We study ancestral structures for the two-type Moran model with mutation and frequency-dependent selection under the nonlinear dominance or fittest-type-wins scheme. Under appropriate conditions, both lead, in distribution, to the same type-frequency process. Reasoning through the mutations on the ancestral selection graph (ASG), we develop the corresponding killed and pruned lookdown ASG and use them to determine the present and ancestral type distributions. To this end, we establish factorial moment dualities to the Moran model and a relative. We extend the results to the diffusion limit and present applications for finite population size as well as moderate and weak selection.
We consider a population of haploid individuals reproducing sexually, i.e. for which the genome of each individual is a random mixture of the genome of its two parents. We assume that initially one individual carries a mutation at one locus, and that individuals carrying this mutation have an advantage regarding genome transmission. Our aim is to study the long time effect of this mutation on the genetic composition of the population, when population size is large.
The appearance of cancer in a tissue is thought to be the result of two or more successive mutations. We propose a stochastic model that allows for an exact computation of the distribution of the waiting time for a second mutation. This models the time of appearance of the first cancerous cell in a tissue. Our model is an alternative to the Moran model with mutations.
The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, and the applications to the study of the speed of convergence to stationarity. The Moran model we consider is a non-reversible in general, continuous-time Markov chain with an unknown stationary distribution. Specifically, we consider $N$ individuals such that each one of them is of one type among $K$ possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of indiv…
We consider the Moran process with two populations competing under an iterated Prisoners' Dilemma in the presence of mutation, and concentrate on the case where there are multiple Evolutionarily Stable Strategies. We perform a complete bifurcation analysis of the deterministic system which arises in the infinite population size. We also study the Master equation and obtain asymptotics for the invariant distribution and metastable switching times for the stochastic process in the case of large but finite population. We also show that the stochastic system has asymmetries in the form of a skew f…
We study the common ancestor type distribution in a $2$-type Moran model with population size $N$, mutation and selection, and in the deterministic limit regime arising in the former when $N$ tends to infinity, without any rescaling of parameters or time. In the finite case, we express the common ancestor type distribution as a weighted sum of combinatorial terms, and we show that the latter converges to an explicit function. Next, we recover the previous results through pruning of the ancestral selection graph (ASG). The notions of relevant ASG, finite and asymptotic pruned lookdown ASG permi…
We consider a Moran model with two allelic types, mutation and selection. In this work, we study the behaviour of the proportion of fit individuals when the size of the population tends to infinity, without any rescaling of parameters or time. We first prove that the latter converges, uniformly in compacts in probability, to the solution of an ordinary differential equation, which is explicitly solved. Next, we study the stability properties of its equilibrium points. Moreover, we show that the fluctuations of the proportion of fit individuals, after a proper normalization, satisfy a uniform c…
The goal of this article is to study the limit of the empirical distribution induced by a mutation-selection multi-allelic Moran model, whose dynamic is given by a continuous-time irreducible Markov chain. The rate matrix driving the mutation is assumed irreducible and the selection rates are assumed uniformly bounded. The paper is divided into two parts. The first one deals with processes with general selection rates. For this case we are able to prove the propagation of chaos in $\mathbb{L}^p$ over the compacts, with speed of convergence of order $1/\sqrt{N}$. Further on, we consider a speci…
We study a population model of fixed size undergoing strong selection where individuals accumulate beneficial mutations, namely the Moran model with selection. In a specific setting with strong selection, Schweinsberg showed that the genealogy of the population is described by the so-called Bolthausen-Sznitman's coalescent. In this paper we sophisticate the model by splitting the population into two adversarial subgroups, that can be interpreted as two different alleles, one of which has a weak selective advantage over the other. We show that the proportion of disadvantaged individuals converg…
Genetic contribution of advantaged ancestors in the biparental Moran model -- finite selection
Université Paris-Saclay, AgroParisTech, INRAE, UMR MIA Paris-Saclay（巴黎萨克雷大学、农业巴黎技术学院、法国国家农业食品环境研究院）；Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay（巴黎萨克雷大学、法国国家科学研究中心）
We extend the Moran model with single-crossover recombination to include general recombination and mutation. We show that, in the case without resampling, the expectations of products of marginal processes defined via partitions of sites form a closed hierarchy, which is exhaustively described by a finite system of differential equations. One thus has the exceptional situation of moment closure in a nonlinear system. Surprisingly, this property is lost when resampling (i.e., genetic drift) is included.
The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of dire…
Some population is made of n individuals that can be of p possible species (or types). The update of the species abundance occupancies is from a Moran mutational model designed by Karlin and McGregor in 1967. We first study the equilibrium species counts as a function of n, p and the total mutation probability $ν$ before considering various asymptotic regimes on n, p and $ν$. Running title: KMG Model with Mutations.
We consider the classical Wright-Fisher model with mutation and selection. Mutations occur independently in each locus, and selection is performed according to the sharp peak landscape. In the asymptotic regime studied in [3], a quasispecies is formed. We find explicitly the distribution of this quasispecies, which turns out to be the same distribution as for the Moran model.
We study the genealogical distance of two randomly chosen individuals in a population that evolves according to a two type Moran model with mutation and selection. We prove that this distance is stochastically smaller than the corresponding distance in the neutral model, when the population size is large. Moreover, we prove convergence of the genealogical distance under selection to the distance in the neutral case, when the system is in equilibrium and the selection parameter tends to infinity.
We consider the classic Moran process modeling the spread of genetic mutations, as extended to structured populations by Lieberman et al.\ (Nature, 2005). In this process, individuals are the vertices of a connected graph $G$. Initially, there is a single mutant vertex, chosen uniformly at random. In each step, a random vertex is selected for reproduction with a probability proportional to its fitness: mutants have fitness $r>1$, while non-mutants have fitness 1. The vertex chosen to reproduce places a copy of itself to a uniformly random neighbor in $G$, replacing the individual that was ther…
Scaling limits for Moran processes on metric strategy spaces: an Eulerian derivation of pure replicator and Fleming--Viot measure-valued PDEs
Department of Mathematics and Applications “R. Caccioppoli”, University of Naples Federico II（那不勒斯费德里科二世大学应用数学系）；Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino（都灵理工大学数学科学系）；Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari（巴里理工大学机械、数学与管理系）；Dipartimento di Scienze e Tecnologie Biologiche ed Ambientali Centro Ecotekne（莱切大学生物与环境科学与技术学院生态技术中心）
The Fleming-Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algeb…
We reconsider the deterministic haploid mutation-selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (random) structures inherent in this deterministic model. In a first step, we obtain a representation of the deterministic equation's solution (and, in particular, of its equilibrium) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error thresho…
We consider a multi-type Moran model (in continuous time) with selection and type-dependent mutation. This paper is concerned with the evolution of genealogical information forward in time. For this purpose we define and analytically characterize a path-valued Markov process that contains in its state at time $t$ the extended ancestral lines (adding genealogical distances) of the population alive at time $t$. The main result is a representation for the conditional distribution of the extended ancestral lines of a subpopulation alive at a fixed time $T$ (present time) given the type information…
We review recent progress on ancestral processes related to mutation-selection models, both in the deterministic and the stochastic setting. We mainly rely on two concepts, namely, the killed ancestral selection graph and the pruned lookdown ancestral selection graph. The killed ancestral selection graph gives a representation of the type of a random individual from a stationary population, based upon the individual's potential ancestry back until the mutations that define the individual's type. The pruned lookdown ancestral selection graph allows one to trace the ancestry of individuals from …
We consider the Moran model in continuous time with two types, mutation, and selection. We concentrate on the ancestral line and its stationary type distribution. Building on work by Fearnhead (J. Appl. Prob. 39 (2002), 38-54) and Taylor (Electron. J. Probab. 12 (2007), 808-847), we characterise this distribution via the fixation probability of the offspring of all individuals of favourable type (regardless of the offspring's types). We concentrate on a finite population and stay with the resulting discrete setting all the way through. This way, we extend previous results and gain new insight …
We consider an infinite-dimensional system of stochastic differential equations describing the evolution of type frequencies in a large population. The type of an individual is the number of deleterious mutations it carries, where fitness of individuals carrying k mutations is decreased by αk for some α>0. Along the individual lines of descent, new mutations accumulate at rate λper generation, and each of these mutations has a probability γper generation to disappear. While the case γ=0 is known as (the Fleming-Viot version of) Muller's ratchet, the case γ>0 is associated with compensatory mut…
The site frequency spectrum (SFS) is a popular summary statistic of genomic data. While the SFS of a constant-sized population undergoing neutral mutations has been extensively studied in population genetics, the rapidly growing amount of cancer genomic data has attracted interest in the spectrum of an exponentially growing population. Recent theoretical results have generally dealt with special or limiting cases, such as considering only cells with an infinite line of descent, assuming deterministic tumor growth, or taking large-time or large-population limits. In this work, we derive exact e…
