The eukaryotic cell cycle is the repeated sequence of events that enable the division of a cell into two daughter cells. It is divided into four phases: G1, S, G2, and M. Passage through the cell cycle is strictly regulated by a molecular interaction network, which involves the periodic synthesis and destruction of cyclins that bind and activate cyclin-dependent kinases that are present in nonlimiting amounts. Cyclin-dependent kinase inhibitors contribute to cell cycle control. Budding yeast is an established model organism for cell cycle studies, and several mathematical models have been proposed for its cell cycle. An area of major relevance in cell cycle control is the G1 to S transition. In any given growth condition, it is characterized by the requirement of a specific, critical cell size, PS, to enter S phase. The molecular basis of this control is still under discussion. The authors report a mathematical model of the G1 to S network that newly takes into account nucleo/cytoplasmic localization, the role of the cyclin-dependent kinase Sic1 in facilitating nuclear import of its cognate Cdk1-Clb5, Whi5 control, and carbon source regulation of Sic1 and Sic1-containing complexes. The model was implemented by a set of ordinary differential equations that describe the temporal change of the concentration of the involved proteins and protein complexes. The model was tested by simulation in several genetic and nutritional setups and was found to be neatly consistent with experimental data. To estimate PS, the authors developed a hybrid model including a probabilistic component for firing of DNA replication origins. Sensitivity analysis of PS provides a novel relevant conclusion: PS is an emergent property of the G1 to S network that strongly depends on growth rate.
During the life cycle of eukaryotic cells, DNA replication is restricted to a specific time window, called the S phase. Several control mechanisms ensure that each DNA sequence is replicated once, and only once, in the period from one cell division to the next. Following S phase, replicated chromosomes separate during mitosis (M phase) and segregate in two nuclei that eventually will be endowed to each newborn daughter cell at cell division. Two gap phases, called G1 and G2, separate cell birth from S phase and S phase from M phase, respectively. Typical pie chart representation of the cell cycle (Figure 1A) stresses the discontinuous events that have to take place only once per cell cycle (i. e. , S and M phases), but fails to show that proliferating somatic cells are continuously increasing in their mass throughout the cell cycle (Figure 1B). As pointed out as early as 1971 by Mitchinson [1], the “continuous events of the growth cycle” (i. e. , increase in cell mass) and the “discontinuous events of the DNA division cycle” (i. e. , DNA replication, mitosis, and cell division) need to be tightly coordinated in order to maintain cell size homeostasis. It has been proposed that coordination of mass accumulation with cell cycle progression relies on a sizer mechanism, so that DNA replication and/or cell division start only when cells have reached a critical cell size (see [2] for a review). In this way, tiny newborn cells will have to grow more than mother cells before being able to overcome the cell size checkpoint. Conversely, a larger cell will overcome the cell size checkpoint earlier than the “normal, average” cell. As a result, both small and large cells will stabilize cell size to the “normal, average” value (Figure 1C). Although evidence for the occurrence of a sizer mechanism (s) has been collected in different eukaryotes from unicellular microorganisms to flies to mammalian cells [3,4], cell size control has been best studied and come to be generally accepted in unicellular fungi, such as the distantly related fission and budding yeasts. In the fission yeast Schizosaccharomyces pombe, a size control is operative over mitosis in wild-type cells, while a cryptic control over S phase is revealed in wee mutants [5] or under conditions of nutritional limitation [6]. In the budding yeast S. cerevisiae, the major size control is operative over DNA replication and bud emergence (G1/S transition, often referred to as “Start” in this organism). We refer to the critical cell size required for the G1/S transition in budding yeast as PS, which represents the protein content per cell at the onset of DNA replication. Ploidy, nutrients, and growth rate modulate PS that is smaller in slow-growing cells than in fast-growing cells [2,7]. Despite extensive efforts, the molecular basis of the cell sizer mechanism controlling the G1/S transition in budding yeast has long remained elusive. While gene dosage data strongly argued for a role for Cln3 in the mechanism [8,9], physiological data gave apparently contradictory results [10–12]. We propose that a growth-dependent threshold lies at the core of the cell sizer mechanism. In its essence, a threshold entails the interplay of two molecules, one acting as an activator and a second acting as an inhibitor [13,14]. When the number of molecules of activator is low and that of inhibitor is high, the activator is below the threshold; if the activator increases with growth, the threshold is overcome when enough molecules of activator are made to exceed the inhibitor. This mechanism is schematically reported in Figure 1D under the idealized assumption that the equilibrium is totally shifted toward the formation of the complex. We propose that the activator and inhibitor molecules are, respectively, Cln3 and Far1 [14,15]. Understanding of the molecular basis of the critical cell size–controlling onset of S phase (PS) would be of great relevance for cell biology. To this end, we present here a mathematical model of the G1 to S transition that integrates the different regulatory links proposed so far. The model takes into consideration data from the literature and various models (not necessarily mathematical models) that were proposed for the G1 to S transition [2,3, 16], together with our own recent experimental results [15,17]. The model was implemented by a set of ordinary differential equations (ODEs) [18], an approach successfully used to describe the cell cycle control in budding yeast [19,20] and the cellular response of yeast to hyperosmotic shock [21]. These equations describe the temporal change of the concentrations of the involved proteins and complexes. Our model explicitly considers the localization of components in different cellular compartments (cytoplasm or nucleus) and cell growth during the G1 phase. We investigate the dynamics of the G1/S transition in various growth conditions, in several mutants, and in response to different signaling pathways. Sensitivity analysis in the form of time-dependent response coefficients [22] is used to estimate the influence of parameter values on the dynamics of key components and to investigate the relevance of the nucleo/cytoplasmic localization of Sic1 on the regulation of the G1/S transition, which has been neglected by earlier cell cycle models. To arrive at experimentally testable predictions, we developed two extended versions of the model. To compare the model output with data on budding, stochasticity in parameter values was taken into account. To estimate the PS value, we constructed a hybrid model regarding the firing of DNA replication origins where the probabilistic model uses as input the output of the deterministic model, here the nuclear concentration of Cdk1-Clb5,6. Our results newly indicate that PS is an emergent property of the network that depends on growth rate, thereby rationalizing the observed increase of PS at fast growth rates.
In an effort to cope with the recently released recommendations for annotation of biochemical models [23], we first list experimental data used for designing the network structure. In Materials and Methods we describe the modeling principles [18,21] and give details on parameter estimation. The full set of equations is given in Table 1, while values of parameters are summarized in Tables 2–5. To study the G1 to S transition network, we assembled all the essential elements into a concise mathematical model that captures the logic of the underlying processes. The main goal is to represent as much complexity as possible through a small number of quantities that have direct experimental interpretation. We drew the model with CellDesigner [44], a structured diagram editor for drawing gene-regulatory and biochemical networks that are stored using the Systems Biology Markup Language (SBML), a standard for representing models of biochemical and gene-regulatory networks. The essential elements we considered are (Figure 2): (1) production and degradation of mRNAs and proteins; (2) formation of dimeric and trimeric protein complexes; (3) nucleo/cytoplasmic localization of the compounds, transport processes being described like reactions, (e. g. , converting Cdk1cyt into Cdk1nuc); (4) cell growth in terms of volume increase with proportional rate constants for growth and protein production; and hence (5) concentration changes in the nuclear and cytoplasmic compartments. We use these elements to develop a mathematical model based on ODEs that describes the dynamics of both how different molecular species transform into each other, and how they traffic between the cytoplasmic and nuclear compartments (Table 1). In its essence, the time course of the network can be summarized as follows (see paragraphs above for references). The first threshold is based on the growth-dependent cyclin Cln3, and on the Cki Far1. Cln3 concentration remains approximately constant during G1, so that its total amount in the cell increases proportional to cell mass. Given that Far1 is mostly endowed to the newborn cell at the end of the previous cycle, the cell sizer threshold will be activated when Cln3 overcomes Far1. The ensuing Cdk1-Cln3–catalyzed release of the inhibitory effect exerted by Whi5—a protein with a role similar to that of pRb in mammalian cell cycle—on SBF/MBF will promotes synthesis of Cln1,2 and Clb5,6. Far1 is degraded after priming by Cdk1-Cln1,2, thus introducing a reinforcing loop in the network that guarantees irreversibility to the transition. The second threshold Cdk1-Clb5,6/Sic1 is overcome when Sic1 gets degraded following Cdk1-Cln1,2–primed phosphorylation. The model was constrained to fit the observed experimental behavior of the G1 to S transition of small, elutriated G1 cells growing on glucose [15,17]. First, we constrained features that play important regulatory roles in this phase of the cell cycle, such as the Cln3/Far1 threshold that controls the release of the SBF and MBF transcription factors from the Whi5 transcriptional repressor, and the complex formation of Sic1 with Cdk1-Clb5,6. To simulate cell growth at a low growth rate (in ethanol medium), we used both parameters' values derived from experimental data (such as the initial levels of Cln3 and Far1 and the growth rate), and a parameter (the binding constant of Sic1 for the Cdk1-Clb5,6 complex) inferred from the experimental dynamics of Clb5 and Sic1 as indicated in the following. A sensitivity analysis was performed for the fine-tuning of the parameters used to simulate the biochemical network. Then, we tested the mathematical model with the dynamics of (1) a large number of mutants, implemented by overexpression or deletion of key regulatory genes; (2) time course of budding obtained both in glucose and in ethanol media; and (3) estimation of the critical cell size PS for wild-type cells grown in both media. Finally, a sensitivity analysis indicates that PS is an emergent property of the G1 to S network. Figure 3 displays the simulated time courses of the concentration of several cell cycle players in small newborn cells growing in glucose using the following input parameters: growth rate characteristic of glucose; cell size; and Cln3 and Far1 levels as detected in small, elutriated cells grown in the same medium [15]. Since we are simulating the G1/S transition and not a full cell cycle with all relevant proteins, time courses of some variables become meaningless after overcoming the second threshold. At the beginning, the Far1, Cln3, and Cdk1 present in the cytoplasm are imported in the nucleus. The Cdk1-Cln3-Far1 complex reaches its maximal value in the nucleus after 30 min, and then it starts to be degraded upon the overcoming of the Cln3/Far1 threshold. Such degradation is mostly dependent on the execution of the first threshold, being strongly delayed in simulated cln3Δ cells (unpublished data). Cdk1-Cln3 starts to build up in the nucleus, the major factor driving accumulation of Cdk1-Cln3 being Cdk1-Cln–primed Far1 degradation (Figure 3A; T1), since little if any Cdk1-Cln3 complex forms in cln1,2Δ cells (unpublished data). At about 50 min, the accumulation of active SBF/MBF reaches its half-maximal value (Figure 3B). Cln and Clb cyclins are produced (Figure 3C), and Cdk1-Cln1,2 is accumulated both in cytoplasm and in nucleus (Figure 3D), while Cdk1-Clb5,6 accumulates preferentially in the nucleus (Figure 3E and 3F). The half-maximal value of Cdk1-Clb5,6 in the nucleus is reached at around 80 min (Figure 3F; T2). Thus, T1 and T2 represent the times when the first and second thresholds, respectively, are overcome. The coherence of this timing with experimental dynamics of cell cycle is discussed below. All data presented above refer to simulation of an idealized single cell. By taking into account biological variability, it is possible to simulate a cohort of synchronous cells that more closely resembles, for instance, a population of newborn elutriated cells placed to grow in a fresh medium. To this end, we modeled a population by a probabilistic approach that simulates cell-to-cell variability through repeated simulations with noisy parameters (see Materials and Methods for further details) (i. e. , all parameters were sampled from a normal distribution with the original model values [Tables 2 and 3] as mean value). Figure 7 shows representative time courses for the Cdk1-Cln1,2cyt and Cdk1-Clb5,6nuc complexes for cells growing in glucose (Figure 7A and 7B) and in ethanol media (Figure 7D and 7E). The simulated time course of budding obtained for a population growing in glucose (black dots) is very closed to the experimentally observed one (gray curve), both in the time required for the onset of budding and in the initial slope (Figure 7C). The same simulation run for ethanol-grown cells (Figure 7F) indicates a close agreement in the timing of the onset of budding and a satisfactory slope for three subsequent points. Although the maximum value of experimental budding is not reached in both simulations, we observe a fairly good correspondence between experimental and simulated behaviors. Taken together, these data show that the model correctly predicts properties of the cells related to the G1 to S transition that have not been taken into account during model construction, thereby offering support to the overall consistency between input data and output performance. Cell viability requires the coordination between cell growth and cell division, which in budding yeast is achieved by the attainment of a nutritionally modulated critical cell size (PS) to trigger budding and DNA replication [2,16]. Since the present model monitors cell growth, it should be possible in principle to estimate PS. To do so, we need to simulate the onset of DNA replication, since operationally PS is defined as the protein content of cells that enter S phase [52]. Therefore we correlate one of the outputs of the model, namely nuclear concentration of Cdk1-Clb5,6, with the onset of DNA replication. To this end, we constructed a hybrid model that uses the time course of Cdk1-Clb5,6nuc as input to a probabilistic model. Given the large number of DNA replication origins present in a yeast nucleus [42,43], and the reported role of the Cdk1-Clb5,6 complex in inducing firing [39], the probability of firing for each DNA replication origin could then be related to the nuclear concentration of the Cdk1-Clb5,6 complex, as explained in more detail in Materials and Methods. We performed simulations of the onset of DNA replication for cells grown in glucose and in ethanol media (Figure 8). In glucose (Figure 8A), activation of DNA replication origins takes place in a coordinated fashion roughly within a period between 70–90 min, in agreement with experimental data. In ethanol (Figure 8B), the Cdk1-Clb5,6 complex should be inefficiently imported into the nucleus, resulting in a longer S phase. This behavior agrees with reported data showing that a very poor carbon source such as ethanol [53], or a nitrogen source limitation, yield elongation of the S phase [54,55]. At this point, we could estimate PS as the cell size when 50% of replication origins were activated in a single cell. The values of PS obtained for various conditions and/or mutants, shown in Table S4 compare well with the data present in literature. In fact, overexpression of FAR1 in glucose-grown cells results in about a doubling of Far1 level and in a modest increase in cell size and PS, while overexpression in ethanol-grown cells yields a larger increase in both Far1 level and cell size (PS) [15]. Appropriate simulations of the modulation of FAR1 overexpression yield PS values that are very consistent with our experimental data [15]. Notably, moderate overexpression of FAR1 in a situation simulating growth in glucose results in a minor increase of PS (Table S4). Having obtained a good agreement between predicted and experimental PS under a set of conditions, we moved to analyze the effects of various parameters of the model on the setting of PS. Sensitivity analysis (Figure 9A) shows that several parameters affect its value: the initial level of Cln3 and Far1, the binding value of Sic1 to the Cdk1-Clb5,6 complex, and the growth rate. These results allow us to reconcile different, apparently conflicting, experimental results that pointed either to the increase of Cln3 concentration in the nucleus, or to the activation of transcription factors SBF/MBF, or to the regulation of Sic1 as the event setting the critical cell size. Besides, they highlight the significant role of the growth rate in determining the PS value, a particularly noteworthy result that could not be obtained by visual inspection of the model reported in Figure 2.
First of all, our model considers that the control over the entrance into S phase is distributed over two sequential thresholds that involve Cdk1, cyclins, and two distinct inhibitors: the Ckis Far1, in the first threshold, and Sic1, which acts on the second threshold. While Sic1 has long been recognized as regulating initiation of DNA replication, the involvement of Far1 in the control of the Cdk1-Cln3 activity is not present in earlier models [19,20]. A role for an inhibitory molecule (either a Cki or a phosphatase) in setting of PS has been proposed independently by different groups [10,14]. Such an inhibitor was expected to present a peak in late M phase, a basal, low synthesis during the other phases of the cycle, and have a Cdk1-Cln–dependent degradation [10]. This pattern, initially suggested for Sic1, fits well also for Far1 [31,67]. Since inhibition of Cln3-Cdk1 by Far1 is a major innovative feature of the network proposed in this paper, let us briefly review relevant experimental evidence about this item in addition to that reported in the section describing the construction of the model, regarding physical interaction, biochemical activity, and genetic interaction. Direct identification of the Cdk1-Cln3-Far1 complex in mitotic cells has not yet been reported. The lack of genome-wide two-hybrid or mass spectrometry data to support the interaction of Far1 and Cdk1-Cln3 is not surprising, since the high unreliability of these data is well-known. In fact, comparison between two-hybrid datasets (obtained independently from two laboratories) and mass spectrometry data (also obtained by two different groups) showed a surprisingly small overlap: about 10% and 14%, respectively (as reviewed by Ito et al. [68]). Moreover, false signals have been estimated to be as high as 50% [69]. On the other hand, in α-factor–treated cells, coimmunoprecipitation experiments showed that Far1 became a sizable co-precipitated substrate of Cdk1-Cln3 [70], and biochemical assays indicate that Far1 inhibits the activity of Cdk1-Cln3 in immunoprecipitates after α-factor treatment [71]. Since activation of Fus3 protein kinase determined by pheromone treatment promotes increased interaction of Far1 to Cdk1-Cln3, it would be reasonable to hypothesize that a basal binding between Cdk1-Cln3 and Far1 is present also in the absence of α-factor. It must be stressed that neither the report of Tyers and Futcher [70] nor that of Jeoung et al. [71] were designed to address Cdk1-Cln3-Far1 interaction in mitotic cells, but only in the presence of pheromone. More compelling experimental evidence for the involvement of Far1 in the G1/S transition of mitotic cycles has recently been accumulated. By analyzing mutants in CDC48, a cell cycle gene essential in the protein degradation pathway that acts as an ubiquitin-selective chaperone, Fu et al. [29] showed that blocking of Far1 degradation results in cell cycle arrest in G1 in mitotic cells. Alberghina et al. [15] independently showed that both FAR1 deletion and its overexpression affect the critical cell size PS as expected for a Cln3 inhibitor, and that deletion of either CLN3 or FAR1 in a sic1Δ background has an almost identical effect on nutritional modulation of cell size and PS. These findings are consistent with the notion that Far1 is acting in the same pathway as Cln3 with regard to nutritional modulation of cell size. To prove more formally that Far1 is acting on Cln3 and not on the closely related Cln1,2 cyclins, the effect of Far1 overexpression on the cell size of cln3Δ cells in exponential growth on ethanol-supplemented medium—chosen because these are the growth conditions in which the effect of Far1 overexpression on cell size is larger [15]—was tested and compared with that of wild-type cells. Figure 10 shows that, as previously reported [15], in wild-type cells overexpression of FAR1 leads to a large increase in cell size (66%). Such an increase is highly statistically significant (Student t-test, p < 0. 01). The increase in cell size is much lower (∼15%) in cells devoid of the CLN3 gene. Such an increase in cell size is at the borderline of statistical significance (Student t-test, p = 0. 09). Put in another way, ∼80% of the increase in size brought about by Far1 overexpression is lost in the absence of the CLN3 gene. These data appear even more relevant when one takes into account that the amount of Cln3 in growing yeast cells (which, according to these newly presented data, contributes to the majority—and possibly all—of the effect of Far1 on cell size) is much lower than the amount of Cln1,2 protein [24–27]. Taken together, literature results independently obtained by several laboratories and experimental findings newly presented here indicate that the assumption made in our model (i. e. , that Far1 plays a role in the control of entrance into S phase of mitotic cells by inhibiting Cdk1-Cln3), has a firm base, although our understanding of its mode of action is not yet complete. The effect that the introduction of the Cln3/Far1 threshold has on cell cycle dynamics must be stressed: it allows the cells to detect the reaching of a given size (set by the amount of Far1 present in a cell) when the amount of Cln3, which increases in proportion to cell mass, overcomes Far1. The actual critical cell size is reached when the cells overcome the second threshold dependent on the Cki Sic1. Recent evidence showed that alterations in nucleo/cytoplasmic localization of cyclins [28] and Ckis [17] play a relevant regulatory role in cell cycle progression. Nuclear localization of Cdk1-Cln3 and Cdk1-Clb5 complexes is assumed throughout recent papers by Tyson and coworkers [19,66], but is in fact modeled in a simplified way by multiplying cyclin synthesis rates with the parameter mass. This implementation gives different patterns of accumulation to cyclins as compared with other components. However, compartmentalization involves much more than that, since: (1) proteins are synthesized in the cytoplasm, starting from mRNA migration from the nucleus; (2) both import and export can be independently regulated; and (3) controlled partitioning affects binding equilibria by altering the actual concentration of a given protein available for binding to a given interactor within a subcellular compartment. None of these regulatory issues is addressed in the models of Chen et al. [19,20]. On the contrary, shuttling of proteins and complexes in and out of the nucleus is a major, explicitly modeled feature of our network that allows us to address biological significance of subcompartmentalization of biochemical reactions, avoiding inconsistencies intrinsically present in the Chen et al. model, where “nuclear” Clb proteins are free to interact with “cytoplasmic” Sic1, thus violating a major biochemical consequence of subcellular compartmentalization. It is relevant at this point to mention that it is the explicit modeling of nuclear compartmentalization that allows us to make predictions regarding biochemical properties of the Cdk1-Clb5,6-Sic1 complexes (discussed in the Results section), which have to be verified experimentally in the iterative procedure characteristic of systems biology [72–74]. The suggestion that comes from our simulation analysis indicates that Sic1 should have a higher binding affinity for the complex Cdk1-Clb5,6 in cells grown in glucose in comparison with cells grown in ethanol, and therefore stresses the interest in analyzing the phospho-signature of Sic1 in two conditions. Although we do not want at this stage to be bound to any specific hypothesis, it is also worth remembering that we have shown that a Sic1-derived peptide has a dramatically increased affinity towards a heterologous Cdk-cyclin complex in the CK2-phosphorylated versus the unphosphorylated state [41], as CK2 is a protein kinase whose activity is found to be higher in fast-proliferating cells [75]. The trigger that controls the activation of transcription factors SBF/MBF in Chen et al. [19,20] is modeled according to a zero-order ultrasensitivity switch [76,77], a function that was chosen because it is suitable for the phosphorylation/dephosphorylation reaction assumed to control SBF/MBF activation [19,20]. It was assumed that each transcription factor (both SBF and MBF) exists either in an active or inactive form, and that the transition between the two forms is catalyzed by two competing enzymes: a protein kinase (Cdk1-Cln3) and a protein phosphatase (whose existence was only supposed at the time of the construction of the model), each of which follows Michaelis–Menten kinetics. The transitions are taken to be fast enough to maintain each transcription factor in a steady-state distribution. The kinetic equation describing the behavior of the Goldbeter and Koshland ultrasensitive switch [56] assumes a sharply sigmoidal function when the cell grows through the critical size. The value of cell mass that triggers the SBF/MBF switch has been calculated from a quadratic equation containing half a dozen variables, and the solution has been set to 1. 2 (1 being the cell mass of the newborn cell), the large margin of error being the result of a determination that requires the estimation of a sizable number of kinetic and efficiency parameters. Subsequent experimental work has shown that the activation of SBF/MBF is not due to direct phosphorylation of transcription factors, with a delicate equilibrium between a kinase (Cdk1-Cln3) and a phosphatase (never found) as assumed by the very basic assumptions of the Chen et al. model [20]. Instead, as we modeled, the activation of SBF/MBF is due to the dissociation of an inhibitor (Whi5) that is phosphorylated by Cdk1-Cln3 [32]. The interest of this regulatory link is further stressed by the fact that Whi5 has the same role as the retinoblastoma protein in the control of cell cycle progression in mammalian cells [33]. The predictive ability of the Chen et al. [20] model is due to the built-in control for the onset of S phase given by the estimation of the critical cell size at 1. 2 that fortunately is very close to the experimental one for cells growing in glucose. In fact, it has been common knowledge for many years, derived from quantitative microscopic observations, that cells growing in batch in glucose medium at the onset of budding are only slightly larger than newborn daughter cells [78,79]. A double-tag flow cytometric analysis performed in one of our laboratories on chemostat-grown cells at variable glucose dilution rates has clarified that the ratio of critical cell size at the onset of S phase/cell size of newborn daughters is about 1. 5 at low-growth rates, then decreasing to 1. 24 at fast-growth rates [52]. Therefore, the value 1. 2 is quite close to the actual one if we consider cell growing in glucose, but it should not be used to simulate cell cycles at low-growth rates. Based on recent biochemical and genetic evidence [15,29], here we show that a simpler, better-defined mechanism (i. e. , the Cln3/Far1 threshold) brings about a switch-like accumulation of nuclear SBF and MBF transcription factors not at a prefixed cell mass, but as a result of the dynamics of the regulatory molecules (see Figure 3B), thus effectively coupling cell growth to the onset of DNA replication and budding. The ensuing Cdk1-Cln3 activity phosphorylates Whi5 [32,33], and leads to the activation of transcription factors SBF/MBF, opening up the pathway that leads to the onset of DNA replication. The requirement of a critical cell size to enter into S phase has been known for several decades, but its molecular basis is still under discussion. Cln3 is certainly involved in the setting of the critical cell size, but does not work alone in the mechanism. In fact, while cells growing at faster growth rates are larger than those growing at slower rates [12], in any given medium, cells overexpressing Cln3 are smaller and have a shorter G1 phase than wild-type cells [2,9], a paradox pointed out by Heideman and collaborators [11]. Results of the sensitivity analysis summarized in Figure 9A neatly show that PS does not originate from the properties of a single molecule, but rather is an emergent property of the network structure. Besides the Cln3 and Far1 dosage, the growth rate is a major factor in setting PS. The role of the growth rate in determining the value of PS can be explained as follows. The value of the cell size at the traverse of the first cell sizer threshold is quite similar both in rich and in poor carbon sources (Figure 9B), since the Cln3/Far1 ratio remains almost equimolecular at the various growth rates [15], with both Cln3 [11] and Far1 [15] increasing at faster growth rates. As shown before, a sizable period of time is needed to move from the first to the second threshold both in glucose and in ethanol. Since it is the traverse of the second threshold that actually sets PS, its value clearly will be much larger in cells that grow faster (Figure 9B). We use the term “emergent property” with true systems biology significance: a property that individual components of the G1 to S network do not have but that emerges from their interaction [80]. Of course, since the setting of PS is an emergent property of the entire G1 to S network, other parameters—such as the accumulation rate of Cln3 and/or Far1, or the binding affinity of Sic1 to the Cdk-cyclin complexes—affect PS independently from the growth rate and may become relevant in appropriate growth conditions. In the current Tyson model (s), the onset of DNA replication is controlled by a single event: a cell sizer is taken to be operative only at low-growth rates, while an oscillator mechanism is assumed to be active at fast-growth rates [66]. In our model, a sizer mechanism is operative at all growth rates, and the presence of two distinct—temporally spaced—thresholds acting together to set PS not only introduces a delay, but also makes the delay sensitive to the growth rate (Figure 9B). Taken together, the results presented in this paper offer an example of the usefulness of a systems biology approach for the understanding of complex biological processes.
The model comprises equations for production and degradation of mRNA and proteins and for the formation of dimeric and trimeric protein complexes (Figure 2). It accounts for the nucleo/cytoplasmic localization of the compounds. Transport processes are described like reactions, converting, for example, Cdk1cyt into Cdk1nuc. Cell growth is characterized as exponential increase in volume. All concentration changes are dependent on the volume changes of the respective compartment. The full set of equations and parameters can be found in Tables 1–5, and the impact of the chosen parameters on behavior of the simulated system was studied using sensitivity analysis [22]. We explicitly consider two compartments, the nucleus and the cytoplasm, with volumes Vnuc and Vcyt, respectively. Nuclear and cytoplasmic compounds are denoted by the subscripts nuc or cyt, respectively. The dynamics of the concentration of every compound is determined by three different types of processes: (1) biochemical reactions, (2) transport over the nuclear membrane, and (3) change of the volumes. The dynamics are described by sets of ODEs. The temporal evolution of a nuclear compound reads and the temporal evolution of a cytoplasmic compound is given by for i = 1, . . , m, where m is the number of biochemical species with the concentrations ci (either nuclear or cytoplasmic). The quantity r is the number of biochemical reactions with the rates vj, and t1 and t2 are the number of transport steps from cytoplasm to nucleus and vice versa with the rates wj. The quantities nij denote the stoichiometric coefficients of the compounds in the respective reactions or transport steps. The volume changes are given by and where kV, cyt and kV, nuc are the rate constants of volume change. All individual reaction rates vj and transport rates wj are governed by irreversible mass action kinetics: and where kj are the rate constants and the index x runs over all substrates and modifiers of reaction j, and the index y denotes the compound transported in step j. Rate constants for production. The model comprises 54 rate constants. The only relevant transcriptional control implemented in the model relates to transcription of CLN1,2 and CLB5,6 genes, which is dependent on SBF/MBF. The rate for CLB5,6 transcription (k2) was estimated from kinetics of Clb5 production in elutriated cells (this article). The same value was used for CLN1,2 transcription (k1). The rates of translation (k4 and k3, respectively) were set one order of magnitude higher. Rate constants for production of Far1 (k5) and Cln3 (k6) were estimated from kinetics of production of the appropriate protein in elutriated cells growing in glucose and in ethanol medium [15]. Basal level of production of Whi5 (k8) and Sic1 (k9) was set the same as k5. Since the Cdk1 level is never limiting, the rate constant for Cdk1 production (k7) was set much higher than those for Cdk-regulatory proteins. The Cln3-independent production of SBF/MBF (k35), implemented to mimic the contribution of Bck2, was set low. Rate constants for degradation. In the absence of other relevant information, rate constants for degradation of proteins were set to the same value, 0. 01 min−1. Rate constants for degradation of mRNAs were set one order of magnitude higher. The rate constants for degradation of Cln2 and Clb5, k12 and k13, respectively, were derived from previous cell cycle models [19,20]. The rate constant for degradation of cytoplasmic Cdk1 was derived assuming a steady partitioning of Cdk1 between nucleus and cytoplasm, while degradation of nuclear Cdk1 (k21) was considered negligible and set to 0. Rate constants for association and dissociation of protein complexes. Values of association and dissociation constants for proteins involved in Cdk-cyclin-Cki complexes formation were not available. Rate constants derived from BIAcore (http: //www. biacore. com) data regarding interaction between Sic1 and heterologous Cdk-cyclin [40,41] were thus used, assuming that regardless of the actual Cdk, cyclin, or Cki involved, relative scale of interaction affinity should be conserved. The same association value was then used for the association constant between SBF and Whi5 (k34). These values guided us also in choosing the value for dissociation of Cdk-cyclin complexes (k25, k27, k29) and of these complexes containing Cdk inhibitors (k31, k33). Rate constants for phosphorylation reactions. Michaelis–Menten kinetics for phosphorylation reactions were simplified to mass action (i. e. , substrate concentration was considered low compared with Km). Such simplification did not significantly affect simulation results (unpublished data). The resulting constants are thus near to the kcat/Km ratio. Data for kcat and Km for typical Cdk-catalyzed phosphorylation reactions [81] were thus used. Rate constants for transport between nucleus and cytoplasm. In the absence of more specific experimental information regarding “in vivo” constants, rate constants for nuclear import of single proteins (Far1, Cln3, Cdk1, and Whi5) were set equal (k42–k45). The rate constants for nuclear export were then set to favor nuclear localization. Similarly, nuclear localization was favored for the Cdk1-Cln1,2 binary complex. According to experimental data indicating that Sic1 promotes Clb5 nuclear localization [17], nuclear import of the ternary complex Cdk1-Clb5,6-Sic1 (k47) was favored over transport of the cognate binary complex (k48). Rate constants for nuclear export of CLN1,2 and CLB5,6 mRNAs (k50 and k51, respectively) were set larger than those of corresponding proteins. Rate constants of exponential growth. Values for glucose- and ethanol-grown cells were obtained by averaging literature data [15] and unpublished data from our laboratory. Absolute and relative initial concentration of protein and protein complexes. The average number of molecules in glucose-grown cells was taken from Ghaemmaghami et al. [25], which was related to the maximal values of that compound assumed during time course. The Far1/Cln3 ratio in newborn cells was taken from Alberghina et al. [15]. Sensitivity analysis. Sensitivity analysis was performed to test the influence of the parameter choice on the systems dynamics (Figure S1). To this end, we calculated the time-dependent response coefficients [22], defined as These coefficients indicate the direction and amount of change of the time course for the concentration c (t) upon an infinitesimal change of the parameter (or initial concentration) p. Loosely spoken, one can also interpret this as the percentage change of the concentration over time upon a 1% change of the parameter. During model development, the response coefficients were used to indicate appropriate parameter changes, since there are not enough data available to estimate the parameters by a global approach. Probabilistic model for firing of the DNA replication origins. The influence of Cdk1-Clb5,6nuc on the licensing of replication origins and DNA replication is described by a probabilistic three-step model that does not regard molecular details of this highly complex process. Step 1 lumps all events from free origin to pre-replicative complex. The transition time for each of the 440 replication origins is taken from a normal distribution with mean of 15 min and standard deviation of 2 min. Step 2 is Cdk1-Clb5,6nuc dependent. The probability for performing Step 2 at a certain time is determined by the concentration of Cdk1-Clb5,6nuc at that time. The period of this step is necessary for Cdk1-Clb5,6 to exceed a value taken from a normal distribution with a mean of 0. 03 μM and standard deviation of 0. 01 μM. The transition time for Step 3 to reach the “fired” state is again taken from a normal distribution with a mean of 1 min and standard deviation of 0. 01 min. When the origin has fired, then DNA replication proceeds bidirectionally from multiple replication origins, as experimentally reported [82,83]. If replication reaches the neighboring origin before it fires on its own, that origin is set to the state “fired. ” The distance between DNA replication origins is fixed. We considered that in ethanol-growing cells—with a growth rate about 2-fold lower compared with the glucose-growing cells—the fork rate is about one-half (and the time of origins activation is doubling) than the glucose ones. This assumption agrees with the reported data, in which the longer S phase in yeast cells growing in poor nitrogen medium can be accounted for by a reduction in replication fork rate [55]. Probabilistic model for budding. Assuming that the parameters for the individual cells may vary around their mean values as given in the parameter list in Tables 2–5 (standard deviation of ki*0. 287), we can simulate the time courses of Cdk1-Cln1,2cyt for cell populations. Onset of budding is determined by the time tmax/2 when Cdk1-Cln1,2cyt reaches a critical value equal to the half-maximal concentration obtained for mean parameter values. As time for the appearance of the bud, we use tbudding = tmax/2 (Cdk1Cln1,2cyt) + tdelay, with tdelay = 0 min in glucose and tdelay = 120 min in ethanol. Yeast strains, cell growth, and cell size determination. Yeast cells W303-CF (cln3: : KAN1, pCLN3–15Myc, FAR1–15Myc-URA3), FAR1tet (cln3: : KAN1, pCLN3–15Myc, far1: : HIS3, pTet-FAR1–15Myc), and cln3Δ (cln3: : KAN1) were grown in synthetic complete media supplemented with ethanol as a carbon source. Growth conditions, media, and flow cytometry determinations were conducted as described previously [15]. Simulation tool. Numerical calculations have been carried out using Mathematica Version 5. 1 (Wolfram Research, http: //www. wolfram. com).
A major property of living cells is their ability to maintain mass homeostasis throughout cell divisions. It has been proposed that in order to achieve such homeostasis, some critical event (s) in the cell cycle will take place only when the cell has grown beyond a critical cell size. In the budding yeast Saccharomyces cerevisiae, a widely used model for the study of the eukaryotic cell cycle, a large body of evidence indicates that cells have to reach a critical size before they start to replicate their DNA and to form bud, which will give rise to the daughter cell. This critical cell size is modulated by growth rate, hence by nutritional conditions and the multiplicity of genetic material (i. e. , ploidy). The authors present a mathematical model of the regulatory molecular network acting at the G1 to S transition. The major novel features of this model compared with previous models of this process are (1) the accounting for cell growth (i. e. , the increase in cell volume); (2) the explicit consideration of the fact that cells have a nucleus and a cytoplasm, and that key cell cycle regulatory molecules must move between these different compartments and can only react or regulate each other if they are in the same compartment; and (3) the requirement of sequential overcoming of two molecular thresholds given by a cyclin-dependent kinase/cyclin and a cyclin-dependent kinase inhibitor. The model was tested by simulating the processes during G1 to S transition for different growth conditions or for different mutants and by comparing the results with experimental data. A parameter sensitivity analysis (i. e. , testing the model predictions when parameters are varied), newly indicates that the critical cell size is an emergent property of the G1 to S network. The model leads to a unified interpretation of seemingly disparate experimental observations and makes predictions to be experimentally verified.
biochemistry
cell biology
yeast and fungi
microbiology
computational biology
eukaryotes
genetics and genomics
saccharomyces
Cell Size at S Phase Initiation: An Emergent Property of the G1/S Network