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A Countable-Type Branching Process Model for the Tug-of-War Cancer Cell Dynamics

Ren-Yi Wang, Marek Kimmel

TL;DR

A version of the model in which a driver mutation pushes the type of the cell L -units up, while a passenger mutation pulls it 1-unit down is introduced, leading to the result that under driver dominance, the type transition process escapes to infinity, while under passenger dominance, it leads to a limit distribution.

Abstract

We consider a time-continuous Markov branching process of proliferating cells with a countable collection of types. Among-type transitions are inspired by the Tug-of-War process introduced in McFarland et al. as a mathematical model for competition of advantageous driver mutations and deleterious passenger mutations in cancer cells. We introduce a version of the model in which a driver mutation pushes the type of the cell $L$-units up, while a passenger mutation pulls it $1$-unit down. The distribution of time to divisions depends on the type (fitness) of cell, which is an integer. The extinction probability given any initial cell type is strictly less than $1$, which allows us to investigate the transition between types (type transition) in an infinitely long cell lineage of cells. The analysis leads to the result that under driver dominance, the type transition process escapes to infinity, while under passenger dominance, it leads to a limit distribution. Implications in cancer cell dynamics and population genetics are discussed.

A Countable-Type Branching Process Model for the Tug-of-War Cancer Cell Dynamics

TL;DR

A version of the model in which a driver mutation pushes the type of the cell L -units up, while a passenger mutation pulls it 1-unit down is introduced, leading to the result that under driver dominance, the type transition process escapes to infinity, while under passenger dominance, it leads to a limit distribution.

Abstract

We consider a time-continuous Markov branching process of proliferating cells with a countable collection of types. Among-type transitions are inspired by the Tug-of-War process introduced in McFarland et al. as a mathematical model for competition of advantageous driver mutations and deleterious passenger mutations in cancer cells. We introduce a version of the model in which a driver mutation pushes the type of the cell -units up, while a passenger mutation pulls it -unit down. The distribution of time to divisions depends on the type (fitness) of cell, which is an integer. The extinction probability given any initial cell type is strictly less than , which allows us to investigate the transition between types (type transition) in an infinitely long cell lineage of cells. The analysis leads to the result that under driver dominance, the type transition process escapes to infinity, while under passenger dominance, it leads to a limit distribution. Implications in cancer cell dynamics and population genetics are discussed.
Paper Structure (21 sections, 7 theorems, 76 equations, 3 figures, 1 table)

This paper contains 21 sections, 7 theorems, 76 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model Setup
  3. Continuous-time Markov Chain
  4. Reduced Process: Driver-Passenger Relation
  5. Embedded Branching Process
  6. Results
  7. Extinction Probabilities
  8. Type Transition Process
  9. Driver Dominance $(\nu\leq \mu L)$
  10. Passenger Dominance $(\nu > \mu L)$
  11. Implication of the Original Process
  12. Discussion
  13. Acknowledgements
  14. Declarations
  15. Competing Interests
  16. ...and 6 more sections

Key Result

Proposition 1

$\forall i,j \in \mathbb{Z},\lim_{n\to\infty}[(M^{n})_{i,j}]^{1/n}=2$, which implies partial extinction probabilities coincides with global extinction probabilities, $(\mathbf{q} = \tilde{\mathbf{q}} < \underline{1})$. $\mathbf{q}\leq \tilde{\mathbf{q}}< \underline{1}$.

Figures (3)

  • Figure 1: Computed extinction probabilities computed using the algorithm in hautphenne2013extinction section $3.1$. The set of taboo types used to compute $q_{i}$ is $\mathbb{Z}\setminus\{i-100L,\cdots,i+100L\}$.
  • Figure 2: Type transition process simulations with fitness boxplots and average fitness curves under driver dominance. We conduct $1500$ simulations for each parameter specification and $J$ is the number of jumps in each simulation. $J$ is chosen to be large enough to observe the limiting behavior of the type transition process. Simulations within a parameter specification have identical number of jumps and the final time displayed in the last row of legend is the minimal terminal time of all $1500$ simulations. The terminal times are truncated to avoid long decimal part.
  • Figure 3: Type transition process simulations with fitness boxplots and average fitness curve under passenger dominance. We conduct $1500$ simulations for each parameter specification and $J$ is the number of jumps in each simulation. $J$ is chosen to be large enough to observe the limiting behavior of the type transition process. Simulations within a parameter specification have identical number of jumps and the final time displayed in the last row of legend is the minimal terminal time of all $1500$ simulations. The terminal times are truncated to avoid long decimal part.

Theorems & Definitions (16)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • proof
  • Lemma 2
  • proof
  • proof
  • Lemma 3
  • proof
  • ...and 6 more