# A Model for the Proliferation–Quiescence Transition in Human Cells

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Proliferation–Quiescence Dynamics Model

- 1
- The production of $M\left(t\right)$ is through mass action by extra-cellular growth signals $GF$ and by $E\left(t\right)$, which is modelled using Michaelis–Menten kinetics, whereas its decay is modelled by mass action. Yao et al. [2] considered Michaelis–Menten kinetics for the production of $M\left(t\right)$ through extra-cellular growth signals.
- 2
- The activation of $R\left(t\right)$ is enhanced by $E\left(t\right)$ and its inhibition is intensified by $M\left(t\right)$ and $E\left(t\right)$, which are both modelled using mass action. It is pertinent to note that, in this study, we assume conservation of mass for the $Rb$ family of proteins, which was not considered in [2]. In addition, we assume self-activation and inhibition of $R\left(t\right)$ using the Hill function with $n=2$. On the contrary, Yao et. al. [2] modelled the production and depletion of $R\left(t\right)$ using Michaelis–Menten kinetics and mass action, respectively, and self-activation and de-activation were not considered.
- 3
- $E\left(t\right)$ is synthesised, cf. [40], with the use of Michaelis–Menten kinetics and its synthesis is enhanced by $M\left(t\right)$ with the use of a Hill function with $n=2$, while its decay is enhanced by $R\left(t\right)$ with the use of mass action. On the contrary, the authors in [2] did not consider constitutive synthesis of $E\left(t\right),$ which has been observed experimentally as indicated in [40].

#### 2.2. Mathematical Analysis of the Model

#### 2.3. Non-Dimensionalisation

#### 2.4. Positivity, Boundedness and Existence and Uniqueness of Solutions

**Lemma**

**1**

**Proof.**

**Lemma**

**2**

**Proof.**

**Lemma**

**3**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 2.5. Steady States

#### 2.6. Linear Stability Analysis

**Theorem**

**2.**

- (A).
- ${\omega}_{1}$, ${\gamma}_{2}<0$ and ${\theta}_{2}>0,$
- (B).
- ${\omega}_{1}{\theta}_{2}<{\omega}_{2}{\theta}_{1}$ and ${\omega}_{2}{\gamma}_{1}<{\omega}_{1}{\gamma}_{2},$
- (C).
- ${\omega}_{2}{\theta}_{1}{\gamma}_{3}+{\omega}_{1}{\omega}_{2}{\gamma}_{1}>0$ and ${\gamma}_{2}{\gamma}_{3}{\theta}_{2}+{\omega}_{2}{\gamma}_{1}{\gamma}_{2}>0$, then (M*, E*, R*) is asymptotically stable.

**Proof.**

## 3. Results

#### 3.1. Bifurcation Analysis

**SN**(saddle node). As $GF$ increases further out of the bistable regime, ${M}^{*}$ jumps into its high steady state, where a cell undergoes proliferation. We also plot ${k}_{1}$ against $GF$ and generate a two parameter bifurcation diagram shown in Figure 2c. In the latter figure, the bistable region is coloured green.

#### 3.2. Numerical Simulations

#### 3.3. Sensitivity Analysis

## 4. Discussion

- Based on the concept of first principles, we investigated the dynamical potential of growth factors in the regulation of the cell-cycle entry.
- A mathematical model for the simplified $Rb-E2F$ network was constructed based on the model proposed in Yao et al. [2]. While previous studies modelled all links using Michaelis–Menten functions only [2], we used mass action, and Michaelis–Menten and Hill functions, resulting in a simpler model. In addition, we considered the R species to exist either in hyper-phosphorylated or hypo-phosphorylated form and that their total concentration is conserved.
- By varying the growth factor signal values through bifurcation analysis, numerical simulations illustrated that the magnitude of the value of the growth factor plays a critical role in regulating cell-cycle entry. Through bifurcation analysis, we deduced the existence of three consecutive dynamical behaviours, namely, stability, bi-stability and stability.
- Numerical simulations performed with different growth factor values validated the results derived from the bifurcation analysis. In particular, the biological interpretation of the uniform steady state can be established as follows:
- For $GF<0.3735$, System (1) is asymptotically stable, indicating the regime in which cells are in a quiescent state. In this state, cells feature low levels of Cyclin D, Myc and high levels of R species.
- On the other hand, in the range $0.3735<GF<0.4138$, System (1) exhibits bi-stability, marking the position of the restriction point, as deduced in Yao et al. [42]. This point sets a high threshold separating quiescence from proliferation and acts as a barrier against unregulated and accidental cell growth. In addition, it provides a low-maintenance mechanism ensuring that the cell cycle proceeds, albeit later due to changes in the extracellular environment which is crucial for maintaining genome integrity.
- For values of $GF>0.4138$, the system generates a stable dynamical behaviour where a cell is in the proliferation mode marking the higher steady state value. This state features high levels of Cyclin D, Myc and low levels of the $Rb$ family of proteins.

However, it remains to investigate the conditions under which the system exhibits excitable and oscillatory dynamics as observed in a different model proposed in [62], but that would be investigated in a subsequent work. While Yao et al. [42] identified a basic gene circuit underlying resettable $Rb-E2F$ bi-stable switch controlling cell-cycle entry, we obtained a range of values of the growth factor concentration for the three dynamical regimes.

## 5. Limitations

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**An activator-inhibitor network for the $Rb-E2F$ signalling network [42]. The solid arrows represent activation mechanism while the broken lines represent inhibition. Here, the growth factors $GF$ activate M (cyclin D and $Myc$) and, in turn M activates E (cyclin $A,$ cyclin E and $E2F$ transcription factors) while E activates M, forming a positive feedback loop. R ($Rb$ family of proteins) inhibits E, while E activates and inhibits R and M inhibits R.

**Figure 2.**Numerical bifurcation diagrams for the system of ODEs (1), with parameter values listed in Table 1. (

**a**) One parameter bifurcation diagram for ${M}^{*}$, with the growth factor $GF$ concentration value as a bifurcation parameter. The saddle-node (SN) bifurcation points for $GF$ are 0.3735 and 0.4138 and the corresponding ${M}^{*}$ are 3.826 and 1.225, respectively. The red line represents the stable steady states whereas the thin black line represent saddle points. The black dots labelled

**SN**are the saddle-node bifurcation points, which span the region characterised by bistable dynamics. (

**b**) A one-parameter bifurcation diagram showing a zoom of (

**a**) around the two saddle-node bifurcation points. (

**c**) Two-parameter bifurcation diagram with ${k}_{1}$ and $GF$ as bifurcation parameters. The green coloured region is characterised by bistable dynamics, whereas the colourless region is characterised by stable dynamics.

**Figure 3.**Numerical simulations illustrating the time evolution dynamics of $\left(M\right(t)$, $R\left(t\right)$, $E\left(t\right))$ for system (1) corresponding to different dynamic regimes. As the value of $GF$ increases, the system transitions from stable ((

**a**) with $GF=0.3$ and initial conditions ${M}_{0}=0.2$, ${E}_{0}=1.2$, ${R}_{0}=3$) through bistable ((

**b**) with $GF=0.39$ and two sets of initial conditions ${M}_{0}=0.2$, ${E}_{0}=1.2$, ${R}_{0}=3$ and ${M}_{0}=20$, ${E}_{0}=5$ and ${R}_{0}=5$) then back to stable ((

**c**) with $GF=0.45$ and initial conditions ${M}_{0}=20$, ${E}_{0}=5$ and ${R}_{0}=5$).

**Figure 4.**Numerical bifurcations and simulations illustrating the dynamics of System (1) where we explore the conditions of Theorem 2. (

**a**–

**c**) bifurcation diagrams for $M\left(t\right)$, $E\left(t\right)$ and $R\left(t\right)$, respectively, with respect to ${\alpha}_{5}$; other parameter values are fixed as shown in Table 1 and $GF=0.3$. (

**d**) shows unbounded solutions of E and M with ${\alpha}_{5}=0.0079$, $\delta =1$ and $GF=0.3$, while keeping all the other parameters fixed as in Table 1 and initial conditions ${M}_{0}=1.2$, ${E}_{0}=3$, ${R}_{0}=1.2$ so that the conditions set in Theorem 2 and 3 are not met. To be more specific, ${\gamma}_{2}>0$, which is against conditions (A), (B) and (C) of Theorem 2. The $M\left(t\right)$ curve was amplified by a factor of 10 to visualize its behaviour with time and show that it grows unbounded with time while the R curve remain bounded because of conservation. (

**e**) illustrates a zoom of the unstable dynamics of (

**d**) for a short time $t\in [0,2000]$ without scaling. (

**f**) shows the stable time evolution of $M\left(t\right)$, $E\left(t\right)$ and $R\left(t\right)$ with ${\alpha}_{5}=0.0079$, $\delta =1$ and $GF=0.3$ with initial condition (${M}_{0},{E}_{0},{R}_{0}$) = (0.4782, 0.594, 1.8616). With these parameters, the model converges to the steady state (${M}^{*},{E}^{*},{R}^{*}$) = (0.7255, 1.4184, 2.0449).

**Figure 5.**Parameter sensitivities for system (1) corresponding to 5% increase and decrease in parameters. (

**a**,

**b**) show sensitivity of ${M}^{*}$ after 5 % increase and decrease in parameters, respectively; (

**c**,

**d**) show sensitivity of ${E}^{*}$ corresponding to 5% increase and decrease, respectively. (

**e**,

**f**) show sensitivity ${R}^{*}$ to 5% increase and decrease in parameters, respectively.

**Figure 6.**Parameter sensitivities for system (1) corresponding to 10% increases and decrease in parameters. (

**a**,

**b**) show sensitivity of ${M}^{*}$ after 10 % increase and decrease in parameters, respectively, (

**c**,

**d**) show sensitivity of ${E}^{*}$ corresponding to 10% increases and decrease, respectively. (

**e**,

**f**) show sensitivity ${R}^{*}$ to 10% increases and decrease in parameters, respectively.

**Figure 7.**Correlation between parameter sensitivity of $M\left(t\right)$, $E\left(t\right)$ and $R\left(t\right)$ corresponding to $10\%$ increase in model parameters. (

**a**) shows the correction between $M\left(t\right)$ and $E\left(t\right)$, (

**b**) shows the correlation between $M\left(t\right)$ and $R\left(t\right)$ and (

**c**) shows the correlation between $E\left(t\right)$ and $R\left(t\right)$.

Parameter | Description | Value | Units | Reference |
---|---|---|---|---|

${\alpha}_{1}$ | Growth factors activation rate | 1 | ${\mathrm{s}}^{-1}$ | [17] |

$GF$ | Growth factors concentration | varies | U | [42] |

$\delta $ | Inhibition rate of M | 1.001 | ${\mathrm{s}}^{-1}$ | [17] |

${k}_{1}$ | Michaelis–Menten constant | 1 | U | [42] |

${\beta}_{1}$ | Activation rate of M | 1 | ${\mathrm{s}}^{-1}$ | [42] |

${\beta}_{2}$ | Activation of R protein family | 1 | U${}^{-1}{\mathrm{s}}^{-1}$ | [42] |

${\alpha}_{3}$ | Inhibition rate of R by E | 1 | U${}^{-1}{\mathrm{s}}^{-1}$ | Estimate |

${\alpha}_{2}$ | Inhibition rate of R by M | 1 | U${}^{-1}{\mathrm{s}}^{-1}$ | Estimate |

${\beta}_{3}$ | R baseline inhibition rate | 1 | U${\mathrm{s}}^{-1}$ | Estimate |

${R}_{T}$ | Total concentration of R | 5 | U | [40] |

${\alpha}_{4}$ | E self activation rate | 0.02 | U${\mathrm{s}}^{-1}$ | [17] |

${\beta}_{4}$ | Activation rate E by M | 0.02 | ${\mathrm{s}}^{-1}$ | [17] |

${\beta}_{5}$ | R baseline inhibition rate | 1 | U${\mathrm{s}}^{-1}$ | Estimate |

${\alpha}_{6}$ | E constitutive activation rate | 0.001 | ${\mathrm{s}}^{-1}$ | [40] |

${\alpha}_{8}$ | Michaelis–Menten constant | 0.92 | U | [42] |

${\alpha}_{5}$ | Inhibition rate of E by R | 0.01 | U${}^{-1}{\mathrm{s}}^{-1}$ | [42] |

${k}_{{r}_{1}}$ | Michaelis–Menten constant | 0.05 | U | [42] |

${k}_{{r}_{2}}$ | Michelis–Menten constant | 1 | U | [42] |

${k}_{{r}_{3}}$ | Michaelis–Menten constant | 1 | U | [42] |

**Table 2.**Percentage changes in the steady-state values ${M}^{*}$, ${E}^{*}$ and ${R}^{*}$ after a 5% increase in parameter values of the non-linear model (1).

5% Increase in Parameter | % Change in ${\mathit{M}}^{*}$ | % Change in ${\mathit{E}}^{*}$ | % Change in ${\mathit{R}}^{*}$ |
---|---|---|---|

${k}_{1}$ | −10.330 | −10.186 | 0.7016 |

$\delta $ | −2.5797 | −2.6158 | −0.0973 |

${\alpha}_{6}$ | −2.3573 | −6.6279 | 1.7049 |

${\alpha}_{4}$ | 1.6213 | 4.5563 | −1.1907 |

${\alpha}_{8}$ | 1.3212 | 3.7135 | −0.96101 |

${\beta}_{4}$ | −1.4816 | −4.1652 | 1.0916 |

${k}_{{r}_{3}}$ | −3.6855 | −10.361 | 2.78 |

${\alpha}_{5}$ | 3.6754 | 10.331 | 0.9486 |

${\beta}_{2}$ | 0.9689 | 2.7222 | 0.1993 |

${R}_{T}$ | 1.1586 | 3.2559 | −0.8912 |

${\beta}_{3}$ | 0.4682 | 1.3150 | 0.0671 |

${k}_{{r}_{1}}$ | −2.1313 | −5.9929 | −0.73 16 |

${\alpha}_{2}$ | −4.4144 | −12.411 | −1.5102 |

${\alpha}_{3}$ | 0.0008 | 0.00217 | −0.0044 |

${\beta}_{5}$ | −0.50358 | −1.4171 | 0.2834 |

${k}_{{r}_{2}}$ | −1.3792 | −3.8783 | −0.5012 |

$GF$ | 8.2181 | 8.6153 | −0.4477 |

${\beta}_{1}$ | 4.8355 | 5.0458 | −0.30391 |

**Table 3.**Percentage changes in the steady-state values ${M}^{*}$, ${E}^{*}$ and ${R}^{*}$ after a 5% decrease in parameter values of the non-linear model (1).

5% Decrease in Parameter | % Change in ${\mathit{M}}^{*}$ | % Change in ${\mathit{E}}^{*}$ | % Change in ${\mathit{R}}^{*}$ |
---|---|---|---|

${k}_{1}$ | 15.379 | 16.574 | −0.74247 |

$\delta $ | 3.0549 | 3.1573 | −0.23624 |

${\alpha}_{6}$ | 2.7989 | 7.8667 | −1.9380 |

${\alpha}_{4}$ | −1.5130 | −4.2539 | 1.0604 |

${\alpha}_{8}$ | −1.3083 | −3.6785 | 0.91636 |

${\beta}_{4}$ | 1.4856 | 4.1750 | −1.1118 |

${k}_{{r}_{3}}$ | 4.6941 | 13.194 | −3.0811 |

${\alpha}_{5}$ | −3.3974 | −9.5522 | −1.1635 |

${\beta}_{2}$ | −0.90100 | −2.5340 | −0.34861 |

${R}_{T}$ | −1,1643 | −3.2747 | 0.77670 |

${\beta}_{3}$ | −0.47058 | −1.3243 | −0.22096 |

${k}_{{r}_{1}}$ | 2.2750 | 6.3943 | 0.56813 |

${\alpha}_{2}$ | 5.6528 | 15.889 | 1.4395 |

${\alpha}_{3}$ | −0.00076 | −2.0707 | 0.00434 |

${\beta}_{5}$ | 0.50506 | 1.4196 | −0.37425 |

${k}_{{r}_{2}}$ | 1.7225 | 4.8407 | 0.41722 |

$GF$ | −7.5303 | −7.4886 | 0.46275 |

${\beta}_{1}$ | −3.9715 | −4.0111 | 0.25848 |

**Table 4.**Percentage changes in the steady-state values ${M}^{*}$, ${E}^{*}$ and ${R}^{*}$ after a 10% increase in parameter values of the non-linear model (1).

10% Increase in Parameter | % Change in ${\mathit{M}}^{*}$ | % Change in ${\mathit{E}}^{*}$ | % Change in ${\mathit{R}}^{*}$ |
---|---|---|---|

${k}_{1}$ | −18.792 | −18.005 | 1.5896 |

$\delta $ | −4.9970 | −5.0238 | 0.25366 |

${\alpha}_{6}$ | −4.5627 | −12.828 | 3.5291 |

${\alpha}_{4}$ | 3.5450 | 9.9639 | −2.4070 |

${\alpha}_{8}$ | 2.7918 | 7.8469 | −1.9470 |

${\beta}_{4}$ | −3.1078 | −8.7368 | 2.3944 |

${k}_{{r}_{3}}$ | −6.9382 | −19.505 | 5.7536 |

${\alpha}_{5}$ | 8.0605 | 22.659 | 1.9911 |

${\beta}_{2}$ | 2.1256 | 5.9736 | 0.52836 |

${R}_{T}$ | 2.4269 | 6.8214 | −1.7297 |

${\beta}_{3}$ | 0.9800 | 2.7539 | 0.2105 |

${k}_{{r}_{1}}$ | −4.3112 | −12.122 | −1.4863 |

${\alpha}_{2}$ | −8.2964 | −23.324 | −3.0332 |

${\alpha}_{3}$ | 0.0017 | 0.0047 | −0.0093 |

${\beta}_{5}$ | −1.0545 | −2.9658 | 0.7076 |

${k}_{{r}_{2}}$ | −2.6078 | −7.3322 | −0.8934 |

$GF$ | 18.296 | 19.8 | −0.7814 |

${\beta}_{1}$ | 11.815 | 12.677 | −0.6242 |

**Table 5.**Percentage changes in steady-state values ${M}^{*}$, ${E}^{*}$ and ${R}^{*}$ after a 10% decrease in parameter values of the non-linear model (1).

10% Decrease in Parameter | % Change in ${\mathit{M}}^{*}$ | % Change in ${\mathit{E}}^{*}$ | % Change in ${\mathit{R}}^{*}$ |
---|---|---|---|

${k}_{1}$ | 43.284 | 51.402 | −0.90841 |

$\delta $ | 6.340764 | 6.6298 | −0.40674 |

${\alpha}_{6}$ | 5.8216 | 16.364 | −3.7086 |

${\alpha}_{4}$ | −2.7912 | −7.8475 | 2.0483 |

${\alpha}_{8}$ | −2.4764 | −6.9631 | 1.7939 |

${\beta}_{4}$ | 2.8264 | 7.9439 | −1.9852 |

${k}_{{r}_{3}}$ | 10.236 | 28.774 | −5.9032 |

${\alpha}_{5}$ | −6.2295 | −17.514 | −2.2045 |

${\beta}_{2}$ | −1.6616 | −4.6717 | −0.56234 |

${R}_{T}$ | −2.2183 | −6.2377 | 1.5914 |

${\beta}_{3}$ | −0.89676 | −2.5224 | −0.34685 |

${k}_{{r}_{1}}$ | 4.44444 | 12.49 | 1.1354 |

${\alpha}_{2}$ | 12.394 | 34.839 | 2.8958 |

${\alpha}_{3}$ | −0.0014110 | −0.0038423 | 0.0081792 |

${\beta}_{5}$ | 0.95936 | 2.6969 | −0.73757 |

${k}_{{r}_{2}}$ | 3.6812 | 10.3 47 | 0.94703 |

$GF$ | −13.836 | −13.470 | 1.0361 |

${\beta}_{1}$ | −7.0325 | −7.0261 | 0.41178 |

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**MDPI and ACS Style**

Mapfumo, K.Z.; Pagan’a, J.C.; Juma, V.O.; Kavallaris, N.I.; Madzvamuse, A.
A Model for the Proliferation–Quiescence Transition in Human Cells. *Mathematics* **2022**, *10*, 2426.
https://doi.org/10.3390/math10142426

**AMA Style**

Mapfumo KZ, Pagan’a JC, Juma VO, Kavallaris NI, Madzvamuse A.
A Model for the Proliferation–Quiescence Transition in Human Cells. *Mathematics*. 2022; 10(14):2426.
https://doi.org/10.3390/math10142426

**Chicago/Turabian Style**

Mapfumo, Kudzanayi Z., Jane C. Pagan’a, Victor Ogesa Juma, Nikos I. Kavallaris, and Anotida Madzvamuse.
2022. "A Model for the Proliferation–Quiescence Transition in Human Cells" *Mathematics* 10, no. 14: 2426.
https://doi.org/10.3390/math10142426