Ancestral Selection Graph Research Articles

The mutation process on the ancestral line under selection

We consider the Moran model of population genetics with two types, mutation, and selection, and investigate the line of descent of a randomly-sampled individual from a contemporary population. We trace this ancestral line back into the distant past, far beyond the most recent common ancestor of the population (thus connecting population genetics to phylogeny), and analyse the mutation process along this line.To this end, we use the pruned lookdown ancestral selection graph (Lenz et al., 2015), which consists of a set of potential ancestors of the sampled individual at any given time. Relative to the neutral case (that is, without selection), we obtain a general bias towards the beneficial type, an increase in the beneficial mutation rate, and a decrease in the deleterious mutation rate. This sheds new light on previous analytical results. We discuss our findings in the light of a well-known observation at the interface of phylogeny and population genetics, namely, the difference in the mutation rates (or, more precisely, mutation fluxes) estimated via phylogenetic methods relative to those observed in pedigree studies.

The ancestral selection graph for a [formula omitted]-asymmetric Moran model

Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanisms, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Λ-reproduction here means that a whole fraction of the population is replaced at a reproductive event. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Provided the measure are ordered stochastically, we can couple them. This allows us to construct the central object of this paper, the Λ−asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinuous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.

Lines of descent in a Moran model with frequency-dependent selection and mutation

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 ASGs 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 populations, as well as for moderate and weak selection limits.

Moran models and Wright–Fisher diffusions with selection and mutation in a one-sided random environment

AbstractConsider a two-type Moran population of size N with selection and mutation, where the selective advantage of the fit individuals is amplified at extreme environmental conditions. Assume selection and mutation are weak with respect to N, and extreme environmental conditions rarely occur. We show that, as $N\to\infty$ , the type frequency process with time sped up by N converges to the solution to a Wright–Fisher-type SDE with a jump term modeling the effect of the environment. We use an extension of the ancestral selection graph (ASG) to describe the genealogical picture of the model. Next, we show that the type frequency process and the line-counting process of a pruned version of the ASG satisfy a moment duality. This relation yields a characterization of the asymptotic type distribution. We characterize the ancestral type distribution using an alternative pruning of the ASG. Most of our results are stated in annealed and quenched form.

Lines of descent in the deterministic mutation–selection model with pairwise interaction

We consider the mutation--selection differential equation with pairwise interaction (or, equivalently, the diploid mutation--selection equation) and establish the corresponding ancestral process, which is a random tree and a variant of the ancestral selection graph. The formal relation to the forward model is given via duality. To make the tree tractable, we prune branches upon mutations, thus reducing it to its informative parts. The hierarchies inherent in the tree are encoded systematically via tripod trees with weighted leaves; this leads to the stratified ancestral selection graph. The latter also satisfies a duality relation with the mutation--selection equation. Each of the dualities provides a stochastic representation of the solution of the differential equation. This allows us to connect the equilibria and their bifurcations to the long-term behaviour of the ancestral process. Furthermore, with the help of the stratified ancestral selection graph, we obtain explicit results about the ancestral type distribution in the case of unidirectional mutation.

General selection models: Bernstein duality and minimal ancestral structures

$\Lambda$-Wright--Fisher processes provide a robust framework to describe the type-frequency evolution of an infinite neutral population. We add a polynomial drift to the corresponding stochastic differential equation to incorporate frequency-dependent selection. A decomposition of the drift allows us to approximate the solution of the stochastic differential equation by a sequence of Moran models. The genealogical structure underlying the Moran model leads in the large population limit to a generalisation of the ancestral selection graph of Krone and Neuhauser. Building on this object, we construct a continuous-time Markov chain and relate it to the forward process via a new form of duality, which we call Bernstein duality. We adapt classical methods based on the moment duality to determine the time to absorption and criteria for the accessibility of the boundaries; this extends a recent result by Gonz\'alez Casanova and Span\`o. An intriguing feature of the construction is that the same forward process is compatible with multiple backward models. In this context we introduce suitable notions for minimality among the ancestral processes and characterise the corresponding parameter sets. In this way we recover classic ancestral structures as minimal ones.

Asymptotic behaviour of sampling and transition probabilities in coalescent models under selection and parent dependent mutations

The results in this paper provide new information on asymptotic properties of classical models: the neutral Kingman coalescent under a general finite-alleles, parent-dependent mutation mechanism, and its generalisation, the ancestral selection graph. Several relevant quantities related to these fundamental models are not explicitly known when mutations are parent dependent. Examples include the probability that a sample taken from a population has a certain type configuration, and the transition probabilities of their block counting jump chains. In this paper, asymptotic results are derived for these quantities, as the sample size goes to infinity. It is shown that the sampling probabilities decay polynomially in the sample size with multiplying constant depending on the stationary density of the Wright-Fisher diffusion and that the transition probabilities converge to the limit of frequencies of types in the sample.

A dual process for the coupled Wright\u2013Fisher diffusion

The coupled Wright–Fisher diffusion is a multi-dimensional Wright–Fisher diffusion for multi-locus and multi-allelic genetic frequencies, expressed as the strong solution to a system of stochastic differential equations that are coupled in the drift, where the pairwise interaction among loci is modelled by an inter-locus selection. In this paper, an ancestral process, which is dual to the coupled Wright–Fisher diffusion, is derived. The dual process corresponds to the block counting process of coupled ancestral selection graphs, one for each locus. Jumps of the dual process arise from coalescence, mutation, single-branching, which occur at one locus at the time, and double-branching, which occur simultaneously at two loci. The coalescence and mutation rates have the typical structure of the transition rates of the Kingman coalescent process. The single-branching rate not only contains the one-locus selection parameters in a form that generalises the rates of an ancestral selection graph, but it also contains the two-locus selection parameters to include the effect of the pairwise interaction on the single loci. The double-branching rate reflects the particular structure of pairwise selection interactions of the coupled Wright–Fisher diffusion. Moreover, in the special case of two loci, two alleles, with selection and parent independent mutation, the stationary density for the coupled Wright–Fisher diffusion and the transition rates of the dual process are obtained in an explicit form.

Haldane’s formula in Cannings models: the case of moderately weak selection

We introduce a Cannings model with directional selection via a paintbox construction and establish a strong duality with the line counting process of a new \emph{Cannings ancestral selection graph} in discrete time. This duality also yields a formula for the fixation probability of the beneficial type. Haldane's formula states that for a single selectively advantageous individual in a population of haploid individuals of size $N$ the prob\-ability of fixation is asymptotically (as $N\to \infty$) equal to the selective advantage of haploids $s_N$ divided by half of the offspring variance. For a class of offspring distributions within Kingman attraction we prove this asymptotics for sequences $s_N$ obeying $N^{-1} \ll s_N \ll N^{-1/2} $, which is a regime of "moderately weak selection". It turns out that for $ s_N \ll N^{-2/3} $ the Cannings ancestral selection graph is so close to the ancestral selection graph of a Moran model that a suitable coupling argument allows to play the problem back asymptotically to the fixation probability in the Moran model, which can be computed explicitly.

The Wright–Fisher model with efficiency

In this article, we propose a Wright–Fisher model with two types of individuals: the inefficient individuals, those who need more resources to reproduce and can have a higher growth rate, and the efficient individuals. In this model, the total amount of resource N is fixed, and the population size varies randomly depending on the number of efficient individuals. We show that, as N increases, the frequency process of efficient individuals converges to a diffusion which is a generalization of the Wright–Fisher diffusion with selection. The genealogy of this model is given by a branching–coalescing process that we call the Ancestral Selection/Efficiency Graph, and that is an extension of the Ancestral Selection Graph (Krone and Neuhauser, 1997a,b). The main contribution of this paper is that, in evolving populations, inefficiency can arise as a promoter of selective advantage and not necessarily as a trade-off.

A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent.

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 equilibria) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error threshold phenomenon from a genealogical point of view. Next, we characterise the ancestral type distribution by means of the pruned lookdown ancestral selection graph and study its properties at equilibrium. We also provide an alternative characterisation in terms of a piecewise-deterministic Markov process. Throughout, emphasis is on the underlying dualities as well as on explicit results.

Lines of Descent Under Selection

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 a stationary distribution back into the distant past, thus leading to the stationary distribution of ancestral types. We illustrate the results by applying them to a prototype model for the error threshold phenomenon.

Branching Brownian motion and selection in the spatial $\Lambda$-Fleming–Viot process

We ask the question when will natural selection on a gene in a spatially structured population cause a detectable trace in the patterns of genetic variation observed in the contemporary population?. We focus on the situation in which 'neighbourhood size', that is the effective local population density, is small. The genealogy relating individuals in a sample from the population is embedded in a spatial version of the ancestral selection graph and through applying a diffusive scaling to this object we show that whereas in dimensions at least three, selection is barely impeded by the spatial structure, in the most relevant dimension, $d=2$, selection must be stronger (by a factor of $\log(1/\mu)$ where $\mu$ is the neutral mutation rate) if we are to have a chance of detecting it. The case $d=1$ was handled in Etheridge et al. (2015). The mathematical interest is that although the system of branching and coalescing lineages that forms the ancestral selection graph converges to a branching Brownian motion, this reflects a delicate balance of a branching rate that grows to infinity and the instant annullation of almost all branches through coalescence caused by the strong local competition in the population.

Common ancestor type distribution: A Moran model and its deterministic limit

In a Moran model with population size N, two types, mutation and selection, let hkN be the probability that the common ancestor is fit, given that the current number of fit individuals is k. First, we express hkN in terms of the tail probabilities of an appropriate random variable LN. Next, we show that, when N tends to infinity (without any rescaling of parameters or time), LN converges to a geometric random variable. We also obtain a formula for h(x), the limit of hkN when k/N tends to x∈(0,1). In a second step, we describe two ways of pruning the ancestral selection graph (ASG) leading to the notions of relevant ASG and of pruned lookdown ASG (pruned LD-ASG). We use these objects to provide graphical derivations of the aforementioned results. In particular, we show that LN is distributed as the asymptotic number of lines in the relevant ASG and as the stationary number of lines in the pruned LD-ASG. Finally, we construct an asymptotic version of the pruned LD-ASG providing a graphical interpretation of the function h.

The fixation time of a strongly beneficial allele in a structured population

For a beneficial allele which enters a large unstructured population and eventually goes to fixation, it is known that the time to fixation is approximately $2\log (\alpha )/\alpha $ for a large selection coefficient $\alpha $. For a population that is distributed over finitely many colonies, with migration between these colonies, we detect various regimes of the migration rate $\mu $ for which the fixation times have different asymptotics as $\alpha \to \infty $. If $\mu $ is of order $\alpha $, the allele fixes (as in the spatially unstructured case) in time $\sim 2\log (\alpha )/\alpha $. If $\mu $ is of order $\alpha ^\gamma , 0\leq \gamma \leq 1$, the fixation time is $\sim (2 + (1-\gamma )\Delta ) \log (\alpha )/\alpha $, where $\Delta $ is the number of migration steps that are needed to reach all other colonies starting from the colony where the beneficial allele appeared. If $\mu = 1/\log (\alpha )$, the fixation time is $\sim (2+S)\log (\alpha )/\alpha $, where $S$ is a random time in a simple epidemic model. The main idea for our analysis is to combine a new moment dual for the process conditioned to fixation with the time reversal in equilibrium of a spatial version of Neuhauser and Krone’s ancestral selection graph.

The common ancestor type distribution of a $\Lambda$-Wright-Fisher process with selection and mutation

Using graphical methods based on a `lookdown' and pruned version of the {\em ancestral selection graph}, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from Lenz, Kluth, Baake, and Wakolbinger (Theor. Pop. Biol., 103 (2015), 27-37) to the case of heavy-tailed offspring, directed by a reproduction measure $\Lambda$. The representation is in terms of the equilibrium tail probabilities of the line-counting process $L$ of the graph. We identify a strong pathwise Siegmund dual of $L$, and characterise the equilibrium tail probabilities of $L$ in terms of hitting probabilities of the dual process.

Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution

In a (two-type) Wright–Fisher diffusion with directional selection and two-way mutation, let x denote today’s frequency of the beneficial type, and given x, let h(x) be the probability that, among all individuals of today’s population, the individual whose progeny will eventually take over in the population is of the beneficial type. Fearnhead (2002) and Taylor (2007) obtained a series representation for h(x). We develop a construction that contains elements of both the ancestral selection graph and the lookdown construction and includes pruning of certain lines upon mutation. Besides being interesting in its own right, this construction allows a transparent derivation of the series coefficients of h(x) and gives them a probabilistic meaning.

The ancestral selection graph under strong directional selection

The ancestral selection graph (ASG) was introduced by Neuhauser and Krone (1997) in order to study populations of constant size which evolve under selection. Coalescence events, which occur at rate 1 for every pair of lines, lead to joint ancestry. In addition, splitting events in the ASG at rate α, the scaled selection coefficient, produce possible ancestors, such that the real ancestor depends on the ancestral alleles. Here, we use the ASG in the case without mutation in order to study fixation of a beneficial mutant. Using our main tool, a reversibility property of the ASG, we provide a new proof of the fact that a beneficial allele fixes roughly in time (2logα)/α if α is large.

A coalescent dual process in a Moran model with genic selection, and the lambda coalescent limit

The genealogical structure of neutral populations in which reproductive success is highly-skewed has been the subject of many recent studies. Here we derive a coalescent dual process for a related class of continuous-time Moran models with viability selection. In these models, individuals can give birth to multiple offspring whose survival depends on both the parental genotype and the brood size. This extends the dual process construction for a multi-type Moran model with genic selection described in Etheridge and Griffiths (2009). We show that in the limit of infinite population size the non-neutral Moran models converge to a Markov jump process which we call the Λ-Fleming–Viot process with viability selection and we derive a coalescent dual for this process directly from the generator and as a limit from the Moran models. The dual is a branching-coalescing process similar to the Ancestral Selection Graph which follows the typed ancestry of genes backwards in time with real and virtual lineages. As an application, the transition functions of the non-neutral Moran and Λ-coalescent models are expressed as mixtures of the transition functions of the dual process.

The conditional ancestral selection graph with strong balancing selection

Using a heuristic separation-of-time-scales argument, we describe the behavior of the conditional ancestral selection graph with very strong balancing selection between a pair of alleles. In the limit as the strength of selection tends to infinity, we find that the ancestral process converges to a neutral structured coalescent, with two subpopulations representing the two alleles and mutation playing the role of migration. This agrees with a previous result of Kaplan et al., obtained using a different approach. We present the results of computer simulations to support our heuristic mathematical results. We also present a more rigorous demonstration that the neutral conditional ancestral process converges to the Kingman coalescent in the limit as the mutation rate tends to infinity.