The cell division cycle orchestrates cellular growth and division. The machinery underpinning the cell division cycle is well characterized, but the actual cue(s) driving the cell division cycle remains unknown. In rapidly growing and dividing yeast cells, this cue has been proposed to be cell size. Presumably, a mechanism communicating cell size acts as gatekeeper for the cell division cycle via the G1 network, which triggers G1 exit only when a critical size has been reached. Here, we evaluate this hypothesis with a minimal core model linking metabolism, growth and the cell division cycle. Using this model, we (a) present support for coordinated regulation of G1/S and G2/M transition in Saccharomyces cerevisiae in response to altered growth conditions, (b) illustrate the intrinsic antagonism between G1 progression and cell size and (c) provide evidence that the coupling of growth and division is sufficient to allow for size homeostasis without directly communicating or measuring cell size. We show that even with a rudimentary version of the G1 network consisting of a single unregulated cyclin, size homeostasis is maintained in populations during autocatalytic growth when the geometric constraint on nutrient supply is considered. Taken together, our results support the notion that cell size is a consequence rather than a regulator of growth and division.
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The cell division cycle coordinates the duplication of all essential cellular components and their separation into two distinct cells. Accurate regulation of cell division is critical and imposed by an intricate regulatory machinery. The eukaryotic cell division cycle is primarily driven by sequential accumulation and destruction of cyclins, which act as activators and targeting subunits for a constitutively present cyclin dependent kinase (CDK) [1, 2]. The cell division cycle's core regulator CDK is itself regulated by external and internal signals, including nutrients, hormones and physical contact with neighbouring cells. This regulation is essential, as inaccurate decisions either to proliferate or to remain quiescent may prove fatal for unicellular and multicellular organisms alike. The mechanistic architecture of the machinery that executes the decision to divide has been studied intensely, giving a good understanding of how duplication and division is orchestrated [3, 4]. In contrast, the actual cue(s) triggering the decision to divide is still unknown. In microorganisms such as yeast where cells have been observed to maintain a certain size, cell size itself, or an equivalent, has been proposed to be the critical cue [5, 6]. However, the mechanistic basis for linking cell cycle events and cell growth is unknown, and hence the nature of the cue(s) driving the progression through the cell division cycle remains a matter of debate .
In the premier eukaryotic model organism Saccharomyces cerevisiae, cells divide asymmetrically . At the START of the cell division cycle, the mother cell initiates DNA replication and spindle pole duplication and polarizes her actin cytoskeleton to grow a bud by polar secretion through a narrow bud neck. The S phase is followed by a G2 phase of comparatively constant duration during which the continued polar growth leads to a volume increase almost exclusively in the bud . Thereafter, the cell enters M phase and initiates rapid mitotic division in which the bud receives a nucleus and separates from the mother. Daughter cells are born at a fraction of the mothers' cell volume and hence need to acquire more mass before they are ready to pass through a subsequent cell division cycle. This is reflected in a prolonged duration of G1, which is recognized as the primary phase for growth regulation of the cell division cycle in S. cerevisiae [5, 9]. This differential G1 duration is instrumental to maintain the size homeostasis within the population and has been attributed to the regulatory network upstream of START (reviewed in ). The earliest undisputed activator of this G1 network is the cyclin Cln3, which binds to and activates the CDK Cdc28. Cdc28-Cln3 phosphorylates Whi5 to relive Whi5's repression of the heterodimeric transcriptional activators SBF and MBF . This triggers expression of a large set of genes which includes the remaining two G1 cyclins CLN1 and CLN2 , resulting in a positive feedback loop that stabilizes once a critical threshold of Cln1/2 and hence Cdc28-Cln1/2/3 activity is reached . High Cdc28-Cln1/2/3 activity directly or indirectly triggers polarization and bud emergence, as well as spindle pole duplication and DNA replication . Mutations in the G1 network frequently lead to alterations in cell size at division , which is why the network has been proposed to impose size regulation, i.e. coordinate cell growth and division. However, the underlying mechanism driving the network is elusive and the role of proposed upstream components such as Whi3 and Ydj1 remains unclear . Moreover, cell size distribution is known to be influenced by growth rate, ploidy and mutations in genes involved in growth, leading to the suggestion that ribosomal biogenesis or translation rate may determine the critical size threshold [14, 16, 17]. While this links the apparent size regulation of the cell division cycle to factors other than size per se, it has also been shown that additional constraints such as active size regulation are needed to provide for size homeostasis during autocatalytic growth .
The combinatorial regulation by distinct internal and external cues and the complex molecular machinery make the connection between cell size and the cell division cycle a prime target for mathematical modelling. The molecular network of the cell division cycle, its robustness and complex properties emerging from the network have been examined in several modelling efforts. These models range from detailed deterministic models of the regulatory circuitry to more abstract stochastic models [19-25]. However, most of these models directly define a critical size, a division ratio or both, making them unsuitable for studying the mechanisms underpinning the apparent size regulation per se. Furthermore, these models abstract growth to a steady exponential increase in cell volume and/or mass, and generally do not include metabolism at all. Conversely, most existing metabolic models are incompatible with cell division cycle modelling. For example, genome-wide metabolic models are too large to allow population level simulation and tools such as flux balance analysis are too abstract, as they do not account for costs and benefits of specific investments such as into ribosomes and metabolic enzymes [26-28]. In contrast, minimalistic models of basic metabolism in self-replicating systems such as those used to study growth rate related metabolic effects could potentially be integrated with a reduced cell division cycle model . However, deterministic single cell models are not optimal for analysis of population properties such as size distributions. Characterization of populations has been addressed by population balance equations (PBEs) [30, 31] and cell ensemble modelling (reviewed in [32, 33]). PBEs derive population behaviour from analysis of single cell behaviour over time but require that the intracellular state can be described with a small set of variables. In contrast, ensemble modelling includes many individual cells that differ slightly, e.g. due to a stochastic component in the model. This is computationally more costly but allows a more detailed dynamic description, and hence ensemble modelling provides a framework in which to create an integrative model of metabolism and the cell division cycle, to allow the study of population effects. However, this has not yet been done and the current understanding of the mechanistic basis for size regulation builds on a critical size defined by the extensive G1 network  or the shuttling of critical cell cycle regulators between the growing cytoplasm and the slower growing nucleus .
Here, we re-examine the coupling of growth and division with a population level analysis of a model integrating metabolism and cell division. We designed this minimal core model to string together elementary concepts concerning cell cycle regulation and growth, including that cell cycle entry is limited by translation of a critical cell cycle regulator and that cell growth is autocatalytic and limited only by the nutrients in the growth medium . This elementary design of the model also allows re-evaluation of the hypothesis that the G1 regulatory network is essential for cell size regulation. Surprisingly, our results indicate that autocatalytic growth is compatible with cell size homeostasis on the population level when considering geometric constraints on nutrient supply, and that it does not require more than a rudimentary version of the G1 regulatory network. While the full network has a strong effect on the actual size distribution in the cell population, size homeostasis itself can emerge from coupling between growth and division. We observe cell size homeostasis on the population level even when cells in the model lack any means to determine their sizes. Instead, size homeostasis emerges from growth due to the surface-area-to-volume constraint on metabolism. It is noteworthy that a very coarse grained model which only allows for the allocation of resources in two biomass pools – surface area and metabolic capacity – suffices for population level cell size regulation. Importantly, the model has only eight parameters, lacks regulatory circuitry, is stable over a wide range of growth rates and is robust against perturbations, after which the population quickly converges back to a stable attractor characteristic for its growth rate. However, the model fails to validate the hypothesis that cell cycle regulation occurs exclusively at the G1/S boundary. Instead, it requires adaptation of both G1 and G2 duration to account for the empirically observed response to altered growth conditions. Despite this, it accurately describes a number of observations on the cellular level, such as (a) the difference in G1 duration between mother and daughter cells, (b) the cell size increase with genealogical age in the mother line and (c) the increase in average cell size with increasing growth rate. Taken together, our results show that cell size homeostasis can emerge from an elementary cellular design with fundamental physiological constraints and offer new insights into the complex coordination of growth and division.
A model linking growth and division
We employed a core model to assess the minimal requirement for size regulation in living cells (Fig. 1). This core model combines cell growth with metabolic capacity and the cell division cycle. It distinguishes the isotropic and apical growth phases but approximates the transient M phase to an instantaneous event. Furthermore, it excludes DNA replication and hence considers a joint S/G2 phase. The START transition from G1 to S/G2 is approximated as a threshold level of Cln1/2 and hence Cdc28-Cln1/2 kinase activity, reflecting localized activity on SBF/MBF target promoters (which remain constant outside S-phase). Biomass production is described as a function of present biomass, i.e. a self-replicating system similar to previous work . The model links metabolism to the decision to divide via accumulation of a regulatory protein: the G1 cyclin Cln1/2. This accumulation is determined solely by the biosynthetic capacity of the cell and there is neither regulatory feedback nor active size sensing. The model is based on two widely accepted high level assumptions: (a) that to grow, cells need to (i) take up nutrients and (ii) incorporate nutrients in biomass, and (b) that metabolic efficiency decreases with decreasing surface-area-to-volume ratio. Hence, the growth rate is proportional to the cell's metabolic capacity multiplied by the uptake capacity (surface area) divided by the supply requirements (volume).
The model accounts for this proportionality with two qualitatively different forms of biomass (and independent model variables): structural biomass (BA) and internal biomass (BR). The structural biomass is proportional to the area (A); it includes cell wall and cell membrane and it determines the cellular uptake capacity for nutrients. The internal biomass corresponds to the cellular metabolic capacity (R, comprising ribosomes and metabolic enzymes) and determines the ability to incorporate intracellular nutrients into new biomass. The cell allocates resources between structural or internal biomass depending on the cell cycle phase (Fig. 1B). Furthermore, structural biomass is allocated to either the mother (BAm) or the daughter (BAd) part of the cell and stays with that part, while the internal biomass (BR) is assigned to mother and daughter proportionally to volume at the time of division. The biomass allocation changes between the cell cycle phases G1 and S/G2. In G1 the unbudded mother grows and the allocation is biased towards internal biomass. In S/G2, the cell switches to polar secretion and growth of the bud, a relatively small allocation to internal biomass, and no growth of the mother. The phases are interspaced by START – at which cells decide to divide – and an instantaneous M phase after a set time at which cells divide. We only consider regulation of the START transition, which is triggered by Cln1/2 accumulation in the nucleus. Importantly, CLN1/2 transcription is entirely stochastic to reflect a rudimentary version of the upstream regulatory network. We do not implement the strong positive feedback loop explicitly, but implicitly as it stabilizes the START transition. This implementation isolates the effect of the metabolic gating from any influence from the G1 network, which allows us to study their contribution separately. In summary, the model comprises seven species (Table 1), five ordinary differential equations, a function for stochastic transcription, five algebraic equations (all in Table 2) and eight parameters (Table 3) and rests on a set of explicit assumptions (Table 4).
Table 1. List of species, summarizing the model species and their initial values
Cyclin Cln1/2 precursor (mRNA)
Cdc28 is always available and therefore not explicitly modelled but implied. Cln1/2 represents the active kinase complex
Structural biomass (mother)
Structural biomass (daughter)
Table 2. List of model equations. The model consists of five differential equations (1)–(5), one stochastic function (8) and five algebraic equations (9)–(13)
Table 3. List of parameters. The model has eight adjustable parameters, some of which change between the G1 and S/G2 phases
Degradation rate mCLN1/2
Degradation rate Cln1/2
Production rate Cln1/2
Fraction of biomass allocation to internal biomass
Fraction of biomass allocation to structural biomass, mother
Fraction of biomass allocation to structural biomass, daughter
Conversion factor structural biomass to membrane
Table 4. List of assumptions, summarizing the assumptions made in the model creation
Nutrient supply is defined by uptake, which is proportional to cell area
Transcription is stochastic
Nutrient incorporation into biomass relies on metabolic capacity (R)
Thus, production reactions are implemented with second-order kinetics, dependent on a precursor and the internal biomass (BR, eqns (2)-(5))
Efficiency of nutrient incorporation decreases with volume
Cell area is sum of mother and daughter area
Cell volume is sum of mother and daughter volume
Mothers and daughters are approximated as separate spheres; thus V ∝ A3/2
Cells may allocate their resources according to cell cycle stage to either structural or internal biomass
Structural biomass can go into area mother (Am) and area daughter (Ad) separately
Increase of metabolic capacity is strong in G1 – less during S/G2
During G1 there is no bud growth
During S/G2 there is only bud growth
Threshold for nuclear kinase activity (zero-order sensitivity)
There is targeted Cln1/2 destruction/nuclear exclusion during S/G2
S/G2 duration is constant
Mitotic cell division event is instantaneous
Biomass precursors are always available (constant)
Old cells go quiet after ~ 24 divisions
The model suffices to reproduce characteristic aspects of the cell cycle
The core model was calibrated manually using high resolution literature data on cell growth that distinguish between mother and bud growth . Figure 2 shows the fit as well as the trajectories of each of the five time-dependent variables in a single cell over two cell division cycles, following only the mother in the second cycle. In this particular case, the newborn daughter spends a long time (~ 100 min) in her first G1 phase before increasing translation of the stochastically expressed CLN1/2 leads to sufficient accumulation of Cln1/2 proteins to trigger START. In the S/G2 phase, growth is redirected to structural growth in the bud leading to an accelerated area and volume increase and a biphasic growth pattern within one cell division cycle. The phase-specific alteration of the growth rate is in accordance with experimental observations [15, 37, 38]. Additionally, CLN1/2 transcription ceases and existing Cln1/2 proteins are actively degraded. After a set time delay, the cell passes through mitosis and volume (V), area (A) and metabolic capacity (R) split between the mother and the daughter (resulting in a drop as only the mother line is followed). The cell enters her second G1 larger and with higher metabolic capacity, resulting in a faster accumulation of the G1 cyclins and hence a much shorter G1 phase – again in accordance with empirical data [37, 39].
Size regulation on the single cell level is not needed for population size regulation
We proceeded to evaluate the population level predictions of the core model. Virtual cultures were seeded with 10 identical cells and simulated over time to generate a complete pedigree stemming from these cells using a tailor-made multiscale simulation environment (MSE) (see Materials and methods). The individual cell lines were followed over several generations on both the mother and daughter lines (Fig. 3A). The individual cell sizes increase until division, split in two to follow the mothers and daughters individually, and resume their increase until the next cell division. For an individual cell this leads to an increase in cellular size over generations, which is consistent with in vivo observations (Fig. 3B) . However, the average and variance in cell size quickly stabilizes despite the fact that the size of individual cells grows strictly monotonously (Fig. 3A). The simulation predicts that the older (and larger) mothers will progress faster through G1. This is consistent with empirical data, although the in vivo decrease in G1 duration stagnates after only a few generations (Fig. 3C) . Finally, the model accurately predicts that older mothers will retain a larger fraction of the total volume, although again the in vivo effect saturates after the first few generations (Fig. 3D) . Note that none of the in vivo data presented in Fig. 3(B)–(D) was used for model fitting. Taken together, the core model accurately predicts several empirically observed aspects of G1 regulation and suffices to provide population level size homeostasis without any size regulation on the individual cell level.
The apparent size regulation emerges due to the surface-area-to-volume scaling of metabolism
To resolve the apparent contradiction between the intrinsic size homeostasis in our model and theoretical proof that auto catalytically growing cells require active size regulation , we performed a deeper analysis of the model to identify the underlying mechanism. First, we ruled out that the implementation of Cln1/2 as amount rather than concentration functions as an implicit size sensing mechanism. As can be seen in Fig. 4(A), a concentration based implementation accounting for the nuclear growth during G1 retains intrinsic size regulation (for calculations of nuclear volumes, see Doc. S1). Second, we proceeded to evaluate the impact of the geometric constraints on metabolism. The effect of the surface-area-to-volume ratio on metabolic rate is well established and has been observed over a wide range of organisms [41, 42], but it is not included in previous models of the cell division cycle nor in the theoretical proof mentioned above [19-25]. We lifted this constraint by removing the factor A/V from eqns (2) to (5) (Table 2), which places the model in a very similar regime as in the proof. As seen in Fig. 4(B), cell division can no longer balance the growth and the average cell size is no longer stable, despite retaining the threshold level of a critical cell cycle regulator produced in a translation rate (metabolism or size) dependent manner. Further examination of the core model revealed that the R/V ratio is stable on the population level, which would mean that the growth is proportional to A (i.e. V2/3; Fig. 4C). As growth is split between BA (proportional to A) and BR (proportional to A3/2), the metabolic rate increases with mass to the power of x, where 2/3 < x <1. As shown in Fig. 4(D), x =0.8 in this model, which is strikingly close to the coefficient x =3/4, or Kleiber's law . Thus, if nutrient availability is the limiting factor for single budding yeast cell growth, they neither grow strictly linearly (x =0) nor fully exponentially (x =1) but in between (x =0.8) due to an autocatalytic dependence on their current metabolic capacity, scaled by their dynamic ability to take up nutrients and how those are diluted (current surface-area-to-volume ratio). Hence, the model shows that surface-area-to-volume scaling of metabolism suffices to place budding yeast in a growth regime where size regulation occurs intrinsically, and the relationship between growth and size within the model is consistent with that observed between organisms over 18 orders of magnitude in body size .
The core model qualitatively predicts the effect of altered growth conditions without nutrient signalling, and suggests that G1 and G2 regulations occur in parallel
A realistic growth model should also be able to capture the change in growth rate and cell size distribution associated with different growth media. It is well known that an increased growth rate gives larger cells in vivo [43, 44]. While the core model is too abstract to account for the exact media composition, it includes a ‘nutrient quality’ term (kgrowth) that allows for different growth rates. We simulated four different nutritional conditions with mass doubling times ranging from 2 to 4 h and compared the size distribution (Fig. 5A,B). Note that the model predicts both a larger size and decreased size variability in the faster growing cultures. The faster growth rate also leads to a shorter G1 duration (Fig. 5C). However, at very fast growth rates, the core model loses the asymmetry between mothers and daughters. It seems that the assumption of a constant S/G2 duration over different growth rates is too simplistic, for simulations show that the mother/daughter asymmetry can be partly restored by altering the S/G2 duration (Fig. 5D). This observation is supported by empirical data showing in vivo differences in G2 duration between growth conditions . Taken together, the core model qualitatively captures the effect of altered nutrient conditions, and it appears that regulation of the S/G2 phase duration is required to fully account for the in vivo adaptation to altered growth conditions.
The average cell size converges to a point attractor, which is characteristic for a given growth rate and robust against perturbations in initial conditions and noisy S/G2 duration
The results above show that an initially synchronous population rapidly desynchronizes and the population average and variance converge to a growth-rate-specific level. Furthermore, shifting between qualitatively different media resets the size that is specific for the growth rate (Figs 6A and S2), as experimentally observed . These results are consistent with average cell size converging to a stable point attractor. To explore the nature of this attractor, we proceeded to test the impact of initial conditions. Unlike growth media composition, initial conditions should not affect the final size distribution. As shown in Fig. 6B, the point attractor is stable against perturbation in initial levels of structural (Fig. 6B, upper panel) and internal (Fig. 6B, middle panel) biomass by at least two orders of magnitude. In all cases, the proliferating population converges to the size average determined by the growth rate within a few generations. Furthermore, we examined to what extent this size regulation could be tied to the set S/G2 time. To simulate a noisy S/G2 duration, we sampled from a log normal distribution with increasing variance around a mean of 90 min (Fig. 6B, lower panel). The resulting simulation shows that, while the variability increases, the attractor remains stable although the point of attraction changes somewhat in every simulation. Intriguingly, we see that increasing variability in the S/G2 duration leads to decreased average cell size. This may be related to an altered population distribution, as the simulated culture's population structure deviates from the theoretical population structure with respect to percentage of cells in different genealogical ages (Fig. S3), but also suggests that the position of the attractor is sensitive to S/G2 duration. Finally, we assessed to what extent the model behaviour is determined by the choice of parameters. While individual parameters clearly influence the position and the shape of the final size distribution, the point attractor itself is stable against parameter variations over at least one order of magnitude in both directions (Figs 6C and S4). Overall, the model validation shows that the average size converges to a stable point attractor that is robust against altered parameters, growth conditions, perturbations in initial conditions and noise in S/G2 duration.
Experimental model validation fails to confirm the assumption of constant G2 duration
To validate the model predictions of larger size and decreased size variability in faster growing cultures, we empirically measured the size distribution in cultures growing at different growth rates (Fig. 7A). The in vivo distributions show a clear difference in average size between cells growing at high and low growth rates (P =0.006; Student's t test), but no significant difference between the size variability, despite a tendency (P =0.08; Student's t test). Hence, these results confirm the increase in size but fail to confirm the decrease in size variability with increasing growth rate, which was predicted by the simulations (Fig. 7B). The simulation results above repeatedly suggested that it was necessary to adapt G2 duration to explain the differences between different growth rates (Figs 5C,D and 6D). Hence, we performed simulations over a range of growth rates and S/G2 durations (Fig. 7C). The results clearly show that for each given nutritional condition the average size and size variability depend on the S/G2 duration. Moreover, the empirical pattern of increasing size but relatively constant size variability can be conserved if higher growth rates are accompanied by decreased S/G2 duration (Fig. 7D), which has been observed also in vivo . Hence, the cell division cycle progression must be regulated by growth conditions at both the G1/S and G2/M transitions in S. cerevisiae.
As a contribution to the long-standing discussion about how cells can sense their size necessary for cell division, we show that geometric scaling of metabolism suffices to place budding yeast in a growth regime where size homeostasis is an intrinsic property. Under all conditions tested, the average population size converges on an apparent point attractor. These results are fully compatible with the data on the G1 network, which has a strong influence on cell size (i.e. the position of the attractor), but also show that at least in theory the attractor emerges already at a more fundamental level. This result is in agreement with the view that cell division must predate the evolution of the G1 network. The minimal core model lacks mechanistic size regulation but suffices to reproduce the cell growth and division pattern on the single cell level as well as on the population level over a range of growth conditions. The core model abstracts the cell division cycle to two phases separated by two events, cellular composition to two qualitatively distinct types of biomass, growth to uptake and metabolism, and it links growth to cell division by stochastic transcription and translation of a single regulatory protein. We chose this level of abstraction to include the basic relationship of growth and division, while keeping the computational cost at a minimum. All of these abstractions except constant S/G2 duration over different growth rates survived the model validation. The assumptions of the metabolic model (Table 4) are consistent with a self-replicating system , although with ribosomes and metabolic enzymes merged in metabolic capacity (R). While there is convincing evidence that the allocation between metabolic enzymes and ribosomes alters with growth rate , we found the distinction between the two superfluous for this model. The difference in allocation over the cell division cycle builds on empirical observations and the allocation parameters have been adjusted to fit experimental data . While the zero allocation to the mother in S/G2 is likely to be an approximation, there is no significant size difference between mother cells with large and small buds, strongly arguing that mother growth during S/G2 is insignificant . Similarly, the allocation of 70–80% to R may be an underestimation at higher growth rates, as up to 80% of the transcriptional machinery in S. cerevisiae is dedicated to synthesis of ribosomal components alone . Finally, stochastic transcription of CLN1/2 leads to a faster loss of synchrony, but does not alter the behaviour of the cell cultures (Fig. S5). Also, implementation of nuclear growth and Cln1/2 dilution during G1 also yields population level size homeostasis (Fig. 4A). To further validate the model, we compared it with a list of criteria summarizing the cell cycle literature (Table 5). Despite its simplicity, the growth model realistically describes growth on the single cell level, both over time within a cell division cycle and over generations (Figs 2 and 3B, respectively). Furthermore, the model accurately predicted key properties on the population level, including convergence on a stable average size despite constant growth of single cells (Fig. 3) and the effects of increased size of cells that grow on more favourable nutrient sources (Fig. 5A,B).
Table 5. List of yeast cell division cycle and growth characteristics. We compared the model to a number of qualitative statements about growth and the cell division cycle related characteristics in budding yeast and indicated which statements our minimal model can account for. We found these facts useful and they may serve as a seed for a more comprehensive summary and for further modelling endeavours
While a model on this level of abstraction is clearly insufficient for detailed molecular conclusions, it allows us to re-evaluate a number of conclusions. First, the assumption that G2 duration is constant over different growth rates is an approximation that needs reconsideration. Without adaptation of the S/G2 duration, our core model predicts that the mother/daughter asymmetry diminishes with increasing growth rates (Fig. 5D). Part of this might be explained by the simplicity of the core model, as it lacks the daughter-specific transcription factors Ace2 and Ash1 which suppress CLN3 transcription and thus provoke a daughter-specific delay in G1 even at high growth rates . However, a substantial part of the size variability is due to the divergence between mother and daughter lines (Fig. 7B,D), and we would expect an active delay in G1 of daughters to diminish this difference. Instead, our results point towards an active adaptation of the G2 duration (Fig. 7C). This regulation of the G2/M transition is likely to integrate metabolic capacity with additional cues, such as cell morphology (as proposed by Lew and Reed ) or even bud size (as proposed by Harvey and Kellogg ). Consistently, experimental data show that G2 length indeed varies in different media , which leads us to conclude that both the G1/S and G2/M transitions are actively regulated in normally growing yeast cells.
Second, we can address the apparent paradox that CLN3 overexpressing cells are smaller but pass START earlier, in contrast to wild type cells where improved nutrient conditions lead to both increased cell size and earlier START transition [34, 51, 52]. While this remains an apparent paradox when the two growth parameters (size and doubling time) are independent, it may be resolved when considering that the growth parameters are intrinsically antagonistic. In other words, given a set nutrient availability, a decreased time in G1 will always lead to a decreased cell size unless the birth size of the daughters is large enough to compensate for the decreased growth in each subsequent G1 phase (as mother growth is negligible in S/G2 even in vivo ). In contrast to empirical data , the core model predicts a decrease not only in cell size but also in the generation time (Fig. S6). This discrepancy again points towards a dynamics regulation of the G2/M transition, which would need to be delayed to compensate for the small size of the cells passing START. However, the decreased size in Cln3 overexpressing cells is captured also in our core model, reinforcing the conception that the metabolic power rather than size triggers START and that cells grow larger on richer media because the increase in growth supersedes the decrease in G1 duration .
Third and most interestingly, we find that size regulation as observed in vivo can be explained without any ability to sense or regulate size on the single cell level. The results here show that the size homeostasis can emerge from metabolic gating when the dynamic surface-area-to-volume effect on metabolic efficiency is taken into account. The metabolic scaling with size is well characterized between organisms, although the power (x) to which metabolic rate depends on mass remains an issue of debate [41, 42]. The simulation results here suggest that this scaling may be important also on a cellular level, and we note that the simulation result x =0.8 falls very close to the favoured x =0.75. These results are therefore in line with existing proofs that autocatalytic growth (only) requires active size sensing when x ≥1. Furthermore, these observations provide a possible new perspective on the discussion on whether single cell growth is exponential or not [7, 38, 53-55], since metabolic scaling may suffice to place exponentially growing cells in an in vivo regime between linear (x =0) and full exponential (x =1) growth.
Taken together, these results favour the hypothesis that the apparent size regulation of the cell division cycle emerges as a result of the scaled metabolic capacity of a cell while still allowing for the strong modulation of cell size by the G1 network [14-16]. The concept that metabolic power gates the cell division cycle would explain observations such as partial cell division cycle synchronization in cell cultures with strong metabolic oscillations . Consistently, biosynthetic capacity (R) is a positive regulator of growth and division (Fig. 6C). Gating through metabolic power would ensure that cells possess the resources required to successfully complete S phase before they pass START.
The question remains why such a sophisticated regulatory network upstream of START has evolved, if the rudimentary version of the G1 network we employ here suffices for size control. While this could simply be a noise reduction mechanism as has been proposed earlier , our results suggest that such regulation would be more efficient at the G2/M boundary (Fig. 7). Moreover, our results indicate that the noise stabilizes together with the mean after relatively few generations, and that the diverging extremes are too few to have significant impact on the population. A more appealing hypothesis is that the network evolved to allow the cell cycle to be regulated by additional factors beyond nutrients, most importantly pheromones and environmental perturbations such as dehydration. The G1 arrest is critical to synchronize mating cells and to allow the time necessary for mating, which occurs only during G1 . Likewise, loss of turgor is devastating for the cell's ability to grow and expand . Both these signals have dedicated MAP kinase pathways acting on dedicated CDK inhibitors; the pheromone response pathway arrests the cell cycle in G1 via Far1 , while the high osmolarity glycerol pathway arrests the cell cycle in G1 via Sic1, in addition to transcriptional inhibition of CLN2 and CLB5 [61, 62]. While mutations in such a control network would be expected to show premature or delayed START transition, which could lead to large effects on the average cell size, the main purpose of the G1 regulatory network may not be to advance the cell division cycle but to allow for stable cell cycle arrest in G1 and to ensure that the arrest can be lifted when conditions are suitable.
In conclusion, we have developed a core model with a minimal regulatory network to study size regulation in S. cerevisiae. The core model is integrated in an MSE, providing a framework to further study the function of the G1 regulatory network and other cell division cycle questions in a population oriented manner. Using this framework, we present support for coordinated regulation of G1/S and G2/M transition in S. cerevisiae in response to altered growth conditions. Furthermore, we illustrate the intrinsic antagonism between G1 progression and cell size. Finally, we provide evidence for the dispensability of the G1 network for cell size regulation and show that size homeostasis can – in theory – emerge when cell division is coupled to metabolism via passive accumulation of a single regulatory protein and biophysical constraints are taken into account. This allows us to support the hypothesis that size homeostasis in a budding yeast population might not require an active sizer on the single cell level, and advocates a more careful interpretation of the G1 network's function beyond modulating cell size.
Materials and methods
The core model has variable G1 duration, constant S/G2 duration and an instantaneous mitotic event. It consists of ordinary differential equations (ODEs) joined with a stochastic function and is embedded into an MSE. The MSE is implemented using the programming language python . The model contains a total of five ODEs (Fig. 1, Table 2) and a total of eight parameters that may assume different values in G1 and S/G2 phase (Table 3). The system of ODEs was solved using the lsoda algorithm from the FORTRAN library odepack . The function odeint from SciPy's integrate package is a wrapper around this algorithm. The following parameters were used for solving the system: relative and absolute tolerance rtol = 10−5 and atol = 10−6 respectively; maximal function evaluations mxstep = 104; for all others we used default values. In addition to the ODE system, we implemented a stochastic function for species 1 (mCLN1/2) such that the model can account for stochastic transcription (transcriptional bursts, as has been shown for single cells ) that occurs with a certain probability (40% min−1, unless specified). Other precursor molecules are maintained constant (eqns (6), (7)), but could potentially change.
The model assumes that a cell produces two types of biomass, structural and internal biomass. The structural biomass (BAm and BAd) is proportional to the area (A) of a cell (eqns (9), (10)) and constitutes cell components that define the outer area of the cell, i.e. the membrane and the cell wall. The total area of a cell is assumed to be the sum of the area of the mother (Am) and the area of the bud (Ad), if present (eqn (11)). The total area determines the cellular uptake capacity for nutrients and therefore positive influence reactions. The internal biomass (BR) is the soluble biomass that is proportional to cellular metabolic capacity (R). It determines the ability to incorporate intracellular nutrients into new biomass. The total biomass production is dependent on the internal biomass. Therefore, all production reactions (e.g. Cln1/2 translation) are implemented with second-order kinetics, dependent on some precursor and the internal biomass (eqns (2)-(5)). All entities are computed as amounts (not concentrations) and thus parameters are scaled with the volume V of the cell to account for volume changes in the calculations. This phenomenon does not affect first-order reactions. This formulation conveniently results in incorporation of the positive effect of A and negative effect of V in the equations as the scaling factor A/V. Cells are approximated as spheres, which is why we calculated the volume of a cell according to V = A3/2 (in arbitrary units). Consistent with the area calculations, we assumed that the total volume of the cell is the sum of the volume of the mother cell (Vm) and the volume of the daughter cell (Vd). We neglected the error that is imposed by volume calculations of two intersecting spheres in the case of a budding cell. Note that the metabolic capacity (R), the surface area (A) and the volume (V) are variables that change as the cell grows.
Furthermore, cells may allocate their resources according to cell cycle phase and distribute them to either structural or internal biomass. As the total amount incorporated per time is defined by the metabolic capacity, increase in one leads to decrease of the others. Only the mother grows during the G1 phase, hence kAd = 0, and the majority of metabolic capacity flows into the internal biomass kR = 4.75 versus kAm = 1, reflecting the high investment in ribosomal RNA and proteins [47, 66]. In accordance with empirical data, this changes for the S/G2 phase allocation . Here, much less energy goes into the internal biomass (kR = 2) and a considerable amount is allocated to bud growth (kAm = 0, kAd = 1). We reasoned that the nutritional setup that we exposed our cells to (kgrowth) should have a strong effect on the long-term growth but not directly on Cln1/2 production, to avoid direct nutrient regulation of the cell cycle; hence the growth parameter occurs exclusively in eqns (3)-(5). Finally, START occurs when the cells accumulate 150 arbitrary units of Cln1/2. This reflects localized activity on several distinct targets which require multiple phosphorylations (in accordance with ). The limitation of Cln1/2 and hence active kinase and the excess of substrate sets the stage for zero-order ultrasensitivity and hence rapid transition that is abstracted as a threshold level . The threshold is given in amount to reflect that the active kinase is targeted to specific subcompartments that expand slower than the total cell volume (or not at all) during G1 [34, 68]. Once the threshold is reached, the phase transition is considered irreversible despite the lack of positive feedback due to the inhibition of Sic1 and release of Cdc28-Clb5/6.
The multiscale simulation environment
In order to simulate complex cell cultures, we integrated the model into an MSE, where every cell is implemented as an object that contains predefined attributes. This enabled us to record the total duration of G1 phase (from birth until Cln1/2 rises above the threshold), the cell size (over time, at birth and division), division ratios between mother and daughter and all of the components of the wiring diagram in Fig. 1 over time for every individual cell. Thus, a simulation generates a complete pedigree of growing and dividing mother and daughter cells . At initialization of the simulation, an adjustable but fixed number (10 000) of potential cells is set up in a matrix (100 × 100) and a number of in silico cells is initialized (10 cells in all simulations). The initial conditions of these cells (Table 1) are chosen such that they resemble newborn cells that emerge during the simulations. The array represents a virtual culture dish with the initial number of cells at time 0. With a resolution of 1 min time steps, the cells now grow and divide, as the algorithm computes the state of each cell for each time step and stores the information.
The core model presented has a low computational cost allowing simulation of relatively large cell cultures, which will facilitate further analysis of the cell division cycle and the regulatory networks surrounding it. The MSE that we present here runs independently of the type of single cell model that is used for the ensemble modelling. In this study, we use a minimal ODE model, which could easily be extended in the future to contain other, more detailed cell cycle regulatory circuits or metabolic components. Also, other already published models could potentially be plugged into the MSE to broaden their scope from single cell to population behaviour. The core model and the MSE simulation platform are provided as a tool for the cell division cycle modelling community.
Parameter fitting and validation
For the parameter adjustment of the model we used published single cell experimental data . During the fitting procedure values for the parameters kR, kAm and kAd were tuned such that in the simulations the qualitative behaviour of the volume increase of a single cell resembles the data (Fig. 2). In addition, kp and kgrowth values were chosen such that the mean G1 duration of daughter cells is approximately 100 min in our reference conditions. On a different note, we chose for kd2 to display a 100-fold difference between the two different phases, to assure complete Cln1/2 protein destruction/inactivation for the subsequent cell division cycle of mother and daughter. However, this is only a precautionary measure since changing kd2 in S/G2 phase has little or no effect on the simulations (Fig. S4). Finally, the constant S/G2 phase is set to 90 min by default. Thus, our reference in vitro condition resembles an in vivo condition in which newborn daughters take 190 min to reach mitosis, and the culture has a generation time of 163 min. The parameterized model was validated by comparison with independent data that had not been used in the fitting procedure, including data generated for this specific purpose (see below for experimental procedure). The results of this validation are displayed in Figs 3-6. In summary, the model was fitted to single cell data  and validated on a population level with population data from several sources as indicated in these figures.
Experimental size distribution measurements
The S. cerevisiae strain BY4741 (MATa, ura3Δ0, leu2Δ0, met15Δ0, his3Δ1) was used to analyse the influence of different carbon sources on cell volume and doubling time. Cells were grown in synthetic YNB media with 2% (w/v) glucose or 2% (v/v) ethanol as indicated. The YNB media contained 0.17% yeast nitrogen base without amino acids, 5% ammonium sulfate, nucleotide bases (adenine, tyrosine and uracil, 55 mg·L−1 each) and amino acids (l-arginine 20 mg·L−1, l-histidine 10 mg·L−1, l-isoleucine 60 mg·L−1, l-leucine 60 mg·L−1, l-lysine 40 mg·L−1, l-methionine 10 mg·L−1, l-phenylalanine 60 mg·L−1, l-treonin 50 mg·L−1 and l-tryptophan 40 mg·L−1). Yeast cells were precultured in medium containing indicated carbon sources at 30 °C in a shaking incubator, transferred to fresh media and followed from early log to stationary phase. Samples were taken for D600 (Shimadzu UV 1202 photometer) and cell size distribution measurements (Cell Counter CASY1 TTC; Schärfe System GmbH, Reutlingen, Germany). The cell counter was calibrated to measure particle sizes up to 10 μm in diameter and equipped with a 60 μm capillary. Cell culture samples were diluted in CASYton (Innovatis AG, Reutlingen, Germany; dilution factor ranging from 100 to 1600) and size distribution measurements were performed in triplicate. Counts per sample ranged from 4204 to 43 616. Three biological replicates with two technical replicates each were analysed for each condition. Data (cell counts per diameter) were recalculated to volume assuming a perfect sphere. Particles below 8 fL were not considered as cells. Cell volumes were log transformed and new histograms were computed (Fig. 7A). Mean and standard deviation were calculated on the log transformed values.
This work was supported by grants from the German Research Foundation (IRTG 1360 ‘Genomics and Systems Biology of Molecular Networks’ and CRC 618 ‘Theoretical Biology’) and a grant from the European Commission 7th Framework Programme (UNICELLSYS, Contract No. 201142), all to EK.
EK conceived and initiated the study. TWS and MK designed the model with advice from EK. TWS implemented the model and performed simulations. CM and GS performed the experimental analysis. TWS, MK and EK analysed the data, planned the work and wrote the paper.