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Methods

Recent human effective population size estimated from linkage disequilibrium

¹ Colon Cancer Genetics Group, University of Edinburgh, Western General Hospital, Edinburgh EH4 2XU, United Kingdom; ² MRC Human Genetics Unit, Western General Hospital, Edinburgh EH4 2XU, United Kingdom; ³ Institute of Evolutionary Biology, University of Edinburgh, Edinburgh EH9 3JT, United Kingdom; ⁴ Victorian Institute of Animal Science, DPI, Attwood 3049, Australia; ⁵ Queensland Institute of Medical Research, Royal Brisbane Hospital, Brisbane 4006, Australia; ⁶ The Wellcome Trust Centre for Human Genetics, The University of Oxford, Oxford OX3 7BN, United Kingdom; ⁷ Institute of Land and Food Resources, University of Melbourne, Parkville 3010, Australia

	Abstract

Top Abstract Methods Results Discussion Acknowledgments References

 
Effective population size (N_e) determines the amount of genetic variation, genetic drift, and linkage disequilibrium (LD) in populations. Here, we present the first genome-wide estimates of human effective population size from LD data. Chromosome-specific effective population size was estimated for all autosomes and the X chromosome from estimated LD between SNP pairs <100 kb apart. We account for variation in recombination rate by using coalescent-based estimates of fine-scale recombination rate from one sample and correlating these with LD in an independent sample. Phase I of the HapMap project produced between 18 and 22 million SNP pairs in samples from four populations: Yoruba from Ibadan (YRI), Nigeria; Japanese from Tokyo (JPT); Han Chinese from Beijing (HCB); and residents from Utah with ancestry from northern and western Europe (CEU). For CEU, JPT, and HCB, the estimate of effective population size, adjusted for SNP ascertainment bias, was 3100, whereas the estimate for the YRI was 7500, consistent with the out-of-Africa theory of ancestral human population expansion and concurrent bottlenecks. We show that the decay in LD over distance between SNPs is consistent with recent population growth. The estimates of N_e are lower than previously published estimates based on heterozygosity, possibly because they represent one or more bottlenecks in human population size that occurred 10,000 to 200,000 years ago.

In this study we estimated genome-wide N_e from LD using data from 1,000,000 SNPs (HapMap project [The International HapMap Consortium 2003], data release #16 [http://www.hapmap.org/genotypes/2005-03_16a_phaseI/]) in four different human populations of African, Asian, and European ascent. Ours is the first example of using LD to estimate the effective population size of human chromosomes.

LD between each pair of SNPs depends on both N_e and the recombination rate between the SNPs. The distances between SNPs that we used (5–100 kb) are too small to estimate recombination rate using pedigree-based linkage analysis, so we have used other methods. Since errors in estimates of recombination rates from population data might bias the estimate of N_e, we have used three different methods to estimate these recombination rates. Each method resulted in very similar estimates of effective population size.

	Methods

Top Abstract Methods Results Discussion Acknowledgments References

 
All our analyses were based on the known approximate relationship between LD, as measured by r², the squared correlation of allele frequencies at a pair of loci, and N_e. In particular, we used E(r²) 1/( + 4N_ec) + 1/n for markers on the same autosome, where c is the recombination rate between the SNPs and n is the chromosome experimental sample size. The constant = 1 in the absence of mutation (Sved 1971) and 2 if mutation is taken into account (Hill 1975; Weir and Hill 1980; McVean 2002). We first describe how these formulae were derived and then how this theory was applied to the estimation of N_e from SNP data from multiple population samples. Although formulae for the expectation of r² have been published, for completeness we include succinct derivations.

Relationship between N_e and E(r²) without mutation
Given the correlation of the frequency of alleles at two autosomal loci at generation t (r_t), the mean and variance of the correlation at generation t + 1 is

(1)

The latter expression uses the general expression for the sampling variance of an estimate of a correlation coefficient r with sample size n; i.e., var(r) = [1 – E²(r)]/n. Using E(x²) = E(x)² + var(x) for a random variable x (for example, see Lynch and Walsh 1998) results in

(2)

At equilibrium, E(r_t+1²) = E(r_t²) = E(r²), so E(r²) = (1 – c)²E(r²) + [1 – E(r²)(1 – c)²]/2N_e. Rearranging and approximating (1 – c)² by (1 – 2c) gives E(r²) = 1/(1 + 4N_ec). This result was first reported by Sved (1971).

For the X chromosome, recombination occurs only in females. The X chromosome in males may have recombined (since it is a maternal chromosome), and only the maternal X chromosomes in females may have recombined. Hence, two-thirds of X chromosomes may have recombined and one-third may not. The sample size for the disequilibrium (correlation) coefficient is (3/2)N_e because females produce N_e X gametes and males produce (1/2)N_e gametes. Hence,

(3)

At equilibrium, and ignoring the smaller terms, E(r²) = 1/(1 + 2N_ec).

Relationship between N_e and E(r²) with mutation
For autosomal loci, Hill (1975) showed that, in the presence of mutation, E(r²) (10 + )/(22 + 13 + ²), with = 4N_ec. Since (22 + 13 + ²) factors into (11 + )(2 + ), a further approximation is E(r²) 1/(2 + ) 1/(2 + 4N_ec). For the X chromosome, following the same logic as before, E(r²) 1/(2 + 2N_ec).

Chance LD due to finite experimental sample size
Weir and Hill (1980) showed that experimental sampling introduces chance disequilibrium of var(r) = 1/n and they suggested the adjustment of E(r²) for chromosome sample size. Taking both experimental and evolutionary sampling effects into account, we can summarize the relationship between LD and N_e in the general expression

(4)

where = 1 in the absence of mutation, = 2 if mutation is taken into account, k = 4 for autosomes, and k = 2 for the X chromosome. In data applications, we observe r² and, assuming that we know c or have a good estimate thereof, N_e can be estimated for autosomes and the X chromosome.

Data
HapMap samples from four different populations were available (http://www.hapmap.org/genotypes/2005-03_16a_phaseI/). Samples were 30 trios from CEPH (CEU) families; 30 trios from Yoruba in Ibadan, Nigeria (YRI); and 45 unrelated Japanese (JPT) and Han Chinese (HCB) individuals. Chromosome sample size (n) was 120 for the CEU and YRI samples and 90 for the JPT and HCB samples. For more information, refer to The International HapMap Consortium (2003).

To compare LD across two samples from approximately the same population, data generated by Perlegen Sciences for European Americans (n = 46) were also obtained (http://genome.perlegen.com/browser/download.html).

Haplotype frequency and r² estimation
For each chromosome, pairwise r² was calculated (Hill and Robertson 1968) only for SNP pairs between 5 kb and 100 kb apart both to avoid the influence of gene conversion on observed LD at SNPs that are closer (Frisse et al. 2001) and to minimize the effect of a very recent expansion of the effective population size on LD (Hayes et al. 2003). A combined EM/Lander-Green algorithm to estimate pairwise haplotype frequencies (as implemented in Haploview; Barrett et al. 2005; http://www.broad.mit.edu/mpg/haploview/) was used to estimate r² for all autosomes for the HapMap data. For the X chromosome, an EM algorithm combining phase known and unknown data was used. The software for the X chromosome calculations is freely available at http://homepages.ed.ac.uk/eanv63/. SNPs were rejected if their P-value for Hardy-Weinberg equilibrium (HWE) was <0.001 (the default setting in Haploview) or if their minor allele frequency (MAF) was <0.05.

For the Perlegen data, standard EM algorithms were applied to estimate haplotype frequencies and these used to estimate r² for all autosomes. We filtered out markers with a minor allele frequency <0.05 and estimated r² for all pairs of markers formed by markers that were between 5 kb and 100 kb apart. A total of 866,949 pairwise r² estimates were in common with the CEU HapMap sample.

Estimation of recombination rates using three methods
The estimation of the recombination rate for pairs of markers is important because given the value of the recombination rate, effective population size can be estimated from the relationship between r² and N_ec. To verify the robustness of our estimates of N_e, we estimated recombination rates using three different methods.

Method 1
We obtained estimates of recombination rate from LD and the known map length of the chromosome. For each chromosome the pairwise r² was calculated for all pairs of SNP in sliding windows of 100 kb with a 50 kb overlap. The average recombination rate for each 100 kb window was estimated from the average LD within that window. These average recombination rates were scaled so that over a whole chromosome they add up to the known map length of the chromosome. The recombination rate for any pair of SNPs within a 100 kb window was estimated to be proportional to the physical distance between the SNPs and the average recombination rate over the window. This approach ignores variation in recombination rate within a 100 kb window.

More specifically, to obtain estimates of the recombination rate for any pair of SNPs, we fitted for each window the nonlinear model y_ij = 1/(_i + _id_ij) + e_ij with y_ij = (r² – 1/n), that is, r² adjusted for chromosome sample size, for SNP pair j in window i at physical distance d_ij. Parameters _i and _i were estimated iteratively using least squares. A restriction was imposed that _i 1 and _i 0. If the recombination rate within window i (c_i) is constant, then _i = 4N_ec_i/d = _i/d is the scaled recombination rate per unit of physical distance. For each window, we calculated the scaled recombination rate of the entire window as _i = 0.05_i, where 0.05 corresponds to the length (in Mb) of the nonoverlapping part of the window. This quantity is summed over all windows and equated to the known map length (L in Morgan, from pedigree data; Kong et al. 2004; http://compgen.rutgers.edu/maps/index.shtml). A calibration constant, x, was estimated using the number of pairs in a window as weight, i.e., = m[n_ii]/[Ln_i], with m the number of windows and n_i the number of pairs in a window. For each window, the recombination rate per Mb was estimated as = _i/. At least 25 pairs were required to estimate local recombination rate in a window.

Given these estimates of local recombination rates, a nonlinear least squares regression method (details below) was subsequently used to estimate N_e from recombination distance between all pairs of markers. For a given pair of markers, the recombination distance was calculated from the estimated recombination rate per unit of physical distance of the window that was the midpoint of the location of the pair and the physical distance between the pair (i.e., _i(jk) = d_jk). A pair was included only if the intermarker distance was <100 kb and if the number of pairs of observations that were used to estimate the local recombination rate for the window was at least 25. The number of pairs of SNPs that were used to estimate the recombination rate in the window was used as a weight in the regression analysis.

Method 2
The recombination fraction between all pairs of markers was estimated using the method described by Clarke and Cardon (2005), and the Kosambi map function was used to convert it to map distance.

Method 3
We obtained fine-scale recombination rates from two independent sets of publicly available data (Phase I HapMap [Altshuler et al. 2005] data [http://www.hapmap.org/downloads/recombination/latest/] and data generated by Perlegen Sciences [Myers et al. 2005; http://www.stats.ox.ac.uk/mathgen/Recombination.html]). We extracted marker pairs for which estimates of recombination rates were available in both data sets. This yielded 98,399 pairs that were formed from 169,545 individual markers. We used these data in four different ways to estimate N_e, using (1) r² estimated from HapMap data (HM) and recombination rates estimated from Perlegen data (PL), (2) r² and recombination rates estimated from HM, (3) r² and recombination rates estimated from PL, and (4) r² estimated from PL and recombination rates estimated from HM.

Estimation of chromosome effective population size
Given the formulae described in the theory section, and knowing r² and c, we estimated N_e for each chromosome by fitting the nonlinear regression model

(5)

with y_i = (r² – 1/n), that is, r² adjusted for chromosome sample size, for SNP pair i at recombination distance c_i (in Morgans). Parameters and were estimated iteratively using least squares.

Heterozygosity and LD in a population depend on N_e over the history of the population. However, LD between SNPs a large distance apart reflects more recent N_e than LD between SNPs closer together (Hayes et al. 2003). We therefore investigated the change in population size over time, as LD between loci with a recombination rate of c reflects the ancestral effective population size 1/(2c) generations ago (Hayes et al. 2003) (under the assumption of linear growth). We compared N_e estimated by recombination estimation method 1 from SNPs 5–100 kb apart. This corresponds to a time between 10,000 and 500 generations ago, i.e., between 200,000 and 10,000 yr ago (Hayes et al. 2003). For each chromosome, we estimated N_e from the mean r² for average SNP recombination distances in the range of 0.01–0.5 cM. For each distance, the corresponding number of generations in the past was calculated.

Estimates of the scaled recombination rate and effective population size per chromosome obtained by method 1 were compared with the number of genes and length of the chromosome (NCBI build 35; http://www.ncbi.nlm.nih.gov/mapview/map_search.cgi?taxid=9606&build=previous) using correlation and linear regression.

The frequency distribution of the SNPs ascertained by the HapMap project is different from the frequency distribution of SNPs that have been completely ascertained (Nielsen et al. 2004). To determine if the HapMap SNP ascertainment procedure was biasing our estimates of N_e, we carried out coalescent (Hudson 1983) simulations for a neutral model and constant population size. Simulations were performed using the program "ms" (Hudson 2002; http://home.uchicago.edu/~rhudson1/source/mksamples.html). For a segment length of 25 Mb and an effective population size of 2000, the chosen input parameters equate to a mutation rate of 10^–8 per nucleotide and a recombination rate of 0.01 per Mb. For each replicate, the average r² was calculated for bins of 1 kb spacing and adjusted for chromosome sample size. A nonlinear regression model was used to relate the adjusted r² to physical distance. Under this model, the estimate of the regression coefficient is an estimate of the scaled recombination rate . One thousand replicates were run.

	Results

Top Abstract Methods Results Discussion Acknowledgments References

 
Figure 1 shows, as expected, a near linear relationship between 1/r² and physical distance between pairs of SNPs. Linearity of the reciprocal of r² with physical distance at small values, and a concave relationship at larger values, is clearly shown, consistent with a population that has increased over time.

View larger version (40K): [in this window] [in a new window]   Figure 1. Reciprocal of r2 plotted against physical distance for each of the 22 autosomes for the CEU population. The X-axis shows the physical distance in kilobases and the Y-axis shows 1/mean(r2). The mean value of r2 was calculated in 1 kb bins.

View this table: [in this window] [in a new window]   Table 1. Ne estimates from the nonlinear model (Method 1)

View larger version (12K): [in this window] [in a new window]   Figure 2. Comparison of the estimate of Ne from each of the labeled 23 chromosomes for the JPT and HCB population (Method 1). The X-axis shows the effective population size for each autosome and the X chromosome estimated from the JPT sample using a nonlinear regression of r2 on physical distance between pairs of SNPs. The Y-axis shows the estimates of Ne for each chromosome from HCB sample. The fitted linear regression line fits nearly perfectly, presumably reflecting common ancestry of the two populations.

The second method, which estimates recombination rates between each pair of adjacent HapMap markers from a model-free method that detects recombination hotspots from LD (Clarke and Cardon 2005; Visscher and Hill 2006), changed the estimate of N_e between +33% and –45% with an average reduction of 27% (mean N_e = 1901) when compared with that obtained from the first method (results not shown in Tables).

The third method used estimates of fine-scale recombination rates (rather than using physical distance as a proxy for recombination rate) from coalescent models from either Phase I HapMap (Altshuler et al. 2005) data or data generated by Perlegen Sciences (Myers et al. 2005). These estimates are shown in Table 2 and are consistent with estimates presented in Table 1.

View this table: [in this window] [in a new window]   Table 2. Ne estimates from the European ancestry samples from HapMap (HM, n = 120) and Perlegen (PL, n = 46) estimates of fine-scale recombination rates (Method 2)

From the simulation study we found that the estimation method was not biased when SNPs were simulated as if they had been completely ascertained (data not shown). We then simulated SNPs to mimic the SNP frequency distribution from the HapMap data. For this, SNPs with minor allele frequencies between 0.05 and 0.5 were ascertained with equal probability; i.e., the frequency distribution of the SNPs was uniform. The estimates of N_e obtained from mimicking the HapMap data were biased downward by 18% compared with the complete ascertainment data. Hence, the HapMap ascertainment strategy may bias our estimates by approximately one-fifth, giving adjusted estimates of effective population sizes of 3100 (non-African populations) and 7500 (African population).

For the CEU and YRI samples we estimated effective population size as a function of time in the past. Results for the CEU data support recent dramatic population growth (Fig. 3A). This is in agreement with the likely demographic history of the ancestral population of the non-African samples; a population bottleneck, following an out-of-Africa expansion, followed by rapid growth (Watkins et al. 2001). Results for the YRI data indicate an ancestral population size of 7000, followed by expansion in the last 20,000 yr (Fig. 3B).

View larger version (22K): [in this window] [in a new window]   Figure 3. Ancestral population size estimated from 22 autosomes of the CEU (A) and YRI (B) data (Method 1). For each chromosome and for each SNP recombination distance bin (0.001 cM spacing), Ne was estimated from the mean r2, using E(r2) = 1/( + 4Nec). The number of generations in the past was calculated as 1/(2c), and was truncated at 5000. Effective population size has increased dramatically in the last 1000 generations (20,000 yr), from a fairly constant ancestral size of 2500 (CEU) and 7000 (YRI).

	Discussion

Top Abstract Methods Results Discussion Acknowledgments References

Our estimate of N_e for the X chromosome was 30%–50% larger than that for the autosomes. The X chromosome in humans has a number of unusually long haplotypes (Altshuler et al. 2005). However, estimates from coalescent methods using the same data also give an increase of 50% in the estimate of N_e for the X chromosome compared with autosomes, although this difference disappears when using HapMap Phase II data (G. McVean, pers. comm.). It is not clear why the average LD, when adjusted for the absence of recombination in males, is smaller for the X chromosome.

We determined by simulation how the approximate ascertainment of SNPs in HapMap Phase I could bias our estimates of N_e, and adjusted these accordingly. Recently, Pe’er et al. (2006) have reported small upward biases in the estimation of LD from HapMap I data, consistent with a downward bias in the estimate of N_e. These biases would also affect our analyses and would not have been fully corrected for by our adjustment, which was based upon the allele frequencies of the ascertained SNPs. In addition, if there is variation in recombination rate that has not been reflected in our estimates of c using the three different methods, then our estimate of N_e would be biased downward.

In populations in which effective population size has changed over time, such as human populations, it is not meaningful to discuss effective population size without reference to a point in time (Hayes et al. 2003). For example, assuming a constant N_e when it has increased over time and estimating it from data on marker heterozygosity will result in an estimate of an average N_e over long periods of time, before bottlenecks if these have occurred recently. Methods, including the coalescent-based ones, that fail to take into account that N_e has changed over time will produce biased population parameter estimates, in particular when inference depends on the observed relationship between recombination distance and linkage disequilibrium.

We have used a relatively small sample of individuals, combined with high-density genome-wide marker genotyping, to infer ancestral population size based upon the observed amounts of LD. Our study has shown that human effective population size estimated from entire human chromosomes is considerably lower than previously suggested, at least during a bottleneck up to 20,000 yr ago when a large expansion began.

	Acknowledgments

Top Abstract Methods Results Discussion Acknowledgments References

 
We thank W.G. Hill for helpful discussions and help in the derivation of X-linked N_e, and N. Barton, B. Weir, J. Taylor, G. McVean, T. Johnson, and A. Morris for helpful discussions. A.T. acknowledges Cancer Research UK; P.N. was supported by the Genes to Cognition Project; and P.M.V. acknowledges the UK Biotechnology and Biological Sciences Research Council, the Wellcome Trust, and the Australian National Health and Medical Research Council for funding.

	Footnotes

 
⁸ Corresponding author.

E-mail peter.visscher{at}qimr.edu.au; fax +61-7-3362-0101.

Article published online before print. Article and publication date are at http://www.genome.org/cgi/doi/10.1101/gr.6023607

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Received October 9, 2006; accepted in revised format January 17, 2007.

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