This monograph studies a series of mathematical models of the evolution of a population under mutation and selection. Its starting point is the quasispecies equation, a general non-linear equation which describes the mutation-selection equilibrium in Manfred Eigen’s famous quasispecies model. A detailed analysis of this equation is given under the assumptions of finite genotype space, sharp peak landscape, and class-dependent fitness landscapes. Different probabilistic representation formulae are derived for its solution, involving classical combinatorial quantities like Stirling and Euler numbers.
It is shown how quasispecies and error threshold phenomena emerge in finite population models, and full mathematical proofs are provided in the case of the Wright–Fisher model. Along the way, exact formulas are obtained for the quasispecies distribution in the long chain regime, on the sharp peak landscape and on class-dependent fitness landscapes.
Finally, several other classical population models are analyzed, with a focus on their dynamical behavior and their links to the quasispecies equation.
This book will be of interest to mathematicians and theoretical ecologists/biologists working with finite population models.
Author(s): Raphaël Cerf, Joseba Dalmau
Series: Probability Theory and Stochastic Modelling, 102
Publisher: Springer
Year: 2022
Language: English
Pages: 242
City: Cham
Tags: Population Geenetics, Quasispecies Equation, Wright-Fisher Model, Moran-Kingman Model, Galton-Watson Model, Eigen Model
Foreword
Contents
Chapter 1 Introduction
Part I Finite Genotype Space
Overview of Part I
Chapter 2 The Quasispecies Equation
2.1 The Equilibrium Equation
2.2 The Perron–Frobenius Theorem
2.3 Solutions
Chapter 3 Non-Overlapping Generations
3.1 The Moran–Kingman Model
3.2 The Galton–Watson Model
3.3 The Wright–Fisher Model
Chapter 4 Overlapping Generations
4.1 The Eigen Model
4.2 The Continuous Branching Model
4.3 The Moran Model
Chapter 5 Probabilistic Representations
5.1 Stopped RandomWalk
5.2 Stopped Branching Process
Part II The Sharp Peak Landscape
Overview of Part II
Chapter 6 Long Chain Regime
6.1 Genotypes and Mutations
6.2 Sharp Peak Fitness
6.3 Hamming Classes
6.4 Limit Equation
Chapter 7 Error Threshold and Quasispecies
7.1 The Error Threshold
7.2 The Distribution of the Quasispecies
Chapter 8 Probabilistic Derivation
8.1 Asymptotics of c*
8.2 Limit of the Mutant Walk Representation
8.3 The Poisson Random Walk
8.4 Formal Derivation
Chapter 9 Summation of the Series
9.1 Stirling Numbers
9.2 Eulerian Numbers
9.3 Combinatorial Identities
Chapter 10 Error Threshold in Infinite Populations
10.1 The Moran–Kingman Model
10.2 The Eigen Model
Part III Error Threshold in Finite Populations
Overview of Part III
Chapter 11 Phase Transition
11.1 The Moran Model
11.2 The Wright–Fisher Model
Chapter 12 Computer Simulations
Chapter 13 Heuristics
13.1 A Simplified Process
13.2 A Renewal Argument
13.3 Persistence Time
Chapter 14 Shape of the Critical Curve
14.1 Critical Curve for the Moran Model
14.2 Critical Curve for the Wright–Fisher Model
Chapter 15 Framework for the Proofs
15.1 Candidate Limits for Moran
15.2 Candidate Limits for Wright–Fisher
Part IV Proof for Wright–Fisher
Overview of Part IV
Chapter 16 Strategy of the Proof
16.1 Main Ideas
16.2 Invariant Probability Measure
16.3 Upper Bounds
Chapter 17 The Non-Neutral Phase M
17.1 Large Deviations Principle
17.2 Perturbed Dynamical System
17.3 Time away from the Fixed Points
17.4 Reaching the Quasispecies
17.5 Escape from the Quasispecies
Chapter 18 Mutation Dynamics
18.1 Binary Process of Differences
18.2 Hamming Class Dynamics
18.3 Time away from the Equilibrium
18.4 Reaching the Equilibrium
18.5 Escape from the Equilibrium
Chapter 19 The Neutral Phase N
19.1 Ancestral Lines
19.2 Monotonicity and Correlations
19.3 Time away from the Disorder
19.4 Reaching the Disorder
19.5 Escape from the Disorder
Chapter 20 Synthesis
20.1 The Quasispecies Regime
20.2 The Disordered Regime
Part V Class-Dependent Fitness Landscapes
Overview of Part V
Chapter 21 Generalized Quasispecies Distributions
21.1 Class-Dependent Fitness Landscapes
21.2 Up-Down Coefficients
21.3 Re-Expansion
Chapter 22 Error Threshold
22.1 Eventually Constant Fitness Functions
22.2 Error Threshold
22.3 Further Solutions
Chapter 23 Probabilistic Representation
23.1 Asymptotics of Perron–Frobenius
23.2 Mutant Walk Representation
23.3 Computation of the Limit
23.4 Rearranging the Sums
Chapter 24 Probabilistic Interpretations
24.1 Poisson Random Walk
24.2 The Branching Poisson Walk
Chapter 25 Infinite Population Models
25.1 The Moran–Kingman Model
25.2 The Eigen Model
Part VI A Glimpse at the Dynamics
Overview of Part VI
Chapter 26 Deterministic Level
26.1 The Moran–Kingman Model
26.2 The Eigen Model
Chapter 27 From Finite to Infinite Population
27.1 From Moran’s to Eigen’s Model
27.2 From Wright–Fisher’s to Moran–Kingman’s Model
Chapter 28 Class-Dependent Landscapes
28.1 Moran Model
28.2 The Wright–Fisher Model
Appendix A Markov Chains and classical results
A.1 Monotonicity
A.2 Construction of Markov Processes
A.3 Lumping
A.4 The FKG Inequality
A.5 Hoeffding’s Inequality
References
Index
