Effective population size when fertility is inherited

What is N e, anyway?

Few doubt that effective population size (Ne) is one of the most important parameters in evolutionary biology, but how many can say they really understand the concept? Ne is the evolutionary analogue of the number of individuals (or adults) in the population, N. Whereas ecological consequences of population size depend on N, evolutionary consequences (rates of loss of genetic diversity and increase in inbreeding; relative effectiveness of selection) depend on Ne. Formal definitions typically relate effective size to a key population genetic parameter, such as loss of heterozygosity or variance in allele frequency. However, for practical application to real populations, it is more useful to define Ne in terms of three demographic parameters: number of potential parents (adult N), and mean and variance in offspring number. Defined this way, Ne determines the rate of random genetic drift across the entire genome in the offspring generation. Other evolutionary forces (mutation, migration, selection)—together with factors such as variation in recombination rate—can also affect genetic variation, and this leads to heterogeneity across the genome in observed rates of genetic change. For some, it has been convenient to interpret this heterogeneity in terms of heterogeneity in Ne, but unfortunately this has muddled the concepts of genetic drift and effective population size. A commonly-repeated misconception is that Ne is the number of parents that actually contribute genes to the next generation (NP). In reality, NP can be smaller or larger than Ne, and the NP/Ne ratio depends on the sex ratio, the mean and variance in offspring number, and whether inbreeding or variance Ne is of interest.

Revealing life-history traits by contrasting genetic estimations with predictions of effective population size

Effective population size, a central concept in conservation biology, is now routinely estimated from genetic surveys and can also be theoretically predicted from demographic, life-history, and mating-system data. By evaluating the consistency of theoretical predictions with empirically estimated effective size, insights can be gained regarding life-history characteristics and the relative impact of different life-history traits on genetic drift. These insights can be used to design and inform management strategies aimed at increasing effective population size. We demonstrated this approach by addressing the conservation of a reintroduced population of Asiatic wild ass (Equus hemionus). We estimated the variance effective size (Nev ) from genetic data (N ev =24.3) and formulated predictions for the impacts on Nev of demography, polygyny, female variance in lifetime reproductive success (RS), and heritability of female RS. By contrasting the genetic estimation with theoretical predictions, we found that polygyny was the strongest factor affecting genetic drift because only when accounting for polygyny were predictions consistent with the genetically measured Nev . The comparison of effective-size estimation and predictions indicated that 10.6% of the males mated per generation when heritability of female RS was unaccounted for (polygyny responsible for 81% decrease in Nev ) and 19.5% mated when female RS was accounted for (polygyny responsible for 67% decrease in Nev ). Heritability of female RS also affected Nev ; hf2=0.91 (heritability responsible for 41% decrease in Nev ). The low effective size is of concern, and we suggest that management actions focus on factors identified as strongly affecting Nev, namely, increasing the availability of artificial water sources to increase number of dominant males contributing to the gene pool. This approach, evaluating life-history hypotheses in light of their impact on effective population size, and contrasting predictions with genetic measurements, is a general, applicable strategy that can be used to inform conservation practice.

Prediction and estimation of effective population size

Effective population size (Ne) is a key parameter in population genetics. It has important applications in evolutionary biology, conservation genetics and plant and animal breeding, because it measures the rates of genetic drift and inbreeding and affects the efficacy of systematic evolutionary forces, such as mutation, selection and migration. We review the developments in predictive equations and estimation methodologies of effective size. In the prediction part, we focus on the equations for populations with different modes of reproduction, for populations under selection for unlinked or linked loci and for the specific applications to conservation genetics. In the estimation part, we focus on methods developed for estimating the current or recent effective size from molecular marker or sequence data. We discuss some underdeveloped areas in predicting and estimating Ne for future research.

Applications of population genetics to animal breeding, from wright, fisher and lush to genomic prediction

Although animal breeding was practiced long before the science of genetics and the relevant disciplines of population and quantitative genetics were known, breeding programs have mainly relied on simply selecting and mating the best individuals on their own or relatives' performance. This is based on sound quantitative genetic principles, developed and expounded by Lush, who attributed much of his understanding to Wright, and formalized in Fisher's infinitesimal model. Analysis at the level of individual loci and gene frequency distributions has had relatively little impact. Now with access to genomic data, a revolution in which molecular information is being used to enhance response with "genomic selection" is occurring. The predictions of breeding value still utilize multiple loci throughout the genome and, indeed, are largely compatible with additive and specifically infinitesimal model assumptions. I discuss some of the history and genetic issues as applied to the science of livestock improvement, which has had and continues to have major spin-offs into ideas and applications in other areas.

Impact of fertility transmission and other sociodemographic factors on reproductive success and coalescent trees

Summary Fertility transmission (FT) is a phenomenon with a cultural and/or genetic basis, whereby a positive correlation exists between the number of offspring of an individual and that of his/her parents. Theoretical studies using a haploid individual-based model have shown that FT increases the variance and intergenerational correlation in reproductive success and results in an imbalance in the coalescent tree of sampled genes. This phenomenon has been documented in several demographic studies conducted on the correlation in fertility between generations, or through the reconstruction of the genealogical trees of mitochondrial DNA sequences. However, as mtDNA is a single locus, potentially subject to other forces (e.g. natural selection), it is of interest to extend the theory of FT to nuclear loci. We show that because random mating between individuals leads to a mixing of their fertility profiles, FT in these cases will have less influence on the variance and intergenerational correlation of reproductive success. This, in turn, results in less impact on the shape of the coalescent trees. Nevertheless, in the presence of FT, high heterogeneity in reproductive success and homogamy for family size will increase the imbalance in the coalescent tree. Thus, FT should be easier to detect when occurring in conjunction with these other factors. We also show the utility of analysing different kinds of loci (X-linked, Y-linked, mitochondrial and autosomal) to assess whether FT is matrilineal, patrilineal or biparental. Finally, we demonstrate that the shape of the coalescent tree depends upon population size, in contrast to the classical Kingman's model.

The dynamics of HIV-1 adaptation in early infection

Human immunodeficiency virus type 1 (HIV-1) undergoes a severe population bottleneck during sexual transmission and yet adapts extremely rapidly to the earliest immune responses. The bottleneck has been inferred to typically consist of a single genome, and typically eight amino acid mutations in viral proteins spread to fixation by the end of the early chronic phase of infection in response to selection by CD8(+) T cells. Stochastic simulation was used to examine the effects of the transmission bottleneck and of potential interference among spreading immune-escape mutations on the adaptive dynamics of the virus in early infection. If major viral population genetic parameters are assigned realistic values that permit rapid adaptive evolution, then a bottleneck of a single genome is not inconsistent with the observed pattern of adaptive fixations. One requirement is strong selection by CD8(+) T cells that decreases over time. Such selection may reduce effective population sizes at linked loci through genetic hitchhiking. However, this effect is predicted to be minor in early infection because the transmission bottleneck reduces the effective population size to such an extent that the resulting strong selection and weak mutation cause beneficial mutations to fix sequentially and thus avoid interference.

Genetic variability and effective population size when local extinction and recolonization of subpopulations are frequent

If a population (species) consists of n haploid lines (subpopulations) which reproduce asexually and each of which is subject to random extinction and subsequent replacement, it is shown that, at equilibrium in which mutational production of new alleles and their random extinction balance each other, the genetic diversity (1 minus the sum of squares of allelic frequencies) is given by 2N(e)v/(1 + 2N(e)v), where [Formula: see text] in which N is the harmonic mean of the population size per line, n is the number of lines (assumed to be large), lambda is the rate of line extinction, and v is the mutation rate (assuming the infinite neutral allele model). In a diploid population (species) consisting of n colonies, if migration takes place between colonies at the rate m (the island model) in addition to extinction and recolonization of colonies, it is shown that effective population size is [Formula: see text] If the rate of colony extinction (lambda) is much larger than the migration rate of individuals, the effective population size is greatly reduced compared with the case in which no colony extinctions occur (in which case N(e) = nN). The stepping-stone type of recolonization scheme is also considered. Bearing of these results on the interpretation of the level of genetic variability at the enzyme level observed in natural populations is discussed from the standpoint of the neutral mutation-random drift hypothesis.

Good genes drive female choice for mating partners in the lek-breeding European treefrog

Investigating the mechanisms underlying female mate choice is important for sexual-selection theory, but also for population-genetic studies, because distinctive breeding strategies affect differently the dynamics of gene diversity within populations. Using field-monitoring, genetic-assignment, and laboratory-rearing methods, we investigated chorus attendance, mating success and offspring fitness in a population of lek-breeding tree-frogs (Hyla arborea) to test whether female choice is driven by good genes or complementary genes. Chorus attendance explained approximately 50% of the variance in male mating success, but did not correlate with offspring fitness. By contrast, offspring body mass and growth rate correlated with male attractiveness, measured as the number of matings obtained per night of calling. Genetic similarity between mating partners did not depart from random, and did not affect offspring fitness. We conclude that females are able to choose good partners under natural settings and obtain benefits from the good genes, rather than compatible genes, their offspring inherit. This heritability of fitness is likely to reduce effective population sizes below values previously estimated.

Opportunity for sexual selection and effective population size in the lek-breeding European treefrog (Hyla arborea)

Sexual selection in lek-breeding species might drastically lower male effective population size, with potentially important consequences for evolutionary and conservation biology. Using field-monitoring and parental-assignment methods, we analyzed sex-specific variances in breeding success in a population of European treefrogs, to (1) help understanding the dynamics of genetic variance at sex-specific loci, and (2) better quantify the risk posed by genetic drift in this species locally endangered by habitat fragmentation. The variance in male mating success turned out to be markedly lower than values obtained from other amphibian species with polygamous mating systems. The ratio of effective breeding size to census breeding size was only slightly lower in males (0.44) than in females (0.57), in line with the patterns of genetic diversity previously reported from H. arborea sex chromosomes. Combining our results with data on age at maturity and adult survival, we show that the negative effect of the mating system is furthermore compensated by the effect of delayed maturity, so that the estimated instantaneous effective size broadly corresponded to census breeding size. We conclude that the lek-breeding system of treefrogs impacts only weakly the patterns of genetic diversity on sex-linked genes and the ability of natural populations to resist genetic drift.

Effective population size/adult population size ratios in wildlife: a review

The effective population size is required to predict the rate of inbreeding and loss of genetic variation in wildlife. Since only census population size is normally available, it is critical to know the ratio of effective to actual population size (Ne/N). Published estimates ofNe/N(192 from 102 species) were analysed to identify major variables affecting the ratio, and to obtain a comprehensive estimate of the ratio with all relevant variables included. The five most important variables explaining variation among estimates, in order of importance, were fluctuation in population size, variance in family size, form ofNused (adults υ. breeders υ. total size), taxonomic group and unequal sex-ratio. There were no significant effects on the ratio of high υ. low fecundity, demographic υ. genetic methods of estimation, or of overlapping υ. non-overlapping generations when the same variables were included in estimates. Comprehensive estimates ofNe/N(that included the effects of fluctuation in population size, variance in family size and unequal sex-ratio) averaged only 0·10–0·11. Wildlife populations have much smaller effective population sizes than previously recognized.

Matrilineal fertility inheritance detected in hunter-gatherer populations using the imbalance of gene genealogies

Fertility inheritance, a phenomenon in which an individual's number of offspring is positively correlated with his or her number of siblings, is a cultural process that can have a strong impact on genetic diversity. Until now, fertility inheritance has been detected primarily using genealogical databases. In this study, we develop a new method to infer fertility inheritance from genetic data in human populations. The method is based on the reconstruction of the gene genealogy of a sample of sequences from a given population and on the computation of the degree of imbalance in this genealogy. We show indeed that this level of imbalance increases with the level of fertility inheritance, and that other phenomena such as hidden population structure are unlikely to generate a signal of imbalance in the genealogy that would be confounded with fertility inheritance. By applying our method to mtDNA samples from 37 human populations, we show that matrilineal fertility inheritance is more frequent in hunter-gatherer populations than in food-producer populations. One possible explanation for this result is that in hunter-gatherer populations, individuals belonging to large kin networks may benefit from stronger social support and may be more likely to have a large number of offspring.

A populationwide coalescent analysis of Icelandic matrilineal and patrilineal genealogies: evidence for a faster evolutionary rate of mtDNA lineages than Y chromosomes

Historical inferences from genetic data increasingly depend on assumptions about the genealogical process that shapes the frequencies of alleles over time. Yet little is known about the structure of human genealogies over long periods of time and how they depart from expectations of standard demographic models, such as that attributed to Wright and Fisher. To obtain such information and to examine the recent evolutionary history of mtDNA and Y-chromosome haplotypes in the Icelandic gene pool, we traced the matrilineal and patrilineal ancestry of all 131,060 Icelanders born after 1972 back to two cohorts of ancestors, one born between 1848 and 1892 and the other between 1798 and 1742. This populationwide coalescent analysis of Icelandic genealogies revealed highly positively skewed distributions of descendants to ancestors, with the vast majority of potential ancestors contributing one or no descendants and a minority of ancestors contributing large numbers of descendants. The expansion and loss of matrilines and patrilines has caused considerable fluctuation in the frequencies of mtDNA and Y-chromosome haplotypes, despite a rapid population expansion in Iceland during the past 300 years. Contrary to a widespread assumption, the rate of evolution caused by this lineage-sorting process was markedly faster in matrilines (mtDNA) than in patrilines (Y chromosomes). The primary cause is a 10% shorter matrilineal generation interval. Variance in the number of offspring produced within each generation was not an important differentiating factor. We observed an intergenerational correlation in offspring number and in the length of generation intervals in the matrilineal and patrilineal genealogies, which was stronger in matrilines and thus contributes to their faster evolutionary rate. These findings may have implications for coalescent date estimates based on mtDNA and Y chromosomes.

Effective size of populations with heritable variation in fitness

The effective size of monogamous populations with heritable variation in fitness is formulated, and the expression obtained is compared with a published equation. It is shown that the published equation for dioecious populations is inappropriate for most animal and human populations, because the derivation is implicitly based on the assumption that zygotes are produced by random union of gametes, each from conceptual male and female gametic pools. A convenient equation for practical use is proposed, and the application is illustrated with the estimation of the effective size of a rural human community in Japan.

Genetic effective size of a wild primate population: influence of current and historical demography

A comprehensive assessment of the determinants of effective population size (N(e)) requires estimates of variance in lifetime reproductive success and past changes in census numbers. For natural populations, such information can be best obtained by combining longitudinal data on individual life histories and genetic marker-based inferences of demographic history. Independent estimates of the variance effective size (N(ev), obtained from life-history data) and the inbreeding effective size (N((eI), obtained from genetic data) provide a means of disentangling the effects of current and historical demography. The purpose of this study was to assess the demographic determinants of N(e) in one of the most intensively studied natural populations of a vertebrate species: the population of savannah baboons (Papio cynocephalus) in the Amboseli Basin, southern Kenya. We tested the hypotheses that N(eV) < N < N(eI) (where N = population census number) due to a recent demographic bottleneck. N(eV) was estimated using a stochastic demographic model based on detailed life-history data spanning a 28-year period. Using empirical estimates of age-specific rates of survival and fertility for both sexes, individual-based simulations were used to estimate the variance in lifetime reproductive success. The resultant values translated into an N(eV)/N estimate of 0.329 (SD = 0.116, 95% CI = 0.172-0.537). Historical N(eI), was estimated from 14-locus microsatellite genotypes using a coalescent-based simulation model. Estimates of N(eI) were 2.2 to 7.2 times higher than the contemporary census number of the Amboseli baboon population. In addition to the effects of immigration, the disparity between historical N(eI) and contemporary N is likely attributable to the time lag between the recent drop in census numbers and the rate of increase in the average probability of allelic identity-by-descent. Thus, observed levels of genetic diversity may primarily reflect the population's prebottleneck history rather than its current demography.

Genetic hitchhiking

Selection on one or more genes inevitably perturbs other genes, even when those genes have no direct effect on fitness. This article reviews the theory of such genetic hitchhiking, concentrating on effects on neutral loci. Maynard Smith and Haigh introduced the classical case where the perturbation is due to a single favourable mutation. This is contrasted with the apparently distinct effects of inherited variation in fitness due to loosely linked loci. A model of fluctuating selection is analysed which bridges these alternative treatments. When alleles sweep between extreme frequencies at a rate lambda, the rate of drift is increased by a factor (1 + E[1/pq]lambda/(2(2lambda + r))), where the recombination rate r is much smaller than the strength of selection. In spatially structured populations, the effects of any one substitution are weaker, and only cause a local increase in the frequency of a neutral allele. This increase depends primarily on the rate of recombination relative to selection (r/s), and more weakly, on the neighbourhood size, Nb = 4(pi rho sigma)2. Spatial subdivision may allow local selective sweeps to occur more frequently than is indicated by the overall rate of molecular evolution. However, it seems unlikely that such sweeps can be sufficiently frequent to increase significantly the drift of neutral alleles.

The coalescent time in the presence of background fertility selection

Selection ultimately entails differential reproductive success over several generations. This can be measured as a correlation of the number of progeny an individual has with the number of progeny its parent had. This correlation could have a genetic or a cultural basis. The effect of such a correlation is to multiply the single generation sampling variance (Vdeltap) in the diffusion approximation for fixation time by (1-b)+bx(1+r)/(1-r), where bxrn is the correlation between the number of progeny of an individual and its ancestor n generations ago (e.g., b is the heritability and br is the resultant parent-offspring progeny number correlation if the progeny number is genetically determined). This results in a reduction of the fixation or coalescent time by division by this factor. Sex differences in this correlation have been observed, and this provides an explanation for the difference of coalescent times of y-chromosomes and mitochondria.

Social transmission of reproductive behavior increases frequency of inherited disorders in a young-expanding population

The observation of high frequencies of certain inherited disorders in the population of Saguenay-Lac Saint Jean can be explained in terms of the variance and the correlation of effective family size (EFS) from one generation to the next. We have shown this effect by using the branching process approach with real demographic data. When variance of EFS is included in the model, despite its profound effect on mutant allele frequency, any mutant introduced in the population never reaches the known carrier frequencies (between 0.035 and 0.05). It is only when the EFS correlation between generations is introduced into the model that we can explain the rise of the mutant alleles. This correlation is described by a c parameter that reflects the dependency of children's EFS on their parents' EFS. The c parameter can be considered to reflect social transmission of demographic behavior. We show that such social transmission dramatically reduces the effective population size. This could explain particular distributions in allele frequencies and unusually high frequency of certain inherited disorders in some human populations.

The effect of recombination on background selection

An approximate equation is derived, which predicts the effect on variability at a neutral locus of background selection due to a set of partly linked deleterious mutations. Random mating, multiplicative fitnesses, and sufficiently large population size that the selected loci are in mutation/selection equilibrium are assumed. Given these assumptions, the equation is valid for an arbitrary genetic map, and for an arbitrary distribution of selection coefficients across loci. Monte Carlo computer simulations show that the formula performs well for small population sizes under a wide range of conditions, and even seems to apply when there are epistatic fitness interactions among the selected loci. Failure occurred only with very weak selection and tight linkage. The formula is shown to imply that weakly selected mutations are more likely than strongly selected mutations to produce regional patterning of variability along a chromosome in response to local variation in recombination rates. Loci at the extreme tip of a chromosome experience a smaller effect of background selection than loci closer to the centre. It is shown that background selection can produce a considerable overall reduction in variation in organisms with small numbers of chromosomes and short maps, such as Drosophila. Large overall effects are less likely in species with higher levels of genetic recombination, such as mammals, although local reductions in regions of reduced recombination might be detectable.

Developments in the prediction of effective population size

Effective population size is a key parameter in evolutionary and quantitative genetics because it measures the rate of genetic drift and inbreeding. Predictive equations of effective size under a range of circumstances and some of their implications are reviewed in this paper. Derivations are made for the simplest cases, and inter-relations between different formulae and methods are discussed.

Mutation load depending on variance in reproductive success and mating system

Equalization of reproductive success of individuals, although it results in an increase of effective population size, leads also to an increase of the mutation load. The magnitude of this increase depends highly on the mode of fitness interactions between deleterious mutations, and is higher in the case of inbreeding. Recommended practices in conservation genetics must be evaluated in regards to these differing consequences of an increase of effective population size. To keep a balance between retaining genetic variability and minimizing the increase of the mutation load, equalization of reproductive success of a set of individuals rather than of every individual might be more advantageous.

Prediction of rates of inbreeding in selected populations

A method is presented for the prediction of rate of inbreeding for populations with discrete generations. The matrix of Wright's numerator relationships is partitioned into 'contribution' matrices which describe the contribution of the Mendelian sampling of genes of ancestors in a given generation to the relationship between individuals in later generations. These contributions stabilize with time and the value to which they stabilize is shown to be related to the asymptotic rate of inbreeding and therefore also the effective population size, Ne approximately 2N/(mu 2r + sigma 2r), where N is the number of individuals per generation and mu r and sigma 2r are the mean and variance of long-term relationships or long-term contributions. These stabilized values are then predicted using a recursive equation via the concept of selective advantage for populations with hierarchical mating structures undergoing mass selection. Account is taken of the change in genetic parameters as a consequence of selection and also the increasing 'competitiveness' of contemporaries as selection proceeds. Examples are given and predicted rates of inbreeding are compared to those calculated in simulations. For populations of 20 males and 20, 40, 100 or 200 females the rate of inbreeding was found to increase by as much as 75% over the rate of inbreeding in an unselected population depending on mating ratio, selection intensity and heritability of the selected trait. The prediction presented here estimated the rate of inbreeding usually within 5% of that calculated from simulation.

Effects of linkage on response to directional selection from new mutations

The influence of linkage on the rates of response to continued directional selection of quantitative traits deriving from variation contributed by new mutations in finite populations is investigated. Mutant genes are assumed to have additive effects which are randomly sampled from a symmetric distribution, and to be randomly located on the chromosome. Results were obtained by Monte Carlo simulation.

The expected steady rate of response, when variability from new mutations is balanced by that lost by drift and selection, is reduced the tighter the linkage, but the reduction is small unless there are few, short chromosomes. For a given rate of new mutational variance per haploid chromosome set per generation, greater effects of linkage are obtained in large than in small populations, because more mutants segregate. The response and influence of linkage are essentially the same whether the new variance is due to many genes of small effect or few of large effect.

The variability of response between replicates and generations was investigated, and the contribution to this of new mutants or recombination of existing mutants compared. Usually most genetic variability was due to the occurrence of a new favourable mutant of large effect.

A note on increasing the limit of selection through selection within families

It is shown that, in a finite population, by ensuring an equal contribution of offspring from each family we achieve a higher selection limit than by using mass selection, given that the correlation between selection criterion and additive genetic value is sufficiently high. The difference between the selection limits of the two schemes increases with more intense selection. The theoretical results were verified by Monte Carlo Simulation, and the influences of several factors were investigated. It is shown that such a breeding scheme might be useful in dairy cattle.