Neutral additive genetic variance in a metapopulation

For neutral, additive quantitative characters, the amount of additive genetic variance within and among populations is predictable from Wright's FST, the effective population size and the mutational variance. The structure of quantitative genetic variance in a subdivided metapopulation can be predicted from results from coalescent theory, thereby allowing single-locus results to predict quantitative genetic processes. The expected total amount of additive genetic variance in a metapopulation of diploid individual is given by 2Ne sigma m2 (1 + FST), where FST is Wright's among-population fixation index, Ne is the eigenvalue effective size of the metapopulation, and sigma m2 is the mutational variance. The expected additive genetic variance within populations is given by 2Ne sigma e2(1-FST), and the variance among demes is given by 4FSTNe sigma m2. These results are general with respect to the types of population structure involved. Furthermore, the dimensionless measure of the quantitative genetic variance among populations, QST, is shown to be generally equal to FST for the neutral additive model. Thus, for all population structures, a value of QST greater than FST for neutral loci is evidence for spatially divergent evolution by natural selection.     

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.     

Genetic consequences of polygyny and social structure in an Indian fruit bat, Cynopterus sphinx. II. Variance in male mating success and effective population size

Variance in reproductive success is a primary determinant of genetically effective population size (Ne), and thus has important implications for the role of genetic drift in the evolutionary dynamics of animal taxa characterized by polygynous mating systems. Here we report the results of a study designed to test the hypothesis that polygynous mating results in significantly reduced Ne in an age-structured population. This hypothesis was tested in a natural population of a harem-forming fruit bat, Cynopterus sphinx (Chiroptera: Pteropodidae), in western India. The influence of the mating system on the ratio of variance Ne to adult census number (N) was assessed using a mathematical model designed for age-structured populations that incorporated demographic and genetic data. Male mating success was assessed by means of direct and indirect paternity analysis using 10-locus microsatellite genotypes of adults and progeny from two consecutive breeding periods (n = 431 individually marked bats). Combined results from both analyses were used to infer the effective number of male parents in each breeding period. The relative proportion of successfully reproducing males and the size distribution of paternal sibships comprising each offspring cohort revealed an extremely high within-season variance in male mating success (up to 9.2 times higher than Poisson expectation). The resultant estimate of Ne/N for the C. sphinx study population was 0.42. As a result of polygynous mating, the predicted rate of drift (1/2Ne per generation) was 17.6% higher than expected from a Poisson distribution of male mating success. However, the estimated Ne/N was well within the 0.25-0.75 range expected for age-structured populations under normal demographic conditions. The life-history schedule of C. sphinx is characterized by a disproportionately short sexual maturation period scaled to adult life span. Consequently, the influence of polygynous mating on Ne/N is mitigated by the extensive overlap of generations. In C. sphinx, turnover of breeding males between seasons ensures a broader sampling of the adult male gamete pool than expected from the variance in mating success within a single breeding period.     

Predicting the annual effective size of livestock populations

Effective population size (Ne) is an important parameter determining the genetic structure of small populations. In natural populations, the number of adults (N) is usually known and Ne can be estimated on the basis of an assumed ratio Ne/N, usually found to be close to 0.5. In farm animal populations, apart from using pedigrees or genetic marker information, Ne can be estimated from the number N of breeding animals, and a value of 1 is commonly assumed for the ratio Ne/N. The purpose of this paper is to show the relation between effective population size and breeding herd size in livestock species. With overlapping generations, Ne can be predicted knowing the number of individuals entering the population per generation and the variance of family size, the latter being directly related to the survival pattern (or replacement policy) in the breeding herd. Assuming an ideal survivorship leading to a geometric age distribution, it can be shown that the number of breeding animals tends to overestimate effective size, particularly in early-maturing species. The ratio of annual effective size to the number of breeding animals is shown to be equal to [1 + (a- 1)(1 - s)]2/(1 - s2), where a is the age at first offspring and s is the survival rate (including culling) of the parents between successive births. This expression shows to what extent inbreeding may be determined by demography or culling policy independently of the actual herd size. In many situations a fast replacement or an early culling will increase annual effective size. Consequences for the management of small populations are discussed.     

Estimation of effective population size for the long-lived darkblotched rockfish Sebastes crameri

We report the variance effective population size (Ne) in darkblotched rockfish (Sebastes crameri) utilizing the temporal method for overlapping generations, which requires a combination of age-specific demography and genetic information from cohorts. Following calculations of age-specific survival and reproductive success from fishery data, we genotyped a sample (n = 1087) comprised by 6 cohorts (from 1995 to 2000) across 7 microsatellite loci. Our Ne estimate (Ne) plus 95% confidence interval was (Ne) = 9157 [6495-12 215], showing that the breeding population number could be 3-4 orders of magnitude smaller than the census population size (N) = 24 376 210). Our estimates resemble closely those found for fishes with similar life history, suggesting that the small (Ne)/(N) ratio for S. crameri is most likely explained by a combination of high variance in reproductive success among individuals, genetic structure, and demographic perturbations such as historical fishing. Because small (Ne)/(N) ratios have been commonly associated with potential loss of genetic variation, our estimates need careful consideration in rockfish management and conservation.     

Effects of Partial Inbreeding on Fixation Rates and Variation of Mutant Genes

Diffusion methods were used to investigate the fixation probability, average time until fixation and extinction, and cumulative heterozygosity and genetic variance for single mutant genes in finite populations with partial inbreeding. The critical parameters in the approximation are the coefficient of inbreeding due to nonrandom mating (F) and the effective population size (Ne), which also depends on F and the variance of family size. For large Ns, the fixation probability (u) is u = 2(Ne/N)s (F + h - Fh), where N is the population census, s is the coefficient of selection of the mutant homozygote and h is the coefficient of dominance. For Poisson family size (independent Poisson distributions of selfed and nonselfed offspring with partial selfing, and independent Poisson distributions of male and female numbers with partial sib mating), Ne = N/(1 + F), and the time until fixation is approximately equal to Ne/N times the time to fixation with random mating, but this relation does not hold, however, for other distributions of family size. The cumulative nonadditive variance until fixation or loss for dominant genes is reduced with increasing F while for recessive genes it is increased with intermediate values of F. The average time until extinction of deleterious mutations is reduced by increasing F. This reduction, when expressed as a proportion, is approximately independent of the initial gene frequency as well as the selective disadvantage if this is large.     

Demographic and genetic estimates of effective population size (Ne) reveals genetic compensation in steelhead trout

Estimates of effective population size (Ne) are required to predict the impacts of genetic drift and inbreeding on the evolutionary dynamics of populations. How the ratio of Ne to the number of sexually mature adults (N) varies in natural vertebrate populations has not been addressed. We examined the sensitivity of Ne/N to fluctuations of N and determined the major variables responsible for changing the ratio over a period of 17 years in a population of steelhead trout (Oncorhynchus mykiss) from Washington State. Demographic and genetic methods were used to estimate Ne. Genetic estimates of Ne were gained via temporal and linkage disequilibrium methods using data from eight microsatellite loci. DNA for genetic analysis was amplified from archived smolt scales. The Ne/N from 1977 to 1994, estimated using the temporal method, was 0.73 and the comprehensive demographic estimate of Ne/N over the same time period was 0.53. Demographic estimates of Ne indicated that variance in reproductive success had the most substantial impact on reducing Ne in this population, followed by fluctuations in population size. We found increased Ne/N ratios at low N, which we identified as genetic compensation. Combining the information from the demographic and genetic methods of estimating Ne allowed us to determine that a reduction in variance in reproductive success must be responsible for this compensation effect. Understanding genetic compensation in natural populations will be valuable for predicting the effects of changes in N (i.e. periods of high population density and bottlenecks) on the fitness and genetic variation of natural populations.     

The effective size of annual plant populations: the interaction of a seed bank with fluctuating population size in maintaining genetic variation

Many annual plant populations undergo dramatic fluctuations in size. Such fluctuations can result in the loss of genetic variability. Here I formalize the potential for a seed bank to buffer against such genetic loss. The average time to seed germination (T) defines the generation time of "annuals" with a seed bank, and assuming random seed germination, I show that, under otherwise ideal conditions, a population's effective size (Ne) equals NT, where N is the number of adult plants. This result supports the general principle that lengthening the prereproductive period increases Ne. When adult numbers vary, Ne at any time depends on N and on the numbers contributing to the seed bank in previous seasons. Averaging these effects over time gives Ne approximately Nh + (T - 1)Na, where Nh and Na are the harmonic and arithmetic means of the adult population. Thus if T > 1, Ne is determined primarily by Na. Simulations showed that until fluctuations in N are large (>25x) this relationship is accurate. I extended the theory to incorporate a selfing rate (S) and reproductive variance (I) through seed production (k), outcrossed pollen (m), and variation in selfing rate: Ne = NT(1 -S/2)/(1 + I) = NT/[1 + FIS)(1 + I)]. Reproductive variance (I) equals [Ik(1 + S)2 + IM(1 - S)2 + 2(1 - S2)Ikm = S2IS(1 + Ik)]/4, , where Ij is the standardized variance (Vj/j2) of factor j and Ikm is the standardized covariance between k and m. These results are applicable to other organisms with a similar life history, such as freshwater crustaceans with diapausing eggs (e.g., tadpole shrimp, clam shrimp, and fairy shrimp) and other semelparous species with discrete breeding seasons and a variable maturation time (e.g., Pacific salmon).     

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).     

Effective population size of steelhead trout: influence of variance in reproductive success, hatchery programs, and genetic compensation between life-history forms

The effective population size is influenced by many biological factors in natural populations. To evaluate their relative importance, we estimated the effective number of breeders per year (Nb) and effective population size per generation (Ne) in anadromous steelhead trout (Oncorhynchus mykiss) in the Hood River, Oregon (USA). Using demographic data and genetic parentage analysis on an almost complete sample of all adults that returned to the river over 15 years (>15,000 individuals), we estimated Nb for 13 run years and Ne for three entire generations. The results are as follows: (i) the ratio of Ne to the estimated census population size (N) was 0.17-0.40, with large variance in reproductive success among individuals being the primary cause of the reduction in Ne/N; (ii) fish from a traditional hatchery program (Htrad: nonlocal, multiple generations in a hatchery) had negative effects on Nb, not only by reducing mean reproductive success but also by increasing variance in reproductive success among breeding parents, whereas no sign of such effects was found in fish from supplementation hatchery programs (Hsupp: local, single generation in a hatchery); and (iii) Nb was relatively stable among run years, despite the widely fluctuating annual run sizes of anadromous adults. We found high levels of reproductive contribution of nonanadromous parents to anadromous offspring when anadromous run size is small, suggesting a genetic compensation between life-history forms (anadromous and nonanadromous). This is the first study showing that reproductive interaction between different life-history forms can buffer the genetic impact of fluctuating census size on Ne.     

Effective Size of Nonrandom Mating Populations

Nonrandom mating whereby parents are related is expected to cause a reduction in effective population size because their gene frequencies are correlated and this will increase the genetic drift. The published equation for the variance effective size, Ne, which includes the possibility of nonrandom mating, does not take into account such a correlation, however. Further, previous equations to predict effective sizes in populations with partial sib mating are shown to be different, but also incorrect. In this paper, a corrected form of these equations is derived and checked by stochastic simulation. For the case of stable census number, N, and equal progeny distributions for each sex, the equation is [formula: see text], where Sk2 is the variance of family size and alpha is the departure from Hardy-Weinberg proportions. For a Poisson distribution of family size (Sk2 = 2), it reduces to Ne = N/(1 + alpha), as when inbreeding is due to selfing. When nonrandom mating occurs because there is a specified system of partial inbreeding every generation, alpha can be substituted by Wright's FIS statistic, to give the effective size as a function of the proportion of inbred mates.     

Large variance in reproductive success and the Ne/N ratio

The ratio of the effective population size to adult (or census) population size (Ne/N) is an indicator of the extent of genetic variation expected in a population. It has been suggested that this ratio may be quite low for highly fecund species in which there is a sweepstakes-like chance of reproductive success, known as the Hedgecock effect. Here I show theoretically how the ratio may be quite small when there are only a few successful breeders (Nb) and that in this case, the Ne/N ratio is approximately Nb/N. In other words, high variance in reproductive success within a generation can result in a very low effective population size in an organism with large numbers of adults and consequently a very low Ne/N ratio. This finding appears robust when there is a large proportion of families with exactly two progeny or when there is random variation in progeny numbers among these families.     

The effect of migration during the divergence

The mean and variance of the number of nucleotide differences were obtained when the ancestral population diverged with migration. The number of nucleotide differences obtained indicates that not only the migration rate but also the period of migration has influence on a population structure. According to the migration rate and the period of migration, populations behave approximately as a single unit, diverged and isolated populations, two populations under equilibrium, or none of them. When sigma m(t) is about one, the variance of the number of nucleotide differences becomes large, where sigma m(t) is the sum of the migration rate for the period of migration. The distribution of the estimated divergence time was also obtained using computer simulations. It was found that the divergence time can be explained by sigma m(t). That is, the divergence time is mostly estimated as the time when sigma m(t) is less than 1.     

Impact of clonal growth on effective population size in Hymenoxys herbacea (Asteraceae)

By influencing the proliferation of different genotypes, clonal growth can affect the maintenance of genetic variability and magnitude of genetic drift within plant populations. However, estimates of effective population size rarely incorporate the contribution of both asexual and sexual reproduction. We estimated effective size (Ne) for two populations of the clonal, self-incompatible plant, Hymenoxys herbacea, using a stage-structured demographic model for organisms with asexual and sexual recruitment and then examined the impact of reproductive strategy using an elasticity analysis. Plant rosettes monitored in two successive years had high survival rates in both populations (mean 0.94). The mean number of sexually derived recruits per initial ramet was 0.041 (SE 0.039), whereas the mean number of clonal recruits was 0.61 (SE 0.90). Effective size was 1642 and 5769 in the two populations and the Ne/N ratio averaged 0.34, comparable to values for other clonal species. Elasticity analysis indicated that increases in both clonal and sexual recruitment cause an increase in Ne while increasing the variance reduced Ne. However, Ne was more sensitive to changes in the mean and variance of asexual recruitment than sexual recruitment. These results highlight the importance of considering asexual modes of reproduction when examining the role of genetic stochasticity in populations.     

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.     

Using Molecular Markers to Estimate Quantitative Trait Locus Parameters: Power and Genetic Variances for Unreplicated and Replicated Progeny

Many of the progeny types used to estimate quantitative trait locus (QTL) parameters can be replicated, e.g., recombinant inbred, doubled haploid, and F3 lines. These parameters are estimated using molecular markers or QTL genotypes estimated from molecular markers as independent variables. Experiment designs for replicated progeny are functions of the number of replications per line (r) and the number of replications per QTL genotype (n). The value of n is determined by the size of the progeny population (N), the progeny type, and the number of simultaneously estimated QTL parameters (q - 1). Power for testing hypotheses about means of QTL genotypes is increased by increasing r and n, but the effects of these factors have not been quantified. In this paper, we describe how power is affected by r, n, and other factors. The genetic variance between lines nested in QTL genotypes (sigma 2n:q) is the fraction of the genetic variance between lines (sigma 2n) which is not explained by simultaneously estimated intralocus and interlocus QTL parameters (phi 2Q); thus, sigma 2n:q = sigma 2n - phi 2Q. If sigma 2n:q not equal to 0, then power is not efficiently increased by increasing r and is maximized by maximizing n and using r = 1; however, if sigma 2n:q = 0, then r and n affect power equally and power is efficiently increased by increasing r and is maximized by maximizing N.r. Increasing n efficiently increases power for a wide range of values of sigma 2n:q.sigma 2n:q = 0 when the genetic variance between lines is fully explained by QTL parameters (sigma 2n = phi 2Q).(ABSTRACT TRUNCATED AT 250 WORDS)     

Within-generation mutation variance for litter size in inbred mice

The mutational input of genetic variance per generation (sigma(m)(2)) is the lower limit of the genetic variability in inbred strains of mice, although greater values could be expected due to the accumulation of new mutations in successive generations. A mixed-model analysis using Bayesian methods was applied to estimate sigma(m)(2) and the across-generation accumulated genetic variability on litter size in 46 generations of a C57BL/6J inbred strain. This allowed for a separate inference on sigma(m)(2) and on the additive genetic variance in the base population (sigma(a)(2)). The additive genetic variance in the base generation was 0.151 and quickly decreased to almost null estimates in generation 10. On the other hand, sigma(m)(2) was moderate (0.035) and the within-generation mutational variance increased up to generation 14, then oscillating between 0.102 and 0.234 in remaining generations. This pattern suggested the existence of a continuous uploading of genetic variability for litter size (h(2)=0.045). Relevant genetic drift was not detected in this population. In conclusion, our approach allowed for separate estimation of sigma(a)(2) and sigma(m)(2) within the mixed-model framework, and the heritability obtained highlighted the significant and continuous influence of new genetic variability affecting the genetic stability of inbred strains.     

Evaluating the effect of stage-specific survivorship on the N(e)/N ratio

Evaluating effective population size (Ne) and the effective size to census size ratio (Ne/N) in species with Type III survivorship curves is complicated when key demographic parameters [mean (k macro) and variance (V(k)) of family size] are measured during early life stages. The method of Crow & Morton (1955) for scaling demographic data collected at a juvenile stage to expected values at adulthood is extended to consider sequential episodes of random and family correlated survival. Results show the following: (i) The order in which the episodes of random and family-correlated survival occur does not affect N(e) or N(e)/N; (ii) If a population experiences an episode of family-correlated survival, N(e)/N scaled to its expected value in a population of constant size (k macro= 2) is simply the survival rate during the family-correlated stage. If multiple such stages occur, scaled N(e)/N is the product of the survivals during all family-correlated life stages; (iii) Under the assumption of random post-enumeration survival, adjusting the variance effective size to its expected value at k macro= 2 is equivalent to computing the inbreeding effective size at the earlier life stage. Application to experimental data for hatchery populations of Pacific salmon (Oncorhynchus spp.) indicates that nonrandom survival during the marine phase led to estimated reductions in effective size of 0-62 (mean 19) in 12 different cohorts. This approach can provide insights into N(e)/N in highly fecund species, including some marine species in which N(e) has been estimated to be several orders of magnitude less than N.     

An Approach to Population and Evolutionary Genetic Theory for Genes in Mitochondria and Chloroplasts, and Some Results

We developed population genetic theory for organelle genes, using an infinite alleles model appropriate for molecular genetic data, and considering the effects of mutation and random drift on the frequencies of selectively neutral alleles. The effects of maternal inheritance and vegetative segregation of organelle genes are dealt with by defining new effective gene numbers, and substituting these for 2N(e) in classical theory of nuclear genes for diploid organisms. We define three different effective gene numbers. The most general is N(lambda), defined as a function of population size, number of organelle genomes per cell, and proportions of genes contributed by male and female gametes to the zygote. In many organisms, vegetative segregation of organelle genomes and intracellular random drift of organelle gene frequencies combine to produce a predominance of homoplasmic cells within individuals in the population. Then, the effective number of organelle genes is N(eo), a simple function of the numbers of males and females and of the maternal and paternal contributions to the zygote. Finally, when the paternal contribution is very small, N( eo) is closely approximated by the number of females, N( f). Then if the sex ratio is 1, the mean time to fixation or loss of new mutations is approximately two times longer for nuclear genes than for organelle genes, and gene diversity is approximately four times greater. The difference between nuclear and organelle genes disappears or is reversed in animals in which males have large harems. The differences between nuclear and organelle gene behavior caused by maternal inheritance and vegetative segregation are generally small and may be overshadowed by differences in mutation rates to neutral alleles. For monoecious organisms, the effective number of organelle genes is approximately equal to the total population size N. We also show that a population can be effectively subdivided for organelle genes at migration rates which result in panmixis for nuclear genes, especially if males migrate more than females.     

On the Theory of Partially Inbreeding Finite Populations. I. Partial Selfing

Some stochastic theory is developed for monoecious populations of size N in which there are probabilities beta and 1 - beta of reproduction by selfing and by random mating. It is assumed that beta much greater than N-1. Expressions are derived for the inbreeding coefficient of one random individual and the coefficient of kinship of two random separate individuals at time t. The mean and between-lines variance of the fraction of copies of a locus that are identical in two random separate individuals in an equilibrium population are obtained under the assumption that there is an infinite number of possible alleles. It is found that the theory for random mating populations holds if the effective population number is Ne = N'/(1 + FIS), where FIS is the inbreeding coefficient at equilibrium when N is infinite and N' is the reciprocal of the probability that two gametes contributing to random separate adults come from the same parent. When there is a binomial distribution of successful gametes emanating from each adult, N' = N. An approximation to the probability that an allele A survives if it is originally present in one AA heterozygote is found to be 2(N'/N)(FISS1 + (1 - FIS)S2), where S1 and S2 are the selective advantages of AA and AA in comparison with AA. In the last section it is shown that if there is partial full sib mating and binomial offspring distributions Ne = N/(1 + 3FIS).