# A Mutation-Selection Model with Recombination for General Genotypes

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*Steven N. Evans; David Steinsaltz; Kenneth W. Wachter*

The authors investigate a continuous time, probability measure-valued
dynamical system that describes the process of mutation-selection balance in
a context where the population is infinite, there may be infinitely many loci,
and there are weak assumptions on selective costs. Their model arises when they
incorporate very general recombination mechanisms into an earlier model of
mutation and selection presented by Steinsaltz, Evans and Wachter in 2005 and
take the relative strength of mutation and selection to be sufficiently small.
The resulting dynamical system is a flow of measures on the space of loci.

Each such measure is the intensity measure of a Poisson random measure on
the space of loci: the points of a realization of the random measure record
the set of loci at which the genotype of a uniformly chosen individual differs
from a reference wild type due to an accumulation of ancestral mutations. The
authors' motivation for working in such a general setting is to provide a basis
for understanding mutation-driven changes in age-specific demographic
schedules that arise from the complex interaction of many genes, and hence to
develop a framework for understanding the evolution of aging.

#### Table of Contents

# Table of Contents

## A Mutation-Selection Model with Recombination for General Genotypes

- Chapter 1. Introduction 110 free
- Chapter 2. Definition, Existence, and Uniqueness of the Dynamical System 1524
- Chapter 3. Equilibria 2736
- 3.1. Introductory example: One-dimensional systems 2837
- 3.2. Introductory example: Multiplicative selective costs 2938
- 3.3. Fréchet derivatives 3342
- 3.4. Existence of equilibria via perturbation 3443
- 3.5. Concave selective costs 3645
- 3.6. Concave selective costs: Existence and stability of equilibria 4251
- 3.7. Iterative computation of the minimal equilibrium 4554
- 3.8. Stable equilibria in the concave setting via perturbation 4554
- 3.9. Equilibria for demographic selective costs 4756

- Chapter 4. Mutation, Selection, and Recombination in Discrete Time 5160
- Chapter 5. Shattering and the Formulation of the Convergence Result 6372
- Chapter 6. Convergence with Complete Poissonization 7584
- Chapter 7. Supporting Lemmas for the Main Convergence Result 8190
- Chapter 8. Convergence of the Discrete Generation System 103112
- Appendix A. Results Cited in the Text 111120
- Bibliography 121130
- Index 125134 free
- Glossary of Notation 127136