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2022 | Book

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The Quasispecies Equation and Classical Population Models

Authors: Raphaël Cerf, Joseba Dalmau

Publisher: Springer International Publishing

Book Series : Probability Theory and Stochastic Modelling

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
We are surrounded by a huge heterogeneity of living beings: insects, plants, animals and humans. Even the creatures belonging to the same species present an extraordinary variability. And yet, what we can see with the naked eye, is but a tiny fraction of the realm of the living.
Raphaël Cerf, Joseba Dalmau

Finite Genotype Space

Frontmatter
Chapter 2. The Quasispecies Equation
Abstract
In this chapter, we first introduce the general quasispecies equation.We then present the classical Perron–Frobenius theorem and apply it to solve the quasispecies equation in the case where the set of genotypes is finite, under some additional assumptions.
Raphaël Cerf, Joseba Dalmau
Chapter 3. Non-Overlapping Generations
Abstract
In this chapter, we present three models of population genetics, namely the Moran– Kingman model, the Galton–Watson model, and the Wright–Fisher model.We show how to relate them with the quasispecies equation. A fundamental feature shared by these three models is that their successive generations are non-overlapping, meaning that the whole population is fully resampled from one generation to the next.
Raphaël Cerf, Joseba Dalmau
Chapter 4. Overlapping Generations
Abstract
In this chapter, we present three further models of population genetics, namely the Eigen model, the continuous branching model and the Moran model. We show how to relate them with the quasispecies equation. A fundamental feature shared by these three models is that their successive generations are overlapping, in fact only one individual changes at a time.
Raphaël Cerf, Joseba Dalmau
Chapter 5. Probabilistic Representations
Abstract
For the six models of evolving populations presented so far, we showed how the quasispecies equation pops up in a suitable asymptotic regime. Typically, the quasispecies equation describes the equilibrium of the models in the limit of infinite population size.
Raphaël Cerf, Joseba Dalmau

The Sharp Peak Landscape

Frontmatter
Chapter 6. Long Chain Regime
Abstract
Ideally, we would like to have explicit formulas for the mean fitness λ and the equilibrium concentrations c* in terms of A and M. There is little hope of obtaining such explicit formulas in the general case, without further assumptions. Therefore, we focus on a particular choice of the set of genotypes E and of the mutation matrix M. Both for practical and historical reasons, we make the same choice as Eigen did.
Raphaël Cerf, Joseba Dalmau
Chapter 7. Error Threshold and Quasispecies
Abstract
In this chapter, we study the infinite system obtained at the limit in the long chain regime (see section 6.4).
Raphaël Cerf, Joseba Dalmau
Chapter 8. Probabilistic Derivation
Abstract
The formula for the quasispecies distribution given in definition 7.1 has been obtained by writing the limiting quasispecies equation in the long chain regime, and then solving the infinite triangular system (7.1) with the help of generating functions.
Raphaël Cerf, Joseba Dalmau
Chapter 9. Summation of the Series
Abstract
So far, we have obtained an explicit formula for the quasispecies distribution in terms of an infinite series. Our goal in this chapter is to compute the sum of this series and to obtain finite formulas for the quasispecies distribution.
Raphaël Cerf, Joseba Dalmau
Chapter 10. Error Threshold in Infinite Populations
Abstract
In the previous chapters, we considered the long chain regime
Raphaël Cerf, Joseba Dalmau

Error Threshold in Finite Populations

Frontmatter
Chapter 11. Phase Transition
Abstract
Our goal in this chapter is to prove convergence results in the long chain regime, which are the counterparts of the convergence results proved in chapter 3.
Raphaël Cerf, Joseba Dalmau
Chapter 12. Computer Simulations
Abstract
The results for the finite population models are supported by simulations.
Raphaël Cerf, Joseba Dalmau
Chapter 13. Heuristics
Abstract
In the Moran model, at most one individual changes at a given time. Therefore the number of master sequences in the population can either decrease by 1, stay stable, or increase by 1.
Raphaël Cerf, Joseba Dalmau
Chapter 14. Shape of the Critical Curve
Abstract
The critical curves described in theorems 11.1 and 11.2 appear from the simulations to be increasing, that is
Raphaël Cerf, Joseba Dalmau
Chapter 15. Framework for the Proofs
Abstract
The proofs of theorems 11.1 and 11.2, although different, share some common arguments. We present here a framework which works for both the Moran and the Wright–Fisher model.
Raphaël Cerf, Joseba Dalmau

Proof for Wright–Fisher

Frontmatter
Chapter 16. Strategy of the Proof
Abstract
The aim of part IV is to prove the theorem 11.2. The heuristics that we have given for the Moran model in the previous chapter do notwork in the case of the Wright–Fisher model, the reason being that theWright–Fisher model replaces the whole population at each time step, thus making the birth and death approximation impossible.
Raphaël Cerf, Joseba Dalmau
Chapter 17. The Non-Neutral Phase M
Abstract
The aim of this chapter is to study the dynamics of the process (Cn)n2N when the master sequence is present in the population.
Raphaël Cerf, Joseba Dalmau
Chapter 18. Mutation Dynamics
Abstract
In this chapter, we focus on the dynamics of a single individual undergoing mutation in the absence of the selection mechanism.
Raphaël Cerf, Joseba Dalmau
Chapter 19. The Neutral Phase N
Abstract
The aim of this chapter is to study the dynamics of the process (Cn)n2N when there is no master sequence present in the population.
Raphaël Cerf, Joseba Dalmau
Chapter 20. Synthesis
Abstract
In this short chapter, we gather the various asymptotic estimates developed in chapters 17 and 19 in the neutral and non-neutral phases in order to complete the proof of theorem 11.2.
Raphaël Cerf, Joseba Dalmau

Class-Dependent Fitness Landscapes

Frontmatter
Chapter 21. Generalized Quasispecies Distributions
Abstract
In chapter 7, we obtained explicit formulas for the distribution of the quasispecies on the sharp peak landscape. To get these formulas, two ingredients played a key role: the Hamming classes and the asymptotic regime.
Raphaël Cerf, Joseba Dalmau
Chapter 22. Error Threshold
Abstract
In the previous sections, we solved the recurrence relation (21.3) for z(0) = 1, and we explored some of the combinatorial properties of the solution.
Raphaël Cerf, Joseba Dalmau
Chapter 23. Probabilistic Representation
Abstract
We follow here the same road as in chapter 8, but we present only conjectures and possible directions for future investigations.
Raphaël Cerf, Joseba Dalmau
Chapter 24. Probabilistic Interpretations
Abstract
In this chapter we give two different formulas for the generalized quasispecies distribution. The first formula expresses the generalized quasispecies distribution as a functional of a Poisson random walk and the second one as a functional of a branching Poisson walk.
Raphaël Cerf, Joseba Dalmau
Chapter 25. Infinite Population Models
Abstract
As for the quasispecies distribution in chapter 10, we explore here the relationship between the infinite population models introduced in chapters 3 and 4 and the limit equation (21.1), as well as its (potentially many) solutions.
Raphaël Cerf, Joseba Dalmau

A Glimpse at the Dynamics

Frontmatter
Chapter 26. Deterministic Level
Abstract
In this chapter we show the relationship between the Moran–Kingman model and Eigen’s model with their infinite counterparts.
Raphaël Cerf, Joseba Dalmau
Chapter 27. From Finite to Infinite Population
Abstract
In this chapter we perform limiting procedures on both the Moran model and the Wright–Fisher model, obtaining law of large numbers type results for the trajectories of both processes. The limiting deterministic processes are the familiar Eigen model and the Moran–Kingman model respectively.
Raphaël Cerf, Joseba Dalmau
Chapter 28. Class-Dependent Landscapes
Abstract
In this chapter, we consider a fitness function that is both class-dependent and eventually constant.
Raphaël Cerf, Joseba Dalmau
Backmatter
Metadata
Title
The Quasispecies Equation and Classical Population Models
Authors
Raphaël Cerf
Joseba Dalmau
Copyright Year
2022
Electronic ISBN
978-3-031-08663-2
Print ISBN
978-3-031-08662-5
DOI
https://doi.org/10.1007/978-3-031-08663-2

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