Background

The study of historical demography is important for understanding the ecology and evolution of species. In particular, timing population size changes allows the discussion of past population patterns in the context of historical geological events such as island formation [1] and climate change [2]. One popular source of information to infer past population dynamics is the genealogical signal contained in linked polymorphic markers [3, 4], such as chloroplast microsatellites (cpSSRs). As highlighted in recent reviews [5, 6], cpSSRs are widely used in plant studies. cpSSRs remain popular despite the ascent of genome wide sequencing tools such as Restriction site-associated DNA sequencing (RADseq) [7], Genotyping-by-sequencing (GBS) [8] and targeted sequencing [9] due to two appealing properties: 1) their high mutation rate, ranging from 10−6 to 10−2 mutations per locus per generation [10], and 2) they can be applied in plant non-model species where few genomic resources have been developed [11].

High mutation rates combined with an approximately step-wise transition between allelic states make cpSSRs prone to homoplasious mutations. Homoplasy takes place in a cpSSR locus when different alleles at the locus are identical by state but are not identical by descent [12]. Two cpSSRs copies of a locus are defined to be identical by state when they have the same size and are defined as identical by descent when there has not been a mutation since their divergence from a common ancestor. Previous studies have quantified the fraction of the homoplasy, called Molecularly accessible size homoplasy (MASH) [12,13,14] by measuring the differences in the DNA sequence of SSRs of identical size. Although that approach can reveal a fraction of the homoplasy in SSRs, it ignores the homoplastic events due to polymorphisms that lead to DNA sequences identical by state but not identical by descent. Therefore, MASH does not provide a direct estimate of homoplasy.

The occurrence of homoplasy is an important limitation of cpSSR based demographic inference in scenarios of population expansions, causing decreased ability to detect population growth [15] and to systematic underestimation of the expansion time [16]. Although some pseudo-likelihood and Bayesian methods of demographic inference [16, 17] successfully correct for homoplasy, they provide little insight into the relationship between homoplasy and the estimation of demographic parameters, nor do they provide estimates of homoplasy itself. In fact, to our knowledge, no formal analysis of the quantitative relation between homoplasy and the underestimation of the expansion time exists to date. Part of the reason is that the concept of homoplasy was developed to describe the proportion of haplotypes or markers that are identical by descent compared to those that are identical by state, while the problem of erroneous demographic inference is linked to an underestimation of the number of mutations between lineages. To illustrate this, the most common measure of homoplasy, the homoplasy index P [12], describes the probability that two cpSSR identical by state are not identical by descent. In the case of haplotypes composed of linked cpSSR, P has been defined as the probability that two haplotypes identical by state are not identical by descent and is dependent on the multi-locus heterozygosity [15]. This is the definition of P we will employ here. While simulation studies show that higher values of P are associated with an underestimation of the expansion time [15], other studies have found that multi-locus heterozygosity is not particularly sensitive to homoplasy [18], suggesting that P may not be the most appropriate measure for describing effects on demographic inference. This motivates the necessity to propose alternative measures of homoplasy that are more directly relevant to demographic inference and that would allow for meaningful quantifications of the effects of homoplasious mutations on the estimation of the expansion time.

In this paper, we propose a new homoplasy metric. We analyze the relationship between three homoplasy metrics, including our proposed metric, and the underestimation of the expansion time. Second, we evaluate the extent to which these homoplasy metrics can be estimated from simulated cpSSR data using Approximate Bayesian Computation (ABC). Finally, we quantify the level of homoplasy in a real dataset from Pinus caribaea, providing an empirical estimate of homoplasy from population genetic data.

Methods

Dataset simulations under a stepwise demographic expansion model

Throughout this study we assume a stepwise demographic expansion model [3]. The model consists on three parameters: θ 0 = 2LN 0 u, θ 1 = 2LN 1 u and τ = 2Ltu, where u is the mutation rate per generation at each linked SSR, N 0 and N 1 are the effective population sizes before and after the expansion, L is the number of linked SSR loci and t is the time in generations since the expansion.

We generated two sets of haplotypes, hISM and hSMM, in the coalescent simulations under the stepwise demographic expansion model used in this study. We used the same genealogy along with the set of mutations falling in each branch of the genealogy to generate hISM and hSMM from each coalescent simulation. hISM represents a set of linked multi-locus SSR haplotypes evolving under the infinite sites model, ISM [19] while hSMM are a set of linked multi-locus SSR haplotypes that evolved under the symmetrical stepwise mutation model, SMM [20]. The haplotypes hISM are free of homoplasy while the haplotypes hSMM can contain homoplasious mutations. These coalescent simulations were performed using a modified a version of the coalescent simulator msHOT [21, 22]. The modified version of msHOT is available at https://github.com/dortegadelv/HomoplasyMetrics.

Measures of homoplasy

We studied the relationship between homoplasy and the underestimation of the population expansion time τ using three different measures.

The first metric is the commonly used homoplasy index (P) [12] as used by [15]:

$$ P=1-\frac{1-{H}_{ISM}}{1-{H}_{SMM}}=1-\frac{F_{ISM}}{F_{SMM}} $$
(1)

Where H ISM and H SMM are the expected heterozygosities [23] per haplotype estimated in a set of haplotypes containing L linked loci evolving under the infinite sites model (hISM) and the stepwise mutation model (hSMM), respectively. F ISM and F SMM are the expected homozygosities in the set of haplotypes hISM and hSMM. Note that F SMM is directly observable from the data, while F ISM is not, in real data of a set of haplotypes hSMM.

We also use the per SSR locus average of P, which we call Mean Size Homoplasy (MSH). It estimates the mean reduction in heterozygosity per SSR locus. This can be interpreted as the mean homoplasy index P per individual loci. It can be expressed as:

$$ MSH=1-\frac{\sum_{i=1}^L\frac{1-{H}_{ISM}^i}{1-{H}_{SMM}^i}}{L}=1-\frac{\sum_{i=1}^L\frac{F_{ISM}^i}{F_{SMM}^i}}{L} $$
(2)

Where L is the number of SSR in the haplotype. \( {H}_{ISM}^i \) and \( {H}_{SMM}^i \) are the expected heterozygosities at the i locus in hISM and hSMM, respectively. \( {F}_{ISM}^i \) and \( {F}_{SMM}^i \) are the expected homozygosities at the i locus in hISM and hSMM.

Inference of demographic growth using haplotypes with linked microsatellites is typically based on the distribution of pairwise differences between multi-locus haplotypes, also known as the mismatch distribution, as the shape of this distribution is determined by the time and magnitude of historical population expansions [4]. Based on this, here we present a new metric, distance homoplasy (DH), which quantifies the proportion of mutations separating two multi-locus haplotypes that are not observed due to homoplasy. Our rationale for using this measure are studies that use the mode of the distribution of pairwise differences as the basis for estimating τ [4]. Therefore, underestimation of the proportion of pairwise differences should impact the mismatch distribution which in turn should alter the inference of τ. DH is expressed as:

$$ DH=\frac{\pi_{ISM}-{\pi}_{SMM}}{\pi_{ISM}} $$
(3)

Where π SMM and π ISM are the mean number of differences between two haplotypes using the haplotypes hSMM and hISM, respectively.

Expected values of π, Fi and F in a stepwise demographic expansion model

We derived the expected values for the diversity statistics π, F i and F as a function of the mutation rate u of each linked SSR, the number of linked simulated SSR’s L and the coalescent time T ij , in number of generations, between a pair of haplotypes i and j present in the sample. We use the following equation \( E\left[\lambda \right]=E\left[E\left[\lambda |{T}_{ij}\right]\right]={\sum}_{x=1}^tE\left[\lambda |{T}_{ij}=x\right]P\left({T}_{ij}=x\right) \) where λ stands for any diversity statistic and t is the time in generations since the expansion. T ij is scaled in units of N generations. We explain how to obtain the values of E[λ| T ij ] for every diversity statistic under the ISM and SMM in the Appendix. The probability distribution of T ij under a stepwise demographic expansion model is equal to:

$$ P\left({T}_{ij}=x\right)=\left\{\begin{array}{cc}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$N$}\right.\left({e}^{-\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$N$}\right.}\right)& \kern1.25em 0\le x<t-1\\ {}1-{\sum}_{x=1}^{t-1}P\left({T}_{ij}=x\right)& \kern1.5em x=t\end{array}\right. $$
(4)

Where N is the effective population time in the present. The probability distribution of T ij is divided into two phases: 1) After the expansion, the population keeps a constant population size and, therefore, \( P\left({T}_{ij}=x\right)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$N$}\right.{e}^{-\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$N$}\right.} \) during that period of time. 2) Before the expansion, all individuals must coalesce quickly at a time very close to the expansion time T ij  = t assuming that the population size is very small. To model that effect, we assume that all individuals coalesce exactly at time T ij  = t if they have not already coalesced going forward in time.

The equations shown above to estimate the expected value of the diversity statistics are used to obtain estimates of the homoplasy parameters P, MSH and DH. As an example, following equation (1) the expected value of P can be calculated if we know the expected value of the diversity statistics F ISM and F SMM .

$$ E\left[P\right]=1-\frac{E\left[{F}_{ISM}\right]}{E\left[{F}_{SMM}\right]} $$
(5)

Where:

$$ E\left[{F}_{SMM}\right]={\sum}_{x=1}^tE\left[{F}_{SMM}|{T}_{ij}=x\right]P\left({T}_{ij}=x\right) $$
(6)
$$ E\left[{F}_{ISM}\right]={\sum}_{x=1}^tE\left[{F}_{ISM}|{T}_{ij}=x\right]P\left({T}_{ij}=x\right) $$
(7)

The same approach was also used to obtain the expected values of MSH and DH. The analytical equations to calculate those expected values are explained in the Appendix.

Simulations to analyze changes in homoplasy measures

We used a simulation framework to test the accuracy of our estimates for the summary statistics π, F and F i under the ISM and the SMM along with our estimates for the homoplasy values P, MSH and DH. We used our modified version of the coalescent simulator msHOT [21, 22] to generate two different sets of haplotypes (hSMM and hISM) for each simulated genealogy. The modified version of msHOT is available at https://github.com/dortegadelv/HomoplasyMetrics. msHOT was used to make simulations under the stepwise demographic expansion model, where we used a value of θ 1 = 30, θ 0 = 0.03 and ten different values of τ {1.5, 3, 4.5, 6, 7.5, 9, 10.5, 12, 13.5, 15}. For each value of τ, we performed 100 simulations of 150 haplotypes with 6 linked SSRs. The ten command lines used for those simulations are shown in the Appendix (Command Line 1).

We also did 100 simulations for 9 different numbers of linked SSRs in the haplotype, going from L = 2 to L = 10 to examine how changes in the value of L affect P, MSH and DH. We simulated 150 haplotypes in each simulation, where the values of the demographic parameters were set to θ 1 = 5L, θ 0 = 0.005L, τ = L = 2tuL, where we kept the parameters t and u fixed to a certain value such that 2tu = 1. Notice that the divergence time t is kept fixed regardless of the number of linked SSRs L in these simulations. The nine command lines used for these simulations are shown in the Appendix (Command Line 2).

Underestimation of expansion time

We quantified the underestimation of expansion time due to homoplasy using a metric called TS, which we define as

$$ TS=\frac{\widehat{\tau_{ISM}}-\widehat{\tau_{SMM}}}{\widehat{\tau_{ISM}}} $$
(8)

Values of the estimated expansion time τ for haplotypes hISM and hSMM, \( \widehat{\tau_{ISM}} \) and \( \widehat{\tau_{SMM}} \) respectively, were obtained using the method by Schneider and Excoffier (1999), implemented in the software Arlequin [24]. This method infers the parameters θ 0, θ 1 and τ based on the observed distribution of pairwise differences between haplotypes, also called mismatch distribution, and its expectation under a stepwise demographic expansion model [25]. This approach assumes that there is no homoplasy in the sample of haplotypes, therefore any differences between \( \widehat{\tau_{ISM}} \) and \( \widehat{\tau_{SMM}} \) are due to homoplasious mutations present in hSMM. Following [15], to use Arlequin for the hSMM analysis we coded the SSRs as binary data, where the number of repeats were coded with ‘1’ and shorter alleles were coded filling the difference in repeats with ‘0’ .

We simulated 100 replicates of 150 haplotypes with 6 linked SSRs for each of 10 different values of τ {1.5, 3, 4.5, 6, 7.5, 9, 10.5, 12, 13.5, 15}. We set a value of θ 1 equal to 30 and 60, which has the same order of magnitude of the value of θ 1 estimated for the Pinus caribaea dataset [26] employed in this study (see Pinus caribaea dataset), and a value of θ 0 which was 1000 smaller than θ 1 for all simulations. The command lines used for the simulations are shown in the Appendix. Command Line 3 and 4 were used for the simulations done where θ 1 = 30 and θ 1 = 60, respectively.

The value of each homoplasy measure and TS was computed for each replicate of each simulation and the relationship of each homoplasy measure with TS was analyzed. In the simulations done with a value of θ 1 = 30, we removed 1 out of the 1000 simulations we performed where that simulation was the only that had a TS value smaller than -10 (see Additional file 1: Figure S1 for details on the removed simulation).

Estimation of homoplasy and expansion time using ABC

We used another modified version of the program msHOT [21, 22], also available at https://github.com/dortegadelv/HomoplasyMetrics, to implement an ABC algorithm that estimates the posterior distribution of demographic parameters θ 0, θ 1 and τ and the posterior predictive distribution of the three measures of homoplasy DH, MSH and P (see Appendix for details about the implementation of the ABC algorithm). We employed three summary statistics previously used [27] to estimate demographic parameters in a model of population growth: The mean of the variance in the size of the SSRs across loci (V), the expected heterozygosity averaged across loci (H) and the number of distinct haplotypes (a). We used the mode of the posterior distribution and posterior predictive distribution as point estimates of the demographic parameters and homoplasy measures, respectively (see Appendix and Additional file 1: Figure S2 for a discussion on why we employed the mode as a point estimate). We also quantified the relative bias and estimated the 50%, 75% and 90% coverage of the demographic parameters and homoplasy measures to ascertain the quality of the point estimates and the inferred posterior distributions (see Appendix). The relative bias is the average difference between the estimated and true value of the parameter divided by its true value [28] . The 50%, 75% and 90% coverage are the proportion of times that the true value is within the 50%, 75% and 90% credible interval.

We compared the real value of the homoplasy measures P, MSH and DH against the estimated values of the homoplasy measures using our ABC approach in 100 simulations of 150 haplotypes with 6 linked SSRs with parameters θ 1 = 30 and θ 0 = 0.03 and 10 different values of τ {1.5, 3, 4.5, 6, 7.5, 9, 10.5, 12, 13.5, 15}, where 10 simulations were done for each τ value. The ten command lines used for the simulations are shown in the Appendix (Command Line 5). We also estimated the three homoplasy measures in the simulations we explain in the next paragraph.

We compared the performance of three different methods to infer τ and θ 1 in three different sets of 100 simulations of 150 haplotypes with 6 linked SSRs done using three different τ values {3,6,9}, a θ 1 = 30 and a θ 0 = 0.03. The three command lines for these simulations are shown in the Appendix (Command Line 6). One of the three methods we used to estimate τ and θ 1 is our ABC approach, and the other two methods use the mismatch distribution to estimate those demographic parameters: 1) One of those methods is the approach taken by [3] as implemented in the software Arlequin [24], which assumes that there is no homoplasy in the data (Least Squares approach without taking Homoplasy into account, LSWH). 2) The other method we used is a maximum-pseudolikelihood estimator that uses a model where it is assumed that homoplasy can occur in the data [16] (Maximum Pseudolikelihood using a model with Homoplasy, MPH). Code for that method was kindly provided by Miguel Navascués.

Estimation of homoplasy and population expansion times in a Pinus caribaea dataset

We used a dataset of 7 SSR loci from 88 individuals of the species Pinus caribaea to estimate τ along with the homoplasy measures MSH and DH. This dataset is a subset of the data previously published in [26], where an analysis of population structure from four species of Pinus subsection Australes, including Pinus caribaea, identified four different groups (groups I-IV). We took the group containing the largest number of individuals distributed in Central America (group II), and retained only the individuals sampled from Central America in that group (88 out of 93 individuals) for our analysis. A hypothesis of population expansion could not be rejected using information from the mismatch distribution in group II [26], making this group suitable for analysis of expansion. We used ABC, LSWH and MPH to estimate τ in that dataset. The estimations of \( \widehat{\tau} \) in the three methods used above were later transformed to years using a mutation rate of 5.5 X 10−5 per SSR per generation [29] and a generation time of 42.5 years [26].

We also report the 95% confidence interval of the estimation of \( \widehat{\tau} \) with LSWH, MPH and ABC using a parametric bootstrap approach as in Arlequin [24]. We report the 95% confidence intervals for the ABC method instead of the 95% credible intervals to compare the 95% confidence intervals created with ABC with those obtained using LSWH and MPH. For each particular inference method (LSWH, MPH or ABC), the approach involves the simulation of 1000 datasets of 88 individuals with 7 SSR using the demographic parameters estimated for the Pinus caribaea data using a particular inference method. The value of the parameter τ · from each of the 1000 datasets was estimated using the inference method under study. Then, for a confidence level of α = 0.05, the approximate limits of the confidence interval were defined as the α/2 and 1 – α/2 percentile values of the 1000 values of τ ·. This parametric bootstrap approach was also used to estimate the 95% confidence interval of MSH and DH using the ABC method and 1000 simulations done using the demographic parameters inferred by ABC.

Results

Response of different measures of homoplasy to differences in expansion time

First we evaluated the response in the three different measures of homoplasy and their components, π, F i and F, to changes in expansion time under the demographic stepwise expansion model in haplotypes containing completely linked SSRs. We thereby corroborate our theoretical expectations for the different metrics and found that our simulations validate their predictions (Fig. 1a-d).

Fig. 1
figure 1

Homoplasy values in the stepwise demographic expansion model for different values of the expansion time parameter τ. The points in each plot are the average values for each statistic across 100 simulations for plots (a-c), those average values were used to calculate the mean values of the homoplasy index (P), mean size homoplasy (MSH) and distance homoplasy (DH) that are plotted as points in (d). The dashed lines are the approximated expected values estimated from our derivations. a π ISM and π SMM ; b \( {F}_{ISM}^i \) and \( {F}_{SMM}^i \); c F ISM and F SMM d P, MSH and DH

Full size image

As can be seen in Fig. 1a-b, the accumulation of homoplasious mutations causes a monotonic increase in the difference between \( {F}_{ISM}^i \) and \( {F}_{SMM}^i \) and between π SMM and π ISM with the expansion time, something that is not observed for the difference between the two measures of haplotype homozygosity F ISM and F SMM (Fig. 1c). This translates to both MSH and DH increasing steadily with expansion time, while P has a parabolic relationship (Fig. 1d) and stays at a constant value close to 0.09 when τ is equal or larger than 8 (Fig. 1d). Therefore, the values of P do not seem likely to relate to underestimation of population expansion time, in contrast with MSH and DH. Additionally, we found that the number of linked SSRs in the haplotype does not influence the values of MSH and DH, but it does change the value of P given a fixed divergence time t (Additional file 1: Figure S3).

The relation between different measures of homoplasy and underestimation of τ

Simulations of demographic expansion under different values of τ reveal that the standard homoplasy index P is not strongly correlated with TS, which measures the underestimation of τ due to homoplasy (Fig. 2a, Pearson’s ρ = −0.1282, p-value = 4.8*10−5). Contrarily, MSH and DH,have a stronger correlation with an underestimation of τ, where MSH has a slightly lower correlation with TS (Fig. 2b, ρ = 0.6903, p-value <2.2*10−16) than DH, which has the strongest correlation with TS of all the homoplasy measures inspected (Fig. 2c, ρ = 0.6989, p-value <2.2*10−16). Correlation between the latter two measures was strong (0.9208), whereas neither of them was strongly correlated with P (ρ between MSH and P = −0.2685; ρ between DH and P = −0.0977). Simulations with a higher value of θ 1 = 60 (Additional file 1: Figure S4), produced a nearly identical relationship between DH, MSH and TS (ρ between P and TS = −0.2617; ρ between MSH and TS = 0.8673; ρ between DH and TS = 0.8777), showing that DH and MSH are robust predictors of an underestimation of τ while P is not.

Fig. 2
figure 2

Linear relationship between TS and three measures of homoplasy: a P (ρ = −0.1282, intercept = 0.3783, slope = −0.2509, p-value = 4.83e−5), b MSH (ρ =0.6903, intercept = 0.0893, slope = 0.9868, p-value <2.2e−16) and c DH (ρ =0.6989, intercept = −0.0541, slope = 1.1424, p-value <2.2e−16) in 999 simulations made with the demographic parameters θ 0 = 0.03, θ 1 = 30 and 10 different values of τ

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ABC estimates of homoplasy and expansion time

We used simulated data to evaluate the estimation of homoplasy metrics on an ABC framework. We performed linear regressions of the estimated values of the homoplasy measures obtained by our ABC approach on their true homoplasy values (Fig. 3) We also measured the relative bias and the correlation between the estimated and true values of the homoplasy measures. On simulations done over a range of τ values, we found that our estimates of MSH and DH were highly correlated with their real values (r = 0.881 and r = 0.740, respectively) and their relative bias was small (relative bias = −0.040 and 0.030, respectively), indicating that MSH and DH values are well estimated by our ABC approach. On the other hand, the estimates of P had a smaller correlation with their true values (r = 0.486) and their relative bias is −0.132, indicating that our ABC approach underestimates P values by approximately 13.2%. The underestimation can also be seen in Fig. 3. Despite differences in the quality of the point estimates of P, MSH and DH, we found that the 50%, 75% and 95% coverage of the homoplasy measures indicate that the inferred posterior distribution of those measures are well estimated (Additional file 1: Table S1 and Appendix).

Fig. 3
figure 3

ABC estimates of a P, b MSH and c DH compared with their true values in 100 simulations done with the demographic parameters θ 0 = 0.03, θ 1 = 30 and 10 different values of τ. A linear model was fitted to analyze the relationship between each homoplasy measure true value and their ABC estimate of P (ρ = 0.4864, intercept = 0.0385, slope = 0.3104, p-value = 2.88e−7), MSH (ρ = 0.8809, intercept = 0.0220, slope = 0.8667, p-value <2.2e−16) and DH (ρ = 0.7399, intercept = 0.0512, slope = 0.8771, p-value <2.2e−16)

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We performed more simulations to analyze the performance of the ABC approach on simulations done using the same demographic parameters. We evaluated this by creating three sets of simulations done over a single value of τ (τ = 3, 6 or 9). We found that the 50%, 75% and 90% coverage indicate that the posterior distributions of the homoplasy measures are correctly inferred (Additional file 1: Table S2 and Appendix). Second, we found that the average relative bias was small for all homoplasy measures (relative bias = 0.053, 0.043 and 0.068 for P, MSH and DH, respectively; Additional file 1: Table S3). This indicates that, on average, the ABC method slightly overestimates the value of the homoplasy measures by approximately 5% on these sets of simulations.

Apart from estimating homoplasy measures, we also used ABC to estimate the value of τ. We found that ABC and the pseudo-likelihood estimator (MPH) perform equally well to obtain estimates of the value of τ, showing that both methods can correct for the effects of homoplasy. As expected, the expansion time is strongly underestimated by the method that does not take homoplasy into account (LSWH), particularly for older expansion events where there are higher values of MSH and DH (Fig. 4). Additionally, we found that ABC gave good estimations of the value of θ 1, compared to LSWH and MPH which gave overestimations of the actual value of θ 1 (Additional file 1: Figure S5), in line with previous studies done using LSWH [3] and MPH [16]. It must be pointed out that in ABC, as in any Bayesian method, the estimates of the parameters depend on the prior distributions used for the parameters. Prior distributions should contain all the possible demographic parameter values [30] and should not be very wide to avoid low acceptance rates in the ABC algorithm (step 5 of the ABC algorithm in the Appendix).

Fig. 4
figure 4

Estimation of τ using three methods (LSWH, MPH and ABC). The boxplots of the estimation of τ were done on simulations where θ 1 = 30, θ 0 = 0.03 and three different values of τ were used, a τ = 3, b τ = 6 and c τ = 9. 100 simulations were performed for each value of τ. The actual value of τ in each plot is displayed with the dashed line

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Population expansions and homoplasy in data of Pinus caribaea populations from central America

The time of expansion for one population of Pinus caribaea in Central America was obtained using LSWH, MPH and ABC (Table 1). We found that the only method where homoplasy is not taken into account, LSWH, produces lower estimates of \( \widehat{\tau} \) compared to ABC and MPH, suggesting that homoplasy may cause the expansion time to be underestimated by approximately 100,000 years in this case. The 95% confidence intervals of \( \widehat{\tau} \) for all methods is large however (Table 1). ABC-based estimates of homoplasy are 0.11 and 0.246 for MSH and DH respectively, which agrees with the theoretical estimates of 0.106 and 0.269 obtained for those homoplasy measures using equations (24, Appendix) and (17, Appendix) given the demographic parameters estimated using ABC.

Table 1 Estimates of homoplasy and times of expansion in the population of Pinus caribaea analyzed
Full size table

Discussion and conclusions

Here we propose a homoplasy metric, DH, which measures the proportion of pairwise differences that are not observed due to homoplasy. Our theoretical estimates and simulations confirm that the mean number of pairwise differences not counted due to homoplasy increase when population expansion times are older, causing a monotonic increase in the value of DH (Fig. 1). We also confirm that DH has a strong, linear relationship with underestimation of population expansion times using classical methods of inference based on pairwise differences (Fig. 2), with older expansion times leading to the expected higher TS [16] and corresponding increase in DH. This in contrast to the standard homoplasy index for multi-locus haplotypes, P, which shows no clear relation to TS and, starting from a certain τ value, actually decreases as a function of expansion time. The latter can be clearly understood from the expected relation between expansion time and homozygosity under the ISM and SSM model. This shows the value of a homoplasy metric that directly captures the underestimation of the number of mutations [18] when trying to capture effects on demographic inference.

Although conceptually and empirically DH most closely relates to the way that homoplasy causes underestimation of expansion time, the average decrease of per-locus heterozygosity, MSH, has a rather similar relation to population expansion time (Fig. 1) and also correlates strongly with the underestimation of the population expansion time (Fig. 2). It has been shown that in constant population sizes the value of MSH is determined by θ [12], the expected number of mutations between a pair of sequences, so the fact that it also increases with τ is not entirely surprising. Our theoretical estimates indeed confirm that MSH increases monotonically with expansion time under the stepwise demographic expansion model (Fig. 1) and also show that there is a relationship between the number of mutations, predicted by older coalescent times due to older population expansions, and MSH on the stepwise population expansion model. Additionally, MSH and DH are not affected by changes in the number of linked SSRs analyzed, while P does depend on the number of linked SSRs studied.

The usefulness of homoplasy metrics such as DH and MSH depends in part on how well they can be estimated from data. We have shown that we can obtain reasonable average estimates of MSH and DH using ABC [31]. Compared to MASH, the ABC method we propose is not biased by the fraction of homoplasy unmeasurable by MASH. It thereby offers a natural solution to the quantification of homoplasy. Additionally, ABC can estimate the posterior distribution of demographic parameters of interest through explicit modeling of SSR evolution under different demographic scenarios. The approach proved successful in correcting for bias in the inference of expansion time, with similar performance to MPH [16] which also explicitly accounts for homoplasy assuming the SSR’s evolve according to a SMM. One advantage of ABC, in addition to allowing for direct estimates of homoplasy, is that more complicated mutational models of SSR evolution can easily be incorporated. This could be important, as many SSR are known not to evolve in a strictly stepwise manner [32].

Given the potential for erroneous demographic inference when using linked SSR, it is important to obtain such homoplasy estimates from empirical data. With our ABC approach, we were able to estimate values of MSH and DH in a published dataset of Pinus caribaea. We found that the underestimation of the expansion time assuming a model that does not take homoplasy into account is of around 80,000 to 100,000 years, a reduction of around 28 to 32% compared to the value estimated with methods that use a more realistic model of SSR evolution where homoplasious events are possible. As with the ABC approach proposed here, other authors have suggested to use model-based approaches to infer past demographic events using linked cpSSR markers in spruces [11]. Since ABC simulation based approaches provide an estimate of homoplasy, we believe that ABC approaches are useful to quantify the effect of homoplasy on demographic parameters and summary statistics of interest. To our knowledge, this is the first time that homoplasy parameters have been inferred using population genetic data.