Ancestral Selection Graph Research Articles

A coalescent dual process in a Moran model with genic selection

A coalescent dual process for a multi-type Moran model with genic selection is derived using a generator approach. This leads to an expansion of the transition functions in the Moran model and the Wright–Fisher diffusion process limit in terms of the transition functions for the coalescent dual. A graphical representation of the Moran model (in the spirit of Harris) identifies the dual as a strong dual process following typed lines backwards in time. An application is made to the harmonic measure problem of finding the joint probability distribution of the time to the first loss of an allele from the population and the distribution of the surviving alleles at the time of loss. Our dual process mirrors the Ancestral Selection Graph of [Krone, S. M., Neuhauser, C., 1997. Ancestral processes with selection. Theoret. Popul. Biol. 51, 210–237; Neuhauser, C., Krone, S. M., 1997. The genealogy of samples in models with selection. Genetics 145, 519–534], which allows one to reconstruct the genealogy of a random sample from a population subject to genic selection. In our setting, we follow [Stephens, M., Donnelly, P., 2002. Ancestral inference in population genetics models with selection. Aust. N. Z. J. Stat. 45, 395–430] in assuming that the types of individuals in the sample are known. There are also close links to [Fearnhead, P., 2002. The common ancestor at a nonneutral locus. J. Appl. Probab. 39, 38–54]. However, our methods and applications are quite different. This work can also be thought of as extending a dual process construction in a Wright–Fisher diffusion in [Barbour, A.D., Ethier, S.N., Griffiths, R.C., 2000. A transition function expansion for a diffusion model with selection. Ann. Appl. Probab. 10, 123–162]. The application to the harmonic measure problem extends a construction provided in the setting of a neutral diffusion process model in [Ethier, S.N., Griffiths, R.C., 1991. Harmonic measure for random genetic drift. In: Pinsky, M.A. (Ed.), Diffusion Processes and Related Problems in Analysis, vol. 1. In: Progress in Probability Series, vol. 22, Birkhäuser, Boston, pp. 73–81].

Duality, ancestral and diffusion processes in models with selection

The ancestral selection graph in population genetics was introduced by Krone and Neuhauser [Krone, S.M., Neuhauser, C., 1997. Ancestral process with selection. Theor. Popul. Biol. 51, 210–237] as an analogue of the coalescent genealogy of a sample of genes from a neutrally evolving population. The number of particles in this graph, followed backwards in time, is a birth and death process with quadratic death and linear birth rates. In this paper an explicit form of the probability distribution of the number of particles is obtained by using the density of the allele frequency in the corresponding diffusion model obtained by Kimura [Kimura, M., 1955. Stochastic process and distribution of gene frequencies under natural selection. Cold Spring Harbor Symposia on Quantitative Biology 20, 33–53]. It is shown that the process of fixation of the allele in the diffusion model corresponds to convergence of the ancestral process to its stationary measure. The time to fixation of the allele conditional on fixation is studied in terms of the ancestral process.

Conditional Gene Genealogies under Strong Purifying Selection

The ancestral selection graph, conditioned on the allelic types in the sample, is used to obtain a limiting gene genealogical process under strong selection. In an equilibrium, two-allele system with strong selection, neutral gene genealogies are predicted for random samples and for samples containing at most one unfavorable allele. Samples containing more than one unfavorable allele have gene genealogies that differ greatly from neutral predictions. However, they are related to neutral gene genealogies via the well-known Ewens sampling formula. Simulations show rapid convergence to limiting analytical predictions as the strength of selection increases. These results extend the idea of a soft selective sweep to deleterious alleles and have implications for the interpretation of polymorphism among disease-causing alleles in humans.

The Common Ancestor Process for a Wright-Fisher Diffusion

Rates of molecular evolution along phylogenetic trees are influenced by mutation, selection and genetic drift. Provided that the branches of the tree correspond to lineages belonging to genetically isolated populations (e.g., multi-species phylogenies), the interplay between these three processes can be described by analyzing the process of substitutions to the common ancestor of each population. We characterize this process for a class of diffusion models from population genetics theory using the structured coalescent process introduced by Kaplan et al. (1988) and formalized in Barton et al. (2004). For two-allele models, this approach allows both the stationary distribution of the type of the common ancestor and the generator of the common ancestor process to be determined by solving a one-dimensional boundary value problem. In the case of a Wright-Fisher diffusion with genic selection, this solution can be found in closed form, and we show that our results complement those obtained by Fearnhead (2002) using the ancestral selection graph. We also observe that approximations which neglect recurrent mutation can significantly underestimate the exact substitution rates when selection is strong. Furthermore, although we are unable to find closed-form expressions for models with frequency-dependent selection, we can still solve the corresponding boundary value problem numerically and then use this solution to calculate the substitution rates to the common ancestor. We illustrate this approach by studying the effect of dominance on the common ancestor process in a diploid population. Finally, we show that the theory can be formally extended to diffusion models with more than two genetic backgrounds, but that it leads to systems of singular partial differential equations which we have been unable to solve.

Integrating genealogy and epidemiology: The ancestral infection and selection graph as a model for reconstructing host virus histories

We model the genealogies of coupled haploid host–virus populations. Hosts reproduce and replace other hosts as in the Moran model. The virus can be transmitted between individuals of the same and succeeding generations. The epidemic model allows a selective advantage for susceptible over infected hosts. The coupled host–virus ancestry of a sample of hosts is embedded in a branching and coalescing structure that we call the Ancestral Infection and Selection Graph, a direct analogue to the Ancestral Selection Graph of Krone and Neuhauser [1997. Theoret. Population Biol. 51, 210–237]. We prove this and discuss various special cases. We show that the inter-host viral genealogy is a scaled coalescent. Using simulations, we compare the viral genealogy under this model to earlier published models and investigate the estimatability of the selection and infectious contact rates. We use simulations to compare the persistence of the disease with the time to the ultimate ancestor.

The structured ancestral selection graph and the many-demes limit.

We show that the unstructured ancestral selection graph applies to part of the history of a sample from a population structured by restricted migration among subpopulations, or demes. The result holds in the limit as the number of demes tends to infinity with proportionately weak selection, and we have also made the assumptions of island-type migration and that demes are equivalent in size. After an instantaneous sample-size adjustment, this structured ancestral selection graph converges to an unstructured ancestral selection graph with a mutation parameter that depends inversely on the migration rate. In contrast, the selection parameter for the population is independent of the migration rate and is identical to the selection parameter in an unstructured population. We show analytically that estimators of the migration rate, based on pairwise sequence differences, derived under the assumption of neutrality should perform equally well in the presence of weak selection. We also modify an algorithm for simulating genealogies conditional on the frequencies of two selected alleles in a sample. This permits efficient simulation of stronger selection than was previously possible. Using this new algorithm, we simulate gene genealogies under the many-demes ancestral selection graph and identify some situations in which migration has a strong effect on the time to the most recent common ancestor of the sample. We find that a similar effect also increases the sensitivity of the genealogy to selection.

Simulation of ancestral selection graphs for Monte Carlo integration

An ancestral selection graph is a realization of an genealogy-process model which incorporates natural selection. The space of ancestral graphs is a countable union of spaces of unequal dimensions. We give a Markov Chain Monte Carlo algorithm simulating ancestral selection graphs. Output can be used to estimate expectations for functions defined on the space of ancestral graphs.

Nonneutral Genealogical Structure and Algorithmic Enhancements of the Ancestral Selection Graph

Nonneutral Genealogical Structure and Algorithmic Enhancements of the Ancestral Selection Graph

Ancestral processes for non-neutral models of complex diseases

We consider non-neutral models for unlinked loci, where the fitness of a chromosome or individual is not multiplicative across loci. Such models are suitable for many complex diseases, where there are gene-interactions. We derive a genealogical process for such models, called the complex selection graph (CSG). This coalescent-type process is related to the ancestral selection graph, and is derived from the ancestral influence graph by considering the limit as the recombination rate between loci gets large. We analyse the CSG both theoretically and via simulation. The main results are that the gene-interactions do not produce linkage disequilibrium, but do produce dependencies in allele frequencies between loci. For small selection rates, the distributions of the genealogy and the allele frequencies at a single locus are well-approximated by their distributions under a single locus model, where the fitness of each allele is the average of the true fitnesses of that allele with respect to the distribution of alleles at other loci.

Perfect Simulation from Population Genetic Models with Selection

We consider using the ancestral selection graph (ASG) to simulate samples from population genetic models with selection. Currently the use of the ASG to simulate samples is limited. This is because the computational requirement for simulating samples increases exponentially with the selection rate and also due to needing to simulate a sample of size one from the population at equilibrium. For the only case where the distribution of a sample of size one is known, that of parent-independent mutations, more efficient simulation algorithms exist. We will show that by applying the idea of coupling from the past to the ASG, samples can be simulated from a general K-allele model without knowledge of the distribution of a sample of size one. Furthermore, the computation involved in generating such samples appears to be less than that of simulating the ASG until its ultimate ancestor. In particular, in the case of genic selection with parent-independent mutations, the computational requirement increases only quadratically with the selection rate. The algorithm is demonstrated by simulating samples at a microsatellite locus.

Simulation of 'hitch-hiking' genealogies.

An ancestral influence graph is derived, an analogue of the coalescent and a composite of Griffiths' (1991) two-locus ancestral graph and Krone and Neuhauser's (1997) ancestral selection graph. This generalizes their use of branching-coalescing random graphs so as to incorporate both selection and recombination into gene genealogies. Qualitative understanding of a 'hitch-hiking' effect on genealogies is pursued via diagrammatic representation of the genealogical process in a two-locus, two-allele haploid model. Extending the simulation technique of Griffiths and Tavare (1996), computational estimation of expected times to the most recent common ancestor of samples of n genes under recombination and selection in two-locus, two-allele haploid and diploid models are presented. Such times are conditional on sample configuration. Monte Carlo simulations show that 'hitch-hiking' is a subtle effect that alters the conditional expected depth of the genealogy at the linked neutral locus depending on a mutation-selection-recombination balance.

Most Recent Common Ancestor Probability Distributions in Gene Genealogies under Selection

A computational study is made of the conditional probability distribution for the allelic type of the most recent common ancestor in genealogies of samples of n genes drawn from a population under selection, given the initial sample configuration. Comparisons with the corresponding unconditional cases are presented. Such unconditional distributions differ from samples drawn from the unique stationary distribution of population allelic frequencies, known as Wright's formula, and are quantified. Biallelic haploid and diploid models are considered. A simplified structure for the ancestral selection graph of S. M. Krone and C. Neuhauser (1997, Theor. Popul. Biol. 51, 210–237) is enhanced further, reducing the effective branching rate in the graph. This improves efficiency of such a nonneutral analogue of the coalescent for use with computational likelihood-inference techniques.

Simulation of Selected Genealogies

Algorithms for generating genealogies with selection conditional on the sample configuration of n genes in one-locus, two-allele haploid and diploid models are presented. Enhanced integro-recursions using the ancestral selection graph, introduced by S. M. Krone and C. Neuhauser (1997, Theor. Popul. Biol. 51, 210–237), which is the non-neutral analogue of the coalescent, enables accessible simulation of the embedded genealogy. A Monte Carlo simulation scheme based on that of R. C. Griffiths and S. Tavaré (1996, Math. Comput. Modelling 23, 141–158), is adopted to consider the estimation of ancestral times under selection. Simulations show that selection alters the expected depth of the conditional ancestral trees, depending on a mutation–selection balance. As a consequence, branch lengths are shown to be an ineffective criterion for detecting the presence of selection. Several examples are given which quantify the effects of selection on the conditional expected time to the most recent common ancestor.

Genealogies and weak purifying selection.

The assumption that selection alters the genealogical tree of a sample of alleles from a population relative to the neutral expectation underlies several "tests of neutrality." Two recent papers have studied the effect of purifying selection; their suggestive but incomplete results indicate that, in the single site case, the shape of a gene genealogy for a locus may differ only from the neutral expectation. We verify this finding for weak selection using the "ancestral selection graph." We consider a wider range of models, including both a four-allele single-site model and an infinite-sites model. Our results confirm the previous claim for the symmetric-mutation single site model. We emphasize, however, that a neutral-seeming genealogy is consistent with detectable effects of selection on the distribution of allele frequences within the sample. With selection operating, the information about a sample cannot be reduced to the genealogy. As a result, a distinction needs to be made between the selected sites themselves, for which the genealogy offers insufficient information, and linked neutral variation. This distinction seems to have been overlooked in previous papers, yet it has significant implications for the interpretation of data on DNA sequence variation. In particular, it predicts that under purifying selection, the frequency spectrum of neutral mutations will not reflect the skew toward rare polymorphisms at replacement sites even if there is no recombination between them. We caution, however, that the effect of weak selection on the genealogy is specific to the model; a (more realistic) model of multiple linked sites could lead to a more distorted genealogy than is observed for a single site.

Ancestral Processes with Selection

In this paper, we show how to construct the genealogy of a sample of genes for a large class of models with selection and mutation. Each gene corresponds to a single locus at which there is no recombination. The genealogy of the sample is embedded in a graph which we call theancestral selection graph. This graph contains all the information about the ancestry; it is the analogue of Kingman's coalescent process which arises in the case with no selection. The ancestral selection graph can be easily simulated and we outline an algorithm for simulating samples. The main goal is to analyze the ancestral selection graph and to compare it to Kingman's coalescent process. In the case of no mutation, we find that the distribution of the time to the most recent common ancestor does not depend on the selection coefficient and hence is the same as in the neutral case. When the mutation rate is positive, we give a procedure for computing the probability that two individuals in a sample are identical by descent and the Laplace transform of the time to the most recent common ancestor of a sample of two individuals; we evaluate the first two terms of their respective power series in terms of the selection coefficient. The probability of identity by descent depends on both the selection coefficient and the mutation rate and is different from the analogous expression in the neutral case. The Laplace transform does not have a linear correction term in the selection coefficient. We also provide a recursion formula that can be used to approximate the probability of a given sample by simulating backwards along the sample paths of the ancestral selection graph, a technique developed by Griffiths and Tavaré (1994).

The genealogy of samples in models with selection.

We introduce the genealogy of a random sample of genes taken from a large haploid population that evolves according to random reproduction with selection and mutation. Without selection, the genealogy is described by Kingman's well-known coalescent process. In the selective case, the genealogy of the sample is embedded in a graph with a coalescing and branching structure. We describe this graph, called the ancestral selection graph, and point out differences and similarities with Kingman's coalescent. We present simulations for a two-allele model with symmetric mutation in which one of the alleles has a selective advantage over the other. We find that when the allele frequencies in the population are already in equilibrium, then the genealogy does not differ much from the neutral case. This is supported by rigorous results. Furthermore, we describe the ancestral selection graph for other selective models with finitely many selection classes, such as the K-allele models, infinitely-many-alleles models. DNA sequence models, and infinitely-many-sites models, and briefly discuss the diploid case.