Social selection in human populations: sufficient conditions for protection of deleterious alleles in a subdivided population

Population dynamics of wild type (A1) and the deleterious genes (A2) under social selection have been studied by considering a subdivided population where the i-th subpopulation consists of Ni individuals with relative size ci (= Ni/sigma i Ni, i = 1,2, ..., n). A social selection model is constructed by assuming that the fitness of an individual is determined by its own as well as the parental phenotypes and that the number of migrants (M) from the ith subpopulation is divided equally into other subpopulations including the ith subpopulation itself. It has been shown that the gene frequency change depends on the loss of fitness of an individual due to the trait (gamma), an affected parent in the ith subpopulation (beta i), the probability that the heterozygote develops the trait (h), and the migration rates mi (= M/Ni). For 0 less than h less than or equal to 1, a sufficient condition for protection of the deleterious allele from extinction also depends on all of these parameters. However, when mi much less than 1 for all i, the condition is beta i less than gamma/(1 - gamma) for some i, whereas when mi much greater than h[gamma + beta i(1 - gamma)] for all i it is given by sigma i ci beta i less than -gamma/(1 - gamma). When h = 0, that condition is given by sigma ici beta i less than - gamma/(1 - gamma). Analyses also show that, when the deleterious alleles in a population are rare, the relative fitnesses of A1A1, A1A2, and A2A2 are given approximately by 1, 1-hS, and 1 - S, respectively, where S is the harmonic mean of Si = gamma + beta i(1 - gamma). Thus, under mutation-selection balance, the equilibrium frequency of deleterious alleles in the entire population is given by alpha/hS for 0 less than h less than or equal to 1 and square root alpha/S for h = 0, where alpha is the irreversible mutation rate from A1 to A2 in each generation. Population dynamics of rare deleterious genes under social selection can readily be studied by considering a finite population size.     

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).     

Linkage disequilibrium under genetic hitchhiking in finite populations

The model of genetic hitchhiking predicts a reduction in sequence diversity at a neutral locus closely linked to a beneficial allele. In addition, it has been shown that the same process results in a specific pattern of correlations (linkage disequilibrium) between neutral polymorphisms along the chromosome at the time of fixation of the beneficial allele. During the hitchhiking event, linkage disequilibrium on either side of the beneficial allele is built up whereas it is destroyed across the selected site. We derive explicit formulas for the expectation of the covariance measure D and standardized linkage disequilibrium sigma 2D between a pair of polymorphic sites. For our analysis we use the approximation of a star-like genealogy at the selected site. The resulting expressions are approximately correct in the limit of large selection coefficients. Using simulations we show that the resulting pattern of linkage disequilibrium is quickly-i.e., in <0.1N generations-destroyed after the fixation of the beneficial allele for moderately distant neutral loci, where N is the diploid population size.     

Probability of fixation under weak selection: a branching process unifying approach

We link two-allele population models by Haldane and Fisher with Kimura's diffusion approximations of the Wright-Fisher model, by considering continuous-state branching (CB) processes which are either independent (model I) or conditioned to have constant sum (model II). Recent works by the author allow us to further include logistic density-dependence (model III), which is ubiquitous in ecology. In all models, each allele (mutant or resident) is then characterized by a triple demographic trait: intrinsic growth rate r, reproduction variance sigma and competition sensitivity c. Generally, the fixation probability u of the mutant depends on its initial proportion p, the total initial population size z, and the six demographic traits. Under weak selection, we can linearize u in all models thanks to the same master formula u = p + p(1 - p)[g(r)s(r) + g(sigma)s(sigma) + g(c)s(c)] + o(s(r),s(sigma),s(c), where s(r) = r' - r, s(sigma) = sigma-sigma' and s(c) = c - c' are selection coefficients, and g(r), g(sigma), g(c) are invasibility coefficients (' refers to the mutant traits), which are positive and do not depend on p. In particular, increased reproduction variance is always deleterious. We prove that in all three models g(sigma) = 1/sigma and g(r) = z/sigma for small initial population sizes z. In model II, g(r) = z/sigma for all z, and we display invasion isoclines of the 'mean vs variance' type. A slight departure from the isocline is shown to be more beneficial to alleles with low sigma than with high r. In model III, g(c) increases with z like ln(z)/c, and g(r)(z) converges to a finite limit L > K/sigma, where K = r/c is the carrying capacity. For r > 0 the growth invasibility is above z/sigma when z < K, and below z/sigma when z > K, showing that classical models I and II underestimate the fixation probabilities in growing populations, and overestimate them in declining populations.     

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.     

Effective size of harvested ungulate populations

The harvest of ungulate populations is often directed against certain sex or age classes to maximize the yield in terms of biomass, number of shot animals or number of trophies. Here we examine how such directional harvest affects the effective size of the population. We parameterize an age-specific model assumed to describe the dynamics of Fennoscandian moose. Based on expressions for the demographic variance     for a small subpopulation of heterozygotes Aa bearing a rare neutral allele a , we use this model to calculate how different harvest strategies influence the effective size of the population, given that the population remains stable after harvest. We show that the annual genetic drift, determined by     , increases with decreasing harvest rate of calves and increasing sex bias in the harvest towards bulls 1 year or older. The effective population size per generation decreased with reduced harvest of calves and increased harvest of bulls 1 year or older. The magnitude of these effects depends on the age-specific pattern of variation in reproductive success, which influences the demographic variance. This shows that the choice of harvest strategy strongly affects the genetic dynamics of harvested ungulate populations.     

On the Genealogy of a Rare Allele

The gene genealogy is derived for a rare allele that is descended from a mutant ancestor that arose at a fixed time in the past. Following Thompson (1976,Amer. J. Human Genet.28, 442–452), the fractional linear branching process is used as a model of the demography of a rare allele. The model does not require the total population size to be constant or the mutant class to be neutral; so long as individuals in the class are selectively equivalent, the class as a whole may have a selective advantage, or disadvantage, relative to other alleles in the population. An exact result is given for the joint probability distribution of the coalescence times among a sample of alleles descended from the mutant. A method is described for rapidly simulating these coalescence times. The relationship between the genealogical structure of a discrete generation branching process and a continuous generation birth–death process is elucidated. The theory may be applied to the problem of estimating the ages of rare nonrecurrent mutations.     

Change in frequency of a rare mutant allele: A general formula and applications to partial inbreeding models

 We deduce and prove a general formula to approximate the change in frequency of a mutant allele under weak selection, when this allele is introduced in small frequency into a population which was previously at a fixation state. We apply the formula to autosomal genes in partial selfing models and to autosomal as well as sex-linked genes in partial sib mating models. It is shown that the fate of a rare mutant allele depends not only on the selection parameters, the inbreeding coefficient and the reproductive values of the sexes in sex-differentiated populations, but also on coefficients of relatedness between mates. This is interpreted as a kin selection effect caused by inbreeding per se. Received: 3 December 2001 / Revised version: 10 April 2002 / Published online: 19 November 2002 Research supported in part by NSERC of Canada and FCAR of Québec. Mathematics Subject Classification (2000): Primary 60J80, Secondary 92D10, 92D25 Keywords or phrases: Adaptive topography – Partial selfing – Partial sib mating – Kin selection     

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.     

Population dynamics of sex-determining alleles in honey bees and self-incompatibility alleles in plants

Mathematical theories of the population dynamics of sex-determining alleles in honey bees are developed. It is shown that in an infinitely large population the equilibrium frequency of a sex allele is 1/n, where n is the number of alleles in the population, and the asymptotic rate of approach to this equilibrium is 2/(3n) per generation. Formulae for the distribution of allele frequencies and the effective and actual numbers of alleles that can be maintained in a finite population are derived by taking into account the population size and mutation rate. It is shown that the allele frequencies in a finite population may deviate considerably from 1/n. Using these results, available data on the number of sex alleles in honey bee populations are discussed. It is also shown that the number of self-incompatibility alleles in plants can be studied in a much simpler way by the method used in this paper. A brief discussion about general overdominant selection is presented.     

The effective population size of an age-structured population with a sex-linked locus

Let a population have the same age distribution and age-specific sex ratios at times 0, 1, 2,..., and let M, F, and L, respectively, be the numbers of males and females in the youngest age group and the generation interval. It can then be shown that if there is a sex-linked locus the fixation probabilities of a neutral allele are respectively 1/3LM or 1/3LF if the allele first appears in one newborn male or in one newborn female. The effective population size can then be derived. It is the same as for a population with discrete generations having the same means, variances, and covariances of male and female progeny during a lifetime and the same number of individuals entering the population per generation.     

Social selection in human populations. I. Modification of the fitness of offspring by an affected parent.

The concept of social selection for deleterious genes has been introduced by considering two alleles at one locus. A social selection model is constructed by assuming that the fitness of an individual is determined by his or her own as well as the parental phenotypes. It is shown that the equilibrium gene frequency depends on the loss of fitness of an individual due to the trait (gamma), due to affected parents (beta), and the probability that the heterozygote develops the trait (h). With mutational changes from the wild-type allele to the deleterious gene at a rate of alpha per generation, the equilibrium frequency of deleterious genes is approximately alpha/hs for 0 less than h less than or equal to 1 and square root alpha/s for h = 0, where s = gamma + beta(1 -- gamma)/2. Implications of the social selection model have been discussed for several diseases in man.     

Estimation of effective population size of HIV-1 within a host: a pseudomaximum-likelihood approach

Using pseudomaximum-likelihood approaches to phylogenetic inference and coalescent theory, we develop a computationally tractable method of estimating effective population size from serially sampled viral data. We show that the variance of the maximum-likelihood estimator of effective population size depends on the serial sampling design only because internal node times on a coalescent genealogy can be better estimated with some designs than with others. Given the internal node times and the number of sequences sampled, the variance of the maximum-likelihood estimator is independent of the serial sampling design. We then estimate the effective size of the HIV-1 population within nine hosts. If we assume that the mutation rate is 2.5 x 10(-5) substitutions/generation and is the same in all patients, estimated generation lengths vary from 0.73 to 2.43 days/generation and the mean (1.47) is similar to the generation lengths estimated by other researchers. If we assume that generation length is 1.47 days and is the same in all patients, mutation rate estimates vary from 1.52 x 10(-5) to 5.02 x 10(-5). Our results indicate that effective viral population size and evolutionary rate per year are negatively correlated among HIV-1 patients.     

The stationary distribution of allele frequencies when selection acts at unlinked loci

We consider population genetics models where selection acts at a set of unlinked loci. It is known that if the fitness of an individual is multiplicative across loci, then these loci are independent. We consider general selection models, but assume parent-independent mutation at each locus. For such a model, the joint stationary distribution of allele frequencies is proportional to the stationary distribution under neutrality multiplied by a known function of the mean fitness of the population. We further show how knowledge of this stationary distribution enables direct simulation of the genealogy of a sample at a single-locus. For a specific selection model appropriate for complex disease genes, we use simulation to determine what features of the genealogy differ between our general selection model and a multiplicative model.     

Population genetics study of the differential fertility in urban populations

In an urban population with widespread birth control practice the distribution of the number of pregnancies, births and abortions was studied in a cohort of women of completed fertility. The mean number of pregnancies per woman was 4.03 +/- 0.08 (sigma = = 2.98); the mean number of births - 1.12 +/- 0.02 (sigma = 0.77). 7.4% of women which had completed their reproductive performance had no pregnancies and 19.5% - no births. The Crow's Index of the Opportunity for Selection and its components connected with differential fertility and differential mortality were estimated. In the population under study two components of selection - selection at the prenatal stages and selection associated with infertility - are shown to be still significant. Such type of selection is exemplified by investigation of couples suffering from repeated spontaneous abortions.     

The influence of spatial inhomogeneities on neutral models of geographical variation IV. Discontinuities in the population density and migration rate

The equilibrium structure of the infinite, one-dimensional stepping-stone model with coincident discontinuities in the population density and migration rate is investigated in the diffusion approximation. The monoecious, diploid population is subdivided into an infinite linear array of equally large, panmictic colonies that exchange gametes isotropically. The population density and the migration rate have a discontinuity at the origin, but are elsewhere uniform. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus without selection; every allele mutates to new alleles at the same rate. The three dimensionless parameters in the theory are alpha=(rho(+)/rho(-))(2) (V(+)/V(-))(3/2), and beta(+/-)=4rho(+/-) 2uV(+/-), where rho(+) (rho(-)) and V(+) (V(-)) designate the population density and variance of gametic dispersion per generation to the right (left) of the discontinuity, respectively, and u denotes the mutation rate. The characteristic length on the right (left) is V(+)/(2u) (V(-)/(2u)). The probability of identity is continuous at the origin, but its partial derivatives have a discontinuity unless migration is conservative (rho(-) V(-)=rho(+) V(+)). At least for nonconservative migration, the probability of identity (including the expected homozygosity) can be nonmonotonic even if the migration rate is uniform and the population density is monotonic. Thus, there can be a nonmonotonic genetic response in a neutral model to a monotonic environment.     

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.     

Why are rare variants hard to impute? Coalescent models reveal theoretical limits in existing algorithms

Genotype imputation is an indispensable step in human genetic studies. Large reference panels with deeply sequenced genomes now allow interrogating variants with minor allele frequency < 1% without sequencing. Although it is critical to consider limits of this approach, imputation methods for rare variants have only done so empirically; the theoretical basis of their imputation accuracy has not been explored. To provide theoretical consideration of imputation accuracy under the current imputation framework, we develop a coalescent model of imputing rare variants, leveraging the joint genealogy of the sample to be imputed and reference individuals. We show that broadly used imputation algorithms include model misspecifications about this joint genealogy that limit the ability to correctly impute rare variants. We develop closed-form solutions for the probability distribution of this joint genealogy and quantify the inevitable error rate resulting from the model misspecification across a range of allele frequencies and reference sample sizes. We show that the probability of a falsely imputed minor allele decreases with reference sample size, but the proportion of falsely imputed minor alleles mostly depends on the allele count in the reference sample. We summarize the impact of this error on genotype imputation on association tests by calculating the r² between imputed and true genotype and show that even when modeling other sources of error, the impact of the model misspecification has a significant impact on the r² of rare variants. To evaluate these predictions in practice, we compare the imputation of the same dataset across imputation panels of different sizes. Although this empirical imputation accuracy is substantially lower than our theoretical prediction, modeling misspecification seems to further decrease imputation accuracy for variants with low allele counts in the reference. These results provide a framework for developing new imputation algorithms and for interpreting rare variant association analyses.