Fig 1.
Schematic representation of the cell-cycle protein regulatory network.
Fig 2.
Simulations cell division within a cell population reveal how simple cell-division rules affect the cell-size distribution.
The simulation begins with a single cell of unit volume, which divides to produce daughter cells. Cells grow exponentially until the specified division condition is satisfied. Cell division is stochastic; at each cell division, a normally distributed random variable is generated to determine the proportion of the cell volume distributed to the daughter cells. A Time course for ideal sizers. B Scatter plot of cell volume at birth against added volume at division (volume at division minus volume at birth) for ideal sizers. C Time course for ideal adders. D Scatter plot for ideal adders. E Time course for ideal timers. F Scatter plot for ideal timers.
Fig 3.
Simulations of the full cell-cycle model reveal limit-cycle dynamics.
A phase plane orbit showing CDKA and KRP. B phase plane orbit showing SMR and CDKB. C-H Time course for each protein as indicated. We note that in panel H, we show both total RBR (RBRT) and unphosphorylated RBR (RBRu). Parameter values used in the simulations are given in Methods section 5.5 with the exception of the cell relative growth rate which is set to rgr = 0 for this simulation.
Fig 4.
Modelling reveals that the network exhibits a hysteresis at the G1/S transition.
A Simulated time course showing the concentrations of CDKA, KRP, un-phosphorylated RBR (uRBR) and active E2FA. The transition is clearly visible at the point when E2FA is suddenly released from RBR inactivation. B Bifurcation diagram showing KRP steady states. At low levels of CDKA, there exists a unique stable steady state of high KRP activity. Eventually, this state is eliminated in a saddle node bifurcation, and the system switches to a new stable state of low KRP activity. Specifically, (36) subject to CDKA/B : CYCBT = MYB3R3T = MYB3R4T = SMRT = FBP = SCF = APC = 0. Parameter values are given in Methods section 5.5.
Fig 5.
Modelling reveals that the network exhibits a hysteresis at the G2/M transition.
A Simulated time course showing the concentrations of CDKB, SMR and un-phosphorylated MYB3R3 (uMYB3R3). B Bifurcation diagram showing SMR steady states. C Relationship between uMYB3R3 concentration and CDKA/B:CYCB concentration. Results are based on numerical simulations of the G2/M Eqs (37)–(49) subject to CDKA : CYCDT = KRPT = E2FAT = E2FB = RBRT = FBL17 = SCFAPC = 0. Parameter values used in the simulations are given in Methods section 5.5 with the exception the synthesis rate of CDKA/B:CYCB is rcb = 0.01 (default value being 0) so that CDKA/B:CYCB is expressed independently of the E2FB transcription factor.
Fig 6.
Cell size control at the G1/S transition, testing 4 hypotheses: 0. All proteins are size-dependent. 1. All proteins are size-dependent, but cells inherit a fixed mass of KRP at birth, irrespective of cell volume at birth. 2. RBR is size-independent and all other proteins are size-dependent. 3. KRP is size-independent and all other proteins are size-dependent. A Time between birth and S phase (i.e. the duration of G1 phase) plotted against birth volume. B Cell volume growth during G1 phase.
Fig 7.
Cell size control at the G2/M transition, testing 4 hypotheses: 0. All proteins are size-dependent. 1. All proteins are size-dependent, but cells inherit a fixed mass of SMR at birth, irrespective of cell volume at birth. 2. SMR is size-independent and all other proteins are size-dependent. 3. MYB3R3 is size-independent and all other proteins are size-dependent. A Time between S phase and M phase (i.e. the duration of G2 phase) plotted against birth volume. B Cell volume growth during G2 phase.
Fig 8.
Cell population simulation under the assumption that all proteins are size-dependent, which fails to produce size control.
A Relationships between G1 duration and birth volume, and between G2/M duration and volume at the G1/S transition. B Relationships between added volume during G1 and birth volume, and between added volume during G2/M and volume at the G1/S transition. C Time course of cell population volumes.
Fig 9.
Cell population simulation under the assumption that all proteins are size-dependent except KRP and SMR, which are size-independent.
Successful cell size homeostasis is established. A Relationships between G1 duration and birth volume, and between G2/M duration and volume at the G1/S transition. B Relationships between added volume during G1 and birth volume, and between added volume during G2/M and volume at the G1/S transition. C Time course of cell population volumes.
Table 1.
Quartile coefficient of dispersion, (Q3 − Q1)/(Q3 + Q1), at birth, G1/S and G2/M for different size control hypotheses.
Abbreviations are: “Eq Inh” = “Equal inheritance”; “SI” = “size-independent”; “SD” = “size-dependent”; “PD” = “phase-dependent” i.e. the synthesis of KRP is restricted to M-phase due to being limited by MYB3R4 activity. “X:Y” means the ratio of the coefficients of dispersion. Values are calculated from all cells that have completed a full cell cycle within the simulated time.
Fig 10.
Cell population simulations under the assumptions that KRP is equally inherited, SMR is size independent and all other proteins are size dependent.
Simulations considered two scenarios: one in which there is continual KRP synthesis at a constant rate, and one in which KRP synthesis is phase-dependent. In the latter case it is assumed that the synthesis of KRP is proportional to the concentration of MYB3R4, i.e. . A The duration of G1 phase against birth volume taken from generated cell populations. The Continual KRP synthesis model is in effect a timer mechanism, as the G1 duration is a constant regardless of cell birth volume. In contrast, in the phase-dependent KRP model, larger cells generally have shorter cell cycles in general, which will theoretically establish a degree of size control at G1/S. B Added volume during G1 phase. In both models the overall trend is positive, although it is less steep in the case of phase-dependent KRP synthesis. C Time course of KRP concentrations, assuming phase-dependent KRP synthesis. KRP is synthesised in late G2/M-phase when MYB3R4 is active, diluted during G1-phase, then rapidly degraded at G1/S. Note that the equal inheritance of KRP mass, causes unequal inheritance of concentration. D Time course of cell population volumes, assuming phase-dependent KRP synthesis.
Fig 11.
Cell population simulation under the assumption that all proteins are size-dependent, but that KRP is equally inherited and has phase-dependent synthesis (due to regulation by MYB3R4).
A Relationships between G1 duration and birth volume, and between G2/M duration and volume at the G1/S transition. B Relationships between added volume during G1 and birth volume, and between added volume during G2/M and volume at the G1/S transition. C Time course of cell population volumes.
Fig 12.
Box plots showing the distribution of cell sizes and phase durations under the assumption of CDKA:CYCD over-expression (OE), wild-type expression (WT), and under-expression (UE).
Wild-type had default parameter values, over-expression a 50% increase in CDKA:CYCD synthesis rate and under-expression a 20% decrease. A Size at G1/S. B Duration of G1 phase. C Size at division. D Duration of the entire cell cycle.
Fig 13.
Box plots showing the distribution of cell sizes and phase durations under the assumption of CDKA/B:CYCB over-expression (OE), wild-type expression (WT), and under-expression (UE).
Wild-type had default parameter values, over-expression a 50% increase in CDKA/B:CYCB synthesis rate and under-expression a 20% decrease. A Size at G1/S. B Duration of G1 phase. C Size at division. D Duration of the entire cell cycle.
Fig 14.
Modelling the interaction between CDKA/B:CYCB and MYB3R3.
A Simulated time course showing the concentrations of CDKB and un-phosphorylated MYB3R3. B Bifurcation diagram showing MYB3R3 steady states. Note that this system exhibits no hysteresis, in contrast with the results in Fig 5 for a network that includes SMR. Although there is mutual inhibition between MYB3R3 and CDKB, this is not sufficient to establish a bistable switch. In this model, MYB3R3 may control entry into G2/M, but only by modulating expression of CDKB.
Table 2.
Table of default parameter values for each protein.
Parameters take these values in model simulations unless otherwise stated. Parameter values marked with an asterisk (*) are reproduced from [37]. As far as we are aware, suggestions for the other parameter values are not currently available in the literature.
Fig 15.
Investigating the robustness of limit cycle solutions to parameter variation.
Each parameter was increased and decreased, one at a time, by at most a factor of 10 in either direction. The range of values in which limit cycle solutions were observed is indicated by bars. Default parameter values (as given in Table 2) are marked with an x.