SNeP: a tool to estimate trends in recent effective population size trajectories using genome-wide SNP data

Effective population size (N_e) is a key population genetic parameter that describes the amount of genetic drift in a population. Estimating N_e has been subject to much research over the last 80 years. Methods to estimate N_e from linkage disequilibrium (LD) were developed ~40 years ago but depend on the availability of large amounts of genetic marker data that only the most recent advances in DNA technology have made available. Here we introduce SNeP, a multithreaded tool to perform the estimate of N_e using LD using the standard PLINK input file format (.ped and.map files) or by using LD values calculated using other software. Through SNeP the user can apply several corrections to take account of sample size, mutation, phasing, and recombination rate. Each variable involved in the computation such as the binning parameters or the chromosomes to include in the analysis can be modified. When applied to published datasets, SNeP produced results closely comparable with those obtained in the original studies. The use of SNeP to estimate N_e trends can improve understanding of population demography in the recent past, provided a sufficient number of SNPs and their physical position in the genome are available. Binaries for the most common operating systems are available at https://sourceforge.net/projects/snepnetrends/.

Introduction

Effective population size (N_e) is an important genetic parameter that estimates the amount of genetic drift in a population, and has been described as the size of an idealized Wright–Fisher population expected to yield the same value of a given genetic parameter as in the population under study (Crow and Kimura, 1970). N_e sizes can be influenced by fluctuations in census population size (N_c), by the breeding sex ratio and the variance in reproductive success.

N_e estimation can be achieved using approaches that fall into three methodological categories: demographic, pedigree-based, or marker-based (Flury et al., 2010). Pedigree data have been traditionally used to obtain N_e estimates in livestock. However, reliable estimates of N_e depend on the pedigree being complete. This state of knowledge is feasible in some domestic populations, the demographic parameters of which have been accurately monitored for a sufficiently large number of generations. However, in practice, the applicability of this approach remains limited to a few cases involving highly managed breeds (Flury et al., 2010; Uimari and Tapio, 2011).

One solution to overcome the limitation of an incomplete pedigree is to estimate the recent trend in N_e using genomic data. Several authors have recognized that N_e could be estimated from information on linkage disequilibrium (LD) (Sved, 1971; Hill, 1981). LD describes the non-random association of alleles in different loci as a function of the recombination rate between the physical positions of the loci in the genome. However, LD signatures can also result from demographic processes such as admixture and genetic drift (Wright, 1943; Wang, 2005), or through processes such as “hitchhiking” during selective sweeps (Smith and Haigh, 1974) or background selection (Charlesworth et al., 1997). In such scenarios alleles at different loci become associated independently of their proximity in the genome. Assuming that a population is closed and panmictic, the LD value calculated between neutral unlinked loci depends exclusively on genetic drift (Sved, 1971; Hill, 1981). This occurrence can be used to predict N_e due to the known relationship between the variance in LD (calculated using allele frequencies) and effective population size (Hill, 1981).

Recent advances in genotyping technology (e.g., using SNP bead arrays with tens of thousands of DNA probes) have enabled the collection of vast amounts of genome-wide linkage data ideal for estimating N_e in livestock and humans among others (e.g., Tenesa et al., 2007; de Roos et al., 2008; Corbin et al., 2010; Uimari and Tapio, 2011; Kijas et al., 2012). However, a software tool that enables estimation of N_e from LD is lacking, and researchers currently rely on a combination of tools to manipulate data, infer LD, and tend to use bespoke scripts to perform the appropriate calculations and estimate N_e.

Here we describe SNeP, a software tool that allows the estimation of N_e trends across generation using SNP data that corrects for sample size, phasing and recombination rate.

Materials and Methods

The method SNeP uses to calculate LD depends on the availability of phased data. When the phase is known the user can select Hill and Robertson (1968) squared correlation coefficient that makes use of haplotype frequencies to define LD between each pair of loci (Equation 1). However, in the absence of a known phase, squared Pearson's product-moment correlation coefficient between pairs of loci can be selected. While these two approaches are not the same, they are highly comparable (McEvoy et al., 2011):

$\begin{array}{cc}{r}^2=\frac{{\left(p_{AB}-p_A{p}_B\right)}^2}{p_A\left(1-p_A\right)p_B\left(1-p_B\right)}\hfill & (1)\end{array}$

$\begin{array}{cc}{r}_{X,Y}^2=\frac{{\left[{\displaystyle {\sum }_{i=1}^n\left(X_i-\overline{X}\right)\left(Y_i-\overline{Y}\right)}\right]}^2}{{\displaystyle {\sum }_{i=1}^n{\left(X_i-\overline{X}\right)}^2}{\displaystyle {\sum }_{i=1}^n{\left(Y_i-\overline{Y}\right)}^2}}& (2)\end{array}$

where p_A and p_B are respectively the frequencies of alleles A and B at two separate loci (X, Y) measured for n individuals, p_AB is the frequency of the haplotype with alleles A and B in the population studied, X and Y are the mean genotype frequencies for the first and second locus respectively, X_i is the genotype of individual i at the first locus and Y_i is the genotype of individual i at the second locus. Equation (2) correlates the genotypic allele counts instead of the haplotype frequencies and is not influenced by double heterozygotes (this approach results in the same estimates as the --r2 option in PLINK).

SNeP estimates the historic effective population size based on the relationship between r², N_e, and c (recombination rate), (Equation 3—Sved, 1971), and enabling users to include corrections for sample size and uncertainty of the gametic phase (Equation 4—Weir and Hill, 1980):

$\begin{array}{cc}E\left(r^2\right)={\left(1+4N_{e}c\right)}^{-1}\hfill & (3)\end{array}$

$\begin{array}{cc}{r}_{adj}^2=r^2-{\left(\beta n\right)}^{-1}\hfill & (4)\end{array}$

where n is the number of individual sampled, β = 2 when the gametic phase is known and β = 1 if instead the phase is not known.

Several approximations are used to infer the recombination rate using the physical distance (δ) between two loci as a reference and translating it into linkage distance (d), which is usually described as Mb(δ) ≈ cM(d). For small values of d the latter approximation is valid, but for larger values of d the probability of multiple recombination events and interference increases, moreover the relationship between map distance and recombination rate is not linear, as the maximum recombination rate possible is 0.5. Thus, unless using very short δ, the approximation d ≈ c is not ideal (Corbin et al., 2012). We therefore implemented mapping functions to translate the estimated d into c, following Haldane (1919), Kosambi (1943), Sved (1971), and Sved and Feldman (1973). Initially SNeP infers d for each pair of SNPs as directly proportional to δ according to d = kδ where k is a user defined recombination rate value (default value is 10⁻⁸ as in Mb = cM). The inferred value of δ can then be subjected to one of the available mapping functions if required by the user.

Solving Equation (3) for N_e and including all the corrections described, allows the prediction of N_e from LD data using (Corbin et al., 2012):

$\begin{array}{cc}{N}_{T\left(t\right)}={\left(4f\left(c_t\right)\right)}^{-1}\left(E{\left[r_{adj}^2|c_t\right]}^{-1}-\alpha \right)& (5)\end{array}$

where N_t is the effective population size t generations ago calculated as t = (2f(c_t))⁻¹ (Hayes et al., 2003), c_t is the recombination rate defined for a specific physical distance between markers and optionally adjusted with the mapping functions mentioned above, r²_adj is the LD value adjusted for sample size and α:= {1, 2, 2.2} is a correction for the occurrence of mutations (Ohta and Kimura, 1971). Therefore, LD over greater recombinant distances is informative on recent N_e while shorter distances provide information on more distant times in the past. A binning system is implemented in order to obtain averaged r² values that reflect LD for specific inter-locus distances. The binning system implemented uses the following formula to define the minimum and maximum values for each bin:

$\begin{array}{cc}{b}_i^{min}=minD+\left(maxD-minD\right){\left(\frac{b_i-1}{totBins}\right)}^x& (6\text{a})\end{array}$

$\begin{array}{cc}{b}_i^{max}=minD+\left(maxD-minD\right){\left(\frac{b_i}{totBins}\right)}^x& (6\text{b})\end{array}$

Where b_i (ℕ¹) is the i^th bin of the total number of bins (totBins), minD, and maxD are respectively the minimum and the maximum distance between SNPs and x is a positive real number (ℝ⁰) When x equals 1, the distribution of distances between the bins is linear and each bin has the same distance range. For larger values of x the distribution of distances changes allowing a larger range on the last bins and a smaller range on the first bins. Varying this parameter allows the user to have a sufficient number of pairwise comparisons to contribute to the final N_e estimate for each bin.

Example Application

We tested SNeP with two published datasets that had been previously used to describe trends in N_e over time using LD, Bos indicus [54,436 SNPs of 423 East African Shorthorn Zebu (SHZ)–Mbole-Kariuki et al., 2014, data available at Dryad Digital Repository: doi:10.5061/dryad.bc598.] and Ovis aries [49,034 SNPs genotyped in 24 Swiss White Alpine (SWA), 24 Swiss Black-Brown Mountain sheep (SBS), 24 Valais Blacknose sheep (VBS), 23 Valais Red sheep (VRS), 24 Swiss Mirror sheep (SMS) and 24 Bundner Oberländer sheep (BOS)–Burren et al., 2014]. The r² estimates for the cattle datasets were obtained by the authors using GenABLE (Aulchenko et al., 2007) using a minimum allele frequency (MAF) < 0.01 and adjusting the recombination rate using Haldane's mapping function (Haldane, 1919). The r² estimates of the sheep data were calculated by the authors using PLINK-1.07 (Purcell et al., 2007), with a MAF < 0.05 and no further corrections. For both autosomal datasets r² estimates where corrected for sample size using equation (4) with β = 2. For these comparative analyses the SNeP command line included the same parameters used for the published data apart from the r² estimates, calculated through genotype count and the use of SNeP's novel binning strategy.

Results

SNeP is a multithreaded application developed in C++ and binaries for the most common operating systems (Windows, OSX, and Linux) can be downloaded from https://sourceforge.net/projects/snepnetrends/. The binaries are accompanied by a manual describing the step-by-step use of SNeP to infer trends in N_e as described here. SNeP produces an output file with tab delimited columns showing the following for each bin that was used to estimate N_e: the number of generations in the past that the bin corresponds to (e.g., 50 generations ago), the corresponding N_e estimate, the average distance between each pair of SNPs in the bin, the average r² and the standard deviation of r² in the bin, and the number of SNPs used to calculate r² in the bin. This file can be easily imported in Microsoft Excel, R or other software to plot the results. The plots shown here (Figures 1, 3) correspond to the columns of generations ago and N_e from the output file. The column with the r² standard deviation is provided for users to inspect the variance in the N_e estimate in each bin, particularly for those bins reflecting older time estimates and which are less reliable as the number of SNPs used to estimate r² becomes smaller.

FIGURE 1

Figure 1. Comparison of N_e trends of six Swiss sheep breeds according to Burren et al. (2014) (dashed lines) and this work (solid lines).

The format required for the input files is the standard PLINK format (ped and map files) (Purcell et al., 2007). SNeP allows the users to either calculate LD on the data as described above, or use a custom precalculated LD matrix to estimate N_e using Equation (5).

The software interface allows the user to control all parameters of the analysis, e.g., the distance range between SNPs in bp, and the set of chromosomes used in the analysis (e.g., 20–23). Additionally, SNeP includes the option to choose a MAF threshold (default 0.05), as it has been shown that accounting for MAF results in unbiased r² estimates irrespective of sample size (Sved et al., 2008). SNeP's multithreaded architecture allows fast computation of large datasets (we tested up to ~100K SNPs for a single chromosome), for example the BOS data described here was analyzed with one processor in 2′43″, the use of two processors reduced the time to 1′43″, four processors reduced the analysis time to 1′05″.

Zebu Example

For the zebu analysis, the shapes of the N_e curves obtained with SNeP and their published data trends showed the same trajectory with a smooth decline until around 150 generations ago, followed by an expansion with a peak around 40 generations ago and ending in a steep decline on the most recent generations (Figure 1). However, while the trends in both curves were the same, the two approaches resulted in different N_e estimates, with SNeP's values being approximately three-fold larger than those in the original paper. While we attempted to use the authors' parameters in our analyses, some differences were inevitable, i.e., the original publication of the cattle data estimated r² with a different approach to that implemented in SNeP. Analyses with SNeP were based on genotypes, while the original analysis was based on inferred two locus haplotypes, which results in the published data showing an expected r² of 0.32 at the minimum distance, while our estimates was 0.23. Similarly, Mbole-Kariuki et al. (2014) obtained a background level r² = 0.013 around 2 Mb, while our estimate at the same distance was 0.0035 (data not shown). Consequently, as our estimates of LD were consistently smaller than Mbole-Kariuki et al. (2014) it is expected that our N_e estimates should be larger. While this observation highlights the importance of a careful choice of the parameters and their thresholds, it is important to highlight that although the absolute magnitude of the N_e values is different, the trends are almost identical.

Swiss Sheep Example

The six Swiss sheep breeds analyzed with SNeP produced comparable results with those from the original paper (Figure 2), with mostly overlapping N_e trend curves (Figure 3). However, the general trend in N_e showed a decline toward the present. SNeP produced slightly larger values of N_e for the more distant past (700–800 generations). This is due to the different binning system used in SNeP, which allows the user to obtain a more even distribution of pairwise comparisons within each bin (i.e., the number of SNP pairwise comparisons within each bin is comparable). For the time span extending beyond 400 generations ago, Burren et al. (2014) used only three bins in their analysis (centered at 400, 667, and 2000 generations ago) while for the same time span SNeP used 5 bins with a number of pairwise comparisons dependent to the range defined with formulae 6a,b. Consequently, Burren and colleagues' approach ends with a higher density of data describing the most recent generations than describing the oldest generations. Therefore, the use of fewer bins tends to increase the presence of smaller values of N_e in each bin, consequently lowering the average N_e value for each bin. The N_e values for the recent past, compared at the 29th generation in the past, gave very similar results. The largest difference (50) was obtained for the SBS breed.

FIGURE 2

Figure 2. Comparison between recent N_e values calculated at the 29th generation in this work and Burren et al. (2014) for six Swiss sheep breeds.

FIGURE 3

Figure 3. Comparison of N_e trends for the last 250 generations in the SHZ data obtained by Mbole-Kariuki et al. (2014) (dashed line) and using SNeP (solid line).

Discussion

Analysis of N_e using LD data was first demonstrated 40 years ago, and has been applied, developed and improved since (Sved, 1971; Hayes et al., 2003; Tenesa et al., 2007; de Roos et al., 2008; Corbin et al., 2012; Sved et al., 2013). The traditionally small number of SNPs analyzed is no longer a limitation, since SNP Chips comprise an extremely large number of SNPs, available in a short time and at a reasonable price. This has boosted the use of the method, which has been applied to humans (Tenesa et al., 2007; McEvoy et al., 2011) as well as to several domesticated species (England et al., 2006; Uimari and Tapio, 2011; Corbin et al., 2012; Kijas et al., 2012). Along with these improvements, methodological limitations have become apparent and have been addressed here, with the majority of the efforts pointing to the correct estimation of recent N_e. Yet, the quantitative value of the estimate is highly dependent on sample size, the type of LD estimation and the binning process (Waples and Do, 2008; Corbin et al., 2012), while its qualitative pattern depends more on the genetic information than on data manipulation.

So far this method has been applied using a variety of software, no standardized approach exists to bin the results and each study has applied a more or less arbitrary approach, e.g., binning for generation classes in the past (Corbin et al., 2012), binning for distance classes with a constant range for each bin (Kijas et al., 2012) or binning per distance classes in a linear fashion but with larger bins for the more recent time points (Burren et al., 2014). To our knowledge the only software available that estimate N_e through LD is NeEstimator (Do et al., 2014), an upgraded version of the former LDNE (Waples and Do, 2008) allowing the analysis of large dataset (as 50k SNPChip). Importantly, while SNeP focuses on estimating historical N_e trends, NeEstimator's aim is to produce contemporary unbiased N_e estimates, the latter should therefore be considered as a complementary tool while investigating demography through LD.

We used SNeP to analyze two datasets where the method was previously applied. The results we obtained for the sheep data were both quantitatively and qualitatively comparable with those obtained by Burren et al. (2014), while for the Zebu data we obtained a N_e trend estimate that closely matched that of Mbole-Kariuki et al. (2014) although our point estimates of N_e were larger than those described for the data (Mbole-Kariuki et al., 2014). The discrepancy between these two results reflects that Burren and colleagues produced their r² estimates using PLINK (the standard software for large scale SNP data manipulation) which uses the same approach used to estimate r² by SNeP, while Mbole-Kariuki et al. followed Hao et al. (2007) for r² estimation. The use of different estimates for LD is critical for the quantitative aspect of the N_e curve, where due to the hyperbolic correlation between N_e and r², a decrease in r² on its range closer to 0 can lead to a very large change in N_e estimates, while differences in estimates are less significant when the r² value is high, i.e., closer to 1. Therefore, although in one of the datasets the N_e values where substantially different, in both cases the N_e curves overlapped with those originally published.

As already suggested by other authors, the reliability of the quantitative estimates obtained with this method must be taken with caution, especially for N_e values related to the most recent and the oldest generations (Corbin et al., 2012) because for recent generations, large values of c are involved, not fitting the theoretical implications that Hayes proposed to estimate a variable N_e over time (Hayes et al., 2003). Estimates for the oldest generations might also be unreliable as coalescent theory shows that no SNP can be reliably sampled after 4N_e generations in the past (Corbin et al., 2012). Further, N_e estimates, and especially those related to generations further in the past, are strongly affected by data manipulation factors, such as the choice of MAF and alpha values. Additionally, the binning strategy applied can interfere with the general precision of the method, for example where an insufficient number of pairwise comparisons are used to populate each bin.

One of the applications of method is to compare breed demographies. In this case the shape of the N_e curves would be the optimal tool to differentiate different demographic histories, more than their numerical values, by using them as a potential demographic fingerprint for that breed or species, yet taking into consideration that mutation, migration, and selection can influence the N_e estimation through LD (Waples and Do, 2010). Additionally, careful consideration of the data analyzed with SNeP (and other software to estimate N_e) is very important, as the presence of confounding factors such as admixture, may result in biased estimates of N_e (Orozco-terWengel and Bruford, 2014).

The aim of SNeP is therefore to provide a fast and reliable tool to apply LD methods to estimate N_e using high throughput genotypic data in a more consistent way. It allows two different r² estimation approaches plus the option of using r² estimates from external software. The use of SNeP does not overcome the limits of the method and the theory behind it, yet it allows the user to apply the theory using all corrections suggested to date.

Author Contributions

MB conceived and wrote the software and the manuscript. MB, MT, and POtW tested the software and performed the analyses. MT, POtW, and MWB revised the manuscript. All authors approved the final manuscript.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

We thank Christine Flury for providing the sheep data and for useful discussion. We also thank the two reviewers for useful suggestions to improve this paper. MB was supported by the program Master and Back (Regione Sardegna).

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