Estimating effective population size from linkage disequilibrium: severe bias in small samples

Effective population size (N _e) is a central concept in evolutionary biology and conservation genetics. It predicts rates of loss of neutral genetic variation, fixation of deleterious and favourable alleles, and the increase of inbreeding experienced by a population. A method exists for the estimation of N _e from the observed linkage disequilibrium between unlinked loci in a population sample. While an increasing number of studies have applied this method in natural and managed populations, its reliability has not yet been evaluated. We developed a computer program to calculate this estimator of N _e using the most widely used linkage disequilibrium algorithm and used simulations to show that this estimator is strongly biased when the sample size is small (<‰100) and below the true N _e. This is probably due to the linkage disequilibrium generated by the sampling process itself and the inadequate correction for this phenomenon in the method. Results suggest that N _e estimates derived using this method should be regarded with caution in many cases. To improve the method’s reliability and usefulness we propose a way to determine whether a given sample size exceeds the population N _e and can therefore be used for the computation of an unbiased estimate.     

An evaluation of the methods to estimate effective population size from measures of linkage disequilibrium

In 1971, John Sved derived an approximate relationship between linkage disequilibrium (LD) and effective population size for an ideal finite population. This seminal work was extended by Sved and Feldman (Theor Pop Biol 4, 129, 1973) and Weir and Hill (Genetics 95, 477, 1980) who derived additional equations with the same purpose. These equations yield useful estimates of effective population size, as they require a single sample in time. As these estimates of effective population size are now commonly used on a variety of genomic data, from arrays of single nucleotide polymorphisms to whole genome data, some authors have investigated their bias through simulation studies and proposed corrections for different mating systems. However, the cause of the bias remains elusive. Here, we show the problems of using LD as a statistical measure and, analogously, the problems in estimating effective population size from such measure. For that purpose, we compare three commonly used approaches with a transition probability‐based method that we develop here. It provides an exact computation of LD. We show here that the bias in the estimates of LD and effective population size are partly due to low‐frequency markers, tightly linked markers or to a small total number of crossovers per generation. These biases, however, do not decrease when increasing sample size or using unlinked markers. Our results show the issues of such measures of effective population based on LD and suggest which of the method here studied should be used in empirical studies as well as the optimal distance between markers for such estimates.     

利用全基因组连锁不平衡估计中国荷斯坦牛有效群体大小

有效群体大小是群体遗传学研究的一个重要内容,有助于我们更清楚地了解群体的遗传变异、进化和复杂性状的遗传机制等。随着高密度SNP标记的出现,越来越多的研究利用SNP标记间连锁不平衡估计有效群体大小。文章采集北京地区中国荷斯坦牛2 093个样本,并利用牛SNP芯片(Illumina BovineSNP50,含5 4001 SNPs)进行基因型测定,估计不同世代中国荷斯坦牛的有效群体大小。质量控制标准设定为SNP检出率0.95,最小等位基因频率>0.05,样本检出率0.95,哈代温伯格平衡检验显著性水平P<0.0001。经过质量控制,共1 968个样本和38 796个SNPs用于连锁不平衡分析。文章选取SNP间距0.1、0.2、0.5、1、2、5、10、15(Mb),估计中国荷斯坦牛在4世代之前有效群体大小。结果表明,中国荷斯坦牛的有效群体呈逐代下降趋势,至4世代前,中国荷斯坦牛平均有效群体为45头左右。     

Assessing performance of single‐sample molecular genetic methods to estimate effective population size: empirical evidence from the endangered Gochu Asturcelta pig breed

Estimating effective population size (N_e) using linkage disequilibrium (LD) information (N_e(LD)) has the operational advantage of using a single sample. However, N_e(LD) estimates assume discrete generations and its performance are constrained by demographic issues. However, such concerns have received little empirical attention so far. The pedigree of the endangered Gochu Asturcelta pig breed includes individuals classified into discrete filial generations and individuals with generations overlap. Up to 780 individuals were typed with a set of 17 microsatellites. Performance of N_e(LD) was compared with N_e estimates obtained using genealogical information, molecular coancestry (N_e(M)) and a temporal (two‐sample) method (N_e(JR)). Molecular‐based estimates of N_e exceeded those obtained using pedigree data. Estimates of N_e(LD) for filial generations F3 and F4 (17.0 and 17.3, respectively) were lower and steadier than those obtained using yearly or biannual samplings. N_e(LD) estimated for samples including generations overlap could only be compared with those obtained for the discrete filial generations when sampling span approached a generation interval and demographic correction for bias was applied. Single‐sample N_e(M) estimates were lower than their N_e(LD) counterparts. N_e(M) estimates are likely to partially reflect the number of founders rather than population size. In any case, estimates of LD and molecular coancestry tend to covary and, therefore, N_e(M) and N_e(LD) can hardly be considered independent. Demographically adjusted estimates of N_e(JR) and N_e(LD) took comparable values when: (1) the two samples used for the former were separated by one equivalent to discrete generations in the pedigree and (2) sampling span used for the latter approached a generation interval. Overall, the empirical evidence given in this study suggested that the advantage of using single‐sample methods to obtain molecular‐based estimates of N_e is not clear in operational terms. Estimates of N_e obtained using methods based in molecular information should be interpreted with caution.     

Interannual variation in effective number of breeders and estimation of effective population size in long‐lived iteroparous lake sturgeon (Acipenser fulvescens)

Quantifying interannual variation in effective adult breeding number (N_b) and relationships between N_b, effective population size (N_e), adult census size (N) and population demographic characteristics are important to predict genetic changes in populations of conservation concern. Such relationships are rarely available for long‐lived iteroparous species like lake sturgeon (Acipenser fulvescens). We estimated annual N_b and generational N_e using genotypes from 12 microsatellite loci for lake sturgeon adults (n = 796) captured during ten spawning seasons and offspring (n = 3925) collected during larval dispersal in a closed population over 8 years. Inbreeding and variance N_b estimated using mean and variance in individual reproductive success derived from genetically identified parentage and using linkage disequilibrium (LD) were similar within and among years (interannual range of N_b across estimators: 41–205). Variance in reproductive success and unequal sex ratios reduced N_b relative to N on average 36.8% and 16.3%, respectively. Interannual variation in N_b/N ratios (0.27–0.86) resulted from stable N and low standardized variance in reproductive success due to high proportions of adults breeding and the species' polygamous mating system, despite a 40‐fold difference in annual larval production across years (437–16 417). Results indicated environmental conditions and features of the species' reproductive ecology interact to affect demographic parameters and N_b/N. Estimates of N_e based on three single‐sample estimators, including LD, approximate Bayesian computation and sibship assignment, were similar to annual estimates of N_b. Findings have important implications concerning applications of genetic monitoring in conservation planning for lake sturgeon and other species with similar life histories and mating systems.     

Inbreeding effective population size under some artificial selection schemes

Summary It is well known that truncation selection is the most efficient form of directional selection in terms of changing gene frequency. In this paper we show circumstances where truncation selection followed by a balanced mating generates inbreeding effective population size smaller than that generated by a selection that assigns mating frequencies to individuals according to their breeding values, where both selection schemes give the same expected performance of selected individuals (selection differential). Breeding values of selected individuals and the weight used to determine mating frequencies are assumed to be linearly distributed on a performance scales, x. To assign mating frequencies to the individuals in the weighting system, the selected individuals are grouped using a constant , and ith group in the interval x_i, x_i + . With small number of groups, say 2 or 3, the weighting system in general generates inbreeding effective population size that is larger than that generated by a truncation selection. As the number of the groups increases, truncation selection generates larger effective numbers.     

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.     

speed‐ne: Software to simulate and estimate genetic effective population size (Ne) from linkage disequilibrium observed in single samples

The genetic effective population size, N_e, can be estimated from the average gametic disequilibrium () between pairs of loci, but such estimates require evaluation of assumptions and currently have few methods to estimate confidence intervals. speed‐ne is a suite of matlab computer code functions to estimate from with a graphical user interface and a rich set of outputs that aid in understanding data patterns and comparing multiple estimators. speed‐ne includes functions to either generate or input simulated genotype data to facilitate comparative studies of estimators under various population genetic scenarios. speed‐ne was validated with data simulated under both time‐forward and time‐backward coalescent models of genetic drift. Three classes of estimators were compared with simulated data to examine several general questions: what are the impacts of microsatellite null alleles on , how should missing data be treated, and does disequilibrium contributed by reduced recombination among some loci in a sample impact . Estimators differed greatly in precision in the scenarios examined, and a widely employed estimator exhibited the largest variances among replicate data sets. speed‐ne implements several jackknife approaches to estimate confidence intervals, and simulated data showed that jackknifing over loci and jackknifing over individuals provided ~95% confidence interval coverage for some estimators and should be useful for empirical studies. speed‐ne provides an open‐source extensible tool for estimation of from empirical genotype data and to conduct simulations of both microsatellite and single nucleotide polymorphism (SNP) data types to develop expectations and to compare estimators.     

Unbroken: RADseq remains a powerful tool for understanding the genetics of adaptation in natural populations

Recently, Lowry et al. addressed the ability of RADseq approaches to detect loci under selection in genome scans. While the authors raise important considerations, such as accounting for the extent of linkage disequilibrium in a study system, we strongly disagree with their overall view of the ability of RADseq to inform our understanding of the genetic basis of adaptation. The family of RADseq protocols has radically improved the field of population genomics, expanding by several orders of magnitude the number of markers available while substantially reducing the cost per marker. Researchers whose goal is to identify regions of the genome under selection must consider the LD of the experimental system; however, there is no magical LD cutoff below which researchers should refuse to use RADseq. Lowry et al. further made two major arguments: a theoretical argument that modeled the likelihood of detecting selective sweeps with RAD markers, and gross summaries based on an anecdotal collection of RAD studies. Unfortunately, their simulations were off by two orders of magnitude in the worst case, while their anecdotes merely showed that it is possible to get widely divergent densities of RAD tags for any particular experiment, either by design or due to experimental efficacy. We strongly argue that RADseq remains a powerful and efficient approach that provides sufficient marker density for studying selection in many natural populations. Given limited resources, we argue that researchers should consider a wide range of trade‐offs among genomic techniques, in light of their study question and the power of different techniques to answer it.     

Genome-wide signatures of population bottlenecks and diversifying selection in European wolves

Genomic resources developed for domesticated species provide powerful tools for studying the evolutionary history of their wild relatives. Here we use 61K single-nucleotide polymorphisms (SNPs) evenly spaced throughout the canine nuclear genome to analyse evolutionary relationships among the three largest European populations of grey wolves in comparison with other populations worldwide, and investigate genome-wide effects of demographic bottlenecks and signatures of selection. European wolves have a discontinuous range, with large and connected populations in Eastern Europe and relatively smaller, isolated populations in Italy and the Iberian Peninsula. Our results suggest a continuous decline in wolf numbers in Europe since the Late Pleistocene, and long-term isolation and bottlenecks in the Italian and Iberian populations following their divergence from the Eastern European population. The Italian and Iberian populations have low genetic variability and high linkage disequilibrium, but relatively few autozygous segments across the genome. This last characteristic clearly distinguishes them from populations that underwent recent drastic demographic declines or founder events, and implies long-term bottlenecks in these two populations. Although genetic drift due to spatial isolation and bottlenecks seems to be a major evolutionary force diversifying the European populations, we detected 35 loci that are putatively under diversifying selection. Two of these loci flank the canine platelet-derived growth factor gene, which affects bone growth and may influence differences in body size between wolf populations. This study demonstrates the power of population genomics for identifying genetic signals of demographic bottlenecks and detecting signatures of directional selection in bottlenecked populations, despite their low background variability.     

Effects of population characteristics and structure on estimates of effective population size in a house sparrow metapopulation

Effective population size (N_e) is a key parameter to understand evolutionary processes and the viability of endangered populations as it determines the rate of genetic drift and inbreeding. Low N_e can lead to inbreeding depression and reduced population adaptability. In this study, we estimated contemporary N_e using genetic estimators (LDNE, ONeSAMP, MLNE and CoNe) as well as a demographic estimator in a natural insular house sparrow metapopulation. We investigated whether population characteristics (population size, sex ratio, immigration rate, variance in population size and population growth rate) explained variation within and among populations in the ratio of effective to census population size (N_e/N_c). In general, N_e/N_c ratios increased with immigration rates. Genetic N_e was much larger than demographic N_e, probably due to a greater effect of immigration on genetic than demographic processes in local populations. Moreover, although estimates of genetic N_e seemed to track N_c quite well, the genetic N_e‐estimates were often larger than N_c within populations. Estimates of genetic N_e for the metapopulation were however within the expected range (<N_c). Our results suggest that in fragmented populations, even low levels of gene flow may have important consequences for the interpretation of genetic estimates of N_e. Consequently, further studies are needed to understand how N_e estimated in local populations or the total metapopulation relates to actual rates of genetic drift and inbreeding.     

RADProc: A computationally efficient de novo locus assembler for population studies using RADseq data

Restriction site‐associated DNA sequencing (RADseq) is a powerful tool for genotyping of individuals, but the identification of loci and assignment of sequence reads is a crucial and often challenging step. The optimal parameter settings for a given de novo RADseq assembly vary between data sets and can be difficult and computationally expensive to determine. Here, we introduce RADProc, a software package that uses a graph data structure to represent all sequence reads and their similarity relationships. Storing sequence–comparison results in a graph eliminates unnecessary and redundant sequence similarity calculations. De novo locus formation for a given parameter set can be performed on the precomputed graph, making parameter sweeps far more efficient. RADProc also uses a clustering approach for faster nucleotide‐distance calculation. The performance of RADProc compares favourably with that of the widely used Stacks software. The run‐time comparisons between RADProc and Stacks for 32 different parameter settings using 20 green‐crab (Carcinus maenas) samples showed that RADProc took as little as 2 hr 40 min compared to 78 hr by Stacks, while 16 brown trout (Salmo trutta L.) samples were processed by RADProc and Stacks in 23 and 263 hr, respectively. Comparisons of the de novo loci formed, and catalog built using both the methods demonstrate that the improvement in processing speeds achieved by RADProc does not affect much the actual loci formed and the results of downstream analyses based on those loci.     

The extent of linkage disequilibrium in a large cattle population of western Africa and its consequences for association studies

Several previous studies concluded that linkage disequilibrium (LD) in livestock populations from developed countries originated from the impact of strong selection. Here, we assessed the extent of LD in a cattle population from western Africa that was bred in an extensive farming system. The analyses were performed on 363 individuals in a Bos indicus x Bos taurus population using 42 microsatellite markers on BTA04, BTA07 and BTA13. A high level of expected heterozygosity (0.71), a high mean number of alleles per locus (9.7) and a mild shift in Hardy-Weinberg equilibrium were found. Linkage disequilibrium extended over shorter distances than what has been observed in cattle from developed countries. Effective population size was assessed using two methods; both methods produced large values: 1388 when considering heterozygosity (assuming a mutation rate of 10(-3)) and 2344 when considering LD on whole linkage groups (assuming a constant population size over generations). However, analysing the decay of LD as a function of marker spacing indicated a decreasing trend in effective population size over generations. This decrease could be explained by increasing selective pressure and/or by an admixture process. Finally, LD extended over small distances, which suggested that whole-genome scans will require a large number of markers. However, association studies using such populations will be effective.     

Variance effective population size is affected by census size in sub-structured populations

Measurement of allele frequency shifts between temporally spaced samples has long been used for assessment of effective population size (N_e), and this ‘temporal method’ provides estimates of N_e referred to as variance effective size (N_eV). We show that N_eV of a local population that belongs to a sub-structured population (a metapopulation) is determined not only by genetic drift and migration rate (m), but also by the census size (N_c). The realized N_eV of a local population can either increase or decrease with increasing m, depending on the relationship between N_e and N_c in isolation. This is shown by explicit mathematical expressions for the factors affecting N_eV derived for an island model of migration. We verify analytical results using high-resolution computer simulations, and show that the phenomenon is not restricted to the island model migration pattern. The effect of N_c on the realized N_eV of a local subpopulation is most pronounced at high migration rates. We show that N_c only affects local N_eV, whereas N_eV for the metapopulation as a whole, inbreeding (N_eI), and linkage disequilibrium (N_eLD) effective size are all independent of N_c. Our results provide a possible explanation to the large variation of N_e/N_c ratios reported in the literature, where N_e is frequently estimated by N_eV. They are also important for the interpretation of empirical N_e estimates in genetic management where local N_eV is often used as a substitute for inbreeding effective size, and we suggest an increased focus on metapopulation N_eV as a proxy for N_eI.     

An empirical test of the estimation of historical effective population size using Drosophila melanogaster

The availability of a large number of high-density markers (SNPs) allows the estimation of historical effective population size (N_e) from linkage disequilibrium between loci. A recent refinement of methods to estimate historical N_e from the recent past has been shown to be rather accurate with simulation data. The method has also been applied to real data for numerous species. However, the simulation data cannot encompass all the complexities of real genomes, and the performance of any estimation method with real data is always uncertain, as the true demography of the populations is not known. Here, we carried out an experimental design with Drosophila melanogaster to test the method with real data following a known demographic history. We used a population maintained in the laboratory with a constant census size of about 2800 individuals and subjected the population to a drastic decline to a size of 100 individuals. After a few generations, the population was expanded back to the previous size and after a few further generations again expanded to twice the initial size. Estimates of historical N_e were obtained with the software GONE both for autosomal and X chromosomes from samples of 17 individuals sequenced for the whole genome. Estimates of the historical effective size were able to infer the patterns of changes that occurred in the populations showing generally good performance of the method. We discuss the limitations of the method and the application of the software carried out so far.     

Estimating genetic drift and effective population size from temporal shifts in dominant gene marker frequencies

Measurement of temporal change in allele frequencies represents an indirect method for estimating the genetically effective size of populations. When allele frequencies are estimated for gene markers that display dominant gene expression, such as, e.g. random amplified polymorphic DNA (RAPD) and amplified fragment length polymorphism (AFLP) markers, the estimates can be seriously biased. We quantify bias for previous allele frequency estimators and present a new expression that is generally less biased and provides a more precise assessment of temporal allele frequency change. We further develop an estimator for effective population size that is appropriate when dealing with dominant gene markers. Comparison with estimates based on codominantly expressed genes, such as allozymes or microsatellites, indicates that about twice as many loci or sampled individuals are required when using dominant markers to achieve the same precision.     

Generation time and effective population size in Polar Eskimos

North Greenland Polar Eskimos are the only hunter-gatherer population, to our knowledge, who can offer precise genealogical records spanning several generations. This is the first report from Eskimos on two key parameters in population genetics, namely, generation time (T) and effective population size (Ne). The average mother-daughter and father-son intervals were 27 and 32 years, respectively, roughly similar to the previously published generation times obtained from recent agricultural societies across the world. To gain an insight for the generation time in our distant ancestors, we calculated maternal generation time for two wild chimpanzee populations. We also provide the first comparison among three distinct approaches (genealogy, variance and life table methods) for calculating Ne, which resulted in slightly differing values for the Eskimos. The ratio of the effective to the census population size is estimated as 0.6-0.7 for autosomal and X-chromosomal DNA, 0.7-0.9 for mitochondrial DNA and 0.5 for Y-chromosomal DNA. A simulation of alleles along the genealogy suggested that Y-chromosomal DNA may drift a little faster than mitochondrial DNA in this population, in contrast to agricultural Icelanders. Our values will be useful not only in prehistoric population inference but also in understanding the shaping of our genome today.     

Small effective population size in the long-toed salamander

The effective population sizes (Ne) of six populations of the long-toed salamander (Ambystoma macrodactylum) from Montana and Idaho, USA were estimated from allozyme data from samples collected in 1978, 1996 and 1997 using the temporal allele frequency method. Five of the six estimates ranged from 23 to 207 (mean = 123 +/- 79); one estimate was indistinguishable from infinity. In order to infer the actual Ne of salamander populations, we compared the frequency distribution of our observed Ne estimates with distributions obtained from simulated populations of known Ne. Our observed Ne estimate distribution was consistent with distributions from simulated populations with Ne values of 10, 25, and 50, suggesting an actual Ne for each of the six salamander populations of less than 100. This Ne estimate agrees with most other Ne estimates for amphibians. We conclude by discussing the conservation implications of small Ne values in amphibians in the context of increasing isolation of populations due to habitat fragmentation.