Q_ST measures the differentiation of quantitative traits between populations. It is often compared to F_ST, which measures population differentiation at neutral marker loci due to drift, migration, and mutation. When Q_ST is different from F_ST, it is usually taken as evidence that selection has either restrained or accelerated the differentiation of the quantitative trait relative to neutral markers. However, a number of other factors such as inbreeding, dominance, and epistasis may also affect the Q_ST – F_ST contrast. In this study, we examine the effects of dominance, selection, and inbreeding on Q_ST – F_ST. We compare Q_ST with F_ST at selected and neutral loci for populations at equilibrium between selection, drift, mutation, and migration using both analytic and simulation approaches. Interestingly, when divergent selection is acting on a locus, inbreeding and dominance generally inflate Q_ST relative to F_ST when they are both measured at the quantitative locus at equilibrium. As a consequence, dominance is unlikely to hide the signature of divergent selection on the Q_ST – F_ST contrast. However, although in theory dominance and inbreeding affect the expectation for Q_ST – F_ST, of most concern is the very large variance in both Q_ST and F_ST, suggesting that we should be cautious in attributing small differences between Q_ST and F_ST to selection.

---

TWO measures of population differentiation are commonly used to assess the genetic structure of populations. The first of these is F_ST, which measures the differentiation of neutral markers between populations. F_ST is generally calculated using information on allele frequencies at neutral loci such as microsatellites and is therefore an estimate of the amount of differentiation that has occurred due to drift acting on the (finite) populations. F_ST is well characterized, with a vast literature on how factors such as mutation, migration, population size, population history and population subdivision affect it (WHITLOCK and MCCAULEY 1999) and how best to calculate and estimate it (WEIR and COCKERHAM 1984; WEIR and HILL 2002). In addition, numerous studies have calculated F_ST for natural populations to investigate, for instance, effective population size, migration rates, and divergence times as well as examine population differentiation itself.

The second widely used measure of population differentiation is that of quantitative traits, termed Q_ST. Because knowledge of the underlying genetic loci (of which there may be dozens) is generally unknown, quantitative traits are measured on individuals and Q_ST is calculated in terms of variation in those traits between and within populations. Assuming an additive model, WRIGHT (1951) derived expressions for the neutral expectation of variance between and within populations, which LANDE (1992) noted may be translated into an F_ST equivalent to measure the differentiation of a quantitative trait. The origin of the term Q_ST is attributed to an article the following year (SPITZE 1993).

In addition to the factors affecting neutral markers, quantitative traits may be under selection. In the absence of selection, the differentiation of an additive trait is equal to the differentiation of neutral markers (that is, Q_ST = F_ST), as both sets of loci are affected equally by drift (LYNCH and SPITZE 1994; LATTA 1998). However, if selection has accelerated the divergence of quantitative traits between the populations (for instance, different phenotypes are selected for in different environments, leading to local adaptation of populations), Q_ST will exceed F_ST. Alternatively, Q_ST will be less than F_ST if quantitative traits are restrained by balancing selection (for instance, if natural selection favors the same phenotype in different populations). The comparison of Q_ST and F_ST, when measured for the same populations (CRNOKRAK and MERILA 2002; WHITLOCK 2008), can therefore be used to infer the type and strength of selection on the quantitative trait. Many dozens of studies have now used the Q_ST – F_ST contrast to predict the effect of selection (see references in LEINONEN et al. 2008), and the conclusion from two meta-analyses is that Q_ST generally exceeds F_ST (MERILA and CRNOKRAK 2001; LEINONEN et al. 2008), supporting a general role for divergent selection driving the differentiation of quantitative traits.

There are two problems when comparing Q_ST and F_ST values to elucidate the role of selection. The first is related to how F_ST and particularly Q_ST are measured and the intrinsic biases in calculating them using samples of natural, finite populations, despite a number of sophisticated methods available to calculate both (O'HARA and MERILA 2005). For example, WHITLOCK (2008, p. 1894) asserts that Q_ST is "difficult to estimate accurately and precisely," while O'HARA and MERILA (2005, p. 1337) conclude from simulations that "the precision of the Q_ST estimates—irrespective of the estimation method used—is very low." Thus, when comparing Q_ST and F_ST, it is not particularly clear how to best assess the significance of an observed difference or correct for biases in their estimates.

The second area of concern when comparing Q_ST and F_ST is in the nature of the quantitative trait itself. In contrast to neutral loci, alleles at quantitative trait loci may exhibit dominance, may interact with other loci via epistasis or extended linkage disequilibrium, or may respond differently than neutral alleles to the effects of mutation. A reasonable research effort has recently been directed toward how these factors affect Q_ST and Q_ST – F_ST, particularly investigating the changes to Q_ST – F_ST when F_ST is calculated using allele frequencies at the quantitative locus itself. Models of epistasis suggest that Q_ST will be depressed by interactions between loci (WHITLOCK 1999; LOPEZ-FANJUL et al. 2003), masking the effect of divergent selection on the trait. Additionally, because multiple loci contribute to a quantitative trait and allele effects sum across loci, mean values of a quantitative trait may not be different between populations even if divergence at individual loci is high. Conversely, low levels of divergence at the allelic level may have significant impacts on the quantitative trait means (LATTA 1998; LE CORRE and KREMER 2003). Thus, the signatures of divergent or uniform selection may be masked if allele frequencies at the quantitative trait are measured (or modeled), rather than the trait values themselves.

Using an island model and a pure drift model respectively, Goudet and colleagues (GOUDET and BUCHI 2006; GOUDET and MARTIN 2007) and Lopez-Fanjul and colleagues (LOPEZ-FANJUL et al. 2003, 2007) have extensively investigated the impact of dominance on Q_ST – F_ST for a neutral trait. For the island model at equilibrium between migration and drift, Q_ST tends to be less than F_ST calculated at the same locus (GOUDET and BUCHI 2006; GOUDET and MARTIN 2007), while if populations diverge via drift from an infinite population, Q_ST tends to be larger than F_ST when the recessive allele is at high frequency (LOPEZ-FANJUL et al. 2007). These results led GOUDET and BUCHI (2006) to suggest that dominance may mask the effects of divergent selection when Q_ST is compared to F_ST calculated from neutral markers. To date, however, there has been no exploration of the combined effect of selection, dominance, and inbreeding on Q_ST and Q_ST – F_ST. In this article, we examine Q_ST values at equilibrium to address whether divergent selection on quantitative traits may indeed be hidden by the effects of dominance. F_ST is calculated both at the selected locus and at neutral loci unlinked to the locus under selection. The numeric results from the analytic approach are checked and extended by simulations, in which multiple loci under divergent and balancing selection are also investigated. The following sections are arranged into (i) the derivation of and results from recursion equations used to calculate F_ST and Q_ST, (ii) a description of and results from an individual-based simulation model, and finally (iii) a general discussion. Parameters and terms for the following F_ST and Q_ST models are summarized in Table 1.

View this table: In this window In a new window   TABLE 1 Parameters of the FST and QST models

 

We consider the simple case of two populations, X and Y, linked by migration. Diploid individuals within a population outcross or self to form the next generation, and some of these offspring migrate so that a proportion m_i of individuals in population i (i = X, Y) are migrants. We assume that migrants are indistinguishable from residents in reproduction, generations are discrete, and population sizes and structure are constant through time.

Let N_i be the number of individuals in population i (i = X, Y) and let the probability of selfing in both populations be f while the probability of parents outcrossing is (1 – f). Selfing and outcrossing are assumed to be independent, and so the number of selfed and outbred offspring per parent is independently binomially distributed with means f and (1 – f), respectively.

Population differentiation at neutral markers (F_ST):

To derive an explicit expression for F_ST, we construct recursion equations for coefficients describing the similarity between and within individuals in a subdivided population. These recursion equations may then be used to solve for F_ST at equilibrium. Full details of the derivation are shown in APPENDIX A. If we make the assumption that migration rates and population sizes are equal in populations X and Y (m_X = m_Y = m and N_X = N_Y = N), the explicit expression for F_ST is

(1)

where the effective population size N_e is defined in APPENDIX A.

If migration rates are small, F_ST is expected to be

(MARUYAMA and TACHIDA 1992; WANG 1997). Interestingly, using overestimates the value of F_ST found in (1). This is likely to be the result of the small number of populations we are considering. F_ST increases with increasing selfing rate and rises sharply with decreasing migration rates and population sizes.

Population differentiation at quantitative traits (Q_ST):

Following previous work (GOUDET and BUCHI 2006), we consider a single locus with two alleles, A and B, at frequencies p_i and 1 – p_i, respectively, in population i (i = X, Y). We set the genotypic value G of an individual to be –1 (G_AA), 1 (G_BB), and d (G_AB) for genotypes AA, BB, and AB, respectively; we assume d > 0 so that allele B is dominant to allele A. Genotype frequencies r of AA, AB, and BB in population i (i = X, Y) with selfing rate f are shown in Table 2.

View this table: In this window In a new window   TABLE 2 Genotypic values, population frequencies, and selection coefficients for a single locus with two alleles under divergent linear selection

 
Following HEYWOOD (2005), we assume that selection acts via a linear fitness function, so that the fitness of an individual in population i with genotypic value G_jk is

(2)

where s_i is the population-specific selection coefficient. The mean fitness of population i is

and the relative fitness of an individual becomes

We assume that selection dominates drift , and so drift is not explicitly modeled in our equations.

Following selection and reproduction, the allele frequencies in the next generation (t) are

Finally, after migration, the frequency of allele A in the next generation for population X becomes

(3)

A similar expression for p_Y(t) can be found by switching X and Y subscripts in the above equation.

To find the equilibrium frequencies, we solve p_X(t) = p_X(t–1) and p_Y(t) = p_Y(t–1). If the parameters are known, the equilibrium values of p_X and p_Y may be easily found. However, it is not possible to find a general expression for p_X and p_Y at equilibrium. Even solving for a reduced set of equations, when we know some of the parameters of the model, proves to be difficult. For a given set of parameter values, the expressions for p_X (3) and p_Y can therefore be iterated over generations until there is no change in allele frequencies and the populations reach equilibrium. A polymorphic equilibrium with both alleles present in both populations requires that selection coefficients act in opposite directions in the two populations, that there is migration between the populations, and additionally that . For example, gives p_X = 0.3823 and p_Y = 0.6181, while and lead to p_X = p_Y = 1 as the high migration rate from population Y to X causes the positive selection in population Y to dominate both populations. Interestingly, if f = 0, d = 0, m_X = m_Y, and s_X = –s_Y, then p_X + p_Y = 1, and both alleles are maintained in populations X and Y [that is, ].

The expression for Q_ST including selfing rate f is

(4)

(BONNIN et al. 1996), where V_B is the variance between populations and V_A is the additive variance within populations. For a single quantitative locus with two alleles, GOUDET and BUCHI (2006) derived a general expression for the trait variance among n populations. For our case of two populations and genotypic values of –1, d, and 1 (see Table 2), this expression becomes

(5)

Similarly, for two populations the expression for the single-locus additive variance V_A within the populations (TEMPLETON 1987; LYNCH and WALSH 1998; GOUDET and BUCHI 2006) becomes

(6)

As demonstrated by GOUDET and BUCHI (2006), the expression for Q_ST reduces to the definition of F_ST (WRIGHT 1951) at the locus when there is no dominance,

(7)

where for the case of two populations X and Y

and

The value of Q_ST at equilibrium may be found by substituting equilibrium allele frequencies and known population parameters into our expression for Q_ST (Equation 4). For example, solving p_X(t) = p_X(t–1) and p_Y(t) = p_Y(t–1) for gives equilibrium allele frequencies and Q_ST = 0.9570, while gives equilibrium allele frequencies and Q_ST = 0.0555.

Effects of dominance, inbreeding, and selection on Q_ST:

When calculated using equilibrium allele frequencies, Q_ST is a strictly increasing function of inbreeding, dominance, and the strength of divergent selection and decreases with increasing migration between populations. Note that equilibrium allele frequencies are themselves functions of the inbreeding, dominance, and selection coefficients. Under divergent selection, dominance has the greatest impact on Q_ST when inbreeding is low. For example, Figure 1 demonstrates that when , equilibrium Q_ST increases dramatically with increasing inbreeding and also increases with increasing dominance at the locus. This nonadditive interaction between dominance and inbreeding was also noted by GOUDET and BUCHI (2006) for a neutral locus.

View larger version (7K): In this window In a new window Download PPT slide   FIGURE 1.— Equilibrium QST analytic values when , shown as a function of increasing selfing rate for three different dominance coefficients.

 
Very small changes in the selection coefficients have a much larger impact on Q_ST than do equivalent changes to the dominance coefficient or selfing rates. This suggests that even a low level of divergent selection will overwhelm the effects of dominance or inbreeding on Q_ST. For example, we investigated the balance between increasing the dominance coefficient and decreasing the selection coefficient to maintain the same value of Q_ST. Assuming a migration rate of , a population size of N = 1000, and an inbreeding coefficient , the initial values for the selection coefficients (s_X = –s_Y) were chosen so that [calculated from (4) and (1), respectively] when d = 0, that is, when selection is approximately the same order of magnitude as drift. For , s_X = (0.0016, 0.0015, 0.0013) to maintain Q_ST (= F_STN) = 0.1169. So an increase in the dominance coefficient from to 1 is offset by a decrease in the strength of selection of only 0.0002. Therefore, although we have seen that dominance and inbreeding indeed influence equilibrium Q_ST values, divergent selection is a much more significant force in determining the level of quantitative trait differentiation.

Q_ST – F_STQ contrast:

For a single locus with two alleles, the value of Q_ST – F_ST is found by substituting equilibrium allele frequencies into Equations 4–6 for Q_ST and Equation 7 for F_ST. The full expression for the difference is shown in APPENDIX B. For a neutral quantitative locus, Q_ST = F_STQ (F_ST for the quantitative locus) only when there is no dominance (GOUDET and BUCHI 2006). If alleles at the locus do not act additively, the difference between Q_ST and F_STQ may be positive or negative, depending on the allele frequencies. For example, in the one-locus, two-allele model with dominance, if the recessive allele is rare in both populations, then Q_ST will be less than F_STQ, and vice versa if the recessive allele is at high frequency in both populations (GOUDET and BUCHI 2006). In general, the expected value for the difference between Q_ST and F_STQ is dependent on assumptions made about the distribution of neutral allele frequencies for a quantitative trait and the mode of population subdivision (see discussion in GOUDET and BUCHI 2006, GOUDET and MARTIN 2007, and LOPEZ-FANJUL et al. 2007 and references therein).

When a dominant quantitative locus is under divergent selection, however, the range of possible allele frequencies in the two populations will be more restricted than in the neutral case. In particular, we no longer expect the frequency of the recessive allele to be positively correlated between populations, as the allele favored in one population is selected against in the other. Therefore, in contrast to the neutral case, it is unlikely that the same allele will be rare in both populations.

To test the impact of dominance with divergent selection on the difference between Q_ST and F_STQ, we iterated the recursion equations for p_X (Equation 3) and p_Y to find equilibrium allele frequencies for 10,000 random combinations of biologically feasible parameter values [ while ] and calculated Q_ST and F_STQ at equilibrium. The distribution of Q_ST – F_STQ for the 10,000 random combinations is shown in Figure 2. We additionally restricted parameters to biologically realistic values [, , and ], which affected the magnitude of Q_ST – F_STQ, but the total frequency of Q_ST – F_STQ values less than zero was similar (data not shown). Significantly, these results demonstrate that for a range of feasible migration, selection, inbreeding, and dominance coefficients Q_ST will most likely exceed F_STQ when both are calculated using equilibrium allele frequencies at the locus under selection. Therefore, we conclude that dominance is unlikely to hide the signature of divergent selection on the Q_ST – F_ST contrast, and in fact the inflation of Q_ST relative to F_ST due to dominance will make it easier to detect divergent selection on the quantitative trait.

View larger version (12K): In this window In a new window Download PPT slide   FIGURE 2.— Distribution of equilibrium QST – FSTQ analytic values, calculated from recursion equations, for 10,000 replicates, using population parameter values sampled from biologically feasible ranges, where and .

 
To assess the impact of individual factors, we calculated the correlation of d, f, s_X, s_Y, m_X, and m_Y with Q_ST – F_STQ and with for the 10,000 random combinations of parameter values described above. Interestingly, there is a significant (P < 0.001) positive correlation between d and both Q_ST – F_STQ and (correlations 0.3449 and 0.5804), a significant negative correlation between Q_ST – F_STQ and f, m_X, and m_Y (correlations –0.3216, –0.2806, and –0.3211), and a significant negative correlation between and f (correlation –0.4750). These correlations indicate that, regardless of the strength of selection and the rates of migration and selfing, as dominance increases the difference between Q_ST and F_STQ also increases. In addition, with other factors constant, increasing the selfing rate or the migration rates decreases the difference between Q_ST and F_STQ. For loci under divergent selection, then, we expect the largest difference between Q_ST and F_STQ when selfing rates are small and the dominance effect is large. This result agrees with the conclusion for neutral loci from GOUDET and BUCHI (2006): the effect of dominance becomes smaller as inbreeding increases.

We also iterated allele frequencies to equilibrium for a range of dominance and inbreeding coefficients given . Figure 3A shows that, as expected from the correlation described above, there is a positive relationship between and the dominance coefficient. Interestingly, however, for the given parameter values, increases as the selfing rate approaches 0.3 and then decreases (Figure 3B). Both the numerator and the denominator in the expression for Q_ST – F_STQ involve f⁴ terms (APPENDIX B), so it is unsurprising to see such a nonlinear effect on the difference. When we plot Q_ST – F_STQ instead of , the relationship with d and with f looks very similar to those seen in Figure 3, A and B, respectively. These graphs indicate that in fact it may not be a low but a moderate level of inbreeding, coupled with strong dominance, that is likely to give the largest positive Q_ST – F_STQ difference.

View larger version (18K): In this window In a new window Download PPT slide   FIGURE 3.— Equilibrium analytic values, calculated using recursion equations, as a function of (A) dominance coefficients (d) and (B) selfing rate (f). Other parameters are set at , , and .

 

Nonequilibrium Q_ST – F_STQ:

Populations may not be at equilibrium when they are sampled, for example, when populations are adapting to a new selection pressure. Therefore, we used our recursion equations (3) to investigate the change in allele frequencies and in Q_ST – F_STQ as two populations adapt to a new selection pressure and stabilize toward equilibrium values. We found that in general, allele frequencies and Q_ST moved initially very rapidly toward equilibrium values, but that the rate of change slowed considerably when populations approach the equilibrium values.

The initial allele frequencies determine how the value of Q_ST – F_STQ changes over time. For example, if , , and , then our expected equilibrium values for p_X, p_Y, and Q_ST – F_STQ are 0.2351, 0.8937, and 0.0249, respectively. The average equilibrium frequency of allele A across populations is 0.5644. Figure 4 demonstrates the change in the value of Q_ST – F_STQ over time when the initial frequencies are (bottom line) and (top line). These graphs are typical of the change in Q_ST – F_STQ when initial (bottom line) and initial (top line). If initial allele frequencies are approximately equal and are less than the average frequency of allele A at equilibrium, Q_ST – F_STQ will initially decrease and become more negative and then increase toward the equilibrium Q_ST – F_STQ. However, if initial allele frequencies are greater than the average equilibrium frequency, Q_ST – F_STQ will always be positive.

View larger version (7K): In this window In a new window Download PPT slide   FIGURE 4.— Change in the value of QST – FSTQ over time, calculated from recursion Equations 3, when the initial frequencies are (bottom line) and (top line). Other parameters are , , and . Expected equilibrium values are pX = 0.2351, pY = 0.8937, and QST – FSTQ = 0.0249.

 
Interestingly, these results suggest that when divergent selection pressure is first applied, the value of Q_ST – F_STQ can become progressively more negative. In addition, a negative value of Q_ST – F_STQ may persist for a substantial number of generations before a positive value is reached. The speed of the transition from negative to positive Q_ST – F_STQ is predominantly determined by the strength of selection. As expected, however, Q_ST is a strictly increasing function of time, so when comparing Q_ST to F_ST measured at neutral loci, the value of Q_ST – F_STN will always be positive and will increase as Q_ST approaches its equilibrium value.

Our numeric results from the analytic approach suggest that dominance is likely to enhance the difference between Q_ST and F_ST for a quantitative trait under divergent selection, and therefore selection will be more easily detected. To test this prediction, and to incorporate the effect of drift, we simulated the genotypes of diploid individuals in two populations linked by migration. For a number l of unlinked loci, genotypes are initially randomly assigned to individuals in each population by assuming that all a alleles at a locus are equifrequent. A number of loci contribute to the quantitative trait values and are therefore selected, while the remainder act as neutral marker loci. Individuals randomly mate or self to produce offspring of the next generation. Offspring migrate with probability m to the other population. These offspring then join the parent pool for the following generation, and the process of mating and migration iterates through generations. This design closely follows the analytic model, with mating followed by migration, discrete generations, and stable population sizes over time.

We ran our simulation program for a number of different scenarios to investigate the impact both divergent and balancing selection have on the Q_ST – F_ST contrast. For each of five simulation sets (A–E), we considered six designs combining high and low levels of inbreeding and low, medium, and high dominance coefficients. In addition, we assume equal migration rates and population sizes (m_X = m_Y = m = and N_X = N_Y = N = 1000). For all loci, mutation from one allele to any other occurs at equal, relatively high, probability µ to avoid fixation of alleles due to drift. Under each of the six designs, simulations were run for 2000 generations and replicated 100 times, and Q_ST and F_ST values were averaged over replicates. In the following sections, we first describe the methods used to calculate Q_ST and F_ST in our simulations and then present the results from the five simulation sets (A–E).

Calculating F_ST and Q_ST in simulations:

Calculating F_ST:

The simulation calculates F_ST separately at neutral loci (F_STN) and at the loci under selection (F_STQ) to allow comparison of the Q_ST – F_ST contrast both at selected and at neutral loci. Using allele frequencies in each population following migration, F_ST at each generation is calculated as

(8)

(WRIGHT 1951), where

(9)

(10)

and

[recall that i indexes the population (i = X, Y), l is the number of loci, and a is the number of alleles at each locus]. p_ijk is the allele frequency of allele k at locus j in population i and p_jk is the average allele frequency of allele k at locus j across populations X and Y. Note the small change of notation from our one-locus, two-allele model (Table 2); we number allele frequencies at locus j in population i as p_ij1, p_ij2, p_ij3, ... , for alleles A₁, A₂, A₃, ... , instead of frequencies p and (1 – p) for alleles A and B.

Calculating Q_ST:

For Q_ST, a small number of diallelic loci are assumed to contribute to the quantitative trait, and genotypic values are assigned by summing over loci. For each of l loci, marginal genotypic values for AA, AB, and BB are –1/l, d/l, and 1/l, where d is the level of dominance. We scale by the number of loci l so that the minimum and maximum genotypic values (–1 and 1) remain constant. On the basis of the genotypic value of an individual, selection is then applied to transform genotypic values into relative fitness. Individuals with higher fitness are more likely to be selected as parents for the next generation. If only one locus is under selection, we may use a linear fitness function as in the analytic approach (2) and model either divergent (s_i of different sign) or uniform (s_i of the same sign) selection. For one or more loci, we also model optimal selection acting on the total genotypic value over loci. Following TURELLI (1984) and LE CORRE and KREMER (2003), we define the absolute fitness of an individual as

(11)

where G_iOPT is the local genotypic optimum in population i and > 0 is the strength of selection. As increases, the strength of selection decreases. Uniform and divergent selection are modeled by setting similar or different values for the local genotypic optima. Note that for the optimal selection regime, more than two populations and more than two alleles at the loci under selection may easily be included in the simulation design, but we restrict our simulations to the two-population, two-allele case for simplicity and comparison with the analytic approach.

After a large number of generations, the within- and between-population variances needed to calculate Q_ST approach asymptotic values. Variances are calculated either from population allele frequencies or from observed genotypic values of individuals in each population. For example, using trait values, V_A may be estimated as twice the covariance between offspring and midparent genotypic values or four times the covariance of half sibs [provided the level of inbreeding is not large and the dominance variance is small (COCKERHAM and WEIR 1984)]. Similarly V_B may be calculated from the difference between the trait means of the populations. However, for the two-allele case we can use the exact expressions for V_A and V_B calculated from allele frequencies [(6) and (5), respectively]. For more than one locus, this expression becomes

(12)

where we sum the between- and within-population variances for each locus across the l loci before calculating Q_ST.

We compared the expectations for quantitative trait differentiation when Q_ST is calculated from (i) allele frequency information [using (5) and (6)] or (ii) phenotypic measures and covariances between parents and offspring as described above. We ran 1000 replicates of our simulation model for the simple case of no inbreeding and no dominance, with selection coefficients and migration rates chosen randomly from their biologically feasible ranges [ and ]. The number of individuals in each population was set to 1000, and the simulation was run for 2000 generations. As expected when d = 0, Q_ST and F_ST values were identical when both are calculated using allele frequencies. The means of Q_ST calculated from allele frequencies (0.4444) and Q_ST calculated from phenotypic values (0.4461) were very similar, as were their variances (0.1376 and 0.1375, respectively). However, calculating Q_ST from phenotypic values introduces additional sampling error to the value. For example, as there is no dominance, the correlation between F_ST and Q_ST calculated from allele frequencies is 1, while the correlation of F_ST and Q_ST calculated from phenotypes was 0.9985. Similarly, if we run the simulation under the same conditions but additionally select dominance and inbreeding coefficients randomly from their biologically feasible ranges and 3000 replicates], the correlation between F_ST and Q_ST calculated from allele frequencies is 0.9929, while the correlation between F_ST and Q_ST calculated from phenotypes is 0.9501. Therefore, using phenotypic values to calculate Q_ST would make the contrast between F_ST and Q_ST more variable and any difference between them less likely to detect. Results of the following simulations are therefore presented with values of Q_ST calculated from allele frequencies.

Simulation set A—differentiation via divergent linear selection:

To verify the predictions of our analytic model, we first ran the simulation with one locus under divergent selection using the linear fitness function described in Table 2. Selection strength was set at . Table 3 compares the expected analytic values (exp), for Q_ST, F_STQ (F_ST at the selected quantitative locus), and F_STN (F_ST for nine neutral loci) with the average results from our simulations (sim) for each design. Exp values are calculated using Equations 1 (F_STN exp), 4 (Q_ST exp), and 7 (F_STQ exp) after iterating recursion equations to equilibrium. Sim values are based on final allele frequencies from the simulations and calculated using Equations 8 (F_STN sim, F_STQ sim) and 12 (Q_ST sim).

View this table: In this window In a new window   TABLE 3 Simulation set A: comparison of QST and FST values under divergent linear selection

 
In general, the simulated values agree well with those from our analytic models. There are some surprisingly large differences between the expected and the simulated values for F_STN for high rates of selfing. This appears to be due to a large proportion of replicates fixing different alleles in the two populations across many of the neutral loci (and hence F_STN 1). Therefore, despite explicitly including drift in our expression for F_ST (Equation 1), the combination of high rates of selfing and low migration has reduced N_e further than we predicted. Setting a higher mutation rate for neutral loci for these particular designs would certainly have prevented such loss of alleles and this elevation in F_STN simulated values.

Q_ST – F_STQ:

The average simulation results for Q_ST – F_STQ are very similar to the expected values, and, as expected, all averages are very close to zero at low dominance (d = ). Figure 5A demonstrates the close correlation between average expected (solid lines) and simulated values (dashed lines) at the three levels of dominance when the selfing rate is low. However, the variation in the value of Q_ST – F_STQ between replicates in many cases exceeds its expectation (data not shown). In addition, we can see that in many cases a high proportion of individual replicates may have negative Q_ST – F_STQ values. For example, for the medium dominance, low-inbreeding design we expect Q_ST – F_STQ = 0.0202, and although the average Q_ST – F_STQ from simulations is close to expectation, nearly a quarter of replicates have Q_ST – F_STQ < 0.

View larger version (12K): In this window In a new window Download PPT slide   FIGURE 5.— Comparison of QST and FSTQ expected (exp) and simulated (sim) values across (A) three levels of dominance (d) and (B) two levels of selfing (f) under divergent linear selection, when f = . See Simulation set A—differentiation via divergent linear selection for details of the simulation design.

 
To investigate the impact of the large variation of Q_ST and F_ST further, we ran a set of 1000 simulations. Dominance, inbreeding and selection coefficients, and migration rates were chosen randomly from their biologically feasible ranges and population sizes fixed (N_X = N_Y = 1000). Figure 6 demonstrates that the distribution of Q_ST – F_STQ differs significantly from the distribution generated from our analytic model (Figure 2). Thus, although we generally expect Q_ST > F_STQ on the basis of our results from the recursion equations, the simulations indicate that in real populations the effect of sampling (i.e., drift) may hide any elevation of Q_ST due to dominance. This result agrees with the discussion in GOUDET and BUCHI (2006) and GOUDET and MARTIN (2007) regarding the large variance of Q_ST and F_STQ for a neutral quantitative locus.

View larger version (12K): In this window In a new window Download PPT slide   FIGURE 6.— Distribution of QST – FSTQ simulated values for 1000 replicates, calculated by sampling population parameters from biologically feasible ranges, where , , NX = NY = 1000, and µ = .

 
As predicted from Figure 2 and the analytic expectations, Table 3 and Figure 5A indicate that there is a positive relationship between Q_ST – F_STQ and the dominance coefficient: within each of the two levels of selfing there is a general trend for increased difference between Q_ST and F_STQ as dominance coefficients increase. This supports our earlier conclusion that dominance increases the likelihood that divergent selection will be detected.

Higher inbreeding levels tend to reduce the Q_ST – F_STQ contrast in both the simulated and the expected analytic values (Figure 5B; plotted for the medium dominance level d = ), although it is difficult to compare these results to Figure 3B because only two selfing rates are modeled. There is no significant difference between the Q_ST – F_STQ contrasts for high, medium, and low mutation rates (data not shown).

Q_ST – F_STN:

Table 3 also shows the value of Q_ST – F_STN for each design, averaged over the 100 replicates. These contrasts are large due to the strong selection coefficients used in the simulations and hence the high Q_ST values compared to neutral marker F_ST. In a meta-analysis of 55 empirical studies comparing Q_ST and F_STN, the average difference between Q_ST and F_STN was 0.12 (LEINONEN et al. 2008), suggesting that on average selection is weak in reality. However, 6 of 62 studies (of which 4 of 55 were included in the meta-analysis) reported Q_ST > 0.8, and of these 3 were also associated with relatively low F_STN values (F_STN < 0.25); therefore such high levels of divergence are not impossible.

Despite the very large average values of Q_ST – F_STN, the large variance in values for Q_ST and F_STN causes the Q_ST – F_STN contrast for simulated values to be negative in some replicates. In particular, designs with a high selfing rate have a reasonable proportion (up to 4%) of negative Q_ST – F_STN values. Interestingly, in a separate set of simulations with a much higher migration rate (m = ), the proportion of negative Q_ST – F_STN values was much larger. In cases where high migration was paired with high selfing rates, even the average Q_ST – F_STN over 100 replicates was negative (data not shown). Although this appeared to be a consequence of a large proportion of replicates fixing alleles at all neutral loci (F_STN 1; see above), these results indicate that high migration and drift may in fact cause Q_ST < F_STN even when the quantitative trait is under divergent selection.

We now assess the effect a different selection regime has on the Q_ST – F_ST contrast for the quantitative and neutral loci in simulation sets B–E. As in simulation set A above, we use a combination of low and high inbreeding coefficients and low, medium, and high dominance as the basis for our six simulation designs.

Simulation set B—differentiation via divergent optimal selection:

Simulation set B assessed whether divergent optimal selection at one locus gives similar results to the simulations using divergent linear selection above (set A). Recalling our optimal selection fitness function (Equation 11), we set divergent local optima for populations X and Y (G_XOPT = –G_YOPT = ) with relatively strong selection strength ( = 5). Again each of the six designs was replicated 100 times and F_STQ, F_STN, and Q_ST were calculated from final allele frequencies using Equations 8 and 12. Average values for Q_ST were 0.06 larger than corresponding simulation values under divergent linear selection (Table 3), suggesting that the strength of selection was slightly higher. All average Q_ST – F_STQ and Q_ST – F_STN values are positive (data not shown). In some cases a reasonable proportion of Q_ST – F_STQ replicates are negative but this fails to affect the average Q_ST – F_STQ values for each design. These results indicate that divergent optimal selection elevates Q_ST still further above F_STQ than does divergent linear selection, making it even more likely that selection will be detected. These simulations further support the conclusion that a dominant locus under divergent selection is likely to increase the value of Q_ST relative to F_STQ.

Simulation sets C and D—differentiation via balancing optimal selection:

Simulation sets C and D represent balancing selection acting on one and two loci, respectively. For both sets, the local optima were set at zero and were the same for populations X and Y (G_iOPT = 0), with strong selection toward this optimum ( = 5).

One selected locus:

Table 4 shows the results of 100 replicates for each design for set C (one locus). Values are based on final allele frequencies from the simulations and calculated using Equations 8 (F_STN, F_STQ) and 12 (Q_ST). Under balancing selection, we expect Q_ST – F_STN < 0 because the differentiation of quantitative traits will be restrained relative to neutral markers. Table 4 shows that indeed in all cases average Q_ST – F_STN values are less than zero.

View this table: In this window In a new window   TABLE 4 Simulation set C: comparison of QST and FST values under balancing optimal selection

 
When allele frequencies at the quantitative trait locus are similar in populations X and Y, we expect Q_ST – F_STQ to be approximately equal to zero. Provided allele frequencies are not identical, a low frequency of the recessive allele in both populations gives Q_ST – F_STQ values slightly less than zero, while a higher frequency gives Q_ST – F_STQ values slightly greater than zero. This difference between Q_ST and F_STQ across different frequencies of the recessive allele in populations X and Y is demonstrated in Figure 7, for and (also see similar contour plots in Figure 2, A and B, in GOUDET and BUCHI 2006). The negative or close to zero values of Q_ST – F_STQ under balancing selection (Table 4) therefore suggest a trend for low frequencies of the recessive allele in both populations. Negative Q_ST – F_STQ values are more likely at medium and high dominance coefficients than for low dominance coefficients. This leads to the exciting conclusion that, as with divergent selection, balancing selection is more likely to be detected when alleles at the locus are not additive. Q_ST values are depressed relative to F_ST at the quantitative locus, and when compared to F_ST for neutral loci, the signal of balancing selection (Q_ST – F_STN < 0) becomes stronger.

View larger version (43K): In this window In a new window Download PPT slide   FIGURE 7.— Contour plot of analytic QST – FSTQ values, plotted across the frequency of the recessive allele in populations X and Y, for and . The line of equality pX = pY and the line intersecting it midplot correspond to QST – FSTQ = 0, contours are spaced in intervals, and values to the right top (contours with lighter shading) become more positive.

 

Two selected loci:

We also simulated 100 replicates of the six designs where two loci contribute to the quantitative trait and eight loci are neutral (set D, data not shown). The F_STN values agree well with those from previous sets, as expected for these neutral loci. However, Q_ST and F_STQ values are remarkably high (Q_ST 0.4) compared to values for one selected locus (Table 4; Q_ST 0.1). This high level of differentiation with low migration is explained by a reasonable proportion of replicates fixing different alleles in populations X and Y (see also GOLDSTEIN and HOLSINGER 1992 and LATTA 1998). For example, for two loci with two alleles A and B at frequencies p and q, the same mean (µ) for populations X and Y can be achieved when allele frequencies are similar in both populations (such as and , giving and ) or different alleles are fixed in both populations (such as while , giving and ) (where p_ij is the frequency of allele A at locus j in population i).

The elevation of Q_ST levels has a large impact on the average Q_ST – F_STN values across these designs (data not shown). Under balancing selection we expect Q_ST – F_STN < 0, but in designs with low levels of inbreeding the average contrast is much greater than zero. This indicates that Q_ST calculated using allele frequencies at the quantitative loci may be very large despite the mean values of the trait of interest being similar between populations (GOLDSTEIN and HOLSINGER 1992; LATTA 1998). Clearly because Q_ST is usually measured on trait values rather than calculated from allele frequencies across loci (as the quantitative loci of interest are unlikely to be known), this is a rather academic conclusion. This result does suggest, however, that knowledge of allele frequencies at loci underlying a quantitative trait will not necessarily help determine whether the trait has been under divergent or balancing selection.

Simulation set E—neutral differentiation:

Finally, we replicated the results of GOUDET and BUCHI (2006) by simulating purely neutral differentiation for both the quantitative and the neutral loci, for two loci coding the quantitative trait and eight neutral loci (set E, data not shown). We set the optimum for both populations X and Y equal to zero, but additionally reduced the strength of selection ( = 5 x 10⁹) so that the fitness of every genotype was effectively equal. Because the quantitative trait is neutral, we expect for each design that Q_ST F_STQ F_STN. Our simulations suggest that average Q_ST – F_STQ values are slightly less than zero, although the proportions of Q_ST – F_STQ greater than and less than zero within replicates of each design are approximately equal. This result agrees with that of LOPEZ-FANJUL et al. (2003, 2007), GOUDET and BUCHI (2006), and GOUDET and MARTIN (2007)—when integrating over the surface of all feasible allele frequencies in populations X and Y, the average Q_ST – F_STQ value tends to be negative when d 0, despite the positive and negative areas of parameter space being approximately equal (Figure 7). Thus, when a trait is evolving neutrally, dominance may deflate the value of Q_ST relative to F_ST and therefore may lead us to conclude that balancing selection is restricting the differentiation of the quantitative trait.

Finally of note, the values of F_STN are reassuringly similar across sets A–E, reflecting pure drift in all cases. Note that the neutral loci are unlinked to the quantitative loci and their differentiation is dependent only upon the effective size of the population. In particular, F_STN values are consistently high when rates of selfing are high, reflecting local fixation of alleles due to the small effective population size of these designs.

There has been growing interest in the evolution of quantitative traits in natural populations, particularly in those traits that have important fitness consequences. Empirical evidence from natural populations suggests that quantitative trait divergence frequently exceeds the divergence seen at neutral markers (MERILA and CRNOKRAK 2001; LEINONEN et al. 2008). In addition to the large variances expected for Q_ST and F_ST, and the difficulty in testing any difference between them (O'HARA and MERILA 2005; WHITLOCK 2008), recent modeling has further suggested that the observation of Q_ST > F_ST may be caused by factors intrinsic to the quantitative loci themselves, such as dominance and epistatic interactions between loci (LATTA 1998; WHITLOCK 1999; LE CORRE and KREMER 2003; LOPEZ-FANJUL et al. 2003; GOUDET and BUCHI 2006).

In this article, we constructed both an analytic and a simulation model to describe the differentiation of dominant quantitative traits and neutral markers between two populations linked by migration. We included dominance, selfing, and a divergent linear selection function into recursive equations to find equilibrium allele frequencies for the two populations and used these allele frequencies to calculate Q_ST and F_ST at the quantitative locus. In addition, we derived expressions and recursive equations to calculate F_ST at equilibrium for neutral loci. Finally, we constructed an individual-based simulation model to investigate the effects of divergent and balancing selection and drift on the Q_ST – F_ST contrast.

These models have given important insight into the behavior of Q_ST and F_ST in the presence of dominance and inbreeding. By using our recursion equations to find expected equilibrium values for Q_ST, we have demonstrated that when a quantitative trait is under divergent selection, the expectation for Q_ST – F_STQ (F_ST calculated at the quantitative locus using equilibrium allele frequencies) is generally greater than zero when the quantitative locus is dominant. Q_ST is likely to be elevated relative to the divergence calculated from allele frequencies at the quantitative trait (i.e., F_STQ), and when comparing Q_ST to F_ST calculated from neutral loci (F_STN), this elevation in Q_ST means that it is more likely that divergent selection will be detected. Results from simulations also indicate that for balancing selection, where we expect Q_ST < F_ST, the presence of dominance may deflate Q_ST relative to F_STQ and also lead to a larger negative value for Q_ST – F_STN. Despite the impact that dominance, and to a certain extent inbreeding, has on enhancing the Q_ST – F_STQ contrast, this should be seen in context: selection has a very much stronger impact on Q_ST values than dominance or inbreeding.

Effect of dominance:

The reason that dominance generally enhances the difference between Q_ST and F_ST for the selected locus can be seen clearly when we plot the increase of expected Q_ST and F_STQ values against increasing dominance coefficients for a "typical" set of parameters (Figure 8). Although Q_ST and F_ST have the same value when there is no dominance (d = 0), when the populations are under divergent selection, Q_ST increases more rapidly than F_ST with increasing dominance coefficients. In contrast to divergent selection, simulation results for balancing selection suggest that the growth in F_ST values across dominance coefficients may be greater than that for Q_ST (Table 4), explaining why the presence of dominance enhances the difference between Q_ST and F_ST for both divergent and balancing selection. We conclude that dominance is very unlikely to mask the effect of selection on the signatures of divergent and balancing selection (Q_ST > F_ST and Q_ST < F_ST, respectively).

View larger version (6K): In this window In a new window Download PPT slide   FIGURE 8.— Plot of analytic QST – FSTQ values calculated from equilibrium allele frequencies against increasing dominance coefficients (d). Other population parameters are , (i = X, Y).

 
There are in addition a number of interesting findings from the sets of simulations. For our simulation results for divergent selection using a linear fitness function (Table 3), the observed difference between Q_ST and F_STQ for d > 0 was frequently negative due to the very large variance in both Q_ST and F_STQ values, despite the relatively large population sizes modeled (N_X = N_Y = 1000). In addition, even with strong selection acting on the quantitative trait, the value of Q_ST was less than F_STN for neutral markers in some cases. Nevertheless, at high levels of dominance it was more probable that Q_ST – F_STQ and Q_ST – F_STN values were positive, suggesting that divergent selection is more likely to be detected in dominant than in additive traits. Interestingly, in simulations with a different divergent selection regime (where genotypic fitnesses are determined by an optimal fitness function), dominance had an even stronger effect on enhancing the difference between Q_ST and F_STQ. Under balancing selection, Q_ST seemed to be less than F_STN in the presence of dominance, suggesting that, as with divergent selection, balancing selection is more likely to be detected in dominant than in additive traits.

The results for one locus under balancing selection are consistent with expectation when comparing quantitative and neutral loci (Q_ST – F_STN < 0). However, when two loci were selected, the observed Q_ST values at low levels of migration and selfing were extremely high compared to the F_STN values. This was a consequence of calculating Q_ST using allele frequencies at these loci rather than the overall trait values; although the mean trait values were similar between populations, allele frequencies could be highly differentiated. Therefore, when interpreting sequence differentiation at loci thought to contribute to quantitative traits, it is important to bear in mind that large sequence differences between two populations may not be reflected in any significant trait differentiation if they are offset by differences in other coding loci throughout the genome.

Finally, as explored by LOPEZ-FANJUL et al. (2003, 2007), GOUDET and BUCHI (2006), and GOUDET and MARTIN (2007), neutral traits show generally very little difference in Q_ST and F_STQ values, with similar proportions of Q_ST – F_STQ greater than and less than zero across designs.

Extensions to more general cases:

Multiple alleles:

In all of our simulations described above, we assumed that the quantitative trait was coded by one or more diallelic loci. To assess whether our conclusions are restricted to the case of two alleles, we ran an additional set of simulations. Inbreeding coefficients, migration rates, and mutation rates were fixed at 1/N with population size N_X = N_Y = N = 1000. To model divergent selection, selection coefficients were chosen from a uniform distribution between 0 and 0.1 (population X) and between –0.1 and 0 (population Y). The simulations were run for 2000 generations with 200 replicates. One set of simulations was run for a single-locus, two-allele model, where the dominance coefficient d was chosen at random from (0, 1), with genotypic values defined as in Table 2. The second set of simulations was run for a single-locus, four-allele model (alleles A_i, i = 1–4). Genotypic values of homozygotes were defined as

where _i is the additive effect of allele A_i and is drawn from a normal distribution of mean 0 and standard deviation of . Genotypic values of heterozygotes were defined as

where d_ij is the dominance term. Each of the six unique dominance terms was drawn from a normal distribution with mean 0 and standard deviation defined by

such that 95% of heterozygotes fall within the range between the two homozygote genotypic values. F_ST at the quantitative locus (F_STQ) was calculated using (8). We calculated Q_ST using Equation 4, where V_A is defined as twice the covariance between parent and offspring genotypic values (averaged over populations), and V_B is the variance in mean genotypic values between populations.

The results from the two sets of simulations strongly support our earlier conclusion that under divergent selection, dominance causes a positive difference between Q_ST and F_STQ. The mean difference between Q_ST and F_STQ was significantly greater than zero whether two or four alleles contribute to the quantitative trait (P < 0.001), and in both cases only a small proportion of replicates had Q_ST – F_STQ values less than zero. Dominance is therefore likely to enhance the difference between Q_ST and F_STQ, whether the quantitative trait locus has two or more alleles, when the quantitative trait is under selection.

Multiple populations:

An additional restriction of our simulations was that we investigated only two populations linked by migration. The variance in Q_ST and F_ST is certainly affected by the number of populations studied (see WHITLOCK 2008 and references therein) and will affect the likelihood that we can detect a difference between them. We therefore performed simulations for two and four populations under divergent selection, to assess the change in variance in Q_ST, F_STQ, and Q_ST – F_STQ. Q_ST and F_ST are calculated by extending Equations 8 and 12 to the case of four populations.

We performed two sets of simulations, the first with low migration and inbreeding, where many parameters were fixed (, d = ). The second set of simulations had a larger proportion of parameters that are allowed to vary, with and f, d, and m chosen at random from their biologically feasible ranges. For both sets, half of the selection coefficients were chosen at random from (–1, 0), and half chosen at random from (0, 1). For two and four populations, each of the simulation sets was run for 2000 generations with 500 individuals in each population, and 1000 replicates were performed.

For both sets of simulations, the variances in Q_ST, F_STQ, and Q_ST – F_STQ decreased substantially for four populations compared to two populations. For the first set of simulations, the variances for Q_ST, F_STQ, and Q_ST – F_STQ were 0.0097, 0.0099, and 0.0017, respectively, for two populations and 0.0022, 0.0021, and 0.0007 for four populations. Results from the second set of simulations showed a similar trend. These findings suggest that increasing the number of populations will not only decrease the variance in Q_ST and F_STQ, but also decrease the variance in the difference between the two. When comparing Q_ST to F_ST from neutral marker loci, the increased precision in Q_ST will make it more likely that we will detect divergent selection on the quantitative trait. The variances in Q_ST and F_ST, and hence the likelihood that we will observe a difference between them, are influenced by many other factors, for example, sample size and how Q_ST and F_ST are estimated (WHITLOCK 2008). These factors need to be taken into account when we determine the significance of any difference (or indeed, lack of difference) between Q_ST and F_ST for empirical data.

We have demonstrated that under divergent selection, dominance is likely to increase the value of Q_ST, relative to the value expected for a purely additive trait. Our simulations also suggest that balancing selection will decrease the value of Q_ST. For neutral loci, the value of Q_ST under dominance may be increased or decreased relative to Q_ST expected for an additive trait. Under divergent, balancing, and no selection, we expect Q_ST greater than, less than, and approximately equal to F_ST, respectively. Thus, the presence of dominance will enhance the difference between Q_ST and F_ST when the quantitative locus is selected and will make it more likely that we will be able to detect these "signatures of selection" in natural populations. For neutral loci, dominance does lead to a difference between Q_ST and F_ST. The difference is, however, small, making a false detection of selection unlikely.

A perhaps unsurprising result from both analytic and simulation models was the very large impact of migration rates on Q_ST and F_ST values. Even with strong directional selection, high levels of migration can severely restrict the differentiation between populations, such that the difference between Q_ST and F_ST measured at neutral markers becomes so low as to be indistinguishable from a neutral model (data not shown). Thus, while we cannot have confidence in the Q_ST – F_ST contrast giving us any reasonable indication of selection when selection is weak and population sizes are small, we similarly are unlikely to detect consistent signatures of selection when migration is high. One major conclusion from these simulations has been previously well stated by GOUDET and MARTIN (2007, p. 1373): compared to dominance (or indeed other factors), "the large variance of Q_ST is certainly more worrisome for the prospect of identifying traits under selection."

Population differentiation at neutral markers (F_ST):

Recall that N_i is the number of individuals in population i (i = X, Y) and that we let the probability of selfing in both populations be f while the probability of parents outcrossing is (1 – f). Selfing and outcrossing are assumed to be independent, and so the numbers of selfed and outbred offspring per parent are binomially distributed with means f and (1 – f), respectively.

We calculate a number of coefficients describing the similarity between and within individuals in a subdivided population, which we use to calculate F_ST. The inbreeding coefficient F describes the probability that two alleles at a given locus within an individual are identical by descent (IBD), while , the coancestry coefficient, is the probability that two alleles chosen at random from two individuals within a population are IBD. Both F and can be calculated separately for the two populations, and overall values found by multiplying each population parameter by its proportional population size. Finally, is the probability that two alleles chosen at random from different populations are IBD.

Recursions for probabilities of identity by descent:

Following the approach of WANG (1997), we derive the full set of recursion equations for F, , and . Individuals within a population outcross or self to give offspring in the next generation, and then a number of these offspring migrate so that population i has a proportion m_i of migrants and (1 – m_i) of nonmigrants. We first derive the expressions for the probability of IBD following reproduction and then following migration. Definitions of terms used in the following derivations may be found in Table 1.

F, , and after reproduction:

F:

Offspring arise from selfing and outcrossing, with expected proportions f and (1 – f). The inbreeding coefficients of offspring from selfing and outbreeding are and _i, respectively. The average F in the next generation is therefore

for population i.

:

When calculating between two individuals I₁ and I₂, we need to consider a number of situations:

1. Both I₁ and I₂ are the result of selfing, with probability f². Assuming all parents have an equal chance of producing selfed offspring, the probability that I₁ and I₂ have the same or different parents is 1/N and (N – 1)/N, respectively. The probability of IBD between I₁ and I₂ with the same parent is and with different parents is .
   
2. I₁ is the result of selfing and I₂ is not (or vice versa), with probability 2f(1 – f). I₁ and I₂ share a parent with probability 2/N, and the probability of IBD between a selfed and an outbred sib is . I₁ and I₂ share no parents with probability , and the probability of IBD between unrelated individuals is .
   
3. Both I₁ and I₂ are the result of outcrossing, with probability (1 – f)². The probability that I₁ and I₂ have the same parent pair is 2/N(N – 1), and the probability of IBD between these full sibs is . I₁ and I₂ share one parent with probability 1/N and do not share the next with probability , and can do so four ways, giving an overall probability of . The probability of IBD between half sibs is . Finally, all parents between and within I₁ and I₂ are different with probability , and the probability of IBD between unrelated individuals is .
   

Multiplying probabilities, summing each of the subcases, and then summing over the three mating types, the overall probability of IBD between two individuals in population i after reproduction is

:

Reproduction does not change the probability of IBD between populations, so

F, , and after migration:

F:

After migration, the probability that alleles within an individual are IBD for population X is a weighted average of and :

A similar expression for population Y is obtained by swapping the X and Y subscripts.

:

After migration, the probability that two alleles from different individuals within X (or Y) are IBD is a weighted average of and . For two individuals taken at random from population X, the probability that they are both nonmigrants is , that they are both migrants is , and that one is migrant and one nonmigrant is . Then the average probability of IBD is

A similar expression for population Y is obtained by swapping the X and Y subscripts.

:

After migration, the probability that two alleles, one from X and one from Y, are IBD is again a weighted average of and : .

By substituting values of and into the equations for F_X(t), F_Y(t), _X(t), _Y(t), and _(t) above, we derive the full set of recursive equations for the probabilities of identity by descent. These recursion equations may be written in matrix notation as

where

(A1)

and M_i = m_iN_i is the number of migrant individuals in population i (i = X, Y). Interestingly, these equations differ from those of WANG (1997) and VITALIS and COUVET (2001), who assume exchange of genes rather than individuals during migration, and hence migration has a different interpretation in the F, , and terms.

If migration and population sizes are the same in the two populations (m_X = m_Y = m and N_X = N_Y = N), then F_X = F_Y = F, _X = _Y = , and the recursions reduce to three equations, where

(A2)

and M = mN.

While the values of F, , and increase monotonically over time, their instantaneous rates of increase [, , and ] converge to the same value if there is some migration between populations (WANG 1997). Their asymptote is equal to half the inverse of the asymptotic effective population size; that is,

(A3)

The expression for N_e may be found from the transition matrix T. Following WANG (1997),

(A4)

where is the leading eigenvalue of T. Although finding is straightforward when the parameters of the model (f, m_X, m_X, N_X, and N_Y) are known, there are five eigenvalues for the 5 x 5 matrix T and it becomes difficult to write a general expression for the leading eigenvalue. For example, Mathematica 4.0 (WOLFRAM RESEARCH 1999) finds the five eigenvalues in terms of root objects that may be evaluated only numerically. For given values of N_X, N_X, m_X, m_Y, and f, we may find a numeric value for N_e either by using the leading eigenvalue of T or by iterating the recursion S_(t) = TS_(t–1) + C until the changes in F, , and become small.

When we make the simplification that migration rates and population sizes are equal for the two populations, and T is reduced to a 3 x 3 matrix [see (A2) above], we can solve for the leading eigenvalue of T. By substituting into (A4), we can derive an explicit expression for N_e,

where

and

and | | is the modulus of the expression.

F_ST at equilibrium:

Using our recursive equations, we may also determine values for Wright's F-statistics, including F_ST. The F-statistics are defined as

(WRIGHT 1969), where F, , and are the probabilities of IBD described above. For given parameter values of f, m_i, and N_i, therefore, we can use recurrent Equation A1 to obtain equilibrium values for F_ST, F_IS, and F_IT. In the simple case of m_X = m_Y = m and N_X = N_Y = N, we obtain

and

Solving these equations for F_ST using (A3) yields

(A5)

where N_e is defined above.

Note that p_i is the frequency of allele A in population i (i = X, Y).

where and .

The authors thank the anonymous reviewers for their extremely helpful suggestions to improve the manuscript.

BONNIN, I., J. M. PROSPERI and I. OLIVIERI, 1996 Genetic markers and quantitative genetic variation in Medicago truncatula (Leguminosae): a comparative analysis of population structure. Genetics 143: 1795–1805.[Abstract]

COCKERHAM, C. C., and B. S. WEIR, 1984 Covariances of relatives stemming from a population undergoing mixed self and random mating. Biometrics 40: 157–164.

CRNOKRAK, P., and J. MERILA, 2002 Genetic population divergence: markers and traits. Trends Ecol. Evol. 17: 501.

GOLDSTEIN, D. B., and K. E. HOLSINGER, 1992 Maintenance of polygenic variation in spatially structured populations: roles for local mating and genetic redundancy. Evolution 46: 412–429.[CrossRef]

GOUDET, J., and L. BUCHI, 2006 The effects of dominance, regular inbreeding and sampling design on Q_ST, an estimator of population differentiation for quantitative traits. Genetics 172: 1337–1347.[Abstract/Free Full Text]

GOUDET, J., and G. MARTIN, 2007 Under neutrality, Q_ST F_ST when there is dominance in an island model. Genetics 176: 1371–1374.[Free Full Text]

HEYWOOD, J. S., 2005 An exact form of the breeder's equation for the evolution of a quantitative trait under natural selection. Evolution 59: 2287–2298.[Medline]

LANDE, R., 1992 Neutral theory of quantitative genetic variance in an island model with local extinction and colonization. Evolution 46: 381–389.[CrossRef]

LATTA, R. G., 1998 Differentiation of allelic frequencies at quantitative trait loci affecting locally adaptive traits. Am. Nat. 151: 283–292.[CrossRef][Medline]

LE CORRE, V., and A. KREMER, 2003 Genetic variability at neutral markers, quantitative trait loci and traits in a subdivided population under selection. Genetics 164: 1205–1219.[Abstract/Free Full Text]

LEINONEN, T., R. B. O'HARA, J. M. CANO and J. MERILA, 2008 Comparative studies of quantitative trait and neutral marker divergence: a meta-analysis. J. Evol. Biol. 21: 1–17.[Medline]

LOPEZ-FANJUL, C., A. FERNANDEZ and M. A. TORO, 2003 The effect of neutral nonadditive gene action on the quantitative index of population divergence. Genetics 164: 1627–1633.[Abstract/Free Full Text]

LOPEZ-FANJUL, C., A. FERNANDEZ and M. A. TORO, 2007 The effect of dominance on the use of the Q_ST – F_ST contrast to detect natural selection on quantitative traits. Genetics 176: 501–511.[Abstract/Free Full Text]

LYNCH, M., and K. SPITZE, 1994 Evolutionary genetics of Daphnia, pp. 109–128 in Ecological Genetics, edited by L. A. REAL. Princeton University Press, Princeton, NJ.

LYNCH, M., and B. WALSH, 1998 Genetics and Analysis of Quantitative Traits. Sinauer Associates, Sunderland, MA.

MARUYAMA, K., and H. TACHIDA, 1992 Genetic variability and geographical structure in partially selfing populations. Jpn. J. Genet. 67: 39–51.[CrossRef]

MERILA, J., and P. CRNOKRAK, 2001 Comparison of genetic differentiation at marker loci and quantitative traits. J. Evol. Biol. 14: 892–903.[CrossRef]

O'HARA, R. B., and J. MERILA, 2005 Bias and precision in Q(ST) estimates: problems and some solutions. Genetics 171: 1331–1339.[Abstract/Free Full Text]

SPITZE, K., 1993 Population-structure in Daphnia obtusa: quantitative genetic and allozymic variation. Genetics 135: 367–374.[Abstract]

TEMPLETON, A., 1987 The general relationship between average effect and average excess. Genet. Res. 49: 69–70.[Medline]

TURELLI, M., 1984 Heritable genetic variation via mutation-selection balance: Lerch's zeta meets the abdominal bristle. Theor. Popul. Biol. 25: 138–193.[CrossRef][Medline]

VITALIS, R., and D. COUVET, 2001 Estimation of effective population size and migration rate from one- and two-locus identity measures. Genetics 157: 911–925.[Abstract/Free Full Text]

WANG, J., 1997 Effective size and F-statistics of subdivided populations. 1. Monoecious species with partial selfing. Genetics 146: 1453–1463.[Abstract]

WEIR, B. S., and C. C. COCKERHAM, 1984 Estimating F-statistics for the analysis of population-structure. Evolution 38: 1358–1370.[CrossRef]

WEIR, B. S., and W. G. HILL, 2002 Estimating F-statistics. Annu. Rev. Genet. 36: 721–750.[CrossRef][Medline]

WHITLOCK, M. C., 1999 Neutral additive genetic variance in a metapopulation. Genet. Res. 74: 215–221.[CrossRef][Medline]

WHITLOCK, M. C., 2008 Evolutionary inference from QST. Mol. Ecol. 17: 1885–1896.[Medline]

WHITLOCK, M. C., and D. E. MCCAULEY, 1999 Indirect measures of gene flow and migration: F-ST not equal 1/(4Nm+1). Heredity 82: 117–125.[CrossRef][Medline]

WOLFRAM RESEARCH, 1999 Mathematica Version 4.0. Wolfram Research, Champaign, IL.

WRIGHT, S., 1951 The genetic structure of populations. Ann. Eugen. 15: 323–354.

WRIGHT, S., 1969 Evolution and the Genetics of Populations. University of Chicago Press, Chicago.

Communicating editor: J. WAKELEY

International Access Link

Online ISSN: 1943-2361  Print ISSN: 0016-6731
Copyright 2009 by the Genetics Society of America
phone: 412-268-1812   fax: 412-268-1813   email: genetics-gsa{at}andrew.cmu.edu