Statistics of natural populations. III. Sequential sampling plans for the estimation of gene frequencies

In previous work several models have been developed for genetic surveys of natural populations. Parents of unknown genotype are collected from a natural population, polymorphic at a single genetic locus. From each of these N cryptic parents a number of offspring are identified for their genotype. Our problem is to select an efficient offspring sampling plan for estimating the frequency of an allele in the cryptic adult population based on the N family profiles of juvenile genotypes. A criterion called the information per unit cost of observation is introduced to evaluate sequential sampling plans, in which the number of offspring per family examined is random. Some simple, practical schemes for stopping the sampling of offspring from a collected parent are introduced; one example is stopping when: (i) the offspring are definitive about the parental genotype(s) for the first time; (ii) a fixed number of one genotype only is seen; or (iii) a fixed maximum feasible number of offspring have been genotyped. This sampling scheme is recommended. For each sampling scheme, the best linear unbiased estimator and the sequential maximum likelihood estimator of the allele frequency are characterized. From the moments of these estimators, it is then possible to tabulate efficient sequential sampling plans, which are better (in the sense of information per unit cost), just as simple, and less costly than corresponding fixed sampling plans in use.     

Prediction of identity by descent probabilities from marker-haplotypes

The prediction of identity by descent (IBD) probabilities is essential for all methods that map quantitative trait loci (QTL). The IBD probabilities may be predicted from marker genotypes and/or pedigree information. Here, a method is presented that predicts IBD probabilities at a given chromosomal location given data on a haplotype of markers spanning that position. The method is based on a simplification of the coalescence process, and assumes that the number of generations since the base population and effective population size is known, although effective size may be estimated from the data. The probability that two gametes are IBD at a particular locus increases as the number of markers surrounding the locus with identical alleles increases. This effect is more pronounced when effective population size is high. Hence as effective population size increases, the IBD probabilities become more sensitive to the marker data which should favour finer scale mapping of the QTL. The IBD probability prediction method was developed for the situation where the pedigree of the animals was unknown (i.e. all information came from the marker genotypes), and the situation where, say T, generations of unknown pedigree are followed by some generations where pedigree and marker genotypes are known.     

Inbreeding effective population size and parentage analysis without parents

An important use of genetic parentage analysis is the ability to directly calculate the number of offspring produced by each parent (k(i)) and hence effective population size, N(e). But what if parental genotypes are not available? In theory, given enough markers, it should be possible to reconstruct parental genotypes based entirely on a sample of progeny, and if so the vector of parental k(i) values. However, this would provide information only about parents that actually contributed offspring to the sample. How would ignoring the 'null' parents (those that produced no offspring) affect an estimate of N(e)? The surprising answer is that null parents have no effect at all. We show that: (i) The standard formula for inbreeding N(e) can be rewritten so that it is a function only of sample size and ∑(k(2)(i)); it is not necessary to know the total number of parents (N). This same relationship does not hold for variance N(e). (ii) This novel formula provides an unbiased estimate of N(e) even if only a subset of progeny is available, provided the parental contributions are accurately determined, in which case precision is also high compared to other single-sample estimators of N(e). (iii) It is not necessary to actually reconstruct parental genotypes; from a matrix of pairwise relationships (as can be estimated by some current software programs), it is possible to construct the vector of k(i) values and estimate N(e). The new method based on parentage analysis without parents (PwoP) can potentially be useful as a single-sample estimator of contemporary N(e), provided that either (i) relationships can be accurately determined, or (ii) ∑(k(2)(i)) can be estimated directly.     

Genetic identity in subdivided populations. I. Two equal-sized subpopulations

Moments of the steady state frequency spectrum (probabilities of identity of samples of genes) are obtained for a subdivided population by using standard recursive identity by state calculations. These moments are used to obtain variances for some measures of genetic identity, including Nei's normalized genetic identity (I) and genetic distance (?log_eI). The results are compared with those obtained from the corresponding undivided population theory, including adjustments to the effective number to try to account for subdivision. Undivided population approximations based on effective number are surprisingly accurate, regardless of the migration rate, when sampling exclusively from one subpopulation.     

Methods to estimate effective population size using pedigree data: Examples in dog,sheep, cattle and horse

Background

Effective population sizes of 140 populations (including 60 dog breeds, 40 sheep breeds, 20 cattle breeds and 20 horse breeds) were computed using pedigree information and six different computation methods. Simple demographical information (number of breeding males and females), variance of progeny size, or evolution of identity by descent probabilities based on coancestry or inbreeding were used as well as identity by descent rate between two successive generations or individual identity by descent rate.

Results

Depending on breed and method, effective population sizes ranged from 15 to 133 056, computation method and interaction between computation method and species showing a significant effect on effective population size (P < 0.0001). On average, methods based on number of breeding males and females and variance of progeny size produced larger values (4425 and 356, respectively), than those based on identity by descent probabilities (average values between 93 and 203). Since breeding practices and genetic substructure within dog breeds increased inbreeding, methods taking into account the evolution of inbreeding produced lower effective population sizes than those taking into account evolution of coancestry. The correlation level between the simplest method (number of breeding males and females, requiring no genealogical information) and the most sophisticated one ranged from 0.44 to 0.60 according to species.

Conclusions

When choosing a method to compute effective population size, particular attention should be paid to the species and the specific genetic structure of the population studied.     

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).     

Gene Conversion, Linkage, and the Evolution of Repeated Genes Dispersed among Multiple Chromosomes

The evolution of the probabilities of genetic identity within and between the loci of a multigene family dispersed among multiple chromosomes is investigated. Unbiased gene conversion, equal crossing over, random genetic drift, and mutation to new alleles are incorporated. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. The linkage map is arbitrary, but the same for every chromosome; the dependence of the probabilities of identity on the location on each chromosome is formulated exactly. The greatest of the rates of gene conversion, random drift, and mutation is epsilon much less than 1. Under the assumption of loose linkage (i.e., all the crossover rates greatly exceed epsilon, though they may still be much less than 1/2), explicit approximations are obtained for the equilibrium values of the probabilities of identity and of the linkage of disequilibria. The probabilities of identity are of order one [i.e., O(1)] and do not depend on location; the linkage disequilibria are of O(epsilon) and, within each chromosome, depend on location through the crossover rates. It is demonstrated also that the ultimate rate and pattern of convergence to equilibrium are close to that of a much simpler, location-independent model. If intrachromosomal conversion is absent, the above results hold even without the assumption of loose linkage. In all cases, the relative errors are of O(epsilon). Even if the conversion rate between genes on nonhomologous chromosomes is considerably less than between genes on the same chromosome or homologous chromosomes, the probabilities of identity between the former genes are still almost as high as those between the latter, and the rate of convergence is still not much less than with equal conversion rates. If the crossover rates are much less than 1/2, then most of the linkage disequilibrium is due to intrachromosomal conversion. If linkage is loose, the reduction of the linkage disequilibria to O(epsilon) requires only O(-ln epsilon) generations.     

The Inbreeding Effective Population Number and the Expected Homozygosity for an X-Linked Locus

Assuming random mating and discrete nonoverlapping generations, the inbreeding effective population number, (see PDF), is calculated for an X-linked locus. For large populations, the result agrees with the variance effective population number. As an application, the maintenance of genetic variability by the joint action of mutation and random drift is investigated. It is shown that, if every allele mutates at rate u to new types, then the probabilities of identity in state (and hence the expected homozygosity of females) converge to the approximate value (see PDF) at the approximate asymptotic rate (see PDF).     

Geneland: a computer package for landscape genetics

Geneland is a computer package that allows to make use of georeferenced individual multilocus genotypes for the inference of the number of populations and of the spatial location of genetic discontinuities between those populations. Main assumptions of the method are: (i) the number of populations is unknown and all values are considered a priori equally likely, (ii) populations are spread over areas given by a union of some polygons of unknown location in the spatial domain, (iii) Hardy–Weinberg equilibrium is assumed within each population and (iv) allele frequencies in each population are unknown and treated as random variable either following the so‐called Dirichlet model or Falush model. Different algorithms implemented in Geneland to perform inferences are first briefly presented. Then major running steps and outputs (i.e. histogram of number of populations and map of posterior probabilities of population membership) are illustrated from the analysis of a simulated data set, which was also produced by Geneland.     

On the probabilities of identity states in permutable populations.

Génin and Clerget-Darpoux recently discussed the derivation of the probabilities of identity states for populations in which there was some degree of kinship, primarily to allow the extension of the classical affected-sib-pair method to such populations. It is argued here that their derivation makes certain assumptions that are valid only for some very restricted population models and that are not needed for an appropriate treatment. Here the probabilities of the identity states of two individuals with a given genealogical relationship are specified in terms of the kinship parameters of the underlying population, from which the founders of the individuals'' genealogy have been randomly selected. It is argued that an appropriate representation for a permutable population, one in which gene identity does not depend on the pattern of genes across individuals, requires three parameters. This representation is related to that of Génin and Clerget-Darpoux and to that of Weir.     

The persistence of vertically transmitted fungi in grass metapopulations

Theory predicts that (i) vertical transmission of parasites (i.e. when they are passed directly from a host to its offspring) selects for benign association with the host and that (ii) vertically transmitted parasites that lower their hosts' fitness cannot persist if they are not able to infect horizontally (i.e. contagiously) other host individuals in the population. In this paper, we develop a mathematical model to examine whether mutualism is a prerequisite for persistence of exclusively vertically transmitted (from maternal plant to offspring via seeds) fungal endophytes in structured grass metapopulations. Interestingly, endophyte survival does not require plant mutualism, even in a metapopulation consisting of qualitatively identical patches, if vertical transmission of the fungus is perfect, i.e. if all established seedlings in offspring of the endophyte-infected plant are infected. In more realistic situations, when the metapopulation consists of qualitatively different patches, endophyte-infected plants may persist at the metapopulation level even if the vertical transmission is imperfect (due to hyphae inviability or failure to grow into all seeds) and the endophyte decreases the host grass fitness in certain environments. These results have biological importance because they (i) question the requirement of a mutualistic nature in exclusively vertically transmitted symbionts and (ii) emphasize the importance of habitat diversity in relation to symbiont success in vertical transmission.     

Parents and offspring in an evolutionary game: the effect of supply on demand when costs of care vary

Current models of parent-offspring communication do not explicitly predict the effect of parental food supply on offspring demand (ESD). However, existing theory is frequently interpreted as predicting a negative ESD, such that offspring beg less when parental supply is high. While empirical evidence largely supports this interpretation, several studies have identified the opposite case, with well-fed offspring begging more than those in poorer condition. Here, we show that signalling theory can give rise to either a negative or a positive ESD depending on the precise form of costs and benefits. Introducing variation among parents in the cost of care, we show that the ESD may change sign depending upon the quantitative relation between two effects: (i) decreased supply leads to increased begging because of an increase in marginal fitness benefit of additional resources to offspring, (ii) decreased supply leads to reduced begging because it is associated with a decrease in parental responsiveness, rendering begging less effective. To illustrate the interplay between these two effects, we show that Godfray's seminal model of begging yields a negative ESD when care is generally cheap, because the impact of supply on the marginal benefits of additional resources then outweighs the associated changes in parental responsiveness to begging. By contrast, the same model predicts a positive ESD when care is generally costly, because the impact of care costs on parental responsiveness then outweighs the change in marginal benefits.     

Adaptive sex allocation in birds: the complexities of linking theory and practice

We review some recent theoretical and empirical developments in the study of sex allocation in birds. The advent of reliable molecular sexing techniques has led to a sharp increase in the number of studies that report biased offspring sex ratios in birds. However, compelling evidence for adaptive sex allocation in birds is still very scant. We argue that there are two reasons for this: (i) standard sex allocation models, very helpful in understanding sex allocation of invertebrates, do not sufficiently take the complexities of bird life histories and physiology into account. Recent theoretical work might bring us a step closer to more realistic models; (ii) experimental field and laboratory studies on sex allocation in birds are scarce. Recent experimental work both in the laboratory and in the field shows that this is a promising approach.     

Population genetic inference using a fixed number of segregating sites: a reassessment

Coalescent theory is commonly used to perform population genetic inference at the nucleotide level. Here, we examine the procedure that fixes the number of segregating sites (henceforth the FS procedure). In this approach a fixed number of segregating sites (S) are placed on a coalescent tree (independently of the total and internode lengths of the tree). Thus, although widely used, the FS procedure does not strictly follow the assumptions of coalescent theory and must be considered an approximation of (i) the standard procedure that uses a fixed population mutation parameter theta, and (ii) procedures that condition on the number of segregating sites. We study the differences in the false positive rate for nine statistics by comparing the FS procedure with the procedures (i) and (ii), using several evolutionary models with single-locus and multilocus data. Our results indicate that for single-locus data the FS procedure is accurate for the equilibrium neutral model, but problems arise under the alternative models studied; furthermore, for multilocus data, the FS procedure becomes inaccurate even for the standard neutral model. Therefore, we recommend a procedure that fixes the theta value (or alternatively, procedures that condition on S and take into account the uncertainty of theta) for analysing evolutionary models with multilocus data. With single-locus data, the FS procedure should not be employed for models other than the standard neutral model.     

Optimal offspring size in a small mammal: an exception to the tradeoff invariant life-history rule

Offspring size and number were examined in a captive population of wild guinea pigs ( Cavia aperea ), and findings were compared with models of optimal offspring size for small litters. Median and modal litter size was two, regardless of maternal size or parity. Females producing their second litter tended to have litters that were larger than average. In contrast, young females that were still growing never had litters that were larger than average. Mean offspring size decreased and variation in offspring size tended to decrease with increasing litter size. Optimal offspring size models, in which offspring survival depended on the amount of resources invested, as well as litter size, predict such a trend. Little support was found for Charnov and Downhower's (1995) tradeoff invariant life-history rule that the range in offspring sizes between litters is inversely proportional to the size of the litter. Cavia aperea may be an exception to this rule because pup mass at birth did not reflect total reproductive investment, because conversion of resources into litter mass may not be linearly related to litter size and because resources were not equally partitioned among offspring within large litters. Experimental data are needed to determine the relevance of these results among mammals in general.     

Dynamic sampling bias and overdispersion induced by skewed offspring distributions

Natural populations often show enhanced genetic drift consistent with a strong skew in their offspring number distribution. The skew arises because the variability of family sizes is either inherently strong or amplified by population expansions. The resulting allele-frequency fluctuations are large and, therefore, challenge standard models of population genetics, which assume sufficiently narrow offspring distributions. While the neutral dynamics backward in time can be readily analyzed using coalescent approaches, we still know little about the effect of broad offspring distributions on the forward-in-time dynamics, especially with selection. Here, we employ an asymptotic analysis combined with a scaling hypothesis to demonstrate that over-dispersed frequency trajectories emerge from the competition of conventional forces, such as selection or mutations, with an emerging time-dependent sampling bias against the minor allele. The sampling bias arises from the characteristic time-dependence of the largest sampled family size within each allelic type. Using this insight, we establish simple scaling relations for allele-frequency fluctuations, fixation probabilities, extinction times, and the site frequency spectra that arise when offspring numbers are distributed according to a power law.     

Fixation probabilities for advantageous mutants: a note on multiplication and sampling

If the average number of gametes produced by the individual is small, as may be the case for haploid organisms, then sampling with and without replacement can lead to considerable differences in the fixation probabilities of mutant alleles. As a function of the population size N, these probabilities converge quickly to the survival probabilities given by branching processes with Poisson or Bernoulli offspring distributions.     

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.     

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.     

Deleterious impact of feto-maternal MHC compatibility on the success of pregnancy in a macaque model

The impact of feto-maternal histocompatibility on reproduction has inspired long-lasting debates. However, after the review of numerous articles, the impact of HLA allele sharing within couples on fecundity remains questionable. We decided to explore the impact of major histocompatibility complex (MHC) feto-maternal compatibility on reproduction in a cynomolgus macaque facility composed of animals of Mauritian descent. The Mauritian-derived macaque population presents a very restricted MHC polymorphism (only seven founding haplotypes) due to a strong founding bottleneck effect. The MHC polymorphism was investigated in 237 trios (male, female and offspring) using 17 microsatellite markers distributed across the MHC. Haplotypes were confirmed by segregation analysis. We evaluated the relative frequencies of MHC-compatible and MHC-semi-compatible offspring with the mothers. Among the 237 trios, we selected 42 trios for which the identity of the father is certain and for which the theoretical probabilities of fully compatible and semi-compatible offspring were equal. We found 11 offspring fully compatible and 31 offspring semi-compatible with their respective mother. The observed proportions were clearly outside the interval of confidence of 99 % and therefore most probably resulted from a selection of the semi-compatible offspring during pregnancy. We concluded that MHC fully compatible cynomolgus macaque offspring have a selective survival disadvantage in comparison with offspring inheriting a paternal MHC haplotype differing from maternal haplotypes.