1 activity is observed with the activation of the different APC/C coactivators Fzr1, Ste9, and Fzr2. Together, our analyses predict that both cyclin-B-Cdk1 and Mes1 are required to restrict the APC/C activity between two divisions.

Figure 2.

Simulation of mutants affecting the MI-to-MII transition. Shown are (a) cdc25-22 at 33.5°C (partial inactivation, V_ac25 = 1.5) and (b) mes1Δ (k_smes1 = 0). To see this figure in color, go online.

Mutations characterizing the relationship between APC/C and Mes1

We simulated the mutation that rescues mes1Δ strain. The double deletion of Mes1 and Fzr1 (mes1Δfzr1Δ) allows cells to complete two divisions similar to the wild-type (8). Fig. 3 a shows that the simulation is consistent with the phenotype of double deletion. The MII exit occurs with the activation of Fzr2 and Ste9. Although Mes1 inhibits multiple APC/C coactivators, the model predicts that the deletion of Ste9 or Frz2 in mes1Δ fails to promote MII entry (Fig. S3, a and b). This confirms that Fzr1 is an important target of Mes1 inhibition in the MI-to-MII transition. On the other hand, the experiment with nonubiquitinable Mes1 shows that degradation of Mes1 is required for the metaphase-to-anaphase transition (8). The simulation of this mutant shows that cyclin-B-Cdk1 and Mes1 levels are high in nonubiquitinable Mes1, resembling the metaphase arrest (Fig. 3 b). Therefore, the mutual antagonism between APC/C coactivators and Mes1 plays a key role in the MI-to-MII transition. The low (MI exit) (Fig. 2 b) and high Cdk1 (metaphase arrest) (Fig. 3 b) states are possible depending on Mes1 levels.

Figure 3.

Simulation capturing the relationship between APC/C and Mes1. Shown are (a) mes1Δfzr1Δ (k_smes1 = k_sfzr1 = 0) and (b) K0-mes1 (nonubiquitinable Mes1 or lysine-less Mes1, k_us1 = k_uf1 = k_us9 = k_uf2 = 0). To see this figure in color, go online.

Mutations of APC/C coactivators

Temperature-sensitive Slp1 mutant (slp1-NF410) exhibits tight metaphase arrest. In meiosis, this mutant shows a phenotype of delayed one division (8). In Fig. 4 a, the simulation of Slp1 inactive mutation shows one peak of cyclin-B-Cdk1 with a delay in the APC/C activation. The exit is dependent on the accumulation of Fzr1 because deletion of it results in the metaphase arrest with high cyclin-B-Cdk1 activity, as observed in the experiment (Fig. 4 b; (8)). Further, the deletion of Mes1 in Slp1 inactive mutation accelerates the exit (Fig. 4 c). The experiment also shows that a decrease in APC-Slp1 activity is accompanied by a further rise in Fzr1 levels, but the mechanism is still unclear (8). Therefore, the delay observed in the simulation can be further reduced by considering this regulation.

Figure 4.

Simulation of APC/C coactivator mutation. Shown are (a) slp1-NF410 (Slp1-inactive mutant, k_aslp1 = 0), (b) slp1-NF410fzr1Δ (k_aslp1 = k_sfzr1 = 0), (c) slp1-NF410mes1Δ (k_aslp1 = k_smes1 = 0), and (d) fzr1Δ (k_sfzr1 = 0). To see this figure in color, go online.

Fzr1 deletion (fzr1Δ) has the striking phenotype that cells enter the third division producing more than four nuclei (11). The simulation of fzr1Δ mutation results in more than two peaks of cyclin-B-Cdk1 activation (Fig. 4 d). This suggests that the degradation of cyclin B by APC-Fzr1 at MII is important to promote the timely exit from meiosis. The deletion of other APC/C coactivators in fzr1Δ background, such as fzr1Δfzr2Δ, leads to sustained cyclin-B-Cdk1 oscillations in the model (Document S1. Figs. S1–S7 and Tables S1–S5, Document S2. Article plus Supporting Material a). A similar characteristic is observed with the deletion of Cuf2 because it is an important transcriptional activator for Fzr1 and Fzr2 synthesis. cuf2Δ cells also enter into the third division but do not show the characteristic of oscillations (11). This difference may emerge because the model does not consider the possibility of transcriptional repression of proteins, which may eventually lead to a delayed exit. In the presence of Fzr1, the model shows that other APC/C coactivators become redundant, with either Fzr2 or Ste9 sufficient for the MII exit (Fig. S4, b and c; (8)). Together, the model predicts that Fzr1 is required for Ste9 and Fzr2 activation, and there is an order of activation of coactivators at the MII exit. An increase in Fzr1 levels during MII helps to overcome the Mes1 and cyclin-B-Cdk1 inhibition, facilitating the activation of APC-Ste9 and APC-Fzr2. APC-Fzr1 reduces the cyclin-B-Cdk1 activity further at the MII exit compared with MI and prevents the reaccumulation of cyclin B. The model also predicts that the ubiquitination of Mes1 is required to promote MII exit. The ratio of APC/C coactivators to inhibitor Mes1 increases because of the increase in the levels of coactivators Fzr1 and Fzr2 and the destruction of Mes1, establishing a robust condition for MII exit.

How sensitive is the model to parametric variations?

Mutant situations probe the parameter space of the model and constrain the parameter values to be in a specific range. The consistency of the model with phenotypes of most mutants provides a reasonable account of meiotic progression in fission yeast. Further, we tested the sensitivity of the model by changing the wild-type parameter values one at a time (see Methods). We found that the model is sensitive to 9 (out of 72) parametric variations in both directions and 12 in either direction (Table S5). We explored how different sensitive parameters affect the MI-to-MII transition and MII exit. Increasing the levels of Fzr1 (k_sfzr1) leads to one division (Document S1. Figs. S1–S7 and Tables S1–S5, Document S2. Article plus Supporting Material a), whereas increasing the levels of Mes1 leads to either three divisions (Fig. S5 b) or MI arrest, depending on the synthesis rate of Mes1 (k_smes1) (Document S1. Figs. S1–S7 and Tables S1–S5, Document S2. Article plus Supporting Material c). These results demonstrate that maintaining the right balance of meiosis-specific APC/C coactivators and Mes1 is required for the meiotic progression. Further, accelerating the accumulation of Fzr2 (decreasing k_hfzr2) can convert the two divisions into one division (Fig. S5 d), and this shows the importance of timely accumulation and activation of different APC/C coactivators in a specific order. Therefore, the sensitive part of the model may represent the real features of the system or the gap in understanding of meiosis regulation, which needs further exploration.

Mei4 influences the Cdk1 activation switch

We performed bifurcation analysis to explore the feedback control of the G2-to-MI transition by Mei4. For this purpose, we eliminated the downstream effect of cyclin-B-Cdk1. Cdk1 exhibits bistable activation with an increase in the total concentration of cyclin B (CycB_T) as a bifurcation parameter (Fig. 5 a). The positive feedback loop between Cdk1 and Cdc25 and the double-negative feedback loop between Cdk1 and Wee1 can make the activation of Cdk1 bistable (25). In the absence of Mei4, we show that the system is locked at a low Cdk1 state in G2 with saddle nodes (SN1 and SN2) at higher CycB_T (Fig. 5 a). This establishes a block despite cyclin B levels being high in G2. In the presence of Mei4, SNs shifts to lower values of CycB_T, which promotes the transition from G2 to MI without requiring further raise in the cyclin B concentration (shown by an arrow in Fig. 5 b). The Mei4-induced transcription of Cdc25 and suppression of Wee1 influence SNs and increase the separation between low and high stable Cdk1 steady states.

Figure 5.

The control of G2 to meiosis I transition by Mei4. The change in Cdk1 activity with respect to cyclin B (CycB_T) is shown. (a) In the absence of Mei4 (Mei4 = 0), Cdk1 activity remains low, corresponding to G2 phase (dark circle), and (b) in the presence of Mei4 (Mei4 = 1), Cdk1 activity jumps to a high value, corresponding to meiosis I (dotted arrow). To see this figure in color, go online.

Feedback loop control of APC/C drives meiotic progression and exit

To study the feedback control of meiotic progression and exit, we used the synthesis rate of Fzr1 (k_sfzr1) by Cuf2 as a bifurcation parameter. The synthesis of Fzr1 increases as cells progress into meiosis. Because there are redundant APC/C coactivators, we first performed the analysis in the absence of coactivators Ste9 and Fzr2. We also eliminated the feedback loop acting upstream of Cdk1 activation (Wee1 and Cdc25). Fig. 6 a shows that for higher values of k_sfzr1, there is a low cyclin-B-Cdk1 state, and for lower values, there are larger-amplitude cyclin-B-Cdk1 oscillations, which emerge because of the absence of other APC/C coactivators Ste9 and Fzr2. The system undergoes an SN infinite period (SNIPER) bifurcation with a decrease in k_sfzr1-value. SNIPER is a special case of SN bifurcation, and at this point, the collision of a saddle point and a stable node makes the remaining steady state unstable surrounded by a limit cycle. This analysis evokes the view of “clock + bistable switch” in meiotic progression. The removal of cyclin-B-Cdk1-dependent regulation of APC-Fzr1, which compromises the double-negative feedback loop between APC-Fzr1 and cyclin-B-Cdk1, converts this view into a Hopf bifurcation and reduces the region of the limit cycle to low k_sfzr1-values (Fig. 6 b). On the other hand, eliminating the negative feedback loop between cyclin-B-Cdk1 and APC-Slp1 (k_aslp1 = 0) makes the system bistable with respect to k_sfzr1 (Fig. 6 c).

Figure 6.

The effect of changing Fzr1 levels on the Cdk1 activity. The synthesis rate of Fzr1 (k_sfzr1) is changed under different conditions: (a) control, (b) in the absence of Fzr1 inactivation by Cdk1 (k_ifzr1 = 0), and (c) in absence of Cdk1-APC-Slp1 negative feedback loop (k_aslp1 = 0). The effect of Fzr1 is studied in the absence of Ste9, Fzr2, and Cdk1 regulation by Wee1 and Cdc25. Solid lines represent the stable steady states, and dashed lines represent unstable steady states. Blue dashed lines represent the minimum and maximum of the oscillations. SN, saddle node; SNIPER, SN infinite period. To see this figure in color, go online.

We also performed a two-parameter bifurcation diagram to study the effect of other APC/C coactivators, Fzr2, and Ste9 regulation. It can be observed that with an increase in the rate of synthesis of Fzr2 (k_sfzr2), both the SNs shift to lower k_sfzr1-values, reducing the requirement of Fzr1 (Fig. 7 a). The bistable region exists only for a narrow range of k_sfzr1-values, whereas it is in a monostable low Cdk1 region for a wide range of k_sfzr1 and k_sfzr2 parameter values. With the increase in k_sfzr1- and k_sfzr2-values, the system leaves the limit cycle region to enter the low cyclin-B-Cdk1 state. However, on decreasing k_sfzr1, the system can lose the low cyclin-B-Cdk1 state, making the MII exit reversible (shown by an arrow in Fig. 7 a). Fig. 7 b shows the effect of increasing the total concentration of Ste9 (Ste9_T) as the second bifurcation parameter. The region of bistability increases, and SN2 shifts to the negative regime with the inclusion of Ste9 regulation. The system can enter a low cyclin-B-Cdk1 state from the limit cycle regime, with an increase in both Ste9_T and k_sfzr1. However, it remains in that state with a decrease in k_sfzr1, making the MII exit irreversible (shown by an arrow in Fig. 7 b). The irreversibility of the MII exit can be due to the consequence of double-negative feedback loops between the APC-Ste9 and its inhibitors cyclin-B-Cdk1 and Mes1. In the absence of Mes1, the double-negative feedback loop between APC-Ste9 and cyclin-B-Cdk1 is sufficient to make the system irreversible for low values of k_sfzr1 (Document S1. Figs. S1–S7 and Tables S1–S5, Document S2. Article plus Supporting Material a). On the other hand, the double-negative feedback loop between APC-Ste9 and Mes1 is insufficient to make the system irreversible in the absence of Cdk1-dependent regulation of APC-Ste9 (Fig. S6 b). We show that Mes1 regulation can also yield bistable behavior provided that Mes1 undergoes polyubiquitination in a distributive manner (Fig. S7, a and b). Here, because Mes1 is an inhibitor and substrate of APC/C, it binds tightly and gets slowly processed in the first step, whereas for later modification(s), it binds weakly and gets rapidly processed. Such regulation is shown to make the cell cycle transitions irreversible (26).

Figure 7.

Two-parameter bifurcation diagrams showing the effect of different APC/C coactivators on the Cdk1 activity. The synthesis rate of Fzr1 (k_sfzr1) is changed with the (a) synthesis rate of Fzr2 (k_sfzr2) and (b) the total concentration of Ste9 (Ste9_T). Solid lines represent SNs, and the dashed line represents Hopf bifurcation. The direction of the dotted arrows represents the transition from the limit cycle regime to the low Cdk1 state with the change in the levels of two coactivators. To see this figure in color, go online.

Discussion

Mathematical models of the core cell cycle regulatory network in different organisms have yielded valuable insights into the dynamics and the network design principles (24, 25, 26, 27, 28). A major challenge is to understand how the cell cycle network is modified by meiosis-specific regulation to order various events of the meiosis program. To address this challenge, a comprehensive mathematical model of the regulatory network that controls MI entry, MI-to-MII transition, and MII exit in the fission yeast was developed. The model captured the phenotypes of several mutants and revealed the molecular logic of meiosis regulation.

An important feature of the cell cycle is the alteration between DNA synthesis (S) and mitosis (M) phases. The oscillation in cyclin-B-Cdk1 drives the progression from G1 to S or G2 to M and back to G1. However, in meiosis, this sequence is altered to yield two meiotic divisions, MI and MII, without intervening the S phase. The Cdk1 activation at the G2-to-MI transition occurs similar to mitosis, but the MI entry depends on the Mei4 control of Cdc25 and Wee1 levels. We showed that Mei4 influences the bistable activation of Cdk1 to eliminate the G2-arrest state. In mouse oocytes, the meiosis-specific regulation of Cdc25 and Wee1 by protein kinase A (PKA) inhibition is known to promote meiotic resumption from germinal vesicle arrest state (G2 arrest) (29,30).

We showed how two peaks of cyclin-B-Cdk1 activation representing the MI and MII emerge because of the difference in the regulation of APC/C coactivators in meiosis. The logic of meiotic progression involves a transition from an oscillatory regime during MI-to-MII transition to a low Cdk1 state at the MII exit with an increase in APC-Fzr1 and other APC/C coactivators (Fig. 7). The negative feedback loop between cyclin-B-Cdk1 and APC-Slp1 is required for the oscillatory dynamics, whereas the other APC/C coactivator Fzr1 is inhibited. We found that both cyclin-B-Cdk1 and Mes1 inhibition of APC-Fzr1 is crucial to keep the cells in a limit cycle regime to aid in the MI-to-MII transition when the synthesis of Fzr1 is ongoing (Figs. 2 b and Document S1. Figs. S1–S7 and Tables S1–S5, Document S2. Article plus Supporting Material c). APC-Slp1-dependent degradation of cyclin B and Mes1 triggers MI-to-MII transition but is insufficient to promote the exit. This is in contrast to fission yeast mitosis, in which APC-Slp1 is primarily responsible for driving the cells from metaphase back to G1 (31).

APC-Fzr1 promotes MII exit by the degradation of cyclin B and by sequestering Mes1 from other APC/C coactivators. This explains why there are more than two divisions in Fzr1 deletion (fzr1Δ), one division in Mes1 deletion (mes1Δ), and the rescue of these by deletion of both (mes1Δfzr1Δ). The APC/C coactivator/inhibitor ratio depends on the Mei4 that forms IC-FFL to control the synthesis of activators (slow arm) and inhibitors (fast arm) of the MII exit. The slow arm of IC-FFL involves transcription factor Cuf2, which is regulated independently of core cell cycle components cyclin-B-Cdk1 and APC-Slp1. Therefore, the accumulation of both Cuf2 and its targets can occur in parallel to meiotic cell divisions and can potentially interfere with divisions. A robust unidentified mechanism to control Fzr1 and Fzr2 at transcriptional or/and translational level might exist in addition to a restraining mechanism involving Mes1 and cyclin-B-Cdk1 (fast arm) to provide enough time for the completion of two divisions. This timer mechanism can trigger exit if MI is delayed, as observed in the case of the inactive Slp1 mutant (8). We propose that Mei4 not only initiates the transcription of components involved in the cell divisions but also triggers a timer mechanism that helps to restrain it. Further, we showed that the APC-Ste9-dependent degradation of both cyclins and Mes1 can lock the system in low Cdk1 state by making the MII exit irreversible.

Although Fzr1 plays a crucial role in terminating meiotic cell divisions, its homolog in budding yeast, Ama1, is only required for sporulation (32). This suggests divergence with respect to how APC/C is activated for the termination of meiosis in fission and budding yeasts. However, we observe the expression of Ama1 also increases in meiosis, similar to Fzr1 (22). APC/C-Ama1 is also regulated by Cdk1-dependent phosphorylation, and it can inhibit its activity in meiosis (33). Increasing the levels of Ama1 is shown to promote the exit from MI arrest (22). In Arabidopsis thaliana, APC/C-interacting protein Tdm1 ensures timely exit from meiosis (34). tdm1Δ cells undergo a third division similar to the fzr1Δ phenotype. Tdm1 is kept inactive in MI by Tam-CdkA-dependent phosphorylation, and its degradation in MII promotes a timely exit from meiosis by Tdm1. Both tamΔ in A. thaliana and a Cdk1-inactive mutant in fission yeast have a similar phenotype of one meiotic division (4,34). Further, the deletion of APC/C inhibitor in A. thaliana (Osd1) and vertebrates (Emi2) also have the same phenotype as mes1Δ (35,36). Our results that both cyclin-B-Cdk1 and Mes1 are required for MI-to-MII transition is consistent with the view that both Tam-CdkA and Osd1 work synergistically to ensure the MI-to-MII transition in A. thaliana. On the other hand, APC/C-dependent ubiquitination of cyclin B and Mes1, in turn, help in the transition back to G1. Thus, the mutual antagonism between APC/C and its inhibitors may represent the underlying principle of meiotic progression and exit.

In summary, our study has made a first, to our knowledge, attempt to develop a mathematical model of the molecular network involved in MI entry, MI-to-MII transition, and MII exit in the fission yeast. The model successfully reproduced wild-type and mutant phenotypes, generated systems-level insights, and highlighted the gap in understanding of meiosis regulation. However, the model also failed to account for the increase in levels of Fzr1 in temperature-sensitive Slp1 mutant situation (slp1-NF410), and the delayed one division on the deletion of Fzr1 in Slp1 mutant that produces a reduced amount of protein (slp1-B05) (21). Further studies are required to understand the relationship between APC/C coactivators in meiotic progression. The model primarily focused on capturing the known regulation of APC/C, which controls the dynamics of meiotic progression and exit. It will be further refined with the accumulation of new experimental data and integrated with our existing model for the mitotic to meiosis transition in the fission yeast (2).

Author Contributions

P.K.V. designed the study. P.D. and N.P. developed the model and performed analysis. P.D., N.P., and P.K.V. interpreted the results and contributed to drafting the manuscript. All authors have read and approved the final submitted manuscript.

Editor: Kevin Janes.