Generic Cell Cycle Model

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Predictions of the model

1. Two ways to homeostasis

Our model predicts that well fed wild-type budding yeasts and many mutants of budding and fission yeast cells have a cell cycle network working in an oscillatory regime.Other cells balance their growth and division cycles by passing a size criteria during their cycle, and after that size is passed then they go through the mitotic oscillations. Whereas fast growing, and some mutant cells always stay in this oscillatory regime reaching balanced growth by matching their cycle time, with the period of mitotic oscillations. Details here.

2. Growth rate dependent lethality

An interesting prediction of the model if the growth rate dependent viability of budding yeast cells perturbed in the mitotic exit module. This phenomena has been observed in two mutants by Cross, 2003 but in that original paper this partial viability of mutants was not highlighted. Here we picked up these results and suggested the same growth rate dependent viability exist in other mutants with problem in mitotic exit. Details here.

3. Two possible oscillations in a fission yeast mutant

With two dimensional bifurcation diagrams we could show what transitions a mutation can cause to the dynamics of the regulatory system. By plotting the genetics (mutation) as a function of physiology (cell mass) we proposed two possible oscillations (endoreplication and normal mitotic cycles) for intermediate Cdc13 activities in fission yeast cells. Details here.

4. Mammalian cells lacking CycE and CycD

Experiments have shown that mouse embryos can proliferate even without any forms of CycE ( Geng et al., 2003 ) or Cdk2 ( Ortega et al., 2003 ) and they are also reach a late developmental stage in the absence of CycD ( Kozar et al., 2004 ) or Cdk4 and Cdk6 ( Malumbres et al., 2004 ). Our simulations fit these experimantal results (without any previous fitting of parameters to mutants) amd predicts that the double deletion of CycD and CuycE related Cdk activity would cause G1 block to the cells. Details here.

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1. Two ways to homeostasis
The different simulated cell cycles at different mass doubling times on the bifurcation diagram of wild-type budding yeast cells show an important phenomena.

Fig. 1: One dimensional bifurcation diagram of wild-type budding yeast cells.

For survival of a population of cells, individual cells have to achieve balanced cell growth: MDT (populations mass doubling time) = individual cell cycle time. This homeostasis can be reached different ways in our system. On weak nutritional conditions, slow growing (high MDT) cells are attracted by stable G1 steady state until they reach a critical cell size, a working sizer determines the mass of transition from G1 to S phase. Faster growing cells can achieve balanced growth when living in the oscillatory regime without a proper size control. In this case the oscillation period is balanced by the mass doubling time and the cell cycle is driven by a oscillator regulator instead of a sizer. This distribution of a sizer and a decreasing period oscillator regulation zones are the consequence of the phenomena of the transition between these regimes which is driven by a SNIPER bifurcation that gives rise to long period oscillations.

Fig. 2: Achieving balanced growth at different growth rates.
Upper panel: bifurcation diagram of the budding yeast cell cycle network with the generic model (same as figure upper on page). Lower panel: period of the oscillatory solutions. Color simulation curves of different MDT are shifted vertically to the corresponding period dashed lines; horizontally they are at the real simulation cell mass position. Background shading shows the sizer or oscillator of cell cycle regulation types at different cell mass.

In more biological terms fast growing cells are already larger then the critical size of G1-S transition when they are born, but these cells still have a short G1 phase as they visit the minima of oscillations. This result can explain the characterized dependence of pheromone induced G1 delay on growth conditions in fission yeast; where only nitrogen starved cells can be stopped in G1 phase by pheromone addition. While in the wee1^- simulation curve balanced growth is achieved in the oscillatory regime this suggest that even thought wee1^- cells has a long G1 phase but they still can not be stopped here by pheromone addition. This correlates with experiments that observed five fold reduced mating efficiency for wee1^- cells compared to wild type cells

2. Growth rate dependent lethality

To bind effectively to Cdc20, proteins of the core APC need to be phosphorylated ( Rudner & Murray, 2000b ). If these phosphorylation sites are mutated to non-phosphorylable alanine residues (the mutant is called APC-A), then Cdc20-mediated degradation of CycB is compromised, although the APC-A cells are still viable. We assume that APC-A has a constant activity that is 10% of the maximum activity of the normally phosphorylated form of APC in conjunction with Cdc20. Furthermore, we assume that APC-A has full activity in conjunction with Cdh1, in accord with the evidence ( Rudner & Murray, 2000b ). In simulations (Fig. 3A), APC-A cells are viable but they oscillate in the 'oscillator' regime even at MDT > 150 min, a consequence of the fact that the mutant control system is delayed in exit from mitosis, so the period of the limit cycle oscillations beyond the SNIPER bifurcation is lengthened, compared to wild type.

Double mutant cells, APC-A cdh1∆, are lethal at fast growth rates but partially viable at slow growth rates ( Cross, 2003 ). Our bifurcation diagram (Fig. 3B) shows a truncated oscillatory regime ending at a cyclic fold bifurcation at cell mass = 3.6. Simulations show that at MDT = 150 min cells stay within the small oscillatory regime, but faster growing cells (MDT = 120 min) grow out of the oscillatory regime and get stuck in mitosis. Mutations of APC core proteins also show growth rate-dependent viability, e.g., apc10-22 is viable in galactose (slow growth rate) but inviable in glucose (fast growth rate).

The same dependence of viability on growth conditions was reported for CLB2db∆ clb5∆ mutant cells (CycB stablized, CycA absent) ( Cross, 2003 ), and is illustrated in our bifurcation diagram (Fig. 3D). In addition to these mutants, which are defective in cyclin degradation, Cross ( Cross, 2003 ) found that the double mutant clb2∆ cdh1∆ also shows growth rate-dependent viability. In our model these cells are viable at MDT = 200 min, but lethal at MDT = 120 min (Fig. 3C).

All of these mutations interfere with the negative feedback loop of CycB degradation (module 3 on the wiring diagram). The bifurcation diagrams explain this effect: weak negative feedback creates long-period oscillations that are stable attractors only at relatively small cell mass. Fast growing cells cannot find a period of oscillation that balances their MDT, so they overgrow the oscillatory region and get stuck in mitosis. These results suggest that other mutants affecting the negative feedback loop should be reinvestigated to see if viability depends on growth rate.

Fig. 3: Bifurcation analysis of budding yeast mutants sensitive for growth conditions.
One dimensional bifurcation diagrams of budding yeast cells with mutations interfering with cyclin degradation.

Cells that show this sensitivity to growth rate are also likely to be sensitive to random noise in the control system. Using a model similar to ours, Battogtokh & Tyson ( Battogtokh & Tyson, 2004 ) showed that, for control systems operating close to a bifurcation to the stable M-like steady state, cells might get stuck in mitosis after a few cycles if a little noise is added to the system. This effect would show up as partial viability of a clone at intermediate growth rates.

3. Two possible oscillations in a fission yeast mutant

The transcription of CycA cyclins is regulated by the E2F related transcription factors (TFE on wiring diagram), at the same time CycA and its fission yeast homologue Cig1 and 2 inactivate this transcription factor by phosphorylation ( Ayte et al., 2001 ). This negative feedback can lead to oscillations if there is a delay in the system. The Cdk inhibitor (CKI) creates this delay by inhibiting CycA until that saturates CKI and initiates its destruction. At this point the negative feedback turns on and CycA disappears. In wild-type cells this system can not create stable oscillations because CycB takes over and keeps CKI low in G2 and M phases. If Cdc13 (CycB) is absent from cells then CKI can come back after CycA killed itself. Hiding in the complex with CKI CycA can build up again until it can free itself from CKI and induce another round of DNA replication. As this oscillation goes on periodic CycA activity induces multiple rounds of DNA replication without ever going into mitosis. This endoreplication phenotype is observed in cdc13∆ mutant ( Hayles et al., 1994 ). On Fig 4 we show the bifurcation diagram of these cells, of course here the respond of the system can be measured only in CycA activity appearing as the bifurcation variable on the diagram. We find a huge regime in cell mass where large amplitude stable oscillations exist. An unstable oscillatory state and a stable steady state can be also noticed on the diagram, we do not relate those attractors to any physiological state while the system tends to stay on the stable limit cycle attractor of endoreplication. With external CycA addition the stable steady state might be noticed in cells as intermediate activity CycA state. While this negative feedback also exists in mammalian cells this might explain the core mechanism of developmental endoreplication.

Fig. 4: One dimensional bifurcation diagram of cdc13∆ mutant fission yeast cells.

With a two dimensional bifurcation diagram of CycB (Cdc13) level as the genetical indicator we can show how the transition happen form mitotic oscillations to endoreplication cycles. Fig 5 shows that Cdc13 synthesis (with this one parameter we can easily change Cdc13 level) greatly influences the critical cell mass for mitotic oscillations (the SNIPER bifurcation at the end of G2 phase). As the synthesis level decreases endoreplication cycles appear (after a couple complicated co-dimension two bifurcations, which we are not focusing on here). The oscillation first appears with long period, as Cdc13 decreases the period goes down and the area of oscillation extends. Interesting to observe that there is a regime in Cdc13 level where the two oscillations coexist.

Fig. 5: Two dimensional bifurcation diagram: Cdc13 activity vs. cell mass.

For instance at half of the wild-type Cdc13 level the cells can oscillate at small mass with endoreplication cycles and at a large mass with normal mitotic cycles (see this one dimensional cut of the bifurcation diagram on Fig 6). This figure predicts that heterozygous cdc13+/- mutant cells might live in normal mitotic lifecycles at a cell mass larger than wild-type size, but small cells might go through periodic endoreplication. One could claim that as these cells are growing they eventually grow out of the endoreplicating oscillatory regime, but as the cell doubles its DNA content periodically the nucleus grows with it so our rule of cell mass treatment (in introduction) is changed. In this mutant case the nucleus volume will not stay constant, but gets larger each cycle, somewhat resetting the system to smaller cell mass/nucleus size ratio. If one could create cells with different levels of Cdc13, then the hypothised phenomena of different oscillations in the same cell type, could be much easier analyzed.

Fig. 6: One dimensional bifurcation diagram at half of wild-type (k_sbp = 0.01) Cdc13 synthesis level.
Endorepliaction type oscillations (small amplitude in CycB, hogy in CycA at small cell mass, and mitotic (large CycB amplitude oscillations at large cell mass.

4. Mammalian cells lacking CycE and CycD

It has been recently discovered that mouse embryos deleted for all forms of CycD ( Kozar et al., 2004 ) or both forms of CycE ( Geng et al., 2003 ) or Cdk4 and Cdk6 ( Malumbres et al., 2004 ) can develop until late stages of embryogenesis and die from causes unrelated to proliferation. Mice lacking Cdk2 are viable ( Ortega et al., 2003 ) and mouse embryo fibroblast from any of these mutants can proliferate normally. Our model is expected to reproduce these results. Indeed the simulation of CycE deleted cells show almost no defect in proliferation with a cell division mass 1.2 times wild-type cells (Fig 7A). The absence of CycD had a bit bigger effect on the system, creating cell cycles with a division mass 3.6 times the original value (Fig 7B). We also tested what happens if we destroy both CycD and CycE activity. We found that the cells can leave G1 phase at a mass equal 5 times wild type division mass (Fig 7C), which might be lethal for cells. These results are related to the corresponding experiments in budding yeast, where cln3 (CycD) and cln1cln2 (CycE) mutants are viable but larger than wild-type ( Dirick et al., 1995 ), while the combined mutation is lethal ( Richardson et al., 1989 ). Our simulations suggest that the average cell size of these mutant mice and fibroblasts from these embryos are larger in size than normally developing mouse cells.

Fig. 7: Simulations of mammalian mutants lacking cyclins.
Numerical simulations with MDT = 14 h with parameters for mammalian cell cycle regulation taken from Novak & Tyson, 2004 changed as: A: ksep = ksep = 0, B: CycD0 = 0, C: combination of A and B.

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