Analysis and modeling of growing budding yeast populations at the single cell level

Abstract

Model organisms and in particular the budding yeast Saccharomyces cerevisiae have been instrumental in advancing our understanding of cell cycle progression. The asymmetric division of the budding yeast and the tight coupling between cell growth and division have challenged the theoretical understanding of the cell size structure of growing yeast populations. Past efforts have centered on modeling the steady-state theoretical age distribution for asymmetric division from which a cell size distribution can be derived assuming dispersion of cell size within each age class. Different developments, especially in the field of flow cytometry, allowed the determination of a number of cellular properties and their joint distributions for the entire population and the different subpopulations as well. A new rigorous framework for modeling directly the dynamics of size distributions of structured yeast populations has been proposed, which readily extends to modeling of more complex conditions, such as transient growth. Literature on the structure of growing yeast populations and modeling of cell cycle progression is reviewed. © 2008 International Society for Advancement of Cytometry

THE budding yeast Saccharomyces cerevisiae is widely used as a model for understanding the cell cycle progression of eukaryotic cells. This article aims to review the current literature about the structure of growing yeast populations and its modeling during cell cycle progression.

In growing populations of S. cerevisiae, the coordination between the continuous accumulation of proteins and RNA and the discontinuous events of the nuclear division cycle represents the most physiologically relevant cell cycle control. The coordination between cell growth and cell division makes it possible to maintain cell size homeostasis, thus preventing cells during balanced exponential growth from becoming too small or too large (1).

In an asynchronously growing S. cerevisiae population, individual cells differ in their position within the cell division cycle, their genealogical age (i.e., daughters, parents of different generations), their size (i.e., cells that have the same size do not necessarily share the same age or the same cell cycle position), and because of clonal variability (i.e., cells of the same age and of the same cell cycle position do not necessarily share the same size). All these variables yield the cell size distribution (i.e., the distribution of cell protein content, generally measured by flow cytometry) of the growing population. It has been shown that the protein distribution of a given population is a stable distinctive feature of each given balanced exponential growth condition, but it quickly changes when the environment is manipulated (2).

One approach to understand the structural heterogeneity of cell size distributions of growing yeast populations, and therefore to better understand the coordination mechanism between cell growth and cell division of this important eukaryotic model organism, is through the modeling of the steady-state theoretical age distribution for asymmetric division from which a cell size distribution can be derived assuming dispersion of cell size within each age class (2–9).

However, experimental developments (10–14), especially in the field of flow cytometry, allowed the determination of a number of cellular properties and their joint distributions for both the entire population and the different subpopulations as well. A new rigorous framework for modeling directly the dynamics of size distributions of structured yeast populations has been proposed, which readily extends to modeling of more complex conditions, such as transient growth (15, 16).

COORDINATION BETWEEN GROWTH AND THE DNA DIVISION CYCLE

In cells dividing by binary fission, cell division originates two cells of equal size. In the budding yeast S. cerevisiae, the cell mass at division is unequally partitioned between a bigger, old parent cell (P) and a smaller, new daughter cell (D). When a yeast cell buds, a chitin ring, called bud scar, builds up at the bud isthmus and remains on the parent cell after the bud has separated. Bud scars on intact cells can be visualized by fluorescent dyes (Calcofluor, Primulin) in fluorescence microscopy. Since each new bud starts at a new site, it is easy to determine the genealogical age of a cell (i.e., the number of cycles it has passed through) from the number of bud scars present on its surface. A cell without bud scars (a daughter cell) has not yet completed a cycle and is of genealogical age 0, while a cell with one bud scar has completed a cycle and is of genealogical age 1, and so on. As it will be described later in detail, cells need to reach a critical cell size to start budding and DNA replication. All daughters have the same critical cell size requirement, whereas for parent cells it increases with the genealogical age (1, 2, 14, 17–21). Each cell undergoing division originates two newborn cells, so that the frequency of newborn cells is twice that of cells immediately before cell division; the frequency of cells of intermediate age is between these extremes. This behavior can be described in mathematical terms by the “age density function” (1) from which it is possible to calculate the frequency of cells of any age in the population (6). Given the age density function of the population and the law of growth of the single cell, the theoretical cell size distribution can be derived, allowing computation of the expected frequency of cells of any size. It has been shown that the size distribution (cell volume distribution and distribution of the cellular protein contents) of a growing population is a stable distinctive feature of a given exponential balanced growth condition (2–5, 8, 22–24). Because of the inherent complexity of the yeast cell cycle, individual cells in an asynchronous S. cerevisiae population differ because of their position within the cell division cycle (G1, S, G2, M), their genealogical age (i.e., daughters, parents of increasing genealogical age), and the fact that cells of the same age and of the same cell cycle position do not necessarily have the same size.

The degree of asymmetry of cell division in S. cerevisiae is modulated by nutritional supply: poor media yield a high level of asymmetry with large parent cells and very small daughter cells, whereas in rich media, parent and daughter cells at division are very close in size. Since cells have to grow to a critical cell size before entering S phase and start budding, small daughter cells have longer cycle time than the corresponding parent cells, most notably in poor media. The fractions of cells of different genealogical age (i.e., age 0 for the daughter cells and age 1–4 for the parent cells) during exponential batch growth on different substrates is plotted in Figure 1 as a function of the specific growth rate of the population. It is clear that the heterogeneity of parent cells is quite significant and that one has to consider at least up to parent cells of genealogical age 4 to have a faithful representation of the parent subpopulation.

Figure 1. Percentage (%) of cells of different genealogical age as a function of the specific growth rate (h⁻¹) of exponentially growing yeast batch cultures. Yeast cells have been batch grown in different media. Specific growth rate and percentage of the different yeast subpopulations have been determined during exponential growth phase. Percentages have been obtained visualizing the bud scars. Data redrawn from Ref.19 . (•) % of parent cells in the whole population; (○) % of daughter cells in the whole population; (□) % of parent cells of generation 1 in the parent subpopulation; (♦) % of parent cells of generation 2 in the parent subpopulation; (⋄) % of parent cells of generation 3 in the parent subpopulation; (▪) % of parent cells of generation 4 in the parent subpopulation.

It has been previously stated that given the population age density function and knowing the fraction of budded and unbudded cells of different genealogical age, it is possible to compute the average value of the cycle time of the daughter cells [TD, Eq. (1)], the average cycle time of the parent cells [TP, Eq. (2)], and the duration of the budded phase [TB, Eq. (3)] for a given growth condition under balanced growth. Because of the asymmetrical division, the parent and the daughter cycle times satisfy Eq. (4). P_S and P_m, the critical cell size required to enter S phase and mitosis, respectively, are important population parameters and can be calculated according to Eqs. (5) and (6), their ratio h being given by Eq. (7). See Refs.2,19, and5 for derivation of Eqs. (1)–(4), (5) and (6), and (7), respectively. Using a modification of the two-dimensional DNA/protein flow cytometry procedure first described by Crissman and Steinkamp (25), the average P_s value for the population can be directly determined (23).

• (1)

• (2)

• (3)

• (4)

• (5)

• (6)

• (7)

where F_B is the fraction of budded cells in the entire population, F_DB is the fraction of budded daughter cells (i.e., frequency of budded daughters/frequency of daughters), F_PB is the fraction of budded parent cells (i.e., frequency of budded parents/frequency of parents), T is the overall duplication time of the population, α is the specific growth rate, P_p is the average protein content of a newborn parent cell, and TΔ is the time from the end of mitosis to cell division (2).

DISSECTING THE COMPLEX STRUCTURE OF A GROWING YEAST POPULATION

The Hartwell and Unger (1) model of the yeast cell cycle was based on two major hypotheses: (i) a yeast cell needs to attain a critical cell size before initiating the DNA division cycle at Start and (ii) budding yeast divides asymmetrically. Their model and the derived cell age distributions were based on some other assumptions, namely: (a) each cell grows according to an exponential law; (b) all parents have the same cycle time; (c) all daughters have the same cycle time; (d) the duplication time of the daughter cells is longer than the duplication time of the parent cells; (e) all cells share the same budded period; and (f) the duplication time of the parent cells is longer that the budded period. In later years, different procedures—namely, synchronization of cellular populations, time-lapse investigations, and new flow cytometric protocols—were developed to study the cell growth of individual cells or cohorts of cells within a yeast population, so allowing experimental confirmation (a, d, e, and f) or denial (b and c) of each assumption. These approaches allowed a deeper understanding of the complex structure of a growing yeast population. In the following sections, we are going to review several experimental results that shed more light on the complex structure of a growing yeast population.

Cohorts of Single Cells of the Same Age Grow Exponentially

A flow cytometry procedure that allows determination of growth properties of homogeneous cell cohorts of S. cerevisiae populations was developed (26) based on the work of May and Mitchison (27). Exponentially growing cells were harvested and their cell wall surface was stained with a fluorochrome-conjugated lectin, ConcanavalinA-fluoresceine isothiocyanate (ConA-FITC). When growth was resumed in a fresh ConA-free culture medium, cells retained the initial amount of fluorescence bound to their older cell wall components, whereas the new cell wall material, whose synthesis was restricted to the bud, was instead not stained. Therefore, this approach allowed an easy differentiation of the newborn daughter cells from the newborn parent ones (8, 9, 26). Coupling this staining procedure with the determination of the protein content of individual cells (9), it was possible to show that the dynamics of cell protein content increase in a cohort of selected cells born at the same time is fitted by an exponential curve (Fig. 2). Furthermore, it has been proved that the specific growth rate of the daughter cell subpopulation (and therefore of the parent one as well) is almost identical to the specific growth rate of the overall population as determined by the increase in cell number (9).

Figure 2. Average cell protein content of a selected cohort of daughter cells as a function of time after birth. A yeast population exponentially growing (specific growth rate: 0.215 h⁻¹) has been stained with Concanavalin A conjugated to fluorescein isothiocyanate (ConA-FITC) and resuspended at time T = 0 h, as described in Ref.9. A cohort of newborn daughter cells has been selected at T = 0.5 h and followed for a period of time equal to the duplication time of the daughter subpopulation. The protein content has been determined staining the cells with tetramethylrodamine isothiocyanate (TRITC). The graph shows the behavior of the average channel number over time. The continuous line represents the exponential fitting of the experimental data. The correlation coefficient of the fit was 0.99 and the calculated specific cell size increase rate was 0.219 h⁻¹. Abscissa, Time (hours); Ordinate, Average Channel Number (i.e., average cell protein content).

It is possible that small deviations from the exponential growth law for individual cells could be detected by more sensitive procedures, for instance by microscopic examination of the cell volume increase of individual cells grown on agar plates (28). Nevertheless, it seems acceptable to affirm that in growing populations, the kinetics of growth that better describes the growth of the “average cell” is the exponential one.

Determination of the Experimental Average Cell Size Distributions of Different Yeast Subpopulations

Given the heterogeneity of budding yeast populations discussed earlier, it would be very informative to gather information on the cell size characteristics of each subpopulation, such as daughter and parent cells of different genealogical age. Table 1 summarizes literature information regarding average size (and dispersion) of different yeast subpopulations. To compare different experimental data, the average cell size of each subpopulation has been normalized to the average cell size of the overall population, taken as 1. Although the data have been obtained with different approaches and from yeast populations growing with somewhat different specific growth rates, a substantial agreement among different data sets can be noted. In particular, the daughter subpopulation has an average cell size of 0.800–0.839, whereas the parent subpopulation stands at 1.232–1.330. Parents of genealogical age 1 are considerably smaller than those of higher genealogical age. Since in exponentially growing populations, cell size homeostasis has to be maintained, the total dividing cells are twice as big, on average, as total newborn cells (columns 2 and 4). Because of its unequal mode of division, a large difference in size can be observed between newborn daughters and newborn parents (0.552 vs. 0.757, column 4). Since the specific growth rate of parents and daughters is the same, i.e., the same as the whole population (see previous paragraph), it is expected that dividing daughters are, on average, more than twice as big as newborn daughters. Consistently the ratio of the cell size of newborn and dividing daughter cells is about 2.5 (column 1).

Table 1. Dissecting the complex structure of a budding yeast populations

Experimental approach	Column 1	Column 2	Column 3	Column 4	Column 5
Data express the ratio between the experimental average cell protein/cell volume content of the relative subpopulation and the experimental average cell protein/cell volume content of the overall distribution, taken as 1.000, respectively. Data in brackets represents the CV values of the relative subpopulation (n.a.: not available). Column 1: Double-tag flow cytometric approach (9). Specific growth rate of the whole population: 0.215 h^−1; Budding index: 53.5%. Column 2: Time-lapse investigations (14). Specific growth rate of the whole population: 0.351 h^−1. Column 3: Centrifugal elutriator as a baby machine (13). Specific growth rate of the whole population: 0.267 h^−1. Column 4: Slit scanning flow cytometric approach (10). Specific growth rate of the whole population: not available. Column 5: Br-Urd flow cytometric approach (12). Specific growth rate of the whole population: 0.426 h^−1; Budding index: 59%.
Overall population	1.000 (0.38)	1.000 (0.36)	1.000 (n.a.)	1.000 (0.24)	1.000 (n.a.)
Yeast subpopulation
Daughters	0.800 (0.29)	0.839 (0.36)
Daughters Newborn	0.500 (0.19)		0.455 (n.a.)	0.552 (0.18)
NB-D. of 1st gen		0.509 (0.15)	0.647 (n.a.)
NB-D. of 5th gen			0.890 (n.a.)
NB-D.of 7th gen			0.850 (n.a.)
NB-D of 10th gen			0.739 (n.a.)
NB-D of 15th gen			0.858 (n.a.)
Daughters in G1	0.600 (0.23)
Daughters at Bud E		0.687 (0.16)	0.795 (n.a.)
Daughters in S + G2 + M + G1*	0.990 (0.22)
Daughters dividing	1.291 (0.18)	1.262 (n.a.)
Parents	1.330 (0.23)	1.232 (0.25)	0.782 (n.a.)
Parents Newborn				0.757 (0.16)
NB-P of 1st gen		0.753 (n.a.)
NB-P of 2nd gen		0.975 (n.a.)
NB-P of 3rd gen		1.039 (n.a.)
NB-P of 4th gen		1.189 (n.a.)
P of 1st gen at Bud E		0.835 (0.16)
P of 2nd gen at Bud E		1.001 (0.15)
P of 3th gen at Bud E		1.141 (0.16)
P of 4th gen at Bud E		1.275 (n.a.)
Parents dividing	1.489 (0.27)
Dividing P of 1st gen		1.411 (n.a.)
Dividing P of 2nd gen		1.471 (n.a.)
Dividing P of 3rd gen		1.587 (n.a.)
Dividing P of 4th gen		1.741 (n.a.)
Total newborns		0.631 (n.a.)		0.654 (0.23)
Total dividing		1.336 (n.a.)		1.303 (0.13)
Total unbudded					0.812 (0.22)
Total in S phase					1.037 (0.19)
Total in G2 + M + G1*					1.326 (0.21)

Simulation of the Average Cell Size Distributions of Different Yeast Subpopulations

The average cell size distribution of the different yeast subpopulations can be easily modeled (2–9, 23). In this respect, the model must take into account the major features of a yeast population so far discussed, namely: (a) individual cells grow exponentially (9); (b) on average, each cell in the population grows with the same specific growth rate that is identical to the specific growth rate of the overall population (9); (c) commitment to DNA replication occurs when the cell reaches a critical cell size at Start (1, 2, 14, 17, 18, 20, 21, 29); (d) for the parents, such a critical value increases at each new generation (2, 30–32); (e) at any given specific growth rate, the budded period is appreciably constant for all cells (2, 30); and (f) at division, the mass synthesized during the budded phase belongs to the new-born daughter (33). Based on such considerations, a simulated yeast population growing with a specific growth rate of 0.215 h⁻¹ and a budding index value of 53.5% is composed of daughter and parents cells having an average size of value 0.790 and 1.320, respectively. A substantial agreement among experimental (Table 1, column 1) and simulated data sets can be noted.

In this respect, Supporting Information Figures S1 and S2 summarize the most relevant information.

ARE METABOLISM AND DNA DIVISION CYCLE COORDINATED?

Detailed metabolic studies should be performed on populations growing under balanced growth conditions, when the distribution of the “single-cell growth rates” do not change over time (34). For a yeast population growing in batch culture, where the physiological conditions change continuously over time, a true balanced exponential growth can be attained only transiently [i.e., for 2–4 duplication times; (2)]. On the other hand, balanced growth conditions can be obtained in a reproducible way and independently from the time course of the experiment in chemostat cultures. Careful analyses of the literature data indicate that in batch cultures, the average cell volume remains low and almost constant during growth on nonfermentable substrates while the average size of the cells linearly increases with the specific growth rate during growth on fermentable substrates (35). Data from glucose-limited continuous cultures validate this observation, since the average cell size starts to increase after the critical dilution rate has been reached, i.e., after cells shift their metabolism from a fully oxidative to an aerobic ethanol fermentation (23).

A more direct evidence putting in relation metabolism and cell cycle-related properties of yeast populations has been obtained (23). In fact, an increase in the critical protein content required for the entrance in S phase was observed under the same conditions that result in a switch from respirative to respiro-fermentative metabolism and in absence of any change in the specific growth rate of the population.

However, more recent experimental evidences obtained in cytostat suggest that it is not a switch of metabolism from respirative to respiro-fermentative, but rather the simple presence of ethanol that assumes a quorum sensing function leading to an increase of the cell size (36).

Furthermore, a clear correlation among viable/dead cell states, glucose uptake affinity and proliferation states during respiratory and/or respiro-fermentative growth has also been well proved (37, 38). It has been shown that cells grown on low-substrate concentrations showed high affinity, whereas cells grown on high-substrate concentration showed low affinity to 2-NBD-glucose. Although already described at the entire population level, this behavior was never observed at the single cell level. Much more interestingly, this approach allowed the identification of two subpopulations of yeast cells with different 2-NBD-glucose accumulation capacity, which cannot be distinguished according to other parameters like cell size or cell cycle phase.

To conclude, all these phenomena are intriguing and intricate as well and more experimental efforts are required for a deeper understanding.

The Supporting Information Figures S3 and S4 show the average cell size of yeast populations during balanced exponential growth in batch and in glucose-limited chemostat cultures at different specific growth rates.

CELL CYCLE REGULATORY EVENTS UNDERLYING CHANGES IN THE POPULATION STRUCTURE: ANALYSIS OF A NUTRITIONAL SHIFT-UP

Nutrient availability plays a key role in controlling cell cycle progression in budding yeast (39). It is therefore interesting to analyze changes in the population structure during transitory states from one medium to another. For instance, during a nutritional ethanol/glucose shift-up, a rapid decrease in the fraction of budded cells has been observed: cells that were before Start at the moment of the shift delay their entry into S phase, whereas cells that had already executed Start delay their exit from the cell cycle (4). Using an integrated approach that relies on bioinformatics and on systems analysis, modeling and simulation, a blueprint of the yeast cell cycle has been proposed, with the aim to describe regulatory circuits of a given function at a molecular scale (40). The blueprint has been used to develop a simulation program incorporating both the protein level required for budding (P_s) and the length of the budded phase (TB). The simulation analysis allowed to derive the transient evolution of TB and P_s, estimated from the best fitting of the simulated behavior to the experimental data.

MULTISTAGED POPULATION MODEL OF YEAST GROWTH

We also proposed a comprehensive framework to model directly the effects of growth and asymmetric division on the size distribution of growing yeast populations, utilizing the rich structure provided by the two-threshold morphologically structured model described earlier and the mathematical framework of “Population Balance Theory” (PBT) (15, 16).

PBT was developed to describe the dynamic behavior of cell populations by acknowledging their segregated nature (41–43). At the heart of PBT is the quantitative definition of the physiological state of each cell in the population specified by the so-called physiological state vector, a collection of properties that sufficiently describe the state of a cell. The PBT supplies equations describing changes in the joint distribution of the physiological state vector of all cells in the population in relation to growth conditions. Recent developments especially in the field of flow cytometry, allowed the direct measurement of a number of cellular properties and their joint distributions, providing the experimental basis to help formulate and validate PBT models. Multistaged population models considered growth to occur smoothly within each stage of a sequence of contiguous stages, whereas the interstage transitions are controlled by checkpoints that are functions of the physiological state of the cells (44).

Data shown in Figure 3 experimentally validates the model developed. First, starting with an elutriated homogenous population of small unbudded daughter cells, we studied the evolution of different yeast subpopulations as well as the development of the stable heterogeneous structure typical of exponentially growing yeast populations. The comparison between the experimental data (left panels) and the corresponding simulations (right panels) showed a striking similarity in the evolution of the size distributions of the population and also pointed out some novel features that we were unable to predict and explain, such as an intriguing trimodal-shaped distribution. The remarkable agreement between the experimental data and the model gives validity to the coordination mechanism between cellular growth and cell division at the single cell level assumed in the model for the different subpopulations.

Figure 3. Time-course protein and simulated content distributions after release of elutriated daughter unbudded cells in fresh medium. Left: Experimental data. During balanced exponential growth, the population showed a specific growth rate of 0.345 h⁻¹. At time T = 0 h, elutriated unbudded daughter cells were released in fresh medium and followed over time. Samples were collected at 20-min intervals. Histograms represent the protein contents distributions (expressed as FITC fluorescence intensity, Channel Number vs. N of events). The x-axes are on a linear scale. Time point: A, T = 0 h; B, T = 2 h; C, T = 2.64 h; D, T = 5 h. Right: Simulated data. Histograms represent the numerical solutions of the morphologically structured multistaged model. Cell size, expressed as protein contents distributions (Channel Number), linear scale and the number of events. Simulated specific growth rate: 0.527 h⁻¹. Time point: A, T = 0 h; B, T = 0.75 h; C, T = 1.5 h; D, T = 4.5 h. This figure is redrawn from Ref.15.

The novelty of this mathematical framework is that it takes into account the morphological structure of the population (previously described by morphologically structured models) together with its corpuscular nature using the population balance theory and combining them in a multistaged dynamic population model. This approach also considers the temporal structure inherent to growth processes describing growth as a sequence of contiguous stages and accounting for the interstage transitions. This yields a model able to predict the dynamic behavior of a yeast population during steady conditions and easily extendable to perturbed and more complex—not balanced—states. In the case presented here, the physiological state of each cell in the population was fully specified by the cell size (expressed as protein content). However, a number of different scenarios can be envisioned in which additional parameters could be taken into account to define more precisely the physiological state of a cell.

CONCLUDING REMARKS: YEAST POPULATIONS IN THE POSTGENOMIC ERA

Recent years have seen an ever increasing use of high-throughput technologies, collectively referred to as “omics” techniques (45).

In this respect, specific citation should be reserved to the omics analyses of single cohorts of yeast cells (46). The recent years have also seen the rise of systems biology approaches (47, 48) aimed to unraveling in molecular terms networks controlling relevant biological phenomena, such as the cell cycle. Omics approaches typically deal with the population as a whole and thus are likely to miss events restricted to specific subpopulations. On the contrary, systems biology models typically deal with a single, idealized cell. There is thus a need to merge these different layers into an integrated view. The information obtained in this way will then allow to refine molecular models of the cell cycle and to expand their scope by allowing the simulation of a clonal expansion of a single cell into a full-fledged asynchronous yeast population.

Acknowledgements

The authors are grateful to Carla Smeraldi for revising the language.