The isolation-with-migration (IM) model is commonly used to make inferences about gene flow during speciation, using polymorphism data. However, it has been reported that the parameter estimates obtained by fitting the IM model are very sensitive to the model’s assumptions—including the assumption of constant gene flow until the present. This article is concerned with the isolation-with-initial-migration (IIM) model, which drops precisely this assumption. In the IIM model, one ancestral population divides into two descendant subpopulations, between which there is an initial period of gene flow and a subsequent period of isolation. We derive a very fast method of fitting an extended version of the IIM model, which also allows for asymmetric gene flow and unequal population sizes. This is a maximum-likelihood method, applicable to data on the number of segregating sites between pairs of DNA sequences from a large number of independent loci. In addition to obtaining parameter estimates, our method can also be used, by means of likelihood-ratio tests, to distinguish between alternative models representing the following divergence scenarios: (a) divergence with potentially asymmetric gene flow until the present, (b) divergence with potentially asymmetric gene flow until some point in the past and in isolation since then, and (c) divergence in complete isolation. We illustrate the procedure on pairs of Drosophila sequences from ∼30,000 loci. The computing time needed to fit the most complex version of the model to this data set is only a couple of minutes. The R code to fit the IIM model can be found in the supplementary files of this article.
THE two-deme, isolation-with-migration (IM) model is a population genetic model in which, at some point in the past, an ancestral population divided into two subpopulations. After the division, these subpopulations exchanged migrants at a constant rate until the present. The IM model has become one of the most popular probabilistic models in use to study genetic diversity under gene flow and population structure. Although applicable to populations within species, many researchers are using it to detect gene flow between diverging populations and to investigate the role of gene flow in the process of speciation. A meta-analysis of published research articles that used the IM model in the context of speciation can be found in Pinho and Hey (2010).
Several authors have developed computational methods to fit IM models to real DNA data. Some of the most-used programs are aimed at data sets consisting of a large number of sequences from a small number of loci. This is the case of MDIV (Nielsen and Wakeley 2001), IM (Hey and Nielsen 2004; Hey 2005), IMa (Hey and Nielsen 2007), and IMa2 (Hey 2010), which rely on Bayesian Markov chain Monte Carlo (MCMC) methods to estimate the model parameters and are computationally very intensive.
In the past decade, the availability of large data sets spanning the entire genome has increased significantly. However, the MCMC-based implementations of the IM model referred to above are computationally expensive even for small numbers of loci, and their running times increase linearly with the number of loci (Wang and Hey 2010). Fitting an IM model also provides a rather simplified picture of the divergence process, which for some research purposes is clearly insufficient (for example, if one wishes to know whether a process of sympatric speciation has been completed, or whether gene flow occurred due to secondary contact). In addition, Becquet and Przeworski (2009) and Strasburg and Rieseberg (2010) showed that inference based on the programs IM and IMa can become unreliable if any of the assumptions made about population structure, recombination, or linkage is severely violated. For these reasons, there has been a significant increase in the demand for methods that not only scale well to genome-sized data, but are also able to estimate increasingly realistic models.
To improve efficiency and scalability, one possible strategy is to work with summary statistics rather than full data patterns. The MCMC-based program MIMAR of Becquet and Przeworski (2007, 2009) uses the four summary statistics studied by Wakeley and Hey (1997) to fit the IM model, and drops the assumption of no intralocus recombination. Gutenkunst et al. (2009) introduced a method based on the joint sample frequency spectrum (JSFS) that is able to fit a range of demographic models incorporating multiple populations, periods of migration and admixture, splits and joins of populations, and changes in population sizes. Based on the same type of data, the more recent implementation of Kamm et al. (2016) can already deal with a large number of individuals and populations, but does not yet include gene flow.
Genome-scale data sets, even when stemming from just a few individuals, tend to be more informative than data sets consisting of many individuals but covering only a relatively short genomic region. In fact, as the sample size for a single locus increases, the probability that an extra sequence adds a deep (i.e., informative) branch to the coalescent tree quickly becomes negligible (see for example Hein et al. 2005, pp. 28–29). Data sets of a small number of individuals per locus are also more suitable for likelihood-based inference: if at each locus the observation consists only of a few sequences, the coalescent process of these sequences is relatively simple and can more easily be used to derive the likelihood for the locus concerned.
Among the methods designed for whole-genome sequence data of only a few individuals are those of Mailund et al. (2012), Schiffels and Durbin (2014), and Steinrücken et al. (2015). The fact that they are designed for full polymorphism data makes these methods computationally more expensive than JSFS-based methods. However, they rely on the coalescent with recombination modeled as a hidden Markov process, i.e., they are able to capture the linkage information present in the data. Presently, complex models of demographic history can already be fitted using this approach (see, for example, Steinrücken et al. 2015).
Arguably the only implementations that can be considered fast are those based on blockwise-likelihood methods. These implementations are also aimed at a small number of sampled individuals, and use the information in each of a large number of relatively short and well separated loci: because recombination within loci is disregarded, it is considerably easier to derive explicitly the likelihood for each locus; and because linkage between loci is assumed to be negligible, the likelihood of a data set is just the product of the likelihoods for the individual loci.
Blockwise-likelihood methods for the standard two-deme IM model have been developed, for example, by Wilkinson-Herbots (2008) and Wang and Hey (2010), for pairs of DNA sequences at a large number of independent loci, and by Lohse et al. (2011) and Andersen et al. (2014) for larger numbers of sequences at each locus. Lohse et al. (2011) also developed a more general Laplace-transform method to calculate blockwise likelihoods for a range of demographic scenarios, which was further extended and efficiently automated in Lohse et al. (2016). Zhu and Yang (2012) developed an implementation, based on triplets of sequences, of an IM model with three species with known phylogeny and symmetric migration between two of them.
Some authors have focused on blockwise-likelihood methods for models of divergence that drop the assumption of constant gene flow until the present, and which are therefore more realistic in the context of speciation. In particular, Innan and Watanabe (2006) considered a model in which the level of gene flow between two subpopulations gradually decreases until they become completely isolated from each other. Their calculation of the likelihood of the number of nucleotide differences between pairs of sequences relies on the numerical computation of the coalescence time density at different points in time, which can be computationally expensive. IM models in which gene flow is allowed to cease at some point in the past—hereafter referred to as isolation-with-initial-migration (IIM) models—have also been considered by, for example, Teshima and Tajima (2002), Becquet and Przeworski (2009), Mailund et al. (2012), Wilkinson-Herbots (2012), and Lohse et al. (2015).
In the present article, we apply matrix eigen-decomposition techniques to expand on the work of Wilkinson-Herbots (2012) on the IIM model, who derived explicit formulas for the distribution of the coalescence time of a pair of sequences, and the distribution of the number of nucleotide differences between them. These analytic results enable a very fast computation of the likelihood under an IIM model, given a data set consisting of observations on pairs of sequences at a large number of independent loci (Lohse et al. 2015; Wilkinson-Herbots 2015; Janko et al. 2016). However, for mathematical reasons, this work adopted two biologically unrealistic assumptions which may affect the reliability of estimates: symmetric migration and equal subpopulation sizes during the migration period.
Here, we study a more general IIM model which allows for asymmetric gene flow during the migration period. It also allows for unequal subpopulation sizes during gene flow, as well as during the isolation stage. Both this model and other simpler models studied in this article assume haploid DNA sequences, which accumulate mutations according to the infinite-sites assumption (Watterson 1975). An extension to the Jukes–Cantor model of mutation is feasible but beyond the scope of this article.
We first describe an efficient method to compute the likelihood of a set of observations on the number of nucleotide differences between pairs of sequences, where each pair comes from a different locus and where we assume free recombination between loci and no recombination within loci. As our method uses an explicit expression for the likelihood, it is very fast, and efficient enough to easily deal with asymmetric bidirectional gene flow, unequal population sizes, mutation rate heterogeneity, and large numbers of mutations. Second, we illustrate how to use this method to fit the IIM model to real data. The data set of Drosophila sequences from Wang and Hey (2010), containing over 30,000 observations (i.e., loci), is used for this purpose. Finally we demonstrate, using this data set, how different models representing different evolutionary scenarios can be compared using likelihood-ratio tests. More specifically, we compare three main scenarios: (a) divergence without gene flow; (b) divergence with potentially asymmetric gene flow until the present; and (c) divergence with potentially asymmetric gene flow until some time in the past, and in isolation since then.
For the purposes of the present article, and from a forward-in-time perspective, the IM model makes the following assumptions: (a) until time ago a population of DNA sequences from a single locus followed a Wright–Fisher haploid model (Fisher 1930; Wright 1931); and (b) at time ago, this ancestral population split into two Wright–Fisher subpopulations with constant gene flow between them. If we take an IM model and add the assumption that, at time ago gene flow ceased, we get an IIM model. Figure 1 illustrates the fullest IIM model dealt with in this article.
In the IIM model of Figure 1, the population sizes are given inside the boxes, in units of DNA sequences. All population sizes are assumed constant and strictly positive. The parameters a, b, and indicate the relative size of each population with respect to subpopulation 1 during the migration stage. For example, if is the number of sequences in the ancestral population, then Between times and ago (two time parameters in units of generations), there is gene flow between the subpopulations: in each generation, a fraction of subpopulation i are immigrants from subpopulation j with i.e., is the migration rate per generation from subpopulation i to subpopulation j backward in time. Within each subpopulation, reproduction follows the neutral Wright–Fisher model and, in each generation, restores the subpopulations to their original sizes, i.e., reproduction undoes any decrease or increase in size caused by gene flow.
Under the IIM model, the genealogy of a sample of two DNA sequences from the present subpopulations can be described by successive Markov chains, working backward in time. We will define these in the simplest possible way, using the smallest state space necessary for the derivation of the coalescence time distribution. Hence, during the isolation stage (until time into the past) and the migration stage (between and the process can only be in state 1—both lineages in subpopulation 1, state 2—both lineages in subpopulation 2, state 3—one lineage in each subpopulation, or state 4—in which lineages have coalesced. After the lineages have either coalesced already—state 4, or have not—state 0. Only states 1, 2, and 3 can be initial states, according to whether we sample two sequences from subpopulation 1, two sequences from subpopulation 2, or one sequence from each subpopulation. When the genealogical process starts in state i the time until the most recent common ancestor of the two sampled sequences is denoted whereas denotes the number of nucleotide differences between them.
If time is measured in units of generations and N is large, the genealogical process is well approximated by a succession of three continuous-time Markov chains; one for each stage of the IIM model (Kingman 1982a,b; Notohara 1990). We refer to this stochastic process in continuous time as the coalescent under the IIM model. During the isolation stage, the approximation is by a Markov chain defined by the generator matrix(1)with being the initial state (Kingman 1982a,b). If 3 is the initial state, the lineages cannot coalesce before During the ancestral stage, the genealogical process is approximated by a Markov chain with generator matrix(2)(Kingman 1982a,b). In between, during the migration stage, the approximation is by a Markov chain with generator matrix(3)(Notohara 1990). In this matrix, represents the rate of migration (in continuous time) of a single sequence when in subpopulation i. The rates of coalescence for two lineages in subpopulation 1 or 2 are 1 and respectively. Note that state 3 corresponds to the second row and column, and state 2 to the third row and column. This swap was dictated by mathematical convenience: the matrix should be as symmetric as possible because this facilitates a proof in the next section.
Distribution of the time until coalescence under bidirectional gene flow (M1 > 0, M2 > 0)
To find the density of the coalescence time of two lineages under the IIM model, given that the process starts in state i and there is gene flow in both directions, we consider separately the three Markov chains mentioned above. We let and denote the times until absorption of the time-homogeneous Markov chains defined by the generator matrices and respectively. Furthermore, we let the corresponding probability density functions (PDFs) [or cumulative distribution functions (CDFs)] be denoted by and (or and Then, can be expressed in terms of the distribution functions just mentioned:(4)for If 3 is the initial state,(5)The important conclusion to draw from these considerations is that to find the distribution of the coalescence time under the IIM model, we only need to find the distributions of the absorption times under the simpler processes just defined.
A Markov process defined by the matrix and starting in state 0, is simply Kingman’s coalescent (Kingman 1982a,b). For such a process, the distribution of the coalescence time is exponential, with rate equal to the inverse of the relative population size:(6)A Markov process defined by is again Kingman’s coalescent, so(7)Finally, with respect to the “structured” coalescent process defined by the matrix we prove in Appendix A that, for (8)where is the entry of the (nonsingular) matrix whose rows are the left eigenvectors of The entry of the matrix is denoted by The are the absolute values of those eigenvalues of which are strictly negative (the remaining one is zero). Since the are real and strictly positive, the density function of is a linear combination of exponential densities.
Substituting the PDFs from Equations 6, 7, and 8 into the Equations 4 and 5, and denoting by the three-by-three matrix with entries we obtain(9)for and(10)If and (i.e., in the case of symmetric gene flow and equal subpopulation sizes during the gene flow period), results 9 and 10 above simplify to the corresponding results in Wilkinson-Herbots (2012)—in this case, the coefficient in the linear combination is zero for
Distribution of the time until coalescence under unidirectional gene flow, and in the absence of gene flow
If either or is equal to zero, or if both are equal to zero, the above derivation of is no longer applicable, as the similarity transformation in Part (ii) of the proof (Appendix A) is no longer defined (see the denominators in some entries of the matrix D). In this section, we derive the density of the absorption time of the Markov chain defined by the matrix given in Equation 3, starting from state i, when one or both the migration rates are zero. Again, this is all we need to fill in Equations 4 and 5 and obtain the distribution of the coalescence time of a pair of DNA sequences under the IIM model. Having gene flow in just one direction considerably simplifies the coalescent. For this reason, we resort to moment-generating functions (MGFs), instead of eigen-decomposition, and derive fully explicit PDFs.
Let again be defined as the absorption time of the Markov chain generated by now with and given that the initial state is We condition on the state of the coalescent after the first transition to obtain the following system of equations for the MGF of where s denotes a dummy variable:(see also more general equations in Wilkinson-Herbots 1998 and Lohse et al. 2011). Solving this system of equations and applying a partial fraction decomposition (analogous to the work done in Griffiths 1981 and Nath and Griffiths 1993, for the case of symmetric migration and equal population sizes), the distributions of and can be expressed as linear combinations of exponential distributions:Thus we obtain the following PDFs:for
The PDF of the coalescence time of a pair of DNA sequences under an IIM model with and can now be easily derived by comparing the above expressions with Equation 8: is given by Equations 9 and 10 above, but now withandIn the opposite case of unidirectional migration we obtained the distribution of the time until coalescence using essentially the same procedure as described above. In addition, for the derivation is trivial. The results for these two cases can be found in Appendix B.
The distribution of the number S of segregating sites
Let denote the number of segregating sites in a random sample of two sequences from a given locus, when the ancestral process of these sequences follows the coalescent under the IIM model and the initial state is state i Recall the infinite-sites assumption and assume that the distribution of the number of mutations hitting one sequence in a single generation is Poisson with mean μ. As before, time is measured in units of generations and we use the coalescent approximation. Given the coalescence time of two sequences, is Poisson distributed with mean where denotes the scaled mutation rate. Since the PDF of is known, the likelihood of an observation from a single locus corresponding to the initial state i can be derived by integrating out :where is the vector of parameters of the coalescent under the IIM model, that is, There is no need to compute this integral numerically: because has been expressed in terms of a piecewise linear combination of exponential or shifted exponential densities, we can use standard results for a Poisson process superimposed onto an exponential or shifted exponential distribution.
The equations 18 and 29 of Wilkinson-Herbots (2012) use this superimposition of processes to derive the distribution of S under a mathematically much simpler IIM model with symmetric migration and equal subpopulation sizes during the period of migration. These equations can now be adapted to obtain the probability mass function (PMF) of S under each of the migration scenarios dealt with in this article. The changes accommodate the fact that the density of the coalescence time during the migration stage of the model is now given by a different linear combination of exponential densities, where the coefficients in the linear combination, as well as the parameters of the exponential densities, are no longer the same. The PMF of S has the following general form:(11)for and(12)for . As defined in the Distribution of the time until coalescence under bidirectional gene flow (M1 > 0, M2 > 0) section, under bidirectional migration is the vector of the absolute values of the strictly negative eigenvalues of and If migration occurs in one direction only, with and the matrix and the vector are those given in the Distribution of the time until coalescence under unidirectional gene flow, and in the absence of gene flow section. In the remaining cases, when and or when there is no gene flow, and are given in Appendix B. In the special case of and Equations 11 and 12 reduce to the results of Wilkinson-Herbots (2012).
The likelihood of a multilocus data set
Recall that, for our purposes, an observation consists of the number of nucleotide differences between a pair of DNA sequences from the same locus. To jointly estimate all the parameters of the IIM model, our method requires a large set of observations on each of the three initial states (i.e., on pairs of sequences from subpopulation 1, from subpopulation 2, and from both subpopulations). To compute the likelihood of such a data set, we use the assumption that observations are independent, so we should have no more than one observation or pair of sequences per locus and there should be free recombination between loci, i.e., loci should be sufficiently far apart.
Let each locus for the initial state i be assigned a label where is the total number of loci associated with initial state i. Denote by the scaled mutation rate at locus where is the mutation rate per sequence per generation at that locus. Let θ denote the average scaled mutation rate over all loci and denote by the relative mutation rate of locus Then, If the relative mutation rates are known, we can represent the likelihood of the observation at locus simply by By independence, the likelihood of the data set is then given by(13)In our likelihood method, the are treated as known constants. In practice, however, the relative mutation rates at the different loci are usually estimated using outgroup sequences (Yang 2002; Wang and Hey 2010).
In the Supplemental Material, File S1 contains the R code to fit the IIM model (and other simpler models) to data sets consisting of observations on the number of segregating sites between pairs of DNA sequences from a large number of independent loci. File S2 contains the R code we used to simulate observations from the IIM model. File S3 contains R functions that are required by File S1 and File S2. The raw Drosophila sequence data used in this article were published by Wang and Hey (2010); the processed Drosophila data to which the models of Figure 7 were fitted are given in File S4.
We generated three batches of data sets by simulation, each batch having 100 data sets. Each data set consists of thousands of independent observations, where each observation represents the number of nucleotide differences between two DNA sequences belonging to the same locus, when the genealogy of these sequences follows an IIM model. Each data set of batches 1, 2, and 3 contains 8000, 40,000, and 800,000 observations, respectively. In each data set, half of the observations correspond to initial state 3, to initial state 1, and to initial state 2.
The data sets shown in this section were generated using the following parameter values: and Each observation in a data set refers to a different genetic locus j, and hence was generated using a different scaled mutation rate for that locus. For batch 1, we first fixed the average mutation rate over all sites to be Then, a vector of 8000 relative-size scalars was randomly generated using a Gamma (15, 15) distribution. The scaled mutation rate at locus j was then defined to be where denotes the relative mutation rate at locus j, that is, the relative size of with respect to the average mutation rate θ. All data sets in batch 1 were generated using the same vector of relative mutation rates. The generation of the mutation rates used in batches 2 and 3 was carried out following the same procedure.
When fitting the IIM model to data sets generated in this manner, the relative mutation rates are included as known constants in the log-likelihood function to be maximized. So, as far as mutation rates are concerned, only the average over all loci is estimated (i.e., the parameter θ). To increase the robustness and performance of the fitting procedure (see also Wilkinson-Herbots 2015, and the references therein), we found the maximum-likelihood estimates for a reparameterized model with parameters θ, and
The boxplots of the maximum-likelihood estimates obtained for the three batches of simulated data are shown in Figure 2 and Figure 3. For each parameter, the boxplots on the left, center, and right-hand side refer to batches 1, 2, and 3, respectively. From the boxplots of time and population size parameters, it is seen that the estimates are centered around the true parameter values. Estimates for the migration rates are skewed to the right for batches 1 and 2, possibly because the true parameter values for these rates are closer to the boundary (zero) than the ones for population sizes and splitting times. For all types of parameters, increasing the sample size will decrease the variance of the maximum-likelihood estimator, as would be expected from using the correct expressions for the likelihood. In the case of the migration rate parameters, increasing the sample size eliminates most of the skewness.
The three quantile-quantile (Q-Q) plots in Figure 4 show the sample quantiles of the maximum-likelihood estimates of (a size parameter) obtained from simulated data, plotted against the theoretical quantiles of the standard normal distribution. Figure 5 and Figure 6 show the corresponding plots for (a time parameter) and (a migration parameter). In each figure, the left-hand side, center, and right-hand-side Q-Q plots are based on simulation batches 1, 2, and 3, respectively. It is clear from Figure 4, Figure 5, and Figure 6 that the distributions of the maximum-likelihood estimates of and become increasingly Gaussian as we increase the number of observations. This is also true for the estimates of the remaining parameters (results not shown). We note also that the distributions of the time and population size estimates already have a reasonably Gaussian shape for a sample size of 8000 loci. Again, this is true for the estimates of the remaining time and size parameters as well. The lack of approximate normality of the migration rate estimates for smaller sample sizes suggests care should be taken when making inferences about these parameters—see Notes on our method and results.
Drosophila DNA sequence data
To illustrate our method, we apply it to a real, multilocus data set from two closely related species of Drosophila: Drosophila simulans and D. melanogaster. The DNA sequence data of Wang and Hey (2010) consist of two subsets: a large subset, which we will call the “Wang subset,” containing 30,247 blocks of intergenic sequence; and a smaller subset, which we will refer to as the “Hutter subset,” consisting of 378 blocks of intergenic sequence. Loci in the Wang subset were sampled by Wang and Hey (2010) from a genome alignment of four inbred lines, two from D. simulans, and one from each of D. melanogaster and D. yakuba. To take into account the assumption of no recombination within loci and free recombination between loci, and based on the findings of Hey and Nielsen (2004) regarding the density of apparent recombination events in Drosophila, Wang and Hey (2010) chose a locus length of ∼500 bp and a space of at least 2000 bp between loci. To build the Hutter subset, they drew 378 pairs of D. melanogaster sequences from the data set of Hutter et al. (2007), which consists of 378 blocks of sequence sampled from 24 inbred lines of D. melanogaster, with an average locus length of 536 bp and an average distance of ∼52 kb between consecutive loci. They then joined each of these sequence pairs with their respective D. yakuba orthologs from the simulans-melanogaster-yakuba genome alignment. Our models are fitted to the D. melanogaster and D. simulans sequences from both subsets. The D. yakuba sequences are only used as outgroup sequences, to estimate the relative mutation rates at the different loci and to calibrate time.
Since our method uses only one pair of sequences at each of a large number of independent loci, and requires observations for all initial states, the following procedure was adopted to select a suitable set of data. According to the genome assembly they stem from, sequences in the Wang subset were given one of three possible tags: “Dsim1,” “Dsim2,” or “Dmel.” To each of the 30,247 loci in the Wang subset, we assigned a letter: loci with positions 1, 4, 7, … in the genome alignment were assigned the letter A; loci with positions 2, 5, 8, … were assigned the letter B; and loci with positions 3, 6, 9, … were assigned the letter C. A data set was then built by selecting observations corresponding to initial states 1 and 3 from the Wang subset (we used the Dsim1-Dsim2 sequences from loci A, the Dmel-Dsim1 sequences from loci B, and the Dmel-Dsim2 sequences from loci C), while observations corresponding to initial state 2 were obtained from the Hutter subset by comparing the two D. melanogaster sequences available at each locus.
To estimate the relative mutation rates we use the ad hoc approach proposed by Yang (2002), which was also used in Wang and Hey (2010) and Lohse et al. (2011). Estimates are computed by means of the following method-of-moments estimator:(14)where J is the total number of loci, and is the average of the numbers of nucleotide differences observed in pairs of one ingroup sequence and one outgroup sequence, at locus
Table 1 contains the maximum-likelihood estimates for the models shown in Figure 7. Note that the parameters of time and population size have been reparameterized as in Simulated data, and recall that and are the scaled migration rates backward in time. In the diagrams, the left and right subpopulations represent D. simulans and D. melanogaster, respectively.
In this section, we use a series of likelihood-ratio tests for nested models to determine which of the models listed in Table 1 fits the data of Wang and Hey (2010) best. The use of such tests in the present situation is not entirely straightforward. We wish to apply a standard large-sample theoretical result which states that, as the number of observations increases, the distribution of the likelihood-ratio test statistic given bywhere(15)approaches a distribution. In Equation 15, denotes the parameter space according to the null hypothesis This space is a proper subspace of the parameter space according to the alternative hypothesis The number of degrees of freedom of the limiting distribution is given by the difference between the dimensions of the two spaces. A list of sufficient regularity conditions for this result can be found, for example, in Casella and Berger (2001, p. 516). One of them is clearly not met in the present case: in the pairwise comparison of some of our models, every point of is a boundary point of In other words, if is true, the vector of true parameters whichever it might be, is on the boundary of This irregularity is present, for example, when according to and according to The problem of parameters on the boundary has been the subject of articles such as Self and Liang (1987) and Kopylev and Sinha (2011). The limiting distribution of the likelihood-ratio test statistic under this irregularity has been derived in these articles, but only for very specific cases. In most of these cases, the use of the naive distribution, with r being the number of additional free parameters according to turns out to be conservative, because the correct null distribution is a mixture of distributions with Our analysis of the data of Wang and Hey (2010) involves two likelihood-ratio tests with parameters on the boundary (ISO vs. IM1, and IM1 vs. IIM1), so we need to check that the naive distribution is also conservative in these cases. This was verified in a short simulation study which we now describe.
We generated 100 data sets from the ISO model, each one consisting of 40,000 observations, and fitted both the ISO model and the IM1 model to obtain a sample of 100 realizations of the likelihood-ratio test statistic. A Q-Q plot (Figure 8, left boxplot) shows that the estimated quantiles of the null distribution are smaller than the corresponding theoretical quantiles of the distribution with two degrees of freedom (the difference between the dimensions of and in this particular case). In other words, the use of the naive distribution is conservative in this case. Using instead of the correct null distribution, at a significance level of 5%, the null hypothesis (i.e., the ISO model) was falsely rejected in only 1 out of the 100 simulations performed.
A similar simulation was carried out with respect to another pair of nested models: the IM1 model (now as H0), in which and the IIM1 model in which Again, the naive distribution (this time with only one degree of freedom) was found to be conservative (Figure 8, right boxplot). And once more, only in 1 out of the 100 simulations performed is the null hypothesis (the IM1 model) falsely rejected at the significance level, if is used instead of the correct null distribution.
To select the model that best fitted the data of Wang and Hey (2010), we performed the sequence of pairwise comparisons shown in Table 2. For any sensible significance level, this sequence of comparisons leads to the choice of IIM2 as the best-fitting model. In fact, assuming the naive as the null distribution, a significance level as low as is enough to reject in each of the three tests. However, since for this model (see Table 1), a final (backward) comparison is in order: one between IIM2 and IIM3 (which corresponds to fixing at zero in IIM2). The nested model in this comparison has one parameter less and, as can be seen in Table 1, has the same likelihood. So, in the end, we should prefer IIM3 to IIM2.
Confidence intervals for the selected model:
The Wald confidence intervals are straightforward to calculate whenever the vector of estimates is neither on the boundary of the model’s parameter space, nor too close to it. In that case, it is reasonable to assume that the vector of true parameters does not lie on the boundary either. As a consequence, the vector of maximum-likelihood estimators is consistent and its distribution will approach a multivariate Gaussian distribution as the sample size grows (see, for example, Pawitan 2001, p. 258). The confidence intervals can then be calculated using the inverted Hessian matrix.
In the case of the data of Wang and Hey (2010), the vector of estimates of the selected model (IIM3) is an interior point of the parameter space. Assuming that the vector of true parameters is also away from the boundary, we computed the Wald 95% confidence intervals shown in Table 3 using the inverted Hessian. In agreement with our assumption, we note that none of the confidence intervals include zero.
For large sample sizes, and for true parameter values not too close to the boundary of the parameter space, the Wald intervals are both accurate and easy to compute. To check how well the Wald intervals for the IIM3 model fare against the more accurate (see Pawitan 2001, pp. 47–48), but also computationally more expensive, profile likelihood intervals, we included these in Table 3. The two methods yield very similar confidence intervals for all parameters except The cause of this discrepancy should lie in the fact that we only had pairs of D. melanogaster sequences available from a few hundred loci is the size of the D. melanogaster subpopulation during the migration stage).
Conversion of estimates:
The conversion of point estimates and confidence intervals to more conventional units is based on the estimates of Powell (1997) of the duration of one generation and the speciation time between D. yakuba and the common ancestor of D. simulans and D. melanogaster (10 MY); see also Wang and Hey (2010) and Lohse et al. (2011). Using these values, we estimated μ, the mutation rate per locus per generation, averaged over all loci, to be
In Table 4, Table 5, and Table 6, we show the converted estimates for the best-fitting model IIM3. The effective population size estimates, in units of diploid individuals, are all based on estimators of the form For example, the estimate of the ancestral population effective size is given by The estimates in years of the time since the onset of speciation and of the time since the end of gene flow are given by and respectively. With respect to gene flow, we use as the estimator of the fraction of subpopulation 1 that migrates to subpopulation 2 in each generation, forward in time; and as the estimator of the number of migrant sequences from subpopulation 1 to subpopulation 2 in each generation, also forward in time.
If g and are treated as constants, then each of the estimators just given can be expressed as a constant times a product—or a ratio—of the estimators of nonconverted parameters. For example, we have thatandSuppose the IIM3 model is reparameterized in terms ofand denotes the maximum-likelihood estimator of Then the estimator of the vector of converted parameterscan be written as where is a diagonal matrix. The random vector is a maximum-likelihood estimator (of a reparameterized model). Hence, for a large enough sample size, its distribution is approximately multivariate Gaussian, with some covariance matrix and the distribution of is approximately multivariate Gaussian with covariance matrix To calculate the Wald confidence intervals of Table 4, Table 5, and Table 6, we used the inverse of the observed Fisher information as an estimate of An estimate of followed trivially.
Profile likelihood confidence intervals were also computed for the parameterization Then, if is the vector of estimated upper (or lower) bounds for the parameters in will be the vector of estimated upper (or lower) bounds for the converted parameters. This follows from the likelihood-ratio invariance—see, for example, Pawitan (2001, pp. 47–48). Confidence intervals for the converted migration parameter (rather than in the procedure above) were obtained analogously, using a slightly different reparameterization of the IIM3 model.
Notes on our method and results
We have described a fast method to fit the IIM model to large data sets of pairwise differences at a large number of independent loci. This method relies essentially on the eigen-decomposition of the generator matrix of the process during the migration stage of the model: for each set of parameter values, the computation of the likelihood involves this decomposition. Nevertheless, the whole process of estimation takes no more than a couple of minutes for a data set of tens of thousands of loci such as that of Wang and Hey (2010), and it does not require high-performance computing resources. The implementation of the simpler IIM model of Wilkinson-Herbots (2012), with R code provided in Wilkinson-Herbots (2015), is even faster than the more general method presented here, since it makes use of a fully analytical expression for the likelihood (avoiding the need for eigen-decomposition of the generator matrix); but it relies on two assumptions which we have dropped here, and which are typically unrealistic for real species: the symmetry of migration rates and the equality of subpopulation sizes during the gene flow period.
Due to the number of parameters, it is not feasible to assess the performance of our method systematically over every region of the parameter space. However, our experience with simulated data sets suggests that there are two cases in which the variances of some estimators become inflated, in particular the variances of the estimators associated with the gene flow period and One of such cases arises whenever V is very small or is very large, making it very unlikely that the genealogy of a pair of sequences under the IIM model is affected by events that occurred during the gene flow period. The second case arises when the values of the scaled migration rates are greater than one, so that the two subpopulations during the period of gene flow resemble a single panmictic population. In either of these cases, the very process of model fitting can become unstable, that is, the algorithm of maximization of the likelihood may have difficulty converging.
Problems can also arise if the number of loci is insufficient. The simulation study in the Simulated data section suggests that convergence to sensible parameter estimates is still possible for a sample size of 8000 loci. However, when we fitted the full IIM model to a simulated sample of 4000 loci (results not shown), outliers started to emerge. It should also be noted that for sample sizes of just a few thousand loci, the distribution of migration rate estimates is still far from Gaussian (Figure 6). In such cases, computation of confidence intervals should be based on bootstrap methods or on the likelihood (profile likelihood confidence intervals) rather than on the Hessian (Wald confidence intervals). How many loci are needed to obtain good estimates and confidence intervals will also depend on the region of the parameter space concerned.
It is not the goal of this article to draw conclusions regarding the evolutionary history of Drosophila species. We used the data of Wang and Hey (2010) with the sole objective of demonstrating that our method can be applied efficiently and accurately to real data. In Table 7, we list both our estimates and those of Wang and Hey (2010) for a six-parameter isolation-with-migration model (the IM1 model—see Figure 7). The same table contains the estimates for our best-fitting IIM model. Our parameter estimates for the IM model agree well with those of Wang and Hey (2010). The reason that they do not match exactly lies in the fact that we have omitted the “screening procedure” described in Wang and Hey (2010) and have therefore not excluded some of the most divergent sequences in the data set. It should also be borne in mind that our model of mutation is the infinite-sites model, whereas Wang and Hey (2010) have worked with the Jukes–Cantor model. Furthermore, our choice of sequence pairs was somewhat different: Wang and Hey (2010) randomly selected a pair of sequences at each locus, whereas we followed the procedure described in the Maximum-likelihood estimation section.
There are some otable differences between the estimates for both IM models and those for the IIM model: under the IIM model, the process of speciation is estimated to have started earlier (3.6 MYA instead of 3.0 or 3.2 MYA), to have reached complete isolation before the present time (1.5 MYA), and to have a higher rate of gene flow (0.064 sequences per generation instead of 0.013 or 0.012 sequences) during a shorter period of time (2.1 MY of gene flow instead of 3.0 or 3.2 MY). As might be expected, the estimates of each descendant population size (D. simulans and D. melanogaster) in the IM models lie in between the estimates of the corresponding current population size and its size during the gene flow period in the IIM model.
The method we used assumes that relative mutation rates are known (see The likelihood of a multilocus data set). In reality, we must deal with estimates of these rates, and this introduces additional uncertainty which is not reflected in the standard errors and confidence intervals obtained. In principle, this uncertainty can be reduced by increasing the number of ingroup and outgroup sequences used to compute the average number of pairwise differences at each locus in Equation 14. Ideally, estimates of the relative mutation rates should be based on outgroup species only (Wang and Hey 2010) to avoid any dependence between the estimates of relative mutation rates and the observations on ingroup pairwise differences, but this was not possible here since the Wang and Hey (2010) data included exactly one outgroup sequence for each locus.
Violation of assumptions
Some assumptions of the IIM model in this article, such as the infinite-sites assumption and the assumption of free recombination between loci and no recombination within loci, may not be sensible for some real data sets. The appropriateness of other assumptions, for example those regarding the constant size of populations or the constant rate of gene flow, will depend on the actual evolutionary history of the species or populations involved. While a systematic, in-depth robustness analysis of our method (similar to, for example, the robustness studies by Becquet and Przeworski 2009 and Strasburg and Rieseberg 2010 for commonly used IM methods) is beyond the scope of this article, we will in this section informally examine the impact of possible violations of some of the main assumptions made.
Misspecification of the demographic model:
To explore the potential effect of misspecification of the demographic model on inference accuracy, we first simulated 20 data sets of 40,000 loci each from a somewhat more complex evolutionary scenario, depicted in the left-hand side diagram of Figure 9, where subpopulation sizes gradually increase and gene flow gradually declines. The precise parameter values assumed for the true model were chosen arbitrarily and are shown in the left-hand side diagram; in accordance with the reparameterization used in Simulated data, divergence times are measured on a mutational scale by twice the expected number of mutations per sequence (as an average over all loci), population sizes are represented by scaled mutation rates, and rates of gene flow by scaled migration rates. We then applied our method to fit isolation, IM, and IIM models to each of the simulated data sets and selected the best-fitting model by means of likelihood-ratio tests—for each of the 20 data sets generated this was found to be the full IIM model. The average point estimates obtained for each parameter are shown on the right-hand-side diagram of Figure 9. In each diagram, the widths of the boxes are proportional to the population sizes and the heights are proportional to the durations of the time periods concerned. It is readily seen that the IIM model reflects the dynamics of the true model quite well. Population sizes, migration rates, and splitting times are all estimated at intermediate values.
We also repeated the simulation and estimation procedure for an evolutionary scenario involving a period of secondary gene flow, depicted in the left-hand side diagram of Figure 10. Again, for each of the 20 simulated data sets, the full IIM model provides the best fit among the models considered (isolation, IM, and IIM). Comparing the two diagrams in Figure 10 (where the IIM parameter values in the right-hand-side diagram are again the averages of the estimates obtained for the 20 simulated data sets), we see that the IIM model obtained provides a reasonable approximation to the true model, though of course our method did not detect the initial period of isolation as this feature was not included in the set of models fitted. The estimates of the time since the onset of speciation and the time since complete isolation are, on average, close to the true values in this case. The average estimates of the migration rate and population size parameters are again at intermediate values, compared to the range of true values over time.
In common with other methods mentioned in this article (for example, Wang and Hey 2010; Lohse et al. 2011), our method assumes that there is no recombination within loci and free recombination between loci. The first of these two assumptions is the most important one, without which our method would not be valid. Recombination within loci mixes up the genealogies of DNA sequences on which our method relies, making pairs of sequences more equidistant: intralocus recombination does not affect the mean number of segregating sites in a pair of sequences but the variance decreases with increasing recombination (Griffiths 1981; Hudson 1983; Schierup and Hein 2000), resulting in data sets which contain more intermediate values and fewer extreme values. This can be expected to lead to overestimation of the current population sizes and underestimation of the ancestral population size, while the effect on estimates of the other parameters is intuitively somewhat less obvious. The impact of intralocus recombination on the variance of the number of pairwise differences, and hence on the accuracy of our method, may be expected to be less severe in cases of recombination rate heterogeneity within loci (see figure 1 in Hudson 1983, for the extreme case of recombination hotspots separating completely linked regions).
A simulation study by Strasburg and Rieseberg (2010) found that even relatively low levels of intralocus recombination can cause substantial bias in estimates of the IM model parameters obtained using the program IMa (Hey and Nielsen 2007), with highest posterior density intervals failing to contain the true parameter values far more often than would be expected by chance. In IM simulations allowing a minimal but realistic amount of intralocus recombination, Lohse et al. (2016) found that the bias in their parameter estimates was small. Although our method and model are different from those of Hey and Nielsen (2007) and Lohse et al. (2016), the effect of recombination on the underlying genealogies remains the same, and therefore similar biases will occur if the assumption of no intralocus recombination is violated.
For the Drosophila data considered in this article, Wang and Hey (2010) assessed the impact of potential intralocus recombination on their estimates of the parameters of an IM model by comparison with the estimates obtained from the same sequences but halved in length (i.e., approximately halving the expected number of intralocus recombination events). Their estimates of the ancestral population size and the migration rate from the half-length data were ∼30% larger than those from the full-length data, while the differences for the other parameter estimates were small. In the same spirit, we repeated our previous analysis of the Drosophila data but now using the trimmed version of the Wang subset prepared by Lohse et al. (2011), in which the average locus length was reduced by approximately a factor of 3; the Hutter subset (∼1% of the total number of loci) was retained in its entirety as we could not afford to further reduce this already very small data set of D. melanogaster pairs. Applying the estimation and model selection procedures described in Drosophila DNA sequence data to this trimmed version of the data, the likelihood-ratio test of the models IIM1 vs. IIM2 was no longer significant, i.e., there was no longer significant evidence of an increase in population size at time and the best-fitting model was a unidirectional version of IIM1 (i.e., with
Table 8 shows the estimates obtained from the trimmed data; the estimates obtained earlier in this article from the full data are also listed again for comparison. In line with our expectations regarding the potential effect of intralocus recombination, it is seen that the full data gave a larger estimate of the current population size of D. simulans and a smaller estimate of the ancestral population size; the estimated size of D. simulans during the gene flow stage was also smaller than that obtained from the trimmed data. The estimated time since the onset of speciation is nearly identical for the two data sets, but the full data placed the end of gene flow substantially further back into the past (1.5 MYA compared to 0.93 MYA) and estimated a somewhat higher number of migrant sequences per generation (0.064 compared to 0.051) during a shorter period of gene flow (2.12 MY compared to 2.68 MY). This suggests that, in addition to the impact on population size estimates already discussed, intralocus recombination may lead to an overestimate of the time since the end of gene flow in an IIM model and (possibly as a consequence) an overestimate of the migration rate. Nevertheless, for both versions of the Drosophila data, the likelihood-ratio tests of nonzero migration rate and nonzero time since the end of gene flow were significant.
The above considerations imply that, when preparing data for use with our method (or any other method relying on the assumption of no intralocus recombination), loci should be chosen carefully to try to keep the amount of intralocus recombination negligible, and some caution may be needed in the interpretation of results. For data sets showing signs of recombination within loci, it may be possible to reduce its effect by trimming or breaking up such loci to form shorter, apparently nonrecombining segments of DNA sequence (Hey and Nielsen 2004; Strasburg and Rieseberg 2010). An extension of our method to account for recombination within loci would be of interest but is challenging. An extension to a finite-sites model for use with shorter fragments of DNA sequence would also be of interest—such an extension is relatively straightforward but is yet to be implemented in our method (but see Wang and Hey 2010 and Andersen et al. 2014 for the IM model).
If the assumption of free recombination between loci does not hold, then loci are not independent, in which case the likelihood in Equation 13 is in fact a composite marginal likelihood (also called the “independence likelihood” in Chandler and Bate 2007) rather than an ordinary full likelihood (see Varin 2008 for an overview of composite marginal likelihood methods; see also the discussion of Lohse et al. 2016). Statistical theory indicates that in that case, the maximum composite likelihood estimator (MCLE) is still consistent (Cox and Reid 2004; Wiuf 2006, with some minor modifications to account for our slightly different assumptions; Varin 2008), provided the relative mutation rates at the different loci are bounded. Thus, if linkage between loci cannot be ignored, the MCLE of the parameters of the IIM model obtained with our method will still be approximately unbiased if the number of loci is sufficiently large, and if all our other assumptions hold (including the assumption of no recombination within loci). However, if linkage between loci is not negligible, then standard errors and confidence intervals computed using the observed Fisher information (as was done in the Results section) will underestimate the true uncertainty about the parameter estimates obtained (Baird 2015); instead, standard errors and confidence intervals should be based on an estimate of the Godambe information (Godambe 1960). For a data set made up of a single string of correlated loci, or a small number of such strings, obtaining an accurate estimate of the Godambe information presents some difficulties (see Varin 2008 and Varin et al. 2011 for a discussion and some possible strategies). A much simpler situation arises if the data consist of a sufficiently large number of “clusters” of loci, where loci within clusters are correlated but where different clusters can be considered independent. This may be the case, for example, if different clusters of loci are chosen from different chromosomes, or are separated by recombination hotspots or by a large enough distance along the genome. For such data, an empirical estimate of the Godambe information can easily be computed as described in Chandler and Bate (2007) or Varin (2008).
To try to quantify the effect of linkage on the standard errors of the IIM parameter estimates, we conducted the following analysis of a suitable subset of the Wang and Hey (2010) data. We partitioned the 30,247 loci of the Wang subset into blocks of 100 consecutive loci and discarded every other block, so that 151 blocks were retained of 100 loci each. Since the individual loci are ∼500 bp in length and separated by at least 2 kb, this leaves a distance of at least 0.25 Mb between different blocks, and we can reasonably assume that any effect of linkage between blocks of loci this far apart is negligible compared to that within blocks. In the Hutter subset, the distance between consecutive loci is on average ∼50 kb, and we retained these 378 loci to enable estimation of the D. melanogaster population size parameters. To examine the effect of linkage, we analyzed this reduced data set in two ways to compare the results: (i) assuming that loci are independent; and (ii) accounting for any linkage between loci within blocks, i.e., accounting for the bulk of the linkage in the data. In case (i), the model selection procedure described in Model selection was carried out on the reduced data set. As was the case for the full data, the model IIM3 also provided the best fit by far for the reduced data set. The P-values computed as part of the model selection procedure were all <10−42 and are shown in Table 9. The parameter estimates for the best-fitting model, IIM3, are shown in Table 10 and are very close to the estimates obtained from the full Wang and Hey (2010) data (see Table 3). Standard errors of the parameter estimates, based on the Fisher information (computed using the inverted Hessian matrix as described in Confidence intervals for the selected model), are also shown in Table 10 for the reduced data set. As expected, these standard errors are larger than those for the full data set by a factor of approximately except those of the D. melanogaster population size parameters, which are largely unchanged. In case (ii), to account for any linkage within blocks of loci, both the model selection procedure and the computation of standard errors were performed using theoretical results for composite marginal likelihoods. The hypothesis tests in the model selection procedure were carried out using result 3.5 and approximation 3.6 of Jesus and Chandler (2011), by which the null distribution of the composite likelihood-ratio test statistic is approximated by a scaled and shifted distribution (see also the comments regarding the distribution of the independence likelihood-ratio test statistic in Chandler and Bate (2007), pp.170–171). The P-values obtained in this way for the tests in the model selection procedure are shown in Table 9. As expected, these P-values are not as small as those obtained when ignoring linkage, and in fact they differ by many orders of magnitude. Nevertheless, these P-values are all still smaller than and the model IIM3 still gives by far the best fit for the reduced Wang and Hey (2010) data (note however that, to the best of our knowledge, it has not been established in the literature whether the approximate null distribution used for the composite likelihood-ratio test statistic is still conservative in the case of tests involving parameters on the boundary, although this would seem plausible). Standard errors of the parameter estimates of the IIM3 model were computed by obtaining an empirical estimate of the inverse of the Godambe information matrix using the method for clustered data described in Chandler and Bate (2007): the covariance matrix of the score vector (the vector of partial derivatives of the log-likelihood) was estimated bywhere the vector is the score of the jth block of loci, evaluated at the MCLE, and the sum is over all blocks; an estimate of the inverse of the Godambe information matrix (also referred to as the “robust” variance estimator) was then computed aswhere H is the Hessian matrix. The resulting standard errors are shown in the right-hand column of Table 10. It is seen that, on average, the standard errors based on the Fisher information account for ∼80% of the uncertainty given by the robust standard errors, though this percentage is different for different parameters. The strongest impact is on the standard error of (the “current size” parameter of D. simulans), for which the standard error ignoring linkage is only 59% of that which does account for linkage between loci within blocks—one would indeed expect the impact of linkage to be strongest on the standard errors of parameters relating to more recent events, as a shorter time allows less opportunity for recombination between loci (no such effect is seen on the standard error of as we continued to treat the Hutter subset as independent loci). To compute standard errors of the parameter estimates obtained from the full Wang and Hey (2010) data, it may be possible to obtain an estimate of the covariance matrix of the score vector, and hence of the Godambe information matrix, by using the method of “window subsampling” (Heagerty and Lele 1998) whereby the data are divided into pseudo-independent subregions, but this would require further investigation. An alternative method to account for linkage disequilibrium is by means of a parametric bootstrap (for example, Lohse et al. 2016), but this is computationally intensive and the results will inevitably depend on the recombination rate assumed, and on any other assumptions made such as homogeneity of the recombination rate along the genome.
The robust standard errors in the right-hand column of Table 10 were derived by accounting for linkage while assuming that all our other assumptions hold. If the latter is not the case, then the individual factors in Equation 13 may be misspecified so that their product no longer defines a composite marginal likelihood. Instead, the derivative of its logarithm can be regarded as an “estimating function” and the corresponding statistical theory applied. In that case, our robust calculations of standard errors and P-values in (ii) above still apply (Jesus and Chandler 2011, Section 3), so that the results in the right-hand columns of Table 9 and Table 10 are still valid. Thus the differences between the left- and right-hand columns of standard errors and P-values in Table 9 and Table 10 should be interpreted as upper bounds on the impact of linkage, since these differences may in part be due to other forms of model misspecification, including model misspecification from any of the potential sources discussed above: inaccurate estimates of the relative mutation rates, misspecification of the mutation model, misspecification of the demographic model, and intralocus recombination.
We are indebted to Yong Wang, Jody Hey, Nick Barton, and Konrad Lohse for kindly providing the Drosophila DNA sequence data. We thank Ziheng Yang for some valuable discussions of the work presented in this article. We are grateful to Yun S. Song, Konrad Lohse, and an anonymous reviewer for their constructive comments and helpful suggestions which led to a much-improved manuscript. This research was supported by the Engineering and Physical Sciences Research Council (grant number EP/K502959/1).
Appendix A: Proof of Equation 8:
This proof has three parts. Part (i) proves the result under two assumptions: (a) has three strictly negative eigenvalues and one zero eigenvalue, all of them real; and (b) is diagonalizable. Part (ii) proves assumption (a). Part (iii) proves assumption (b). To simplify the notation, we denote by throughout the proof.
Consider the continuous-time Markov chain defined by the matrix Let the entry of the matrix be the probability that the process is in state j at time t into the past, given that the process starts in state i. can be calculated by solving the following initial value problem:where is the four-by-four identity matrix. Under the assumptions that is diagonalizable and that its eigenvalues are real, the solution to this initial value problem is given by:where denotes the diagonal matrix containing the real eigenvalues of and is the matrix of left eigenvectors of Note that is the probability that the process has reached coalescence by time t, if it started in state i. In other words, it is the CDF of where is the row vector of and the fourth column vector of Differentiating, we get the PDF:If we denote the eigenvalue equal to zero by and the remaining eigenvalues are strictly negative, this PDF can be written as a linear combination of exponential densities:(A1)where for
As is given by Equation 3, its characteristic polynomial, is of the form where is the three-by-three upper-left submatrix of that is:Thus the eigenvalues of are the solutions to Consequently, one of them is zero and the remaining three eigenvalues are also eigenvalues of
Now consider the similarity transformationBecause is a symmetric matrix, its eigenvalues are real. Therefore, all the eigenvalues of are real (a similarity transformation does not change the eigenvalues). is also a negative definite matrix, since its first, second, and third upper-left determinants are respectively negative, positive, and negative. Hence its eigenvalues are all strictly negative, and so are the eigenvalues of Hence has one zero eigenvalue and three real, strictly negative eigenvalues
Being a symmetric matrix, has three independent eigenvectors. A similarity transformation preserves the number of independent eigenvectors, so has three independent eigenvectors as well. We denote by the matrix whose rows are the left eigenvectors of
By definition, any left eigenvector of satisfies the system of equations where . The first three linear equations of this system are identical to for and which is solved by So this implies that, for any row vector in that has as its first three elements will solve the first three equations of the system, whatever the value of If that vector will be an eigenvector of because it also solves the fourth equation of the system:for . For the case of a row eigenvector is Collecting these row eigenvectors in a single matrix, we get So,If the matrix can be shown to be invertible, then is diagonalizable. This will be the case if the system can only be solved by . Now since the three-by-three upper-left submatrix of is full-ranked, is a necessary condition for But then from the last equation of the system. Thus we have shown that is diagonalizable.
Appendix B: Complementary Results for the Distribution of the Time Until Coalescence Under Unidirectional Gene Flow, and in the Absence of Gene Flow
Migration from Subpopulation 1 to Subpopulation 2 Backward in Time (M1 > 0, M2 = 0)
Using the derivation procedure described in Distribution of the time until coalescence under unidirectional gene flow, and in the absence of gene flow, we find that:As a result, the PDF of the coalescence time of a pair of sequences under the IIM model, is again given by Equations 9 and 10, now withand
Distribution of the Time Until Coalescence Under an IIM Model with M1 = M2 = 0
In this case, the IIM model reduces to a complete isolation model where both descendant populations may change size at time into the past. The distribution of the absorption time corresponding to will now be either exponential, if both sampled sequences are from the same subpopulation (i.e., for or coalescence will not be possible at all until the ancestral population is reached, if we take a sequence from each subpopulation (i.e., if i = 3). It follows that the PDF of the coalescence time of a pair of sequences in the IIM model is given by Equations 9 and 10 withand
Supplemental material is available online at www.genetics.org/lookup/suppl/doi:10.1534/genetics.116.188060/-/DC1.
Communicating editor: Y. S. Song
- Received May 11, 2016.
- Accepted January 25, 2017.
- Copyright © 2017 Costa and Wilkinson-Herbots
Available freely online through the author-supported open access option.
This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.