Immediately following fertilization, the fruit fly embryo undergoes 13 rapid, synchronous, syncytial nuclear division cycles driven by maternal genes and proteins. During these mitotic cycles, there are barely detectable oscillations in the total level of B‐type cyclins. In this paper, we propose a dynamical model for the molecular events underlying these early nuclear division cycles in Drosophila. The model distinguishes nuclear and cytoplasmic compartments of the embryo and permits exploration of a variety of rules for protein transport between the compartments. Numerical simulations reproduce the main features of wild‐type mitotic cycles: patterns of protein accumulation and degradation, lengthening of later cycles, and arrest in interphase 14. The model is consistent with mutations that introduce subtle changes in the number of mitotic cycles before interphase arrest. Bifurcation analysis of the differential equations reveals the dependence of mitotic oscillations on cycle number, and how this dependence is altered by mutations. The model can be used to predict the phenotypes of novel mutations and effective ranges of the unmeasured rate constants and transport coefficients in the proposed mechanism.
After fertilization, the Drosophila embryo undergoes 13 rapid, syncytial cycles of alternating DNA replication and mitosis (Wolpert, 2001). In cycle 14, zygotic transcription turns on, to support further cell cycle progression and development of the embryo. These early embryonic cycles are driven by cyclin‐dependent protein kinases (Cdk)/cyclin complexes, similar to somatic cell cycles in eukaryotes. However, there are characteristic peculiarities in the dynamics of certain regulatory molecules in the Drosophila syncytial cycles. For example, the overall level of mitotic B‐type cyclins (CycB) and the protein‐kinase activity of mitosis promoting Cdk1/CycB complex (M‐phase‐promoting factor (MPF)) stays high and constant during the very first cell cycles (Edgar et al, 1994b). This is in sharp contrast to other eukaryotic cell cycles where CycB gets ubiquinitated and degraded at the end of every mitosis, which results in a drop in Cdk1/CycB activity as well. Although the level of CycB is constant in the early Drosophila embryo, expression of nondegradable CycB blocks mitotic progression, suggesting a role for CycB degradation (Su et al, 1998). These puzzling observations were resolved by the recognizing that CycB degradation occurs only locally in the vicinity of the spindle (Huang and Raff, 2002; Raff et al, 2002). Inhibitory tyrosine phosphorylation of Cdk1 subunit is also not apparent during the very first mitotic cycles in Drosophila and it becomes observable only after cycle 10 (Edgar et al, 1994b).
In this paper, we develop a two‐compartment, dynamic model for the early cycles of a Drosophila embryo, to test whether the hypothesis of local CycB degradation provides an adequate description for the observed dynamic patterns in regulatory molecules. The model, based on the molecular network shown in Figure 1A, distinguishes two compartments in the embryo (nucleus and cytoplasm). CycB is synthesized in the cytoplasm and gets degraded as a consequence of Fizzy/anaphase promoting complex (Fzy/APC)‐dependent ubiquitination in the nucleus. Fzy/APC is in fact activated by the nuclear form of MPF after a time‐delay. After each activation of Fzy/APC (nuclear division), the volume of the nuclear compartment increases at the expense of the cytoplasmic compartment. Cdk1/CycB complexes are also inactivated by Wee1‐dependent phosphorylation and reactivated by String‐dependent dephosphorylation (String is the Drosophila homolog of Cdc25). Both Wee1 and String are regulated by Cdk1/CycB phosphorylation: Wee1 is inhibited whereas String is activated. All these cell cycle regulators are transported between the two cellular compartments, except Fzy/APC, which is localized to the nucleus only. Whereas Wee1 level is assumed to be constant, the String protein is translated from its preexisting maternal mRNA, which gets degraded by a hypothetical zygotic factor.
By writing nonlinear differential equations for the time rate of change in concentration of all these molecular species in both compartments, we develop a quantitative model for the early Drosophila cycles. By fitting the model to the behavior of wild‐type and different mutant embryos, we had assigned specific values to the kinetic parameters.
Numerical simulations with the model reproduce the main dynamic characteristics of early mitotic cycles in wild‐type embryos (Figure 1B): after 13 fast oscillations in the level of nuclear Cdk1/CycB, the control system halts in interphase 14 with low MPF levels in both compartments. During the first 13 cycles, the fluctuations in overall CycB levels and phosphorylation state of String protein are gradually increasing, consistent with the experimental data (Edgar et al, 1994b).
Bifurcation analysis of the nonlinear differential equations reveals that the characteristic of these oscillations depend on the number of cell cycles, which determines the number nuclei in the embryo (Figure 2). The reason for this dependence is the competition among nuclei for cell cycle regulators (CycB, String etc.) present in the cytoplasm. At low number of nuclei (N), there is plenty CycB present in the cytoplasm and mitotic cycles are driven by the time‐delayed‐feedback loop between Cdk1/CycB and Fzy/APC, which creates a limit cycle oscillation. The amplitude of these oscillations are negligible in overall CycB level, but large for nuclear MPF. After cycle 10, there is a qualitative change in the oscillation, which is caused by the engagement of Cdk1 phosphorylation in the mechanism. Finally, at even larger N, the mitotic control systems settles into a stable steady state (interphase arrest), because the CycB level falls below a threshold determined by Wee1 and Sting level. Therefore, the number of cycles executed before interphase arrest is reached depends sensitively on the level of the cell cycle regulators (String, Wee1, CycB etc.)
The model is consistent with known mutations or biochemical interventions that introduce subtle changes in the number of mitotic cycles before interphase arrest (see e.g. Dawson et al, 1993; Edgar and Datar, 1996; Su et al, 1998; Ji et al, 2004; Stumpff et al, 2004; Crest et al, 2007).
A dynamic model was developed to describe the early mitotic cycles in a Drosophila embryo, which are characterized by 13 rapid nuclear divisions driven by maternally supplied gene products.
The model distinguishes two compartments and describes the temporal dynamics of the following cell cycle regulatory molecules in both compartments: Cyclin‐B, Wee1, String, Fizzy/APC and Cdk1/Cyclin‐B complex.
Bifurcation analysis reveals that the amplitude of oscillation in total Cyclin‐B level is low initially and increases slowly with number of nuclear cycles, while the amplitude of nuclear Cdk1/Cyclin‐B complex shows large amplitude oscillations in all cycles.
The model is consistent with many cell cycle mutants and predicts the characteristics of some as yet uncharacterized ones, in terms of the dynamics of regulatory molecules and the number of maternally driven cycles they can achieve.
During oogenesis, an egg grows very large and inherits from its mother all the nutrients and cell components needed to proceed through a series of cell cycles after fertilization (Wolpert, 2001). In many types of embryos, these early division cycles rapidly alternate between DNA replication (S‐phase) and mitosis (M‐phase). Later in development, the synchrony and speed of the first divisions is lost, and gap phases (G1 and G2) are introduced into the somatic cell cycle (Morgan, 2007).
The Drosophila egg is an extreme case. Its nuclei proceed through 13 very rapid (10–12 min) divisions without cell division. As a consequence, three hours following fertilization, 6000 nuclei share the same cytoplasm (syncytium). The rapidity of these early cycles can be explained by an abundance of maternally supplied cell cycle components. After mitosis 13, the nuclei become cellularized. Some cells arrest in G2, whereas other cells continue to divide at a slower and more variable schedule (Edgar and O'Farrell, 1990; Wolpert, 2001). The molecular basis of the first 13 rapid, synchronous nuclear division cycles is the subject of mathematical modeling in this paper.
The master regulators of the eukaryotic cell cycle are the cyclin‐dependent protein kinases (Cdk's). To be active, a Cdk has to be associated with a cyclin partner, which determines the substrate specificity and subcellular localization of the Cdk/cyclin complex. The prototype of a Cdk/cyclin pair is M‐phase‐promoting factor (MPF), first identified in frog eggs. MPF is a complex of Cdk1 and a B‐type cyclin (CycB). Cdk1 subunits are usually present in excess in cells, and therefore do not limit formation of Cdk/cyclin dimers. Cyclin subunits, on the other hand, fluctuate dramatically during the cell cycle and thereby play a major role in determining MPF activity (Morgan, 2007).
In growing cells, cyclin synthesis is regulated at the transcriptional level. In early embryonic cells, transcription is blocked or greatly restricted because they lack G1‐ and G2‐phase (Morgan, 2007). In both growing and embryonic cells, cyclin level is controlled by proteolysis. Ubiquitylation of CycB by the anaphase promoting complex (APC) targets them to the proteasome for degradation (Glotzer et al, 1991). In Drosophila embryos, a protein called Fzy (encoded by the fizzy gene) is responsible for targeting CycB to the APC during exit from mitosis (Dawson et al, 1995). The Fzy/APC complex is activated by phosphorylation by MPF (Peters, 2006), creating a negative‐feedback loop (+/−) in the reaction network. The activation of Fzy/APC by MPF is an indirect process (Felix et al, 1990).
In addition, Cdk/cyclin activity can be controlled by phosphorylation of the Cdk subunit. Phosphorylation of a tyrosine residue (near the N terminus) inhibits the protein kinase activity of Cdk/cyclin complexes (the phosphorylated form of MPF, P‐Cdk1/CycB, is called preMPF). Cdk phosphorylation is catalyzed by the protein kinase Wee1, and the phosphate group is removed by a phosphatase called String in Drosophila (Edgar and O'Farrell, 1990). Wee1 and String proteins are themselves phosphorylated by MPF, creating a pair of positive‐feedback loops. String is activated by MPF (a +/+ loop) and Wee1 is inactivated by MPF (a −/− loop).
In sea urchin and frog embryos, the first 12 cell cycles are known to be driven by a cytoplasmic clock that causes periodic degradation of the CycB subunit of MPF as cells exit mitosis (Gerhart et al, 1984). The resultant oscillations of MPF activity control both nuclear divisions (M‐phase) and the characteristic surface contractions that persist even after enucleation (Hara et al, 1980).
In contrast, in Drosophila, Edgar et al (1994b) observed that total CycB level and MPF activity remain high (not oscillating) during the first eight cycles. After cycle eight, small fluctuations appear in both CycB level and MPF activity with increasing amplitude. Even though CycB degradation might appear negligible during these early cycles, introduction of a nondegradable form of CycB into a Drosophila embryo blocks mitotic cycles, which underlines the importance of CycB degradation at certain stages of the cell cycle (Su et al, 1998; Raff et al, 2002).
The apparent paradox surrounding CycB degradation during Drosophila embryogenesis can be resolved by recognizing that CycB degradation occurs only locally, in the vicinity of the mitotic spindle (Huang and Raff, 2002; Raff et al, 2002). In this paper, we use mathematical modeling to explore whether the hypothesis of local CycB degradation gives an adequate description of CycB patterns during the first 13 nuclear division cycles of the Drosophila embryo. After introducing compartmentalization and local degradation, we show that it is possible to simulate the key features of early embryonic cell cycles in Drosophila. The model also reproduces the effects of alpha‐amanitin treatment, loss‐of‐function mutations and overexpression mutations.
A model for the molecular network controlling embryonic cell cycles in Drosophila
Our model for cell cycle regulation in the early Drosophila embryo, inspired by Figure 7 of Edgar et al (1994b), is diagrammed in full in Figure 1A. The model tracks the interactions of MPF, Fzy, Wee1 and String in the cytoplasm and in the nuclei. In Drosophila, Fzy/APC is localized at centrosomes, kinetochores and spindles (Su et al, 1998; Huang and Raff, 1999; Raff et al, 2002). Consequently, we assume that CycB subunits of MPF undergo regulated destruction only in the nuclear compartments. In our model, MPF activates Fzy/APC through an ‘intermediary enzyme’ (IE), because some evidence indicates that the activation step is indirect (Felix et al, 1990). These interactions comprise a delayed negative‐feedback loop that generates local oscillations of MPF activity in the nuclear compartments. Meanwhile, Wee1 and String are also regulating MPF activity in both compartments by phosphorylating and dephosphorylating (respectively) a tyrosine residue on the Cdk subunit of MPF.
Using the basic principles of biochemical kinetics, we translate the diagram into a set of ordinary differential equations (Table I). The equations, which describe the time‐rates of change of the fluctuating protein species in the diagram, contain a number of unknown rate constants that must be estimated by fitting the model to the available data (mutant phenotypes, responses to drug treatments, and so on). The parameter values reported in Table II are suitable for simulating the experiments described in this paper.
Modeling localized cyclin degradation
We distinguish two compartments in the embryo (Figure 1A): cytoplasm and nuclei. What we call the ‘nuclear compartment’ is not the volume enclosed by the nuclear envelope, because the nuclear envelope breaks down during mitosis. Despite the loss of a barrier between nucleus and cytoplasm, we assume that CycB degradation during M‐phase occurs only in a limited region (our ‘nuclear compartment’) in the vicinity of the mitotic spindle. Hence, in our model, the nuclear compartment persists in separation from the cytoplasmic compartment throughout the nuclear division cycles. At telophase, the number of nuclear compartments doubles (or, nearly so), as described later. We will assume that the transport coefficients for MPF, Wee1 and String between nucleus and cytoplasm do not change as the nuclear envelope breaks down and reassembles each cycle. In the Discussion, we will present evidence that our conclusions are not significantly changed by the more likely assumption that intercompartmental transport increases during mitosis.
The volume of the nuclear compartment is assumed to be the product of the number of nuclei (N) and the volume of a single nuclear compartment (Vn), that is VN=NVn. Consequently, cytoplasmic volume is VC=VT(1−Nε), where VT=total egg volume (constant) and ε=Vn/VT≪1.
Next, we need to specify how individual proteins are distributed between nuclei and cytoplasm. For example, MPF is present in both cytoplasm and nuclei, where its concentrations are denoted [MPFc] and [MPFn], respectively. The average concentration of MPF across the entire cell is given by
The balance equations for transport of MPF between cytoplasm and nucleus are
where Jin and Jout are the fluxes of MPF (number of molecules per unit time) into and out of a single nucleus. We assume that Jin=AnPin[MPFc] and Jout=AnPout[MPFn], where Pin and Pout are permeability constants and An is the surface area of a single nucleus (assumed constant). Defining first‐order rate constants, kin=AnPin/VT and kout=AnPout/VT, we can write the transport laws as
For proteins that are actively accumulated in nuclei, kin≫kout.
The balance equations for MPFc and MPFn can now be written as
where IN, OUT, SYN, and so on stand for the rate expressions for transport and chemical reactions. Similar equations hold for preMPF. For MPF and preMPF, we assume that kout=0.
Wee1 and String are also distributed between cytoplasm and nuclei, and these proteins can be phosphorylated and dephosphorylated in both compartments. As a consequence, we consider four different forms of each protein (unphosphorylated and phosphorylated forms in the cytoplasm and in the nuclei), where the cytoplasmic forms (Stgc, StgPc, Wee1c and Wee1Pc) interact with MPFc and preMPFc, whereas the nuclear forms (Stgn, StgPn, Wee1n and Wee1Pn) interact with MPFn and preMPFn. For String and Wee1, we can write balance equations for the total concentrations :
In some figures we plot the total concentration of phosphorylated String, given by:
By assumption [Wee1]T is constant in the model, so we can compute Wee1Pc from the equation:
In the model, we assume that nuclei enter M‐phase when MPFn activity abruptly increases, and they divide when MPFn activates Fzy/APC (i.e., when Fzy activity increases through 0.5). At each cycle, the N nominally doubles. However, to account for the few nuclei that do not divide, we multiply the existing N at each division by 1.95 rather than by 2.
If we assume that the concentrations of protein species do not change at the point of discontinuity (when N increases to 1.95 N), then there is a sudden ‘creation of matter’ needed to populate the new nuclei with all their protein components. For proteins that turn over (synthesis and degradation), this new material is quickly equilibrated and the excess removed by protein degradation, with no appreciable effect on the solution of the differential equations during the 13 cycles. But for conserved species (e.g., Wee1), there is a steady increase of total concentration, as a little new material is created at each division. The increase starts to become noticeable at the 12th division. The problem can be remedied by rescaling concentrations after nuclear division according to the prescriptions (see Materials and methods section for details):
Figure 1B shows the results of a numerical simulation of the model with localized cyclin degradation. For this simulation, we assume no synthesis or degradation of String, that is, [String]T=constant. The initial concentration for MPF in the cytoplasm is high (one arbitrary unit) and this activity suffices to keep Wee1 inactive and String active in the cytoplasm (at least initially). The first oscillations of the nuclear concentration of MPF are very rapid. However, in later cycles, they slow down, and finally arrest in G2‐phase of the 15th cycle with low MPF activity. The first 13 cycles are so rapid in Drosophila compared to Xenopus, because they are driven by nuclear import of pre‐formed cytoplasmic CycB rather than by de novo synthesis of CycB. In the early cycles, MPF enters the nucleus so quickly that nuclear Wee1 cannot inhibit it and, as a result, the rapid accumulation of MPFn drives the nucleus into mitosis. These early cycles are driven by the negative‐feedback circuit involving MPFn, IE and Fzy/APC.
As the N and the volume of the nuclear compartment nearly double after each mitosis, the concentration of MPF+preMPF in the cytoplasm decreases. The drop in the cytoplasmic concentration slows down the transport of MPF into the nuclei and thus slows down the oscillation. As Wee1 kinase activity in the nucleus becomes more and more comparable to the nuclear entry rate of MPF, MPF complexes can be inhibited through tyrosine phosphorylation by Wee1. MPFn activity must reach a certain threshold to switch Wee1 kinase off and to activate String. As reaching this threshold takes some time, the period of the oscillation increases further. Eventually, total cyclin level in the cytoplasm is so low that nuclear MPF concentration cannot surmount the threshold level to enter mitosis. Hence, Wee1 remains active and String inactive, and MPF oscillations cease. In Figure 1B, the ‘embryo’ has an extended cycle 14 and arrests in cycle 15. (The cycle number (C) for which the model arrests depends sensitively on parameter values.)
Probing the regulatory mechanism and parameter constraints at the origin of cellular oscillations
In this section, we use one‐parameter bifurcation diagrams to characterize the mitotic control system with localized cyclin degradation (Tyson et al, 2002; Csikasz‐Nagy et al, 2006). A bifurcation diagram describes the stability of a dynamical system as some key parameter (the bifurcation parameter) is changed. As bifurcation parameter, we introduce ‘cycle number’ C, defined by N=1.95C−1. With this definition, N is the number of nuclei in the egg during cycle number C. When computing a bifurcation diagram, C is treated as a real number, even though N increases stepwise during simulations. For the bifurcation diagrams in Figure 2, the state of the control system (on the vertical axis) is characterized by either nuclear MPF or total CycB (to give two different views of system state).
For a large N, the control system has a single, stable steady state (solid line) with low MPFn concentration. At this steady state, MPF is tyrosine phosphorylated both in the cytoplasm and in the nuclei. This stable steady state represents a G2 arrest. When N is small, the control system is in a region where a branch of unstable steady states (dashed line) is surrounded by stable limit cycle oscillations (filled circles). These oscillations have large amplitude of [MPF]n, but small amplitude of [CycB]T (Figure 2A). As N increases after every oscillation, we move from left to right (C increasing) on the bifurcation diagram as the embryo develops.
Between cycles 11 and 12 (Figure 2A), the limit cycle oscillations undergo a pair of ‘cyclic fold’ bifurcations, which indicate a qualitative change in the oscillation mechanism. For C⩽11, the oscillations are driven by the negative‐feedback loop (cyclin degradation) alone. For C⩾12, the positive‐feedback loops (involving MPF phosphorylation) contribute to the oscillatory mechanism. These positive‐feedback loops become more and more significant as C increases, eventually creating alternative stable steady states between cycles 14 and 15. The stable steady state with low activity of MPFn blocks the oscillations at a SNIC bifurcation (saddle‐node‐invariant‐circle, where the period of oscillation tends toward infinity) at C=14.33. Hence, the MPF control system completes cycle 14 and arrests at a stable steady state in interphase of cycle 15. This is a problem, because Drosophila eggs arrest in interphase of cycle 14. It could be corrected by adjusting some parameters to move the SNIC point to 13 <C<14, but there are other more serious problems to be solved, along with this one, in the next section.
Interestingly, the bifurcation diagram does not change very much in the absence of String (Figure 2B), that is, a maternal string loss‐of‐function mutation. The reason for this insensitivity is the existence of a second protein phosphatase, Twine, which overlaps the function of String (Edgar and Datar, 1996). (Twine activity is represented in the model by the parameter kstg′.) In the absence of String, the amplitude of MPFn oscillations is slightly reduced, and the SNIC bifurcation point moves to a slightly smaller value of C (=13.22). Consequently, MPFn oscillations stop one cycle earlier (in cycle 14) than for [String]T=0.8.
Including String synthesis and degradation
Although the simulation in Figure 1B is superficially similar to observations of MPF fluctuations in fruit fly embryos (Edgar et al, 1994b), there are significant quantitative differences: (1) the number of division cycles is incorrect (easily fixed), and (2) the total amount of String in the embryo is not constant (less easily fixed). String protein level in embryos is low at fertilization, rises for seven or eight cycles and drops gradually to zero at interphase 14. As String level determines the position of the SNIC bifurcation (where oscillations are suppressed), we need to extend the model to changing levels of String protein. To this end, we take into account that string mRNA is stable until cycle 14 and is degraded abruptly in the first 20 min of interphase 14 (Edgar et al, 1994a).
Two distinct mechanisms seem to be responsible for string mRNA degradation (Edgar and Datar, 1996; Bashirullah et al, 1999), one operating on maternal string mRNA and the other one mediated by zygotically induced genes. To describe String dynamics, we append to the model (Table I) three differential equations: for string mRNA (equation (14)), for the mRNA of an unknown factor X (equation (15)) that is responsible for degrading string mRNA, and for the corresponding protein level (equation (16)). We assume that Xm (mRNA for factor X) is synthesized in the embryo at a rate proportional to N. The protein Xp is produced at a rate proportional to Xm. The rate constants in equations (15) and (16) are chosen so that there is no detectable synthesis of Xp until about the 10th cycle, and then its level rises sharply in cycles 11–13. (Other assumptions might be more reasonable: for example, Xm may not be synthesized during the very rapid cycles 1–6, when the DNA is likely unavailable for transcription. In that case, the rate constants would have to be readjusted to maintain the sharp rise in Xp in cycles 11–13.)
We assume that the string gene is not transcribed during early embryonic cycles, so the differential equation for string mRNA (equation (14)) has only degradation terms. The first term represents degradation by maternal products, whereas the second term is degradation by Xp. Hence, String message level drops slowly at first and then increasingly faster in cycles 11–13. String protein is assumed to be synthesized at a rate proportional to its mRNA (Stgm) and degraded with first order kinetics. Hence, total String protein level rises during the first 9–10 cycles and then decays away after its message is destroyed.
With these amendments to the model, a better description of the early cell cycles in Drosophila is achieved (Figure 3). The levels of String message and protein change as observed in Edgar's experiments (Edgar and Datar, 1996) and the embryo arrests solidly in interphase of cycle 14. In the simulation (Figure 3C), oscillations in String phosphorylation become noticeable around cycle 8, whereas, in experiments, fluctuations in String phosphorylation are already observed in cycle 5 or 6 (Edgar et al, 1994b).
Simulation of gene dosage effects
If this model is basically correct, it should account not only for all features of nuclear division in wild‐type embryos, but also for subtle differences in phenotypes of mutant embryos. In this case, the relevant mutants are deletions and overexpressions of cycB, fizzy, wee1, string and twine. Simulations of these mutants are summarized in Table III.
Deletion of fizzy breaks the negative‐feedback loop on which the oscillations depend and is, of course, lethal to embryonic development (Dawson et al, 1993). The model predicts that a half‐dose of fizzy (fizzy+/− heterozygote) is similar to wild type, whereas overexpressing fizzy may produce an extra nuclear division. Overexpressing cycB speeds up the cycles and adds an extra nuclear division (Ji et al, 2004; Crest et al, 2007). Conversely, a half‐dose of cycB causes delays in cycles 10–13 (Ji et al, 2004; Crest et al, 2007).
Deletion of string has no effect on the early mitotic cycles, as long as twine is in place (see Figure 2B: string deletion halts in cycle 14), but deletion of both genes is lethal to the egg (Edgar and Datar, 1996). Interestingly, deletion of string and a half‐dose of twine cause the early cycles to stop one division earlier than normal (in cycle 13) in the model and in experiments (Edgar and Datar, 1996). In the bifurcation diagram for this mutant, the SNIC bifurcation is moved to C=12.57 (i.e., the egg arrests in cycle 13).
Overexpressing string and/or twine is tricky to interpret. We might expect such mutations to speed up the cycles and perhaps add an extra nuclear division (arrest in cycle 15). An extra division is observed in a small fraction (3–5%) of embryos (Edgar and Datar, 1996). In simulations, extra peaks of MPFn are observed, but the later cycles are of reduced amplitude and it is not clear whether they could effectively drive a complete mitosis or not. The reason for the reduction in amplitude of MPFn oscillations is clear from the bifurcation diagram (Supplementary Figure S1). The SNIC bifurcation is lost and the later oscillations arise from a supercritical Hopf bifurcation (small amplitude limit cycle oscillations). The bifurcations in this case can be very complex, but the end result in simulations is a damped oscillation of MPFn.
The wee1 gene is not essential for early nuclear divisions (Stumpff et al, 2004). In simulations, the wee1‐deletion mutant (like twine and string overexpression mutants) arrests via damped oscillations at a supercritical Hopf bifurcation (Supplementary Figure S2). Simulation of a wee1‐overexpressing mutant (6 × wee1) predicts nuclear‐division arrest in cycle 12 (Table III).
Simulation of the effects of alpha‐amanitin treatment
Edgar and Datar (1996) showed that if an embryo is treated before cycle 6 with alpha‐amanitin (an inhibitor of RNA polymerase), then 88% of the nuclei arrest at interphase of cycle 15 instead of cycle 14. If the same treatment is performed later, then the extra cycle is not observed (Myers et al, 1995; Edgar and Datar, 1996). These results are nicely reproduced by simulations. In Figure 4A, we let the model run up to t=55 min (just before cycle 6), then set ks,xm=0 (i.e., no further synthesis of X message), then continue the simulation. The concentration of Xp never gets very large, and the degradation of string mRNA is slower. As a result, total String protein is slightly higher, allowing an extra cycle to occur. In contrast, when we simulate the same treatment at t=75 min (after cycle 6), Xp concentration is higher, string mRNA drops faster and no extra cycle is observed.
Early development of Drosophila embryos has been thoroughly studied by developmental geneticists and molecular biologists. Prompted by beautiful data on gene expression patterns, many groups have proposed models for pattern formation in post‐blastoderm embryos (von Dassow et al, 2000; Sanchez and Thieffry, 2003; Jaeger et al, 2004). Until now, however, no one has systematically explored the role of protein interactions during the first 13 nuclear division cycles in the syncytial, undifferentiated embryo. These cycles are remarkable because of their great speed (approx 10 min cycle time) and because they are not associated with large changes in observable activity of MPF, as seen for example in the embryos of sea urchins and frogs.
Although the bulk cytoplasm of the syncytial Drosophila egg contains a massive amount of active MPF, the activity of MPF in the vicinity of chromosomes may fluctuate dramatically, first of all because active MPF must be imported into the nuclei during interphase, and secondly because MPF activity may be cleared from chromosomal regions by rapid CycB degradation on anaphase spindles. Many authors have previously forwarded this hypothesis, or something similar, to explain the curious features of nuclear division in the early embryo of Drosophila (Edgar et al, 1994b; Crest et al, 2007). We have explored this hypothesis in terms of a mathematical model that distinguishes nuclear compartments from the bulk cytoplasm. Nuclear import of active MPF drives DNA replication, nuclear envelope breakdown, spindle assembly and congression of chromosomes to the metaphase plate. Local degradation of CycB at the spindle drives chromosome segregation and nuclear envelope reformation. Simulations of the model agree with salient features of wild‐type embryos, as summarized in Figure 7 of Edgar et al (1994b) and our Figure 3. Wild‐type embryos arrest in interphase of cycle 14, but mutant embryos may arrest sooner or later (or may not cycle at all), depending on their genetic make‐up. The model is consistent with these subtle effects on nuclear division cycles, as summarized in Table III. Lastly, we have shown that the model accounts for differential effects of alpha‐amanitin treatment before and after cycle 6.
Our model does not attempt to describe the complex regulatory signals introduced at the mid‐blastula transition, when many zygotic genes are newly transcribed.
Some general principles drawn from the model
Roles of positive and negative feedback.
The delayed negative‐feedback circuit involving Fzy/APC and MPF is primarily responsible for controlling the early nuclear divisions. By contrast, the positive‐feedback circuits, involving MPF, Wee1 and String, are not essential to the early mitotic cycles and play important roles only late in the syncytial phase, contributing to the maintenance of large amplitude oscillations and reliable interphase arrest in cycle 14. Even in the absence of Wee1, the model predicts rather normal looking cycles (see Supplementary Figure 2), which arrest because the embryo runs out of CycB. On closer examination, the cycles are faster than normal and disappear by damped oscillations in MPF activity. The actual number of cycles observed under these conditions is difficult to predict, as discussed previously.
Role of the unreplicated‐DNA checkpoint.
The first 6 cycles of the Drosophila embryo are extremely short (∼10 min), then cycles 7–13 get progressively longer (up to ∼20 min). This lengthening of the mitotic cycle could be due in part to the reduction of CycB level in the embryo and in part to the ‘DNA replication’ checkpoint, which delays MPF activation while DNA is being replicated. For N=2C−1 <64, N in the embryo is too small to have removed much CycB or to generate much of an unreplicated‐DNA signal. For cycles 7–10, the lengthening of cycle time is attributable to CycB reduction only (Ji et al, 2004; Crest et al, 2007), as can be established using embryos with different dosages of cycB with or without grapes. (The grapes gene encodes Chk1 kinase (Sibon et al, 1997). In response to unreplicated DNA, Chk1 phosphorylates String, which causes String to be sequestered in the cytoplasm (Lopez‐Girona et al, 1999).) Cycles 11–13, on the other hand, are lengthened by a combination of CycB depletion and the unreplicated‐DNA checkpoint.
Our model (Figure 3) shows modest lengthening of late mitotic cycles due to CycB depletion and Wee1‐dependent phosphorylation of MPF. We can easily introduce a DNA replication checkpoint into the model by increasing the value of kouts when DNA is unreplicated. The increase should depend on N and should be periodic during normal cycling and steady during a sustained block to DNA replication. Simulations (Supplementary Figure 3) confirm that proper lengthening of mitotic cycles 11–13 depend on engaging the DNA replication checkpoint.
Roles of nuclear transport of regulatory proteins.
To get robust interphase arrest in cycle 14 of wild‐type embryos, it is important that MPF be targeted to the nuclear compartment. In our calculations, kout=0 for MPF; if kout>0.1, then an extra cycle is observed. In addition, Wee1 must accumulate in the nucleus (kin=5 × kout for Wee1 transport, in our simulations); if kin=kout for Wee1, then the cell proceeds through 14 cycles instead of 13. On the other hand, String need not be differentially transported across the nuclear boundary (kin=kout for String transport, in our simulations). String may be targeted to the cytoplasm; the model works quite well even with kin=0 for String.
In fitting the model to the behavior of wild‐type and mutant embryos, we have had to assign specific values to the reaction rate constants and transport coefficients in the model. These values are constrained by the available data and represent predictions of the model. Although tedious, it is possible to obtain direct experimental evidence for some of these kinetic constants, to test the predictive power of the model. In a similar situation years ago, we predicted the rate constants of the MPF‐APC‐Wee1‐Cdc25 control system in Xenopus embryos (Novak and Tyson, 1993), and these predictions were born out by subsequent measurements of the phosphorylation and dephosphorylation of recombinant proteins in frog egg extracts (Kumagai and Dunphy, 1995; Marlovits et al, 1998; Zwolak et al, 2005).
The model can be used to predict the phenotype of any mutant that can be constructed by knocking down and/or overexpressing any of the genes encoding components of the regulatory mechanism. For example, of the mutants described in Table III, some of the Fzy and Wee1 mutants have yet to be characterized experimentally.
Nuclear envelope breakdown
The nuclear ‘compartments’ of the model are localized regions around chromosomes and spindles where MPF activity can periodically build up and be destroyed, thereby governing nuclear division cycles without perturbing much the cytoplasmic ‘compartment’ until there are 1000's of nuclei in the egg. In our model, we allow transport of proteins (MPF, Wee1 and String) between these compartments, but we do not change the transport coefficients during the phase of the division cycle when the nuclear envelope is disassembled. In the crowded environment of the cell for the few minutes when the nuclear envelope is missing, protein movements between spindle environs and bulk cytoplasm may still be restricted, but we may certainly expect intercompartmental transport to be faster than when the nuclear envelope is intact. To study the effects of increased rates of protein transport between nuclear and cytoplasmic compartments when the nuclear envelope is missing, we created an alternative version of the model, as described in the ‘ode’ file in the Supplementary Material. Briefly put, we assign two different values to each transport coefficient (kin and kout for MPF, Wee1 and String): a ‘small’ value when the nuclear envelope is intact and a ‘large’ value when the nuclear envelope is disassembled. We call this the ‘nuclear envelope breakdown’—NEB—model.) For Wee1 and String, we assume that
with an adjustable factor f>1 for each species. For MPF, as koutm,small=0, we assume that
The latter inequality seems appropriate, because MPF dimers are likely bound to mitotic spindles, so they are not easily lost from the nuclear compartment even when the nuclear envelope breaks down. We assume that the nuclear envelope is disassembled when MPF activity is greater than some threshold. (In principle, the thresholds for nuclear envelope breakdown and re‐assembly could be different. We set the thresholds so that the nuclear envelope is missing for about 30% of the mitotic cycle.)
With the ‘NEB’ version of the differential equations, we can test how sensitive the model is to variations in transport rates before and after nuclear envelope breakdown. Reassuringly, we find that Figure 3 is not much changed over a range of f values >1 (see, for example, Supplementary Figure S4). As expected, fm is most restricted; if fm>2.5, then the cycle blocks in M‐phase. Furthermore, if koutm,large is too large (say, >kinm,large), then [MPFn] starts dropping as soon as the nucleus enters mitosis. For this reason, we suppose that nuclear MPF is bound to a structure during mitosis (e.g., the spindle) so that it cannot interchange freely with the cytoplasmic pool. On the other hand, String and Wee1 may move freely between nuclei and cytoplasm during M‐phase (acceptable behavior is observed for fs and fw as large as 10).
Systematic comparison of mathematical models of the networks controlling cell division cycles in various organisms helps to delineate common principles as well as significant variations in the design of these networks along the evolutionary tree (Csikasz‐Nagy et al, 2006). The model presented here of the syncytial mitotic cycles in Drosophila embryos is closely related to earlier models covering cell cycle regulation in frog eggs (Novak and Tyson, 1993)and fission yeast cells (Novak and Tyson, 1995). In contrast to those models, the Drosophila model underscores the importance of nuclear compartmentation of cyclin degradation to achieve very rapid multiplication of nuclei in the fruit fly embryo. The record‐setting pace of nuclear division presumably helps the developing fruit fly to proceed as rapidly as possible through the most vulnerable stage of its life cycle.
Programs and software
The models are defined in terms of ordinary differential equations and simulated with the software XPPAut (http://www.math.pitt.edu/~bard/xpp/xpp.html) (Ermentrout, 2002). An ‘ode’ file for simulating Figure 3 is provided in the Supplementary Material. To compute Figure 1B, some numerical constants must be changed, as described in the ode file. When modified as described therein, the ode file can also implement the NEB model described in the Discussion. A second ode file is supplied for computing bifurcation diagrams (as in Figure 2) using XPPAut. In addition, our model is available as an SBML file from the BioModels database (http://www.ebi.ac.uk/biomodels, temporary accession MODEL1509031628).
Rescaling of concentrations after division
Define Xn_before and Xc_before to be the concentrations of species X in the nucleus and in the cytoplasm before nuclear division (at time t), and Xn_after and Xc_after to be the concentrations after division (at time t+Δt). Then:
Insisting that XTotal_after=XTotal_before, we find that Xn_after and Xc_after must be adjusted at nuclear division as follows:
This rescaling needs to be applied to MPF, Wee1 and String each time the nuclei divide.
This work has been supported by grants from the Defense Advanced Research Project Agency (AFRL No. F30602‐02‐0572), the James S. McDonnell Foundation (21002050), the European Commission (LSHG‐CT‐2004‐503568) and the Action de Recherche Concertée INRIA MOCA (2006).
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