Homozygosity in a population of variable size and mutation rate

A formula is obtained for the probability that two genes at a single locus, sampled at random from a population at time t, are of particular types. The model assumed is a diffusion approximation to a neutral Wright-Fisher model in which mutation is not necessarily symmetric and the population size is a function of time. It is shown that for symmetric mutation in a population undergoing a step-function type bottleneck, homozygosity increases with decreasing population size. A formula is given for the distribution of the number of segregating sites occurring in two randomly sampled sequences of completely linked sites, with general mutation at a site and identical mutation structure between sites.We give similar results for a population of fixed size but for which the mutation rate is a function of time, and not necessarily symmetric. We confirm the intuitively clear effect that increasing the mutation rate decreases homozygosity.     

Estimation of effective population size and migration rate from one- and two-locus identity measures

Standard methods for inferring demographic parameters from genetic data are based mainly on one-locus theory. However, the association of genes at different loci (e.g., two-locus identity disequilibrium) may also contain some information about demographic parameters of populations. In this article, we define one- and two-locus parameters of population structure as functions of one- and two-locus probabilities for the identity in state of genes. Since these parameters are known functions of demographic parameters in an infinite island model, we develop moment-based estimators of effective population size and immigration rate from one- and two-locus parameters. We evaluate this method through simulation. Although variance and bias may be quite large, increasing the number of loci on which the estimates are derived improves the method. We simulate an infinite allele model and a K allele model of mutation. Bias and variance are smaller with increasing numbers of alleles per locus. This is, to our knowledge, the first attempt of a joint estimation of local effective population size and immigration rate.     

Variance of Neutral Genetic Variances within and between Populations for a Quantitative Character

The variances of genetic variances within and between finite populations were systematically studied using a general multiple allele model with mutation in terms of identity by descent measures. We partitioned the genetic variances into components corresponding to genetic variances and covariances within and between loci. We also analyzed the sampling variance. Both transient and equilibrium results were derived exactly and the results can be used in diverse applications. For the genetic variance within populations, sigma 2 omega, the coefficient of variation can be very well approximated as [formula: see text] for a normal distribution of allelic effects, ignoring recurrent mutation in the absence of linkage, where m is the number of loci, N is the effective population size, theta 1(0) is the initial identity by descent measure of two genes within populations and t is the generation number. The first term is due to genic variance, the second due to linkage disequilibrium, and third due to sampling. In the short term, the variation is predominantly due to linkage disequilibrium and sampling; but in the long term it can be largely due to genic variance. At equilibrium with mutation [formula: see text] where u is the mutation rate. The genetic variance between populations is a parameter. Variance arises only among sample estimates due to finite sampling of populations and individuals. The coefficient of variation for sample gentic variance between populations, sigma 2b, can be generally approximated as [formula: see text] when the number of loci is large where S is the number of sampling populations.     

Two-locus identity probabilities and identity disequilibrium in a partially selfing subdivided population

Measures of association of genes at different loci (linkage disequilibrium) are widely used to determine whether the structure of natural populations is clonal or not, to map genes from population data, or to test for the homogeneity of response of molecular markers to background selection, for example. However, the usual definitions of parameters for gametic associations may not be suitable for all these purposes. In this paper, we derive the recursion equations for one- and two-locus identity probabilities in an infinite island model. We study the role of drift, gene flow, partial selfing and mutation model on the expected association of genes across loci. We define the 'within-subpopulation identity disequilibrium' as the difference between the joint two-locus probability of identity in state and the expected product of one-locus identity probabilities. We evaluate this parameter as a function of recombination rate, effective size, gene flow and selfing rate. Within-subpopulation identity disequilibrium attains maximum values for intermediate immigration rates, whatever the selfing rate. Moreover, identity disequilibrium may be very small, even for high selfing rates. We discuss the implications of these findings for the analysis of data from natural populations.     

Patterns of inbreeding depression and architecture of the load in subdivided populations

Inbreeding depression is a general phenomenon that is due mainly to recessive deleterious mutations, the so-called mutation load. It has been much studied theoretically. However, until very recently, population structure has not been taken into account, even though it can be an important factor in the evolution of populations. Population subdivision modifies the dynamics of deleterious mutations because the outcome of selection depends on processes both within populations (selection and drift) and between populations (migration). Here, we present a general model that permits us to gain insight into patterns of inbreeding depression, heterosis, and the load in subdivided populations. We show that they can be interpreted with reference to single-population theory, using an appropriate local effective population size that integrates the effects of drift, selection, and migration. We term this the "effective population size of selection" (NS(e)). For the infinite island model, for example, it is equal to NS(e) = N1 + m/hs, where N is the local population size, m the migration rate, and h and s the dominance and selection coefficients of deleterious mutation. Our results have implications for the estimation and interpretation of inbreeding depression in subdivided populations, especially regarding conservation issues. We also discuss the possible effects of migration and subdivision on the evolution of mating systems.     

Effective size of fluctuating populations with two sexes and overlapping generations

We derive formulas that can be applied to estimate the effective population size N(e) for organisms with two sexes reproducing once a year and having constant adult mean vital rates independent of age. Temporal fluctuations in population size are generated by demographic and environmental stochasticity. For populations with even sex ratio at birth, no deterministic population growth and identical mean vital rates for both sexes, the key parameter determining N(e) is simply the mean value of the demographic variance for males and females considered separately. In this case Crow and Kimura's generalization of Wright's formula for N(e) with two sexes, in terms of the effective population sizes for each sex, is applicable even for fluctuating populations with different stochasticity in vital rates for males and females. If the mean vital rates are different for the sexes then a simple linear combination of the demographic variances determines N(e), further extending Wright's formula. For long-lived species an expression is derived for N(e) involving the generation times for both sexes. In the general case with nonzero population growth and uneven sex ratio of newborns, we use the model to investigate numerically the effects of different population parameters on N(e). We also estimate the ratio of effective to actual population size in six populations of house sparrows on islands off the coast of northern Norway. This ratio showed large interisland variation because of demographic differences among the populations. Finally, we calculate how N(e) in a growing house sparrow population will change over time.     

The effect of dispersal and neighbourhood in games of cooperation

The prisoner's dilemma (PD) and the snowdrift (SD) games are paradigmatic tools to investigate the origin of cooperation. Whereas spatial structure (e.g. nonrandom spatial distribution of strategies) present in the spatially explicit models facilitates the emergence of cooperation in the PD game, recent investigations have suggested that spatial structure can be unfavourable for cooperation in the SD game. The frequency of cooperators in a spatially explicit SD game can be lower than it would be in an infinitely large well-mixed population. However, the source of this effect cannot be identified with certainty as spatially explicit games differ from well-mixed games in two aspects: (i) they introduce spatial correlations, (ii) and limited neighbourhood. Here we extend earlier investigations to identify the source of this effect, and thus accordingly we study a spatially explicit version of the PD and SD games with varying degrees of dispersal and neighbourhood size. It was found that dispersal favours selfish individuals in both games. We calculated the frequency of cooperators at strong dispersal limit, which in concordance with the numerical results shows that it is the short range of interactions (i.e. limited neighbourhood) and not spatial correlations that decreases the frequency of cooperators in spatially explicit models of populations. Our results demonstrate that spatial correlations are always beneficial to cooperators in both the PD and SD games. We explain the opposite effect of dispersal and neighbourhood structure, and discuss the relevance of distinguishing the two effects in general.     

Gene genealogies when the sample size exceeds the effective size of the population

We study the properties of gene genealogies for large samples using a continuous approximation introduced by R. A. Fisher. We show that the major effect of large sample size, relative to the effective size of the population, is to increase the proportion of polymorphisms at which the mutant type is found in a single copy in the sample. We derive analytical expressions for the expected number of these singleton polymorphisms and for the total number of polymorphic, or segregating, sites that are valid even when the sample size is much greater than the effective size of the population. We use simulations to assess the accuracy of these predictions and to investigate other aspects of large-sample genealogies. Lastly, we apply our results to some data from Pacific oysters sampled from British Columbia. This illustrates that, when large samples are available, it is possible to estimate the mutation rate and the effective population size separately, in contrast to the case of small samples in which only the product of the mutation rate and the effective population size can be estimated.     

Hidden Genetic Variability within Electromorphs in Finite Populations

The amount of hidden genetic variability within electromorphs in finite populations is studied by using the infinite site model and stepwise mutation model simultaneously. A formula is developed for the bivariate probability generating function for the number of codon differences and the number of electromorph state differences between two randomly chosen cistrons. Using this formula, the distribution as well as the mean and variance of the number of codon differences between two identical or nonidentical electromorphs are studied. The distribution of the number of codon differences between two randomly chosen identical electromorphs is similar to the geometric distribution but more leptokurtic. Studies are also made on the number of codon differences between two electromorphs chosen at random one from each of two populations which have been separated for an arbitrary number of generations. It is shown that the amount of hidden genetic variability is very large if the product of effective population size and mutation rate is large.     

Gene Flow and Genetic Differentiation

A brief analysis is presented for the effects of gene flow upon genetic differentiation within and between populations generated by mutation and drift. Previous results obtained with the "island" model are developed into a form that lends itself to biological interpretation. Attention is focused upon the effective local population size and the ratio of the genetic identity of two genes in different populations to that of two genes in the same population. The biological significance of this ratio, which is independent of population size, is discussed. Similarities between the results of this model and those of the "stepping-stone" model are noted.     

Population size and identity influence the reaction norm of the rare,endemic plant Cochlearia bavarica across a gradient of environmental stress

Habitat degradation and loss can result in population decline and genetic erosion, limiting the ability of organisms to cope with environmental change, whether this is through evolutionary genetic response (requiring genetic variation) or through phenotypic plasticity (i.e., the ability of a given genotype to express a variable phenotype across environments). Here we address the question whether plants from small populations are less plastic or more susceptible to environmental stress than plants from large populations. We collected seed families from small (<100) versus large natural populations (>1,000 flowering plants) of the rare, endemic plant Cochlearia bavarica (Brassicaceae). We exposed the seedlings to a range of environments, created by manipulating water supply and light intensity in a 2 x 2 factorial design in the greenhouse. We monitored plant growth and survival for 300 days. Significant effects of offspring environment on offspring characters demonstrated that there is phenotypic plasticity in the responses to environmental stress in this species. Significant effects of population size group, but mainly of population identity within the population size groups, and of maternal plant identity within populations indicated variation due to genetic (plus potentially maternal) variation for offspring traits. The environment x maternal plant identity interaction was rarely significant, providing little evidence for genetically- (plus potentially maternally-) based variation in plasticity within populations. However, significant environment x population-size-group and environment x population-identity interactions suggested that populations differed in the amount of plasticity, the mean amount being smaller in small populations than in large populations. Whereas on day 210 the differences between small and large populations were largest in the environment in which plants grew biggest (i.e., under benign conditions), on day 270 the difference was largest in stressful environments. These results show that population size and population identity can affect growth and survival differently across environmental stress gradients. Moreover, these effects can themselves be modified by time-dependent variation in the interaction between plants and their environment.     

The population structure associated with the Ewens sampling formula

Models for selectively neutral mutation, in which mutation always yields a new allele, seem always to lead, in the limit of large population size, to a sampling formula first propounded by Ewens in 1972. It is shown that the asymptotic validity of the Ewens formula is equivalent to a certain limiting joint distribution for the allele proportions in the population, arranged in descending order. The familiar diffusion approximations are corollaries of this limiting distribution, and therefore share the apparent robustness of the sampling formula.     

The distribution of coalescence times and distances between microsatellite alleles with changing effective population size

We investigate the effects of past changes of the effective population size on the present allelic diversity at a microsatellite marker locus. We first derive the analytical expression of the generating function of the joint probabilities of the time to the Most Recent Common Ancestor for a pair of alleles and of their distance (the difference in allele size). We give analytical solutions in the case of constant population size and the geometrical mutation model. Otherwise, numerical inversion allows the distributions to be calculated in general cases. The effects of population expansion or decrease and the possibility to detect an ancient bottleneck are discussed. The method is extended to samples of three and four alleles, which allows investigating the covariance structure of the frequencies f(k) of pairs of alleles with a size difference of k motifs, and suggesting some approaches to the estimation of past demography.     

The Inbreeding Effective Population Number in Dioecious Populations

The inbreeding effective population number in a dioecious population with discrete, nonoverlapping generations is investigated for both autosomal and X-linked loci. The recursion relations for the probabilities of genic identity and the effective population numbers are analyzed and compared in two cases: (i) the offspring identified by sex in the calculation of the probability of common parentage and (ii) the offspring not so identified. Case i gives the correct evolution of the probabilities of identity, but case ii has been more widely studied and applied. A general symmetric framework that reduces the number of parameters is developed and used to examine a wide variety of models of panmixia and monogamy. Cases i and ii agree in many, but not all, models.     

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).     

A general population genetic framework for antagonistic selection that accounts for demography and recurrent mutation

Antagonistic selection--where alleles at a locus have opposing effects on male and female fitness ("sexual antagonism") or between components of fitness ("antagonistic pleiotropy")--might play an important role in maintaining population genetic variation and in driving phylogenetic and genomic patterns of sexual dimorphism and life-history evolution. While prior theory has thoroughly characterized the conditions necessary for antagonistic balancing selection to operate, we currently know little about the evolutionary interactions between antagonistic selection, recurrent mutation, and genetic drift, which should collectively shape empirical patterns of genetic variation. To fill this void, we developed and analyzed a series of population genetic models that simultaneously incorporate these processes. Our models identify two general properties of antagonistically selected loci. First, antagonistic selection inflates heterozygosity and fitness variance across a broad parameter range--a result that applies to alleles maintained by balancing selection and by recurrent mutation. Second, effective population size and genetic drift profoundly affect the statistical frequency distributions of antagonistically selected alleles. The "efficacy" of antagonistic selection (i.e., its tendency to dominate over genetic drift) is extremely weak relative to classical models, such as directional selection and overdominance. Alleles meeting traditional criteria for strong selection (N(e)s > 1, where N(e) is the effective population size, and s is a selection coefficient for a given sex or fitness component) may nevertheless evolve as if neutral. The effects of mutation and demography may generate population differences in overall levels of antagonistic fitness variation, as well as molecular population genetic signatures of balancing selection.     

estim 1.0: a computer program to infer population parameters from one‐ and two‐locus gene identity probabilities

Estimating effective population size is an important issue in population and conservation genetics. Recently, we proposed a new method to infer effective size and migration rate from one‐ and two‐locus identity probability measures. We now announce the release of a user‐friendly Microsoft® Windows program that uses this method to provide joint estimates of local effective population size and immigration rate for each subpopulation in a population genetics data set.     

Genetic neighbourhood and effective population size in the endangered European mink <Emphasis Type="Italic">Mustela lutreola</Emphasis>

Genetic neighbourhood and effective population size (N _e) are critical factors when determining the potential survival of threatened species. Carnivores have intrinsically small effective numbers, because, as top predators, they show low densities. The European mink, Mustela lutreola, is one of the most endangered carnivores in the world and has suffered continual decline and local extinctions. The genetic neighbourhood, area within which adults could randomly mate, averaged N _a = 31.7 km diameter, allowing that population size within the neighbourhood area only ranged from N _b = 6.1 to 22.8 animals. Although the population size was assessed in one of the main mink populations in the world, this neighbourhood size is far below the values regarded as critical in literature. However, in contrast with recent propositions, the ratio N _e /N only ranged between 0.09 and 0.19, estimates close to the average recognised by Frankham [(1995) Genetic Research 66: 95–107] for wildlife populations. In the context of the challenge to conserve this endangered carnivore, the studied neighbourhood provided crucial information suggesting both a low neighbourhood size and severe disturbance of breeding exchanges, emphasising the dramatically threatened status of the European mink.     

The effective number of a population that varies cyclically in size. I. Discrete generations

We consider a dioecious population having numbers of males and females that vary over time in cycles of length k. It is shown that if k is small in comparison with the numbers of males and females in any generation of the cycle, the effective population number (or size), N(e), is approximately equal to the harmonic mean of the effective population sizes during any given cycle. This result holds whether the locus under consideration is autosomal or sex-linked and whether inbreeding effective population numbers or variance effective population numbers are involved in the calculation of N(e). If, however, only two successive generations in the cycle are considered and the population changes in size between these generations, the inbreeding effective population number, N(eI), differs from the variance effective population number, N(eV). The mutation effective population number turns out to be the same as the number derived using calculations involving probabilities of identity by descent. It is also shown that, at least in one special case, the eigenvalue effective population number is the same as N(eV).     

THE EFFECTS OF POPULATION BOTTLENECKS ON CLONAL INTERFERENCE, AND THE ADAPTATION EFFECTIVE POPULATION SIZE

Clonal interference refers to the competition that arises in asexual populations when multiple beneficial mutations segregate simultaneously. A large body of theoretical and experimental work now addresses this issue. Although much of the experimental work is performed in populations that grow exponentially between periodic population bottlenecks, the theoretical work to date has addressed only populations of a constant size. We derive an analytical approximation for the rate of adaptation in the presence of both clonal interference and bottlenecks, and compare this prediction to the results of an individual-based simulation, showing excellent agreement in the parameter regime in which clonal interference prevails. We also derive an appropriate definition for the effective population size for adaptive evolution experiments in the presence of population bottlenecks. This "adaptation effective population size" allows for a good approximation of the expected rate of adaptation, either in the strong-selection weak-mutation regime, or when clonal interference comes into play. In the multiple mutation regime, when the product of the population size and mutation rate is extremely large, these results no longer hold.