Times on trees, and the age of an allele

In this paper we consider the genealogy of a random sample of n chromosomes from a panmictic population which has evolved with constant size N over many generations. We address two related problems. First we describe how genealogical information may be usefully partitioned into information on the events (mutations and coalescences) which occur in the genealogy, and the times between these events. We show that the distribution of the times given information on the events is particularly simple and describe how this can considerably reduce the computational burden when performing inference for these times. Second we investigate the effect on the genealogy of conditioning on a single mutation having occurred during the ancestry of the sample. In particular we use results from the first part of the paper to derive explicit formulae for the density of the age of a mutant allele, conditional on its frequency in either a sample or the population.     

On the genealogy of a sample of neutral rare alleles

This paper concerns the genealogical structure of a sample of chromosomes sharing a neutral rare allele. We suppose that the mutation giving rise to the allele has only happened once in the history of the entire population, and that the allele is of known frequency q in the population. Within a coalescent framework C. Wiuf and P. Donnelly (1999, Theor. Popul. Biol. 56, 183-201) derived an exact analysis of the conditional genealogy but it is inconvenient for applications. Here, we develop an approximation to the exact distribution of the conditional genealogy, including an approximation to the distribution of the time at which the mutation arose. The approximations are accurate for frequencies q<5-10%. In addition, a simple and fast simulation scheme is constructed. We consider a demography parameterized by a d-dimensional vector alpha=(alpha(1), em leader, alpha(d)). It is shown that the conditional genealogy and the age of the mutation have distributions that depend on a=qalpha and q only, and that the effect of q is a linear scaling of times in the genealogy; if q is doubled, the lengths of all branches in the genealogy are doubled. The theory is exemplified in two different demographies of some interest in the study of human evolution: (1) a population of constant size and (2) a population of exponentially decreasing size (going backward in time).     

Site frequency spectra from genomic SNP surveys

Genomic survey data now permit an unprecedented level of sensitivity in the detection of departures from canonical evolutionary models, including expansions in population size and selective sweeps. Here, we examine the effects of seemingly subtle differences among sampling distributions on goodness of fit analyses of site frequency spectra constructed from single nucleotide polymorphisms. Conditioning on the observation of exactly two alleles in a random sample results in a site frequency spectrum that is independent of the scaled rate of neutral substitution (θ). Other sampling distributions, including conditioning on a single mutational event in the sample genealogy or randomly selecting a single mutation from a genealogy with multiple mutations, have distinct site frequency spectra that show highly significant departures from the predictions of the biallelic model. Some aspects of data filtering may contribute to significant departures of site frequency spectra from expectation, apart from any violation of the standard neutral model.     

DNA-PKcs mutations in dogs and horses: allele frequency and association with neoplasia

Previously, spontaneous genetic immunodeficiencies in mice, Arabian foals, and recently in Jack Russell terriers have been ascribed to defects in DNA-PKcs (catalytic subunit of the DNA dependent protein kinase) expression. In severe combined immunodeficiency (SCID) foals, a 5 bp deletion at codon 9480 results in a frameshift and a 967 amino acid deletion from the C terminus (including the entire PI3 kinase domain) and an unstable mutant protein. In SCID mice, a single base pair mutation results in a premature stop codon and deletion of 83 amino acids; as in SCID foals, the mutant protein is unstable. Here, we define the mutation within the canine DNA-PKcs gene that results in SCID. In this case, a point mutation results in a stop codon at nucleotide 10,828 and premature termination at a position 517 amino acids before the normal C terminus resulting in a functionally null allele. Thus, this is the third documentation of a spontaneous germline mutation in the C terminus of DNA-PKcs. Emerging data implicate DNA repair factors as potential tumor suppressors. Here, we have ascertained the carrier frequency of the defective DNA-PKcs genes in Arabian horses and in Jack Russell terriers. Our data indicate (in good agreement with a previous report) that the carrier frequency of the equine SCID allele is approximately 8%; in contrast, the carrier frequency of the canine SCID allele is less than 1.1%. We also assessed the frequency of the equine SCID allele in a series of 295 tumors from Arabian horses. We find a statistically significant correlation between the development of a virally induced tumor (sarcoid) and heterozygosity for the equine SCID allele. These data provide further support for an emerging consensus: that DNA-PK may normally act as a tumor suppressor through its caretaker role in maintaining chromosomal stability.     

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.     

Recombination in Relation to Ultraviolet Sensitivity in CHLAMYDOMONAS REINHARDI

A mutant strain of Chlamydomonas reinhardi which is UV sensitive as a result of a single-gene chromosomal mutation has also been found to exhibit reduced recombination. In crosses homozygous for the mutant allele, a reduction in recombination frequency was demonstrated in two different linkage groups and in three different genetic backgrounds. Thus a single mutation affecting UV sensitivity also has a possible effect on recombination. Such a mutant could be analogous to the rec(-) mutants discovered in E. coli and as such be useful in the study of recombination mechanisms. Three additional UV-sensitive isolates were tested. Recombination was not altered in these mutant strains.     

The joint allele-frequency spectrum in closely related species

We develop the theory for computing the joint frequency spectra of alleles in two closely related species. We allow for arbitrary population growth in both species after they had a common ancestor. We focus on the case in which a single chromosome is sequenced from one of the species. We use classical diffusion theory to show that, if the ancestral species was at equilibrium under mutation and drift and a chromosome from one of the descendant species carries the derived allele, the frequency spectrum in the other species is uniform, independently of the demographic history of both species. We also predict the expected densities of segregating and fixed sites when the chromosome from the other species carries the ancestral allele. We compare the predictions of our model with the site-frequency spectra of SNPs in the four HapMap populations of humans when the nucleotide present in the Neanderthal DNA sequence is ancestral or derived, using the chimp genome as the outgroup.     

Simulating genealogies of selected alleles in a population of variable size

An importance-sampling method is presented that allows the simulation of the history of a selected allele in a population of variable size. A sample path describing the number of copies of an allele that arose as a single mutant is generated by simulating backwards from the current frequency until the allele is lost. The mathematical expectation of a quantity or statistic is then estimated by taking averages over replicate simulations, weighting each replicate by the ratio of its probabilities under the Markov chains for the forward and backwards processes. This method was used to find the average age of a selected allele in an exponentially growing population. In terms of the effect on average allele age, selection in favour of an allele is not equivalent to exponential growth. To generate gene genealogies of a sample of copies of a selected allele, the neutral coalescent model is simulated for the subpopulation containing only the selected allele. From the resulting intra-allelic genealogy, it is possible to calculate the likelihood of the selection intensity as a function of the observed level of variability at marker loci closely linked to the selected allele. This method was used to estimate the intensity of selection affecting the delta 32 allele at the CCR5 locus in Europeans and a mutant at the MLH1 locus associated with colorectal cancer in the Finnish population.     

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.     

Fixation probability for a beneficial allele and a mutant strategy in a linear game under weak selection in a finite island model

The effect of population structure on the probability of fixation of a newly introduced mutant under weak selection is studied using a coalescent approach. Wright's island model in a framework of a finite number of demes is assumed and two selection regimes are considered: a beneficial allele model and a linear game among offspring. A first-order approximation of the fixation probability for a single mutant with respect to the intensity of selection is deduced. The approximation requires the calculation of expected coalescence times, under neutrality, for lineages starting from two or three sampled individuals. The results are obtained in a general setting without assumptions on the number of demes, the deme size or the migration rate, which allows for simultaneous coalescence or migration events in the genealogy of the sampled individuals. Comparisons are made with limit cases as the deme size or the number of demes goes to infinity or the migration rate goes to zero for which a diffusion approximation approach is possible. Conditions for selection to favor a mutant strategy replacing a resident strategy in the context of a linear game in a finite island population are addressed.     

Multiple Allele Approach to the Study of Genes in DROSOPHILA MELANOGASTER That Are Involved in Imaginal Disc Development

The phenotypes of five different lethal mutants of Drosophila melanogaster that have small imaginal discs were analyzed in detail. From these results, we inferred whether or not the observed imaginal disc phenotype resulted exclusively from a primary imaginal disc defect in each mutant. To examine the validity of these inferences, we employed a multiple-allele method. Lethal alleles of the five third-chromosome mutations were identified by screening EMS-treated chromosomes for those which fail to complement with a chromosome containing all five reference mutations. Twenty-four mutants were isolated from 13,197 treated chromosomes. Each of the 24 was then tested for complementation with each of the five reference mutants. There was no significant difference in the mutation frequencies at these five loci. The stage of lethality and the imaginal disc morphology of each mutant allele were compared to those of its reference allele in order to examine the range of defects to be found among lethal alleles of each locus. In addition, hybrids of the alleles were examined for intracistronic complementation. For two of the five loci, we detected no significant phenotypic variation among lethal alleles. We infer that each of the mutant alleles at these two loci cause expression of the null activity phenotype. However, for the three other loci, we did detect significant phenotypic variation among lethal alleles. In fact, one of the mutant alleles at each of these three loci causes no detectable imaginal disc defect. This demonstrates that attempting to assess the developmental role of a gene by studying a single mutant allele may lead to erroneous conclusions. As a byproduct of the mutagenesis procedure, we have isolated two dominant, cold-sensitive mutants.     

Epistasis and frequency dependence influence the fitness of an adaptive mutation in a diversifying lineage

The opportunity for a mutation to invade a population can dramatically vary depending on the context in which this mutation occurs. Such context dependence is difficult to document as it requires the ability to measure how a mutation affects phenotypes and fitness and to manipulate the context in which the mutation occurs. We identified a mutation in a gene encoding a global regulator in one of two ecotypes that diverged from a common ancestor during 1200 generations of experimental evolution. We replaced the ancestral allele by the mutant allele, and vice versa, in several clones isolated during the time course of the evolution experiment, and compared the phenotype and fitness of clones isogenic except for the focal mutation. We show that the fitness and phenotype of the mutation are strongly affected by epistatic interactions between genes in the same genome, as well as by frequency dependent selection resulting from biotic interactions between individuals in the same population. We conclude that amongst the replicate population in which it spread, the mutation we identified is only adaptive when occurring in specific genomes and competing with specific individuals. This study thus demonstrates that the opportunity for an adaptive mutation to spread in an evolutionary lineage can only be understood in the light of its genomic and competitive environments.     

Kin selection and coefficients of relatedness in family-structured populations with inbreeding

We consider family specific fitnesses that depend on mixed strategies of two basic phenotypes or behaviours. Pairwise interactions are assumed, but they are restricted to occur between sibs. To study the change in frequency of a rare mutant allele, we consider two different forms of weak selection, one applied through small differences in genotypic values determining individual mixed strategies, the other through small differences in viabilities according to the behaviours chosen by interacting sibs. Under these two specific forms of weak selection, we deduce conditions for initial increase in frequency of a rare mutant allele for autosomal genes in the partial selfing model as well as autosomal and sex-linked genes in the partial sib-mating model with selection before mating or selection after mating. With small differences in mixed strategies, we show that conditions for protection of a mutant allele are tantamount to conditions for initial increase in frequency obtained in additive kin selection models. With particular reference to altruism versus selfishness, we provide explicit ranges of values for the selfing or sib-mating rate based on a fixed cost-benefit ratio and the dominance scheme that allow the spreading of a rare mutant allele into the population. This study confirms that more inbreeding does not necessarily promote the evolution of altruism. Under the hypothesis of small differences in viabilities, the situation is much more intricate unless an additive model is assumed. In general however, conditions for initial increase in frequency of a mutant allele can be obtained in terms of fitness effects that depend on the genotypes of interacting individuals or their mates and generalized conditional coefficients of relatedness according to the inbreeding condition of the interacting individuals.     

Premeiotic clusters of mutation and the cost of natural selection

Haldane stated that there is a cost of natural selection for new beneficial alleles to be substituted over time. Most of this cost, which leads to "genetic deaths," is in the early generations of the substitution process when the new allele is low in frequency. It depends on the initial frequency and dominance value, but not the selection coefficient, of the advantageous allele. There have been numerous suggestions on how to reduce the cost for preexisting genetic variation that goes from disadvantageous, or neutral, to advantageous with a change in the environment. However, the cost of natural selection for new alleles that arise by mutation is assumed to be high, based on the assumption that new mutant alleles arise in natural populations as single events [1/(2N) of the total alleles]. However, not all mutant alleles arise as single events. Premeiotic mutations occur frequently in individuals (germinal mosaics), giving rise to multiple copies of identical mutant alleles called a "cluster" (C) with an initial allele frequency of C/(2N) instead of 1/(2N). These clusters of new mutant alleles reduce the cost of natural selection in direct proportion to the relative size of the cluster. Hence new advantageous alleles that arise by mutation have the greatest chance of going to fixation if they occur in large clusters in small populations.     

Failure of imprinting at Igf-2: two models of mutation-selection balance.

The failure of maternal imprinting at the insulin-like growth factor II (Igf-2) locus predisposes individuals to several clinical conditions, including Wilms tumor. Having two functional Igf-2 genes, therefore, is selectively disadvantageous, and the condition is probably maintained in human populations by recurrent mutation. We propose two models that predict the expected frequency of functionally diploid individuals in a large population, in terms of a mutation rate, mu, and the selection coefficient against functionally diploid individuals, s. In the first model a mutant Igf-2 allele that cannot be imprinted arises from the standard, imprintable allele at a rate mu. Our second model hypothesizes a second modifier locus at which a recessive allele arises at rate mu. Mothers who are homozygous for this recessive modifier allele fail to imprint their eggs. Both models predict the expected frequency of affecteds to be 2 mu/s(1 + mu), approximately twice that predicted by the standard one-locus model of a recessive allele in mutation-selection balance. This frequency suggests that < or = 25% of the cases of Wilms tumor are due to the failure to imprint the maternal Igf-2 gene.     

SNP calling, genotype calling, and sample allele frequency estimation from new-generation sequencing data

We present a statistical framework for estimation and application of sample allele frequency spectra from New-Generation Sequencing (NGS) data. In this method, we first estimate the allele frequency spectrum using maximum likelihood. In contrast to previous methods, the likelihood function is calculated using a dynamic programming algorithm and numerically optimized using analytical derivatives. We then use a Bayesian method for estimating the sample allele frequency in a single site, and show how the method can be used for genotype calling and SNP calling. We also show how the method can be extended to various other cases including cases with deviations from Hardy-Weinberg equilibrium. We evaluate the statistical properties of the methods using simulations and by application to a real data set.