Coordination of cell growth and cell division: a mathematical modeling study

Cell growth and cell division are two tightly regulated, coupled processes that control cell size, cell number, and ultimately organ size (Coelho and Leevers, 2000; Conlon and Raff, 1999; Neufeld and Edgar, 1998; Potter and Xu, 2001; Saucedo and Edgar, 2002; Stocker and Hafen, 2000). If the two processes become uncoupled, cells would either become progressively larger if the doubling time for cell mass was shorter than the cycle time, or progressively smaller if the mass doubling time was longer than the cycle time. Despite numerous elegant experiments, the details of how cell growth is regulated and coupled to cell division remain poorly understood.

It has been shown in fission yeast experiments that a cell must grow to a critical size for DNA replication and cell division to occur (Fantes and Nurse, 1977; Fantes, 1977; Mitchison and Nurse, 1985; Nurse and Thuriaux, 1977; Sveiczer et al., 1996). A key observation is that when the birth size of a cell is smaller than the critical size, the cycle time consists of two components: a `sizer' phase, which is the time for the cell to reach the critical size, followed by a `timer' phase, which is nearly independent of cell size. Thus, cells born larger than the critical size have a nearly constant cycle time, regardless of their birth size. For cells born below the critical size however, the cycle time lengthens as birth size decreases due to the influence of the sizer phase. Similar size control of the cell cycle also exists in higher eukaryotes such as Xenopus laevis (Wang et al., 2000) and Drosophila (Edgar and Lehner, 1996).

Likewise, another example demonstrating regulation of cell division by growth is restriction point analysis (Zetterberg and Larsson, 1995). If serum is removed or cycloheximide (CHX) applied to inhibit cell growth for a brief interval, cells that have not passed a point called the `restriction point' (located in late G1) will arrest and exhibit a delay in cell cycle progression, even after serum is reintroduced. In these cells, the subsequent cycle time following the delayed cycle is normal. In contrast, if cells have passed the restriction point when growth is inhibited, they will continue through the current cell cycle and undergo mitosis, but their next cell cycle will be delayed. Thus, at the start of the third cell cycle after the growth inhibition, both groups of cells are almost resynchronized.

Recently, we formulated a physiologically based mathematical model of a cell cycle signaling network in higher eukaryotes, in which the sizer and timer phases of the cell cycle arise naturally from a Hopf bifurcation point (Qu et al., 2003). This model also successfully reproduced the restriction point features of the eukaryotic cell cycle described above. However, an incomplete aspect of this model was that cell growth coordinating the cell cycle was treated phenomenologically, as has also been the case for other modeling studies examining cell cycle dynamics (Tyson et al., 2001; Tyson et al., 2002). Moreover, no convincing mechanism has emerged from previous studies (Sveiczer et al., 1996; Wang et al., 2000) examining the relationship between cell growth and cell cycle time. In previous models these relationships have been fitted using empirical functions, without any clearly defined biological rationale. In this paper, we present a mathematical model for cell growth control based on principles established from recent mechanistic biological experiments (Coelho and Leevers, 2000; Saucedo and Edgar, 2002; Stocker and Hafen, 2000; Thomas, 2000). Using this formulation, combined with the concept of sizer and timer phases of the cell cycle, we derive relations between cycle time and cell birth size for several growth conditions. We show that when growth rate is determined by the cell surface area at birth, all experimental data on cycle time versus cell birth size in fission yeast (Sveiczer et al., 1996) and Xenopus laevis (Wang et al., 2000) and cell cycle delay in restriction point experiments (Zetterberg and Larsson, 1995) can be explained by the same model.

A basic understanding of the signal transduction network regulating growth control is now emerging (for reviews, see Coelho and Leevers, 2000; Neufeld and Edgar, 1998; Potter and Xu, 2001; Saucedo and Edgar, 2002; Stocker and Hafen, 2000). Fig. 1 summarizes its key features. Cell growth is stimulated by insulin or insulin-like growth factors (IGF) binding to their receptors. The activated receptor recruits phosphatidylinositol 3-OH kinase to the membrane, which then phosphorylates phosphatidylinositol (4,5)-bisphosphate [PtdIns(4,5)P₂], converting it to phosphatidylinositol (3,4,5)trisphosphate [PtdIns(3,4,5)P₃]. Increased levels of PtdIns(3,4,5)P₃ induce the activation of phosphoinositide-dependent protein kinase 1 (PDK1), which then stimulates 40S ribosomal protein S6 kinase (S6K). S6K mediates the phosphorylation of 40S ribosomal protein S6, which is thought to potentiate the recruitment of ribosomal-protein mRNAs from stores of messenger ribonucleoproteins (mRNps) to actively translating polysomes.

in which the growth rate is simply proportional to the cell surface area.

Assuming that cell cycle time is controlled exclusively by cell growth before the cell reaches a critical size (the sizer period), and is independent of cell size beyond this point (the timer period, T₀), then a relation between cell cycle time T and the birth size can be deduced if the growth rate is known. Here we couple our growth equation (Eqn 4) with the concept of sizer and timer to predict the relationship between cycle time and birth size, using two different assumptions about growth rate. In the first case, we assume that growth rate during the sizer phase is proportional to cell surface area. In the second case, we assume that cell mass during the sizer phase grows at a constant rate determined by the surface area of the cell at birth, as suggested by the experimental observations in fission yeast in which growth rate is constant in the sizer period (Fantes, 1977; Mitchison and Nurse, 1985; Sveiczer et al., 1996) and dependent on birth length (Sveiczer et al., 1996).

where r_B is the cell radius at birth and r_C is the critical radius.

where: α=(ρ/3μ′).

More than two decades ago, Fantes (Fantes, 1977) showed that cell cycle time in fission yeast (Schizosaccharomyces pombe) varied with birth size if the birth size was smaller than a critical size (∼10.5 μm), after which they became almost constant. Twenty years later, the relationship between cycle time and birth size in both wild-type yeast and various cell cycle mutants was re-examined (Sveiczer et al., 1996), confirming the earlier observations. Their data for the cdc2-33 mutant, which exhibits the widest range of birth lengths compared to the wild type and other mutants they studied, is replotted in Fig. 2A.

If growth rate during the sizer phase is proportional to the surface area, then the derived formula of cycle time T versus birth length l_B exhibits a logarithmic relationship (Eqns 7 and 8), which could not be well fitted to the experimental data of either the wild type or mutant fission yeast. However, if the growth rate during the sizer phase is assumed to be constant and determined by the birth size, as supported by experimental findings, then our model (Eqn 11) agrees well with the experimental data (Fig. 2). The measured growth rate for the cdc2-33 mutant can be best fitted by 0.006+0.0046l_B for l_B<12 μm (solid line in the inset of Fig. 2A). Inserting the growth rate into Eqn 11, we then fitted T₀ and l_C to the experimental cycle time data (symbols in Fig. 2A). The first nine points in the data set correspond to shorter birth lengths (i.e. l_B<12 μm) for which the subsequent cell cycle is comprised of both sizer and timer phases. It is well fit by T=105+(12.75-l_B)/(0.006+0.0046l_B) (solid line), in which the first term represents the timer and second term the sizer period. To define the critical length more precisely, the data set was fitted as follows: the first three points were fitted to Eqn 11, then the next data point was added to the fitting set and another best fit carried out. This process was repeated iteratively to obtain the best fit for T₀ and l_C (Fig. 2B) and the χ² value (Fig. 2C) using Origin 6.0 (Microcal Software, Inc.). The χ² (solid triangles in Fig. 2C) grew exponentially after data point 10, corresponding to a birth length of 12-13 μm, at which point the best fit values of T₀ (circles in Fig. 2B) and l_C (diamonds in Fig. 2B) also changed dramatically. Therefore, Eqn 11 predicted a timer with period T₀ ∼105 minutes and a sizer controlling the period for l<l_C∼12.75 μm for cdc-33 mutant cells. For comparison, we also show χ² for the linear fit to the data, as used by Sveiczer et al. (Sveiczer et al., 1996) in their phenomenological analysis; this gave a poorer fit (larger χ²) than our nonlinear fit to Eqn 11.

Data for the cdc2-33 wee1-6 mutant was also fitted to Eqn 11 and the results are shown in Fig. 3A. The first 5 points (l_B<6 μm) were well fitted by T=95+(7.8-l_B/0.004+0.0052l_B) (solid line), but T₀, l_C, and χ² changed suddenly at the data point 6 (Fig. 3B,C), corresponding to a birth size of 6 μm. The χ² for the nonlinear fit to Eqn 11 was also superior to the linear fit (Fig. 3C). The critical size (l_C) and timer (T₀) agree well with the direct experimental observations.

For the wild type and the wee1Δ mutant fission yeast, no growth rate was reported (Sveiczer et al., 1996). We fitted our Eqn 11 to their measured cycle time versus birth length using T₀, l_B, g₀, and β as fitting parameters. The results are summarized in Table 1. It is interesting to note that, using our Eqn 11, the correct timer period and the critical cell size can be recovered even without knowing the growth rate. The χ² for the nonlinear fit to our model, using growth rate, T₀ and l_C as fitting parameters, was comparable to that for the linear fit empirically used by these authors (Table 1).

This analysis demonstrates that a physiologically based mathematical formulation of cell growth relating cycle time to birth size, derived under the assumption that the cell growth rate during the sizer phase is determined by the surface area at birth, agrees well with the experimental fission yeast cell cycle data. Moreover, we can estimate the critical size l_C and the timer period T₀ using the measured relationship of cycle time versus birth size alone.

In Xenopus embryonic cells (Wang et al., 2000), the initial 11-12 cell cycles occur with a fixed cycle length until the birth size of the daughter cells is smaller than a critical size (∼37 μm in radius). After this period, the cycle time T depends on initial cell size. Using a phenomenological analysis, Wang et al., (Wang et al., 2000) showed that the empirical formula (⁠

If one assumes the cell growth during the sizer phase is continuously proportional to surface area, the resulting Eqn 10 fits to the data (Wang et al., 2000) poorly. However, if, as for fission yeast, we assume cell growth during the sizer phase is proportional to the surface area at birth, the resulting Eqn 13 provides a good fit to the experimental data (Fig. 4). In our analysis, we preset T₀ to the constant cycle time portion of the data (open circles in each panel of Fig. 4) and fitted the rest of the data (filled circles in Fig. 4) to Eqn 13 to obtain the growth related constant α and the critical radius r_C for each experiment. All experimental data were uniquely fitted by Eqn 13.

where μ₀=1.49 μm/min and K_d=0.036 μg/ml.

Assuming the protein synthesis rate is proportional to the concentration of the 60S subunit, i.e., substituting k₁ in Eqn 12 by k₁e, we obtain Eqn 14.

As our analysis using the same assumptions we made with in fission yeast demonstrates a good fit to the experimental data for Xenopus cells, it suggests that the concept of a sizer and timer period is also applicable to the Xenopus embryonic cell cycle. In addition, the observation that cells grow at a constant rate proportional to their birth size in fission yeast may also be applicable for cell growth in Xenopus. Wang et al. (Wang et al., 2000) had to assume that the volume became more and more important as [CHX] increased, as n in their fitting equation (⁠

where g(t) is the growth rate. Starting from an initial mass, the cell grows according to Eqn 17 to a critical size (the sizer period), after which it grows for a fixed period T₀ (the timer period) and then divides. At mitosis, we reduce the cell mass in half. To mimic the real cycle time in the Zetterberg and Larsson's experiments (Zetterberg and Larsson, 1995), the growth rate is adjusted to give a normal cycle time of 14 hours, during which the cell mass doubles from 1 to 2 units. The restriction point is set to 6 hours into the cycle, the timer period T₀ to 8 hours, and the non-growing period to 12 hours (4 hours treatment plus 8 hours delay as determined experimentally) (Zetterberg and Larsson, 1995).

Fig. 6A compares the simulated cell cycles for different growth conditions. At the time of treatment, the first cell (Cell 1) has passed the restriction point and is larger than the critical size (cell age 7 hours, solid lines in Fig. 6A), and so undergoes mitosis (M1) without delay. Because its growth has been retarded when it divides into two daughter cells however, their mass is smaller than 1 unit. This smaller mass at birth results in a extension of the time required for the cell to reach the critical mass (sizer period) during the next cycle, and so prolongs the second mitotic cycle (M2). In contrast, the other cell (Cell 2) is smaller than the critical size at the time of treatment (cell age 5 hours), and after the 12 hours of growth arrest, it resumes growing. After resuming growth, it eventually reaches the critical size and passes through the restriction point, followed by the 8-hour timer period required to reach mitosis (M1′). At this point, it has attained a normal mass of 2 units, and divides into two cells, each with mass of 1 unit. Thus, Cell 2 has a 12-hour delay in its first mitosis, caused exclusively by the growth retardation period, but its second mitosis (M2′) occurs after a normal period of 14 hours as its mass at birth is 1 unit.

The key point is that Cell 1, which is not delayed in the first mitosis, becomes delayed in the second mitosis (M2) because its birth mass is less than 1 unit. However, Cell 2, which is delayed in the first mitosis, is not delayed in the second mitosis because of its normal birth mass. With respect to different models of growth rate, the important issue is that for Cell 1, the delay time in the second mitosis depends on which growth rate model is operational. In contrast, for Cell 2, the delay in the first mitosis depends only on the duration of growth stoppage, not on the growth rate model. Thus, for the exponential growth model [g(t)=(mln2/14)], the delay in M2 for Cell 1 precisely compensates for the delay in M1 for Cell 2, and the two cells are resynchronized at the end of M2 (Fig. 6A, upper panel), consistent with Zetterberg and Larsson's interpretation of their restriction point data (Zetterberg and Larsson, 1995). For the birth-size dependent constant growth model {g(t)=[(m_B)^2/3/14], lower panel} however, the delay in the second mitosis in Cell 1 is not long enough to compensate for the delay in the first mitosis in Cell 2 and the cells therefore do not resynchronize at the end of M2. Thus, the birth size-dependent growth model, which is the only model to fit the cell cycle time versus cell size data discussed above, does not predict exact resynchronization of the cells after the second mitosis, as has been commonly interpreted from Zetterberg and Larsson's experimental restriction point data (Cooper, 1998; Zetterberg and Larsson, 1995).

To explore this issue further, we replotted the data from Zetterberg and Larsson (Zetterberg and Larsson, 1995) in Fig. 6B-D. We superimpose our simulation results for three growth models: exponential (open circles), constant [g(t)=(1/14), diamonds], and birth-size dependent (triangles). For the first mitosis (Fig. 6B), all three models give equivalent results, since the growth model does not influence the delay in the first mitosis. For the second mitosis (Fig. 6C), however, the delay for cells whose growth was inhibited after they had reached the restriction point (i.e. cells aged >6 hours, equivalent to Cell 1 in Fig. 6A) is not fixed, as predicted by the exponential growth model (open circles), but has a positive slope of 0.3 (green line, Fig. 6C) with the data points mostly falling below the 26-hour cycle time required for the cells to resynchronize exactly at the end of the second mitosis (as cells in Fig. 6B that were arrested before 6 hours in the cycle all had a consistent M1 lasting 26 hours). Moreover, there are two apparently `bad' data points (enclosed by red circles) in Fig. 6C. If we exclude these two points, the slope increases to 1.1 (red line, Fig. 6C). Therefore, neither analysis supports the conclusion that the delay in the second mitosis is constant and equal to 12 hours (4 hours' treatment plus 8 hours' delay).

To explore whether or not the cells resynchronized exactly at the end of M2, we calculated the average cycle time of M1 and M2 for each cell age and then added them to obtain the time at which M3 began (excluding the four circled points in Fig. 6B and C). Fig. 6D shows that for the first 5 points (aged 0 to 6 hours at treatment), all began their M3 at ∼40 hours, as indicated by a nearly horizontal linear regression slope of 0.15 (cyan line). In contrast, for the last 5 points (aged 7 to 12 hours at treatment), the slope was 1.9 (red line). This indicates that the cells whose growths were arrested before and after restriction point at 6 hours did not precisely resynchronize at the end of M2, in agreement with our mathematical analysis assuming that growth rate is set by cell surface area at birth.

Our analysis shows that the concept of a sizer and timer period in the cell cycle can account for the experimental observation of a restriction point in the cell cycle. However, in contrast to previous claims, the delay in the second mitosis is not constant but rather depends on growth rate and cell age.

We have developed a growth model based on a currently proposed signal transduction network for cell growth, and combined it with the cell cycle concepts of sizer and timer phases to derive relations between cycle time and birth size. We find that the derived equation agrees well with the experimental data from both fission yeast and Xenopus laevis if: (1) cell growth rate during the sizer phase is assumed to be determined by cell surface area at birth when the birth length is below the critical length; (2) cell cycle time is determined by growth before the critical size (the sizer period), but independent of growth after the critical size (the timer period). In addition, we simulated restriction point experiments in HeLa cells and also found good agreement under the same conditions, provided, as supported by our re-analysis of the experimental data, that the delay in the second mitosis depends on growth rate and cell age, rather than being constant.

In previous studies (Tyson et al., 2001; Tyson et al., 2002) examining cell cycle dynamics, growth has been treated as a phenomenological process driving production of key cell cycle elements, without any explicit mechanistic formulation based on biologically plausible concepts. In experimental studies (Sveiczer et al., 1996; Wang et al., 2000) on the other hand, growth rate has been empirically fitted to arbitrary functions which have differed for data sets obtained from different species (e.g. fission yeast versus Xenopus laevis) or conditions. However, the same function does not fit all the data sets. Our physiologically based growth model however, when coupled to a cell cycle with sizer and timer phases, provides a unified biologically plausible mechanism which fits all of the currently available experimental data sets.

A key conclusion from our mathematical analysis is that the best fit to the experimental data from yeast, Xenopus and HeLa cells is achieved when it is assumed that growth rate during the sizer phase is determined by the cell surface area at birth when the cell is born smaller than the critical size. The rationale for this assumption is based on experimental data in fission yeast showing that the growth rate is constant and proportional, on average, to the birth length of the cells (Sveiczer et al., 1996). Although this data was obtained from populations of yeast cells, we assume that it also applies to an individual cell followed through multiple successive divisions. If this hypothesis is true, is there a rationale biological explanation for this phenomenon that could suggest an experimentally testable prediction? One possibility is that following cell division, the rate of synthesis and degradation of growth receptors remains balanced, so that the total number of growth receptor present at birth remains constant even as cell size increases during the sizer phase (i.e. the density of growth receptors on cell membrane decreases). During the timer phase, when DNA content doubles, a higher synthesis rate of growth receptors (relative to degradation rate) might restore the receptor density back to a uniform level prior to cell division. Thus, the birth size would determine the number of growth receptors present and the initial growth rate during the sizer phase. A recent experiment (Tashiro et al., 2003) showed that FGF receptor-2 transcription was constant during early portions of the cell cycle where cell growth would normally occur but was induced in the mid-to-late G1 phase of the cell cycle in serum-starved mouse NIH3T3 cells. Other growth signaling proteins that localize to the cell membrane, such as PI 3-inase and PDK1, are also candidate proteins. An alternative hypothesis is that activation of ribosomal RNA transcription by the signaling proteins is limited by ribosomal RNA production instead of the signaling proteins. In other words, the ribosomal RNA is produced in one constant rate proportional to the initial size in the sizer phase but another constant rate in the timer phase. Direct measurements have provided evidence that ribosomal RNA content increased at a constant rate through S phase and increased to a higher constant rate after DNA duplication (Fujikawa-Yamamoto, 1982). Both hypotheses are consistent with the experimental observations that cells increase their growth rate in the middle of the cell cycle (Killander and Zetterberg, 1965; Mitchison and Nurse, 1985; Sveiczer et al., 1996).

Although our growth model fits the available experimental data sets in fission yeast, Xenopus laevis, and HeLa cells, direct measurement of cell growth rate, particularly during the sizer phase, in higher eukaryotic cells will be necessary to confirm this model more generally. There are several limitations to our present model. There are some subtle features in the experimental data that are not explained by our model. The experimental data shown in Fig. 2A and Fig. 3A indicate that cycle time in yeast is still influenced to a modest extent by size during the timer phase, which violates the basic assumption in our model that cells born larger than the critical size have a constant cycle time. We did not consider the condition in which protein degradation rate is comparable to protein synthesis rate in our analysis, which may result in cell atrophy in the case of serum starvation or poor nutrient environment.

To simplify the mathematical analysis, our model for growth control was based on a simplified signal transduction network. Other non-S6K signaling pathways, such as those related to TOR (Coelho and Leevers, 2000; Prober and Edgar, 2001; Stocker and Hafen, 2000) could be incorporated in future studies. However, despite the complexity of the signaling network for growth control and the cell cycle machinery, the simple mathematical model still predicts most of the key experimental observations and provides a useful first step towards a biologically authentic growth model which can be coupled to cell cycle dynamics.

This study was supported by funds from UCLA Department of Medicine and by the Kawata and Laubisch Endowments.