# Early Evolution in Cancer: A Mathematical Support for Pathological and Genomic Evidence in Clear Cell Renal Cell Carcinoma

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## Abstract

**:**

## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Clinical Context

#### 2.2. The Hawk-Dove Game

^{2}(v − c)/2 + α(1 − α)v + 0(1 − α)α + (1 − α)

^{2}v/2.

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{A}), is composed of A-cells and B-cells; the 3-type population game, denoted by Γ₃(v,c, x

_{A}, x

_{E}), is composed of A-cells, B-cells and E-cells. The parameters are v, c, and the proportions x

_{I}of I-cells (I = A, B, E). In each game, a strategy specifies the probability of behaving aggressively when meeting each type of cell present in the population. Let α

_{I}be the probability of behaving aggressively when facing an I-cell (I = A, B, E), and U

_{I}be the payoff of an I-cell. We give the ESS in each of the three games when the A-cells are the cells with the largest payoff, and the E-cells are the cells with the smallest payoff. That is, U

_{A}> U

_{B}> U

_{E}.

_{A}*= v/c is the only ESS [13]. The expected payoff of an A-cell in the ESS, denoted by U

_{A}*, is given by U

_{A}* = (1 − v/c) v/2.

_{A}), a cell meets A-cells and B-cells. The ESS, denoted (α

_{A}**, α

_{B}**), depends on the proportion of A-cells [15]. It is given by

_{A}

^{**}, α

_{B}

^{**}) = (0, (v/c)(n − 1)/(n − nx

_{A}− 1)) if x

_{A}< z

_{A}

(0,1) if z

_{A}< x

_{A}< z

_{A}+1/n

(((n − 1)v/c − n + nx

_{A})/(nx

_{A}− 1),1) if x

_{A}> z

_{A}+ 1/n,

_{A}=(1 − v/c)(1 − 1/n). The expected payoff of the A-cells in the ESS, denoted by U

_{A}**(x

_{A}), is given by:

_{A}**(x

_{A}) = (v/2)(1 − v/c) + (v

^{2}/2c)((2n(1 − x

_{A}) − 1)/(n − nx

_{A}− 1)) if x

_{A}< z

_{A}

(v/2)(1 − v/c) + (v/2c)((v(n − 1) + cn(1 − x

_{A}))/(n − 1)) if z

_{A}≤ x

_{A}≤ z

_{A}+ 1/n

_{A})/(nx

_{A}− 1)) if x

_{A}> z

_{A}+ 1/n.

_{A}= z

_{A}and starts decreasing thereafter.

_{A}, x

_{E}), the ESS, denoted (α

_{A}***, α

_{B}***, α

_{E}***), exists if and only if x

_{A}< z

_{A}and x

_{E}< z

_{E}[9], where z

_{E}=(v/c)(1 − 1/n). We have (α

_{A}***, α

_{B}***, α

_{E}***) = (0,((n − 1)v/c − nx

_{E})/(n − nx

_{A}-nx

_{E}− 1),1).

_{A}***(x

_{A},x

_{E}), is given by:

_{A}***(x

_{A},x

_{E}) = v/2(1 − v/c) + v

^{2}/2c((2n(1 − x

_{A}− x

_{E}) − 1)/(n − nx

_{A}− nx

_{E}− 1)) − v/2nx

_{E}/((n − nx

_{A}− nx

_{E}− 1)(n − 1)).

_{A}is, the larger the expected payoff of each A-cell is. By contrast, the larger x

_{E}is, the smaller the expected payoff of each A-cell is.

_{A}) and Γ₃(v,c, x

_{A}, x

_{E}) in which all types of cells obtain the same payoff (Result 1).

**Proposition**

**1.**

_{A}), there is no ESS with U

_{A}= U

_{B}.

**Proposition**

**2.**

**Proof.**

_{A},x

_{E}), there is no ESS with U

_{A}= U

_{B}= U

_{C}. □

_{A}**(y

_{A}) > U

_{A}***(y

_{A},x

_{E}) > U

_{A}*. That is, U

_{A}* < U

_{A}***(y

_{A},x

_{E}) and U

_{A}* < U

_{A}**(y

_{A}) (Result 2) and U

_{A}**(y

_{A}) > U

_{A}***(y

_{A},x

_{E}) (Result 3).

**Proposition**

**3.**

**Proposition**

**4.**

**Proof.**

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**Figure 1.**Example of a model of temporal evolution in clear cell renal cell carcinomas. The size of the circles reflects tumor aggressiveness (level of clonal fitness). Early evolution exemplifies the transition from linear (homogeneity (VHL monodriver clone)) to branching (high heterogeneity (VHL, PBRM1, and BAP-1 driven clones)) models. However, late evolution shows the transition from branching (high heterogeneity (VHL, PBRM1, and BAP-1 driven clones)) to punctuated (low heterogeneity (VHL, BAP-1 driven clones)) models due to the expansion of an aggressive clone.

Hawk | Dove | |
---|---|---|

Hawk | (v − c)/2 | v |

Dove | 0 | v/2 |

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**MDPI and ACS Style**

Laruelle, A.; Manini, C.; López, J.I.; Rocha, A.
Early Evolution in Cancer: A Mathematical Support for Pathological and Genomic Evidence in Clear Cell Renal Cell Carcinoma. *Cancers* **2023**, *15*, 5897.
https://doi.org/10.3390/cancers15245897

**AMA Style**

Laruelle A, Manini C, López JI, Rocha A.
Early Evolution in Cancer: A Mathematical Support for Pathological and Genomic Evidence in Clear Cell Renal Cell Carcinoma. *Cancers*. 2023; 15(24):5897.
https://doi.org/10.3390/cancers15245897

**Chicago/Turabian Style**

Laruelle, Annick, Claudia Manini, José I. López, and André Rocha.
2023. "Early Evolution in Cancer: A Mathematical Support for Pathological and Genomic Evidence in Clear Cell Renal Cell Carcinoma" *Cancers* 15, no. 24: 5897.
https://doi.org/10.3390/cancers15245897