The genealogy of a random sample of a population of organisms can be represented as a rooted binary tree. Population dynamics determine a distribution over sample genealogies. For large populations of constant size and in the absence of selection effects, the coalescent process of Kingman determines a suitable distribution. Neuhauser and Krone gave a stochastic model generalising the Kingman coalescent in a natural way to include the effects of selection. The model of Neuhauser and Krone determines a distribution over a class of graphs of randomly variable vertex number, known as ancestral selection graphs. Because vertices have associated scalar ages, realisations of the ancestral selection graph process have randomly variable dimensions. A Markov chain Monte Carlo method is used to simulate the posterior distribution for population parameters of interest. The state of the Markov chain Monte Carlo is a random graph, with random dimension and equilibrium distribution equal to the posterior distribution. The aim of the project is to determine if the data is informative of the selection parameter by fitting the model to synthetic data. / Foundation for Research Science and Technology Top Achiever Doctoral Scolarship

The genealogy of a random sample of a population of organisms can be represented as a rooted binary tree. Population dynamics determine a distribution over sample genealogies. For large populations of constant size and in the absence of selection effects, the coalescent process of Kingman determines a suitable distribution. Neuhauser and Krone gave a stochastic model generalising the Kingman coalescent in a natural way to include the effects of selection. The model of Neuhauser and Krone determines a distribution over a class of graphs of randomly variable vertex number, known as ancestral selection graphs. Because vertices have associated scalar ages, realisations of the ancestral selection graph process have randomly variable dimensions. A Markov chain Monte Carlo method is used to simulate the posterior distribution for population parameters of interest. The state of the Markov chain Monte Carlo is a random graph, with random dimension and equilibrium distribution equal to the posterior distribution. The aim of the project is to determine if the data is informative of the selection parameter by fitting the model to synthetic data. / Foundation for Research Science and Technology Top Achiever Doctoral Scolarship

The genealogy of a random sample of a population of organisms can be represented as a rooted binary tree. Population dynamics determine a distribution over sample genealogies. For large populations of constant size and in the absence of selection effects, the coalescent process of Kingman determines a suitable distribution. Neuhauser and Krone gave a stochastic model generalising the Kingman coalescent in a natural way to include the effects of selection. The model of Neuhauser and Krone determines a distribution over a class of graphs of randomly variable vertex number, known as ancestral selection graphs. Because vertices have associated scalar ages, realisations of the ancestral selection graph process have randomly variable dimensions. A Markov chain Monte Carlo method is used to simulate the posterior distribution for population parameters of interest. The state of the Markov chain Monte Carlo is a random graph, with random dimension and equilibrium distribution equal to the posterior distribution. The aim of the project is to determine if the data is informative of the selection parameter by fitting the model to synthetic data. / Foundation for Research Science and Technology Top Achiever Doctoral Scolarship

The genealogy of a random sample of a population of organisms can be represented as a rooted binary tree. Population dynamics determine a distribution over sample genealogies. For large populations of constant size and in the absence of selection effects, the coalescent process of Kingman determines a suitable distribution. Neuhauser and Krone gave a stochastic model generalising the Kingman coalescent in a natural way to include the effects of selection. The model of Neuhauser and Krone determines a distribution over a class of graphs of randomly variable vertex number, known as ancestral selection graphs. Because vertices have associated scalar ages, realisations of the ancestral selection graph process have randomly variable dimensions. A Markov chain Monte Carlo method is used to simulate the posterior distribution for population parameters of interest. The state of the Markov chain Monte Carlo is a random graph, with random dimension and equilibrium distribution equal to the posterior distribution. The aim of the project is to determine if the data is informative of the selection parameter by fitting the model to synthetic data. / Foundation for Research Science and Technology Top Achiever Doctoral Scolarship

Ce mémoire analyse l’espérance du temps de fixation conditionnellement à ce qu’elle se produise et la probabilité de fixation d’un nouvel allèle mutant dans des populations soumises à différents phénomènes biologiques en uti- lisant l’approche des processus ancestraux. Tout d’abord, l’article de Tajima (1990) est analysé et les différentes preuves y étant manquantes ou incomplètes sont détaillées, dans le but de se familiariser avec les calculs du temps de fixa- tion. L’étude de cet article permet aussi de démontrer l’importance du temps de fixation sur certains phénomènes biologiques. Par la suite, l’effet de la sé- lection naturelle est introduit au modèle. L’article de Mano (2009) cite un ré- sultat intéressant quant à l’espérance du temps de fixation conditionnellement à ce que celle-ci survienne qui utilise une approximation par un processus de diffusion. Une nouvelle méthode utilisant le processus ancestral est présentée afin d’arriver à une bonne approximation de ce résultat. Des simulations sont faites afin de vérifier l’exactitude de la nouvelle approche. Finalement, un mo- dèle soumis à la conversion génique est analysé, puisque ce phénomène, en présence de biais, a un effet similaire à celui de la sélection. Nous obtenons finalement un résultat analytique pour la probabilité de fixation d’un nouveau mutant dans la population. Enfin, des simulations sont faites afin de détermi- nerlaprobabilitédefixationainsiqueletempsdefixationconditionnellorsque les taux sont trop grands pour pouvoir les calculer analytiquement. / The expected time for fixation given its occurrence, and the probability of fixa- tion of a new mutant allele in populations subject to various biological phe- nomena are analyzed using the approach of the ancestral process. First, the paper of Tajima (1990) is analyzed, and the missing or incomplete proofs are fully worked out in this Master thesis in order to familiarize ourselves with calculations of fixation times. Our study of Tajima’s paper helps to show the importance of the fixation time in some biological phenomena. Thereafter, we extend the work of Tajima (1990) by introducing the effect of natural selec- tion in the model. Using a diffusion approximation, the work of Mano (2009) provides an interesting result about the expected time of fixation given its oc- currence. We derived an alternative method that uses an ancestral process that approximates well Mani’s result. Simulations are made to verify the accuracy ofthenewapproach.Finally,onemodelsubjecttogeneconversionisanalyzed, since this phenomenon, in the presence of bias, has a similar effect as selection. We deduce an analytical result for the probability of fixation of a new mutant in the population. Finally, simulations are made to determine the probability of fixation and the time of fixation given its occurrence when rates are too large to be calculated analytically.

Ce mémoire analyse l’espérance du temps de fixation conditionnellement à ce qu’elle se produise et la probabilité de fixation d’un nouvel allèle mutant dans des populations soumises à différents phénomènes biologiques en uti- lisant l’approche des processus ancestraux. Tout d’abord, l’article de Tajima (1990) est analysé et les différentes preuves y étant manquantes ou incomplètes sont détaillées, dans le but de se familiariser avec les calculs du temps de fixa- tion. L’étude de cet article permet aussi de démontrer l’importance du temps de fixation sur certains phénomènes biologiques. Par la suite, l’effet de la sé- lection naturelle est introduit au modèle. L’article de Mano (2009) cite un ré- sultat intéressant quant à l’espérance du temps de fixation conditionnellement à ce que celle-ci survienne qui utilise une approximation par un processus de diffusion. Une nouvelle méthode utilisant le processus ancestral est présentée afin d’arriver à une bonne approximation de ce résultat. Des simulations sont faites afin de vérifier l’exactitude de la nouvelle approche. Finalement, un mo- dèle soumis à la conversion génique est analysé, puisque ce phénomène, en présence de biais, a un effet similaire à celui de la sélection. Nous obtenons finalement un résultat analytique pour la probabilité de fixation d’un nouveau mutant dans la population. Enfin, des simulations sont faites afin de détermi- nerlaprobabilitédefixationainsiqueletempsdefixationconditionnellorsque les taux sont trop grands pour pouvoir les calculer analytiquement. / The expected time for fixation given its occurrence, and the probability of fixa- tion of a new mutant allele in populations subject to various biological phe- nomena are analyzed using the approach of the ancestral process. First, the paper of Tajima (1990) is analyzed, and the missing or incomplete proofs are fully worked out in this Master thesis in order to familiarize ourselves with calculations of fixation times. Our study of Tajima’s paper helps to show the importance of the fixation time in some biological phenomena. Thereafter, we extend the work of Tajima (1990) by introducing the effect of natural selec- tion in the model. Using a diffusion approximation, the work of Mano (2009) provides an interesting result about the expected time of fixation given its oc- currence. We derived an alternative method that uses an ancestral process that approximates well Mani’s result. Simulations are made to verify the accuracy ofthenewapproach.Finally,onemodelsubjecttogeneconversionisanalyzed, since this phenomenon, in the presence of bias, has a similar effect as selection. We deduce an analytical result for the probability of fixation of a new mutant in the population. Finally, simulations are made to determine the probability of fixation and the time of fixation given its occurrence when rates are too large to be calculated analytically.