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Metastability between the clicks of Muller's ratchet

Mauro Mariani, Etienne Pardoux, Aurélien Velleret

Abstract

We prove the existence and uniqueness of a quasi-stationary distribution for three stochastic processes derived from the model of Muller's ratchet. This model was invented with the aim of evaluating the limitations of an asexual reproduction mode in preventing the accumulation of deleterious mutations through natural selection alone. The main considered model is non-classical, as it is a stochastic diffusion evolving on an irregular set of infinite dimension with hard killing on an hyperplane. We are nonetheless able to prove exponential convergence in total variation to the quasi-stationary distribution even in this case. The parameters in this last convergence result are directly related to the core parameters of Muller's ratchet. The speed of convergence to the quasi-stationary distribution is deduced both for the infinite dimensional model and for approximations with a large yet finite number of potential mutations. Likewise, we give uniform moment estimates of the empirical distribution of mutations in the population under quasi-stationarity.

Metastability between the clicks of Muller's ratchet

Abstract

We prove the existence and uniqueness of a quasi-stationary distribution for three stochastic processes derived from the model of Muller's ratchet. This model was invented with the aim of evaluating the limitations of an asexual reproduction mode in preventing the accumulation of deleterious mutations through natural selection alone. The main considered model is non-classical, as it is a stochastic diffusion evolving on an irregular set of infinite dimension with hard killing on an hyperplane. We are nonetheless able to prove exponential convergence in total variation to the quasi-stationary distribution even in this case. The parameters in this last convergence result are directly related to the core parameters of Muller's ratchet. The speed of convergence to the quasi-stationary distribution is deduced both for the infinite dimensional model and for approximations with a large yet finite number of potential mutations. Likewise, we give uniform moment estimates of the empirical distribution of mutations in the population under quasi-stationarity.

Paper Structure

This paper contains 70 sections, 45 theorems, 351 equations.

Table of Contents

  1. Introduction
  2. General presentation
  3. The mathematical model of Muller's ratchet
  4. The individual-based model as a guideline
  5. The stochastic diffusion under consideration
  6. Notations
  7. Exponential quasi-stationarity results
  8. Exponential quasi-stationarity
  9. The discrete population case
  10. The finite dimensional case
  11. The infinite dimensional case
  12. Some crucial sets of conditions ensuring exponential quasi-stationarity
  13. Basic assumptions
  14. General theorems of convergence
  15. Outlook
  16. ...and 55 more sections

Key Result

Corollary 2.1.2

Assume PtCV. Then for any $t\ge 0$ and $\mu \in \mathcal{M}_1(\mathcal{X})$ such that $\langle \mu\, | \, h \rangle>0:$

Theorems & Definitions (75)

  • Remark 1.2.1
  • Remark 1.2.2
  • Definition 2.1.1
  • Corollary 2.1.2
  • Remark 2.1.3
  • Definition 2.1.4
  • Remark 2.1.5
  • Remark 2.1.6
  • Remark 2.1.7
  • Remark 2.2.1
  • ...and 65 more