The author(s) have made the following declarations about their contributions: Conceived and designed the experiments: TJL DCB JRN LY. Performed the experiments: TJL. Analyzed the data: TJL LY. Contributed reagents/materials/analysis tools: GY JRN LY. Wrote the paper: TJL GY DCB LY.
The authors have declared that no competing interests exist.
A new, stochastic model of entry into the mammalian cell cycle provides a mechanistic understanding of the temporal variability observed across populations of cells and reconciles previously proposed phenomenological cell-cycle models.
The transition of the mammalian cell from quiescence to proliferation is a highly variable process. Over the last four decades, two lines of apparently contradictory, phenomenological models have been proposed to account for such temporal variability. These include various forms of the transition probability (TP) model and the growth control (GC) model, which lack mechanistic details. The GC model was further proposed as an alternative explanation for the concept of the restriction point, which we recently demonstrated as being controlled by a bistable Rb-E2F switch. Here, through a combination of modeling and experiments, we show that these different lines of models in essence reflect different aspects of stochastic dynamics in cell cycle entry. In particular, we show that the variable activation of E2F can be described by stochastic activation of the bistable Rb-E2F switch, which in turn may account for the temporal variability in cell cycle entry. Moreover, we show that temporal dynamics of E2F activation can be recast into the frameworks of both the TP model and the GC model via parameter mapping. This mapping suggests that the two lines of phenomenological models can be reconciled through the stochastic dynamics of the Rb-E2F switch. It also suggests a potential utility of the TP or GC models in defining concise, quantitative phenotypes of cell physiology. This may have implications in classifying cell types or states.
Mammalian cells enter the division cycle in response to appropriate growth signals. For each cell, the decision to do so is critically dependent on the interplay between environmental cues and the internal state of the cell and is influenced by random fluctuations in cellular processes. Indeed, experimental evidence indicates that cell cycle entry is highly variable from cell to cell, even within a clonal population. To account for such variability, a number of phenomenological models have been previously proposed. These models primarily fall into two types depending on their fundamental assumptions on the origin of the variability. “Transition probability” models presume that variability in cell cycle entry originates from the fact that entry in each individual cell is random but also governed by a fixed probability. In contrast, “growth-controlled” models assume that the growth rates across a population are variable and result in cells that are out of phase developmentally. While both kinds of models provide a good fit to experimental data, their lack of mechanistic details limits their predictive power and has led to unresolved debate between their practitioners. In this study, we developed a mechanistically based stochastic model of the temporal dynamics of activation of the E2F transcription factor, which is used here as a marker of the transition of cells from quiescence to active cell cycling. Using this model, we show that “transition probability” and “growth-controlled” models can be reconciled by incorporation of a small number of basic cellular parameters related to protein synthesis and turnover, protein modification, stochasticity, and the like. Essentially our work shows that each kind of phenomenological model holds true for describing a particular aspect of the cell cycle transition. We suggest that incorporation of basic cellular parameters in this manner into phenomenological models may constitute a broadly applicable approach to defining concise, quantitative phenotypes of cell physiology.
Cell-to-cell variability in the timing of cell-fate commitment is widely observed in biological settings
To account for the variable transition timing in cell cycle progression, two major types of models have been proposed: the transition probability (TP) models
There has been an active debate between these two lines of thinking since their initial propositions. While never fully resolved, the debate gradually faded as the focus in the field of cell cycle studies moved to identifying the dynamical basis for various cell cycle regulations, including the restriction point (R-point)
The temporal variability described by the GC and TP models is based on the distribution of inter-mitotic times and may differ from temporal variability in E2F activation from quiescence. However, we suggest that the stochastic Rb-E2F model embodies the concepts assumed in the TP models and the GC models. Our model predictions and experiments suggest that stochastic activation of E2F can account for temporal variability in cell cycle entry, and the degree of such variability is determined by environmental cues and the regulatory network parameters. These results suggest that the TP and GC models are not mutually exclusive but rather reflect different aspects of the same temporal dynamics in cell cycle entry, as has been speculated
A population of quiescent cells can undergo the G1/S transition with serum stimulation. The timing of cell cycle entry is highly variable in a cell population, characterized by an exponential drop in the percentage of G0 cells over time (G0 exit curve). To account for such temporal variability, two groups of phenomenological models have been previously proposed: the transition probability (TP) model, which describes the dynamics of cell cycle entry by a transition rate (KT) and a time delay of the cell population (TDP), and the growth-controlled (GC) model with a mean growth rate (
Our recent work has shown that traverse of the R-point is regulated by the Rb-E2F bistable switch
The fluctuations in the bistable switch result in significant discrepancies between stochastic and deterministic simulations
(A) Stochastic simulations (25 events) exhibit variable time delays in E2F activation, as shown in gray lines. Two distinct modes of E2F (low and high) are clear and can be separated by a switching threshold (horizontal dotted red line). The inset shows the distribution of E2F at the end of 5,000 simulations (time = 50 h). The minimum time required to reach this threshold is defined as the switching time (vertical dotted red line). In contrast to the stochastic simulations, the deterministic simulation has the same trajectory for a given set of parameters (black line). (B) The percentage of G0 cells over time (G0 exit curve) is plotted for a population of 5,000 simulated cells stimulated at strong (red line, S = 5) and weak (blue line, S = 0.5) input concentrations. The G0 exit curve for the strong input is fitted with an exponential function (black dotted line),
Based on our simulations and definitions in
Our simulated E2F activation dynamics predict serum-dependence of both transition rate and time delay. For a weak input (KT = 0.029±0.0014 h−1 and TDP = 18.0±1.2 h, blue line in
To validate our model predictions, we measured E2F activity in the E2F-d2GFP cell line, which is derived from a rat embryonic fibroblast REF52 cell line and carries a destabilized green fluorescent protein reporter (d2GFP) under the E2F1 promoter. We have shown that this reporter system can be used to monitor E2F activity in response to serum stimulation previously
(A) The temporal dynamics of a cell population depends on serum concentration. At 0th h E2F-d2GFP cells were synchronized in quiescence by serum-starvation (24 h at 0.02% serum). These cells were then stimulated with either 0.3% or 5% serum, and corresponding GFP levels (reporting E2F activity) were determined by flow cytometry. The cell population treated with 0.3% serum exhibited a bimodal distribution of GFP at the 24th h . In contrast, a monomodal distribution was observed at 24th h in the cell population treated with 5% serum. The dotted lines represent switching threshold, which is used to distinguish the low and high modes of E2F. Here, the switching threshold was defined as 2.5 times the variance from the mean of the GFP distribution of serum-starved cells (or GFP distribution at time 0). (B) Serum concentration modulates the temporal dynamics of E2F activation. The thresholds shown as dotted blue lines in (A) were used to calculate the percentage of cells in the low mode of E2F. The two G0 exit curves showed that transition rate increased (KT = 0.031±0.0036 h−1 at 0.3% serum and 0.16±0.011 h−1 at 5% serum) and time delay decreased (TDP = 5.1±1.1 h at 0.3% and 1.1±0.27 h at 5% serum) with increasing serum concentration. (C) The transition rate increased with serum concentration. (D) The time delay decreased with serum concentration. Data in panels A and B and those in panels C and D were from two independent experiments.
Based on the distribution of E2F in
The temporal dynamics of biological systems often depend strongly on network parameters
To investigate modulation of the transition rate by nodal perturbations in the Myc-Rb-E2F network, we introduced in silico perturbations of one particular node: the CycE/Cdk2 complex, which forms the CycE-mediated positive feedback loop. Our bifurcation analyses predict that weakening of the CycE-mediated positive feedback loop will desensitize the Rb-E2F bistable switch to serum stimulation, requiring a higher critical serum concentration (
(A) The strength of the CycE-mediated positive feedback determines the sensitivity of the system to serum stimulation. Bifurcation analyses of the Rb-E2F switch with weak (Rb phosphorylation rate constant kP4 = 9 h−1, blue), intermediate (kP4 = 14 h−1, black), and strong strength (kP4 = 18 h−1, red) of the positive feedback were performed. For decreasing strength of the positive feedback, the system became less sensitive to the input strength, requiring greater critical input strength for E2F activation. (B) The temporal dynamics can be modulated by adjusting the feedback strength. At the saturating input level (S = 10), the Rb-E2F switch was subjected to varying degrees of feedback strength mediated by CycE. G0 exit curves from 5,000 simulations were constructed for strong (red line,
To test these predictions experimentally, we perturbed the Myc-Rb-E2F network by applying varying concentrations of a cyclin-dependent kinase inhibitor (CVT-313), which has a much higher affinity towards Cdk2 than to other Cdks (
(A) E2F activity, measured by the GFP signal in the E2F-d2GFP cells, was assayed under varying concentrations of Cdk2 inhibitor CVT-313 (0.5, 1, 2, 3, and 5 µM) and serum (0.2%, 0.3%, 0.5%, 0.7%, 1.0%, 2.0%). After 24 h of the inhibitor drug treatment in DMEM supplemented with varying serum concentrations, the E2F-d2GFP cells were collected and their GFP signals were assayed by flow cytometry. For each serum and inhibitor drug condition, the fraction of cells with GFP signals above a threshold level was counted and plotted. For a given serum concentration, increasing drug dose led to a decreasing fraction of cells at the high E2F mode. Increasing serum concentration resulted in an increasing fraction of cells at the high E2F mode. (B) The temporal dynamics of E2F activation is altered when CycE-mediated positive feedback is weakened. At 2% serum, we applied Cdk2 inhibitor CVT-313 at 2 µM (blue curve) and monitored the effect on E2F activation over time by flow cytometry. Compared to the case without drug (red curve), the transition rate decreased from 0.078±0.0073 to 0.058±0.0070 h−1and the time delay increased from TDP = 9.1±0.70 to TDP = 12.0±0.86 h. (C) Targeting the CycE-mediated positive feedback modulates the transition rate. In an independent set of experiments, time-courses of cell populations treated with varying serum concentrations were obtained for a given drug dose, and the transition rate was calculated for each serum condition. The transition rate increased with serum concentration and reached a plateau at saturating serum concentrations in the absence of the inhibitor, but it continued to increase in the presence of the inhibitor. Overall, KT was greater without the inhibitor than with it. (D) Time delay decreased with increasing serum concentration in the absence of the inhibitor and reached a plateau at the saturating serum concentration. In the presence of the inhibitor, however, TDP continued to decrease. Overall, TDP was greater with the inhibitor than without it.
Next, we tested modulation of temporal dynamics by the Cdk2 inhibitor. At 2% serum, we applied the Cdk2 inhibitor (CVT-313, 2 µM) to monitor its effect on E2F activation over time. Our results in
Throughout this study, we have analyzed the temporal dynamics of E2F activation by extracting a set of parameters defining the TP model (transition rate and time delay). This parameter extraction establishes a connection with the mechanistic Rb-E2F model. Similarly, the GC model parameters (mean growth rate
TP Model Parameters | GC Model Parameters | ||||
Serum (%) | KT (h−1) | TDP (h) | |||
Simulations | 0.5 | 0.029±0.0013 | 18±1.3 | 0.21±0.034 | 0.56±0.067 |
1 | 0.073±0.0037 | 9.7±0.49 | 0.82±0.12 | 0.49±0.053 | |
2 | 0.12±0.0053 | 8.5±0.35 | 2.5±0.31 | 0.40±0.037 | |
5 | 0.16±0.0076 | 7.7±0.27 | 4.4±0.72 | 0.36±0.044 | |
Experiments | 0.5 | 0.050±0.0054 | 9.8±0.99 | 0.25±0.19 | 0.36±0.067 |
1 | 0.09±0.0089 | 9.3±0.62 | 1.9±0.45 | 0.80±0.13 | |
2 | 0.14±0.011 | 8.9±0.46 | 4.1±0.92 | 1.4±0.24 | |
5 | 0.16±0.011 | 6.5±0.40 | 3.2±0.70 | 1.3±0.21 |
In both simulations and experiments, we varied serum concentration only, while keeping all else the same. For each G0 exit curve, we extracted the TP and GC model parameters.
In addition, we predicted the dependence of the strength of the CycE-mediated positive feedback on the GC model parameters, as shown in
Simulation results from the stochastic Rb-E2F model are fitted to the GC model with two parameters (adapted from the G1-rate model
Focusing on E2F activation, we have shown that the temporal variability in cell cycle entry from quiescence can be quantitatively modeled by stochastic activation of a bistable Rb-E2F switch
Our model predictions are overall consistent with experimental measurements. In particular, our analysis indicates that serum and a Cdk2 inhibitor drug exert opposite influences on the temporal dynamics of E2F activation: transition rate increases and time delay decreases with increasing serum, but transition rate decreases and time delay increases with increasing Cdk2 inhibitor concentrations. We suggest that such a well-calibrated stochastic model for the Rb-E2F switch may guide further experimental analyses to gain insights into the system-level dynamics underlying cell cycle entry. For example, our model predicts that reducing the CycD/Cdk4,6 activity may have similar effects on temporal dynamics of E2F activation as the Cdk2 inhibitor, while knocking down Rb may increase transition rate (unpublished data). In addition, we can predict stochastic dynamics of E2F activation under combinatorial perturbations including growth factors, inhibitor drugs targeting the Myc-Rb-E2F network, or mutations within this network.
Throughout this study, we have focused on a single transition during cell cycle progression (quiescence to proliferation) due to its experimental and computational tractability. To further simplify analysis, we have chosen not to model cell division or growth explicitly. Instead, the variability associated with these processes is lumped into the extrinsic noise terms in our SDE model. More explicit mechanisms to account for such variability may further improve the quantitative agreement between the modeling and the experiment. For example, our simulation results suggest that the major source of noise is extrinsic noise, while variability in the initial conditions can lead to minor yet discernable change in the temporal dynamics of E2F activation. This is evident when E2F activation dynamics are compared under two conditions: varying initial conditions and varying variance of the extrinsic noise (ω) in the stochastic model (see
Equally important, we further show that these predicted stochastic dynamics of the Rb-E2F model can be quantitatively mapped into two lines of phenomenological models reflecting seemingly conflicting views: the TP model and the GC model. For a given set of parameters defining the stochastic model, the simulated stochastic E2F activation at the population level can be uniquely described by a set of parameters defining the TP model or the GC model (compare
During the mapping from our stochastic model to the TP or GC models, details associated with individual signaling reactions are necessarily lost in the resulting TP or GC models, pointing to their limitations in offering mechanistic insights. However, a by-product of this mapping is a potential, unappreciated utility of the TP and GC models. On one hand, these phenomenological models are simple and are able to provide quantitative description of the population-level dynamics associated with variable cell cycle entry. On the other, specific changes in the underlying reaction networks can be manifested in changes in the parameters in these simple models. As such, together with a mechanistically based model, the TP and GC models can serve as a concise platform to define quantitative phenotypes that facilitate classification of cell types or cell states.
This utility may be particularly useful for cancer diagnosis, since most cancers have defects in the Myc-Rb-E2F signaling pathway
The deterministic version of the Rb-E2F model, developed in our previous work
Actively growing E2F-d2GFP cells
E2F-d2GFP rat embryonic fibroblasts were assayed for a destabilized green fluorescent protein reporter (d2GFP) for E2F activity. The intensity of d2GFP was measured with a flow cytometry system (BD FACSCanto II).
E2F-d2GFP cells were serum-starved (BGS = 0.02%) for 24 h before they were treated with varying concentration of the Cdk2 inhibitor (CVT-313, EMD # 238803) and serum. After 24 h of serum/inhibitor drug treatment, cell lysates were collected and Western blotting was conducted with primary antibodies recognizing Rb phosphorylation at Cdk4-specific serine 780 (Santa Cruz, #sc-12901-R) and at Cdk2-specific threonine 821 (Santa Cruz, #sc-16669-R). These were conjugated with anti-rabbit secondary antibodies (GE Healthcare, #NA934) for detection. As a loading control, actin was measured with actin-recognizing primary antibodies (Santa Cruz, #sc-8432) conjugated with anti-mouse secondary antibodies (GE Healthcare, #NA9310).
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We would like to thank Tony Huang, Shwetadwip Chowdhury, and William Kim for preliminary analysis and discussions, and Jin Wang and Jeff Wong for helpful suggestions.
growth control
transition probability