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1992 | Book

Introduction Read chapter Read first chapter

Branching Processes and Neutral Evolution

Author: Ziad Taïb

Publisher: Springer Berlin Heidelberg

Book Series : Lecture Notes in Biomathematics

Included in: Professional Book Archive

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About this book

The Galton-Watson branching process has its roots in the problem of extinction of family names which was given a precise formulation by F. Galton as problem 4001 in the Educational Times (17, 1873). In 1875, an attempt to solve this problem was made by H. W. Watson but as it turned out, his conclusion was incorrect. Half a century later, R. A. Fisher made use of the Galton-Watson process to determine the extinction probability of the progeny of a mutant gene. However, it was J. B. S. Haldane who finally gave the first sketch of the correct conclusion. J. B. S. Haldane also predicted that mathematical genetics might some day develop into a "respectable branch of applied mathematics" (quoted in M. Kimura & T. Ohta, Theoretical Aspects of Population Genetics. Princeton, 1971). Since the time of Fisher and Haldane, the two fields of branching processes and mathematical genetics have attained a high degree of sophistication but in different directions. This monograph is a first attempt to apply the current state of knowledge concerning single-type branching processes to a particular area of mathematical genetics: neutral evolution. The reader is assumed to be familiar with some of the concepts of probability theory, but no particular knowledge of branching processes is required. Following the advice of an anonymous referee, I have enlarged my original version of the introduction (Chapter Zero) in order to make it accessible to a larger audience. G6teborg, Sweden, November 1991.

Table of Contents

Frontmatter
Chapter Zero. Introduction
Abstract
The last two decades have witnessed an increasing interest in models for the behaviour of what we shall call labelled populations. If we restrict ourselves to asexual populations, then a labelled population can be described as one where every individual carries a label which she does or does not transmit to her offspring. An individual not receiving her mother’s label is assumed to carry a completely new label neither currently nor previously encountered in the population. Such an individual is called a mutant. Individuals carrying distinct labels are assumed to behave in the same way, at least with respect to reproduction and viability. This provides models for the study of the genetical structure of haploid populations subject to neutral mutations at some single locus under the infinite alleles hypothesis of population genetics. A first model of this kind can be traced back, in an embryonic form, to [Kimura & Crow, 1964]. Since then the topic has been subject to intensive research.
Ziad Taïb
Chapter One. The Construction of the Process
Abstract
The modern construction of the general branching process relies heavily upon the Ulam-Harris family history space. The following version is mainly based on [Jagers & Nerman, 1984a] (for another construction cf. Chapter Ten). The idea of using random characteristics was introduced in [Jagers, 1974] and was later on developed in [Nerman, 1981] and [Jagers & Nerman, 1984a] to become a useful tool in the study of various aspects of branching populations counted in different manners.
Ziad Taïb
Chapter Two. Labelled Branching Processes
Abstract
In this chapter we give the formal definition of a labelled branching process and start to study one of its aspects: the fate of the label carried by the ancestor. Using quite well known results we are able to describe the extinction and the growth of the subpopulation of individuals carrying this label. Some of the formulae given in this chapter will be made use of in §4.2.
Ziad Taïb
Chapter Three. The Number of Distinct Labels
Abstract
Let us now consider the number of labels encountered in the population in several senses. Thanks to the idea of counting subpopulations of a general branching process by means of random characteristics, we are able to study not only the total number of distinct labels ever seen in the process up to some time point t, but also the development of the cardinalities of label groups exhibiting some required properties. As a natural example of such counts, we shall consider the set of labels represented by at least one living individual at t. This will be referred to as the set of “present labels” at t. Let Nt stand for the total number of labels and M, for the number of present labels at time t. In the Galton-Watson case [Pakes, 1984] contains results about the convergence of b-nE[M] and the a.s. convergence of Mn/Zn where Zn and Mn stand for the size of and the number of different labels in the nth generation respectively. As usual, b stands for the mean reproduction of an individual. Similar results will be derived in this chapter in the context of our more general model.
Ziad Taïb
Chapter Four. The Label Process
Abstract
An alternative approach to the study of some aspects of labelled branching processes is to consider the embedded unlabelled process formulated directly in terms of the labels themselves thought of as “individuals” in a population of labels. The resulting branching process, let us call it the label process, turns out to be completely characterized by the underlying one. This is not very surprising since nothing new is involved. In Chapter Ten we shall discuss the logical foundation of this and other similar embeddings. In §4.1 we present an attempt to actually construct the label process and to check its “good behaviour” in the sense of Chapter One. In the remaining sections we describe some aspects of the labelled population that are more conveniently formulated and studied in terms of the label process.
Ziad Taïb
Chapter Five. The Limiting Stable Case
Abstract
A well established fact about supercritical branching population processes is that they either become extinct or explode at an exponential rate. Still it is possible to imagine some kind of asymptotic stability even in this case, namely that of the composition, in several senses, of a population according to some appropriate limit composition laws, even though the population size continues to grow. This idea has been exploited by many authors. A very useful way of thinking about such composition laws is to view them as related to some randomly sampled individual chosen in some convenient manner from a very old branching population (see below). Probabilistic statements concerning different aspects of the pedigree of this sampled individual can then be done using the probability measure P the stable pedigree measure, which in a sense summarizes the above mentioned limit laws, and which is induced by the sampling mechanism and the life law. In the present chapter we make some definitions and introduce some notation to be used in the coming chapters as a straightforward tool in the study of retrospective aspects of labelled branching processes. The whole chapter is more or less based on [Jagers & Nerman 1984a] and [Nerman & Jagers, 1984].
Ziad Taïb
Chapter Six. The Process of Mutant Ancestors
Abstract
Under this heading we will be concerned by some applications of what can be called the stable pedigree calculus. By this we mean that we consider some functionals of the pedigree of a randomly sampled individual (from a very old well behaved branching process) written in the terminology of the preceding chapter. Those functionals are such that the same results are valid under P̃ as under P̃ l (see Chapter Five). For that reason it is motivated to call the randomly sampled individual RSI according to the convention of the preceding chapter. Occasionally, the precise limit theorems formulated in terms of the underlying process will be given.
Ziad Taïb
Chapter Seven. A Measure of Relatedness
Abstract
Most of the results in this chapter will be formulated in terms of an RSI (cf. Chapter Five) but we will also describe what happens when the randomly sampled individual is chosen among living individuals.
Ziad Taïb
Chapter Eight. Infinite Sites Labels
Abstract
In this chapter the idea of a label will be made more precise. As has already been pointed out, models with infinitely many labels have their motivation in the fine structure of the gene. This fine structure seems to suggest that a label may be viewed as a very long sequence of specific positions (sites). Each such position can be occupied by some letter chosen from a four—letter alphabet (the four nucleotides A,C,T and G). Models adopting this special point of view go back to [Kimura, 1969] and [Kimura, 1971] and are often named infinite sites models.
Ziad Taïb
Chapter Nine. Related Individuals
Abstract
As pointed out in [Jacquard, 1974], the starting point of all genetical thought must have been the fact that related individuals resemble one another. A question that many geneticists ask is “what information about an individual x can we gain from information about another individual y related to x in a given manner”? From the point of view of the evolutionist it is relevant to ask the inverse question “How is x likely to be related to y, given some knowledge about the genetic constitution of x and y or at least about their relative genetic relationship”? It may, for example, happen that we are in possession of their complete nucleotide sequences and want to use this knowledge to make some inference about the backward time since the two lineages they belong to may have diverged.
Ziad Taïb
Chapter Ten. Embeddings in Branching Processes
Abstract
Some aspects of branching processes are better understood in terms of suitable groups of individuals rather than in terms of the individuals themselves. A convenient name which may be given to such groups is generalized individuals. The label process studied in Chapter Four is but one example of a situation where it is fruitful to think in terms of such generalized individuals. Other such examples can be found in [Doney, 1976] and in [Broberg, 1987].
Ziad Taïb
Chapter Eleven. Extensions
Abstract
In this chapter we present some extensions of the results presented in the preceding chapters as well as some suggestions for further research.
Ziad Taïb
Backmatter
Metadata
Title
Branching Processes and Neutral Evolution
Author
Ziad Taïb
Copyright Year
1992
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-51536-1
Print ISBN
978-3-540-55529-2
DOI
https://doi.org/10.1007/978-3-642-51536-1