Multiallelic models of genetic effects and variance decomposition in non-equilibrium populations

Multiallelic models of genetic effects and variance decomposition in non-equilibrium populations Jose ´M. A ´lvarez-Castro •Rong-Cai Yang Received: 16 May 2011 / Accepted: 31 October 2011 / Published online: 10 November 2011 The Author(s) 2011. This article is published with open access at Springerlink.com Abstract Quantitative genetics stems from the theoretical models of genetic effects, which are re-parameterizations of the genotypic values into parameters of biological (genetic) relevance. Different formulations of genetic effects are adequate to address different subjects. We thus need to generalize and unify them under a common framework for enabling researchers to easily transform genetic effects between different biological meanings. The Natural and Orthogonal Interactions (NOIA) model of genetic effects has been developed to achieve this aim. Here, we further implement the statistical formulation of NOIA with multiple alleles under Hardy–Weinberg departures (HWD). We show that our developments are straightforwardly connected to the decomposition of the genetic variance and we point out several emergent properties of multiallelic quantitative genetic models, as compared to the biallelic ones. Further, NOIA entails a natural extension of one-locus developments to multiple epistatic loci under linkage equilibrium. Therefore, we present an extension of the orthogonal decomposition of the genetic variance to multiple epistatic, multiallelic loci under HWD. We illustrate this theory with a graphical interpretation and an analysis of published data on the human acid phosphatase (ACP1) polymorphism. Keywords Models of genetic effects Hardy–Weinberg disequilibrium Multiple alleles Variance decomposition Acid phosphatase polymorphism Introduction The models of genetic effects are re-parameterizations of the genotypic values into parameters entailing clearer biological (genetic) meanings (Kempthorne 1957; Fisher 1918,1930). A general form of such reparameterizations can be written in matrix notation as G=SE, where the vector of genotypic values, G, is expressed as a linear function of the vector of genetic effects, E(Cockerham and Zeng 1996). Cockerham (1954) expressed statistical models of genetic effects in terms of orthogonal scales, also known as contrasts, whose coefficients are used to form the genetic-effect design matrix, S. Tiwari and Elston (1997) showed that this notation enormously facilitates the extension of the models of genetic effects to several loci under linkage equilibrium (LE) and Zeng et al. (2005) have further shown the convenience of this notation for statistical analyses of two-allele equilibrium populations. The G=SE matrix notation also provides an appropriate theoretical framework to unify different models of genetic effects (A ´lvarez-Castro and Carlborg 2007). Some issues in quantitative genetics need to be addressed using different mathematical formulations and it is thus necessary to unify all different uses of genetic effects under a J. M. A ´lvarez-Castro (&) Department of Genetics, University of Santiago de Compostela, Avda. Carvalho Calero s/n, 27002 Lugo, Galiza, Spain e-mail: [email protected] URL: http://webspersoais.usc.es/persoais/jose.alvarez.castro/ index_en.html R.-C. Yang Research and Innovation Division, Alberta Agriculture and Rural Development, Edmonton, AB T6H 5T6, Canada R.-C. Yang Department of Agricultural, Food and Nutritional Science, University of Alberta, Edmonton, AB T6G 2P5, Canada e-mail: [email protected] 123 Genetica (2011) 139:1119–1134 DOI 10.1007/s10709-011-9614-9 single theoretical perspective (Phillips 2008). The natural and orthogonal interactions (NOIA) model has indeed been developed for this purpose by using the matrix notation (A ´lvarez-Castro and Carlborg 2007). The initial setting of the NOIA model provided a framework under which genetic effects can be defined on the basis of allele substitutions from the reference of individual genotypes (functional effects) or average effects of substitutions in populations (statistical effects) for the case of two alleles per locus. As an example to show the difference between these two formulations, the additive functional effect is always equal to half the difference between the performance of the two homozygotes whereas the additive statistical effect is the coefficient of weighted regression on the allele content—the number of one of the alleles—in the different genotypes present in a population. We have further extended the functional formulation of the NOIA model to the case of multiple alleles and explored the relationship between multiallelic functional and statistical genetic effects (Yang and A ´lvarez-Castro 2008). In that article, we have also shown some new properties of the multiallelic models as compared to the biallelic ones. We here extend the NOIA statistical formulation to the multiallelic framework in non-equilibrium populations. To do this, we first obtain the multiallelic genetic effects as explicit functions of the genotypic values. This enables us to obtain the scales for the multiallelic statistical geneticeffect design matrix, S. Next, we show that this G=SE matrix notation makes it possible to provide the components of the genetic variance with multiple alleles also as explicit functions of the genotypic values through relatively simple expressions. The major focus of this communication is on one multiallelic locus under Hardy– Weinberg dissequilibrium (HWD). Under LE, extensions of one-locus formulations to an arbitrary number of multiallelic, epistatic loci is straightforward. We illustrate the theory developed in this communication with a graphical interpretation and an application to published experimental data on the polymorphism at the human acid phosphatase locus (ACP1) (Greene et al. 2000). The biallelic statistical formulation of NOIA Before embarking on our new multillelic model, we provide a brief overview on the biallelic model as described by A ´lvarez-Castro and Carlborg (2007). For a one-locus twoallele (A 1 and A 2 ) genetic system, the general expression for the statistical formulation as a function of the genotypic frequencies of the population, p ij ,ij =11, 12, 22, can be expressed in matrix notation as G=SE. This matrix expression can be expanded as: G11 G12 G22 0 @1 A¼ 12p2p12p22 2p1p21= 2p12 1p1p2p11p22 p1p21= 4p12 12p1p11p12 2p1p21= 2p12 0 B @1 C A l a d 0 @1 A;ð1Þ where the reference point of the model is the population mean, l,p i ,i=1, 2, are the frequencies of the alleles, p i =p ii ?p ij ,j=i, and aand dare, respectively, the additive and dominant statistical genetic effects. The use of the G=SE matrix notation enables us to readily obtain the genetic effects defined under one reference population as a function of the genetic effects defined under another reference population. A well-known example is the translation of the genetic effects defined under F 2 and F ? populations (Yang 2004;VanderVeen1959). Given two different decompositions (i.e. different formulations and/or from a different reference point) G=S 1 E 1 and G=S 2 E 2 ,wecan obtain E 2 from E 1 by (A ´lvarez-Castro and Carlborg 2007): E2¼S2 ðÞ 1S1E1:ð2Þ For obtaining a vector of genetic effects fitting to a particular biological meaning, E 2 , from another vector of genetic effects or genotypic values, it is necessary to have an expression for its corresponding genetic-effect design matrix, S 2 . In other words, expression (2) can be applied only when the appropriate genetic-effect design matrices are available. It is thus desirable to obtain expressions of the form G=SE describing the properties of all possible genetic systems and populations. Hereafter, we develop genetic-effect design matrices fitting to statistical genetic effects and the decomposition of the genetic variance of multiallelic loci, whether their population frequencies are under HWD or not. The multiallelic statistical formulation of NOIA We will now describe a one-locus multiallelic statistical formulation of genetic effects. We will first use the matrix notation to express the average genetic effects in terms of the genotypic values, particularly in the form E=S -1 G. We will thus provide algorithms to build inverse geneticeffect design matrices that fit to the definitions of the genetic effects for any population frequencies. We start from the decomposition of the genotypic values into statistical genetic effects as expressed by Kempthorne (1957): G¼1lþNagþdG:ð3Þ This expression entails a multiple regression in which the additive (average) effects of the ralleles, a g =(a i ), i= 1,…,r, are the regression coefficients and the dominance deviations of the genotypes, d G =(d ij ), i,j=1,…,r,i[j, are the residuals. Note that we use a capital Gin the subscripts to indicate genotypes and a lower–case gto 1120 Genetica (2011) 139:1119–1134 123 indicate alleles. The vector 1is an n91 vector of ones, where n=r(r?1)/2 is the number of possible genotypes, and Nis the genetic content matrix, as detailed in expression (16), ‘‘Appendix A’’. The product Na g =(a ij )=a G is thus the vector of the additive components of the genotypic values, often called the breeding values. The solution of regression (3) can be obtained from its normal equation (Kempthorne 1957) as we recapitulate in ‘‘Appendix A’’ for completeness. From that solution (18), we have obtained the genetic effects explicitly, as a linear function of the genotypic values. To do so we have factored the vector Gas: ag¼ðNTPdiagNÞ1NTPdiagðI1PT GÞ  G;ð4Þ where Iis the identity matrix of dimension n9n, PG¼p11 p12 p22 p13 ... prr ðÞ T,P diag =Diag(P G ), ‘‘Diag’’ of a vector generates a diagonal matrix with that vector in the diagonal and the superscript ‘‘T’’ stands for the transposition operation. We can thus rewrite expression (4)as: ag¼AG:ð5Þ The additive genetic effects are just subtractions of the allelic effects in a g ,a ij =a j -a i (we justify the use of these superscripts in ‘‘Appendix B’’). Therefore, it is possible to operate from matrix Ato obtain these genetic effects, which are the additive-related rows of the inverse genetic-effect design matrix S -1 . We build an operator Badding a column filled with -1 to the left of an identity matrix so that we can obtain the additive genetic effects as BAG. This product provides (r-1) genetic additive effects, the remaining ones being easily retrieved from them (cf. multiallelic functional genetic-effect design matrices in Yang and A ´lvarez-Castro 2008). We need two additional easy-to-build matrix operators for constructing the inverse genetic-effect design matrix S -1 . The first operator, C, places the rows of BA in the appropriate positions of an inverse genetic-effect design matrix S -1 . The other operator, S ld , adds the rows with the appropriate values that define the mean and dominance deviations in S -1 . We refer to ‘‘Appendix B’’ for a detailed definition of the three operators and we define: S1¼CBA þSld:ð6Þ Finally, the inverse of this full-rank matrix is the genetic-effect design matrix Sand we can obtain the desired statistical decomposition of genotypic values through the expression G=SE. Allelic effects from genetic effects Vectors of genetic effects can be obtained, for instance, as the output of a typical QTL analysis of a line-cross experiment (see e.g. Zeng et al. 2005). With only two alleles, it is straightforward to obtain the additive (average) effects of the alleles from the statistical formulation G=SE.This expression entails the decomposition of genetic values, from which the additive (average) effects can be computed just as a i =()a ii ,i=1,…,r(in general, a ij =a i ?a j ). However, in the multiallelic case, the additive contributions and the dominance deviations are the summation of several products of scalars from Sand genetic effects from E. Thus, we here provide an automated procedure for obtaining the additive (average) effects and average excesses from vectors of genetic effects, E. Expressions (4–6) straightforwardly provide such a procedure. Indeed, combining expression (5) with formulations of the type G=SE it follows: ag¼ASE;ð7Þ The statistical decomposition of the genotypic values, G=SE (1,6), enables us to express each genotypic value as G ij =l?a ij ?d ij , as in expression (3). Cancelling out the coefficients of the mean and the additive effects in S leads to the removal of the terms land a ij in this expression, thereby leaving the dominance effects d ij alone. We thus express that column vector d G =(d ij ) as: dG¼SDiag Di ðÞE;ð8Þ where (D i ) i=1,…,n is a vector of index variables for dominance-related columns of S, which coincide with the positions of the dominance effects in the vector E(an automated algorithm to obtain this vector of indexes is provided in ‘‘Appendix B’’). Hardy–Weinberg equilibrium Under the Hardy–Weinberg proportions, the expressions within the matrices of the statistical formulation can be more easily illustrated than in the general (non-equilibrium) case. In particular, the matrix Afor three alleles can be expressed as: A¼ p1ðp2þp3Þp2ð12p1Þp2 2p3ð12p1Þ2p2p3p2 3 p2 1p1ð12p2Þp2ðp1þp3Þ2p1p3p3ð12p2Þp2 3 p2 12p1p2p2 2p1ð12p3Þp2ð12p3Þp3ðp1þp2Þ 0 @1 A:ð9Þ Genetica (2011) 139:1119–1134 1121 123 Using (7) and (27), the three-allele statistical formulation under Hardy–Weinberg proportions, E=S -1 G,is expanded to: As a check, we get G=SE as: In ‘‘Appendix B’’, we justify the notation of the scalars within the Evector in expressions (10,11). Expressions (9– 11) are, as expected, equivalent to previous expressions of multiallelic genetic effects under the Hardy–Weinberg proportions (Kempthorne 1957; Wang and Zeng 2006, 2009). We here provide them using matrix notation as a particular case of a general formulation including also HWD—expressions (4–6). The decomposition of the genetic variance In expressions (4–8), we have formulated statistical multiallelic genetic effects as explicit functions of the genotypic values. Hereafter we use that formulation for the decomposition of the variance components. We indeed derive a method that provides the classical decomposition of the genetic variance of a trait in a population with the additional advantage of enabling a straightforward extension to genetic systems with multiple epistatic loci under LE, while allowing arbitrary numbers of alleles and arbitrary departures from the Hardy–Weinberg proportions at all loci. The average excess of one allele, a i, is the difference by which the average of genotypes carrying that allele exceeds the average of genotypes carrying the alternative alleles (Fisher 1941). A common way to obtain the additive variance under HWD involves both the allelic additive (average) effects and average excesses of alleles (see e.g. Lynch and Walsh (1998) and (23)in‘‘Appendix A’’). From that expression, but using just vectors, the additive variance can be expressed as VA¼2PT gaga g  , where a g¼ða iÞ,P g is the column vector of the gene frequencies and ‘‘’’ is the Hadamard product (just the pairwise product of the elements at the same position in the two vectors). It is also possible to compute the additive variance with HWD without the need of the use of the average excesses (see e.g. Bu ¨rger 2000). This can be done by using instead the additive genotypic components, often called the breeding values (25 in ‘‘Appendix A’’). This way of computing the additive variance can also be expressed through just vectors as: VA¼PT GaGaG ðÞ;ð12Þ where a G =(a ij )=Na g is the vector of the additive components of the genotypic values. Similarly, from expression (24)in‘‘Appendix A’’ the dominance variance can also be expressed as: VD¼PT GdGdG ðÞ:ð13Þ Now we provide a way to compute expressions (12,13) simultaneously, which will conveniently have a straightforward extension to multiple loci. We first provide the decomposition of the genotypic values in matrix form as: Gdec ¼SDiag(EÞH;ð14Þ where His an operator that sums the additive-related and the dominance-related columns of the matrix to the left of l a12 d12 a13 d13 d23 0 B B B B B B @ 1 C C C C C C A¼ p2 12p1p2p2 22p1p32p2p3p2 3 p1p1p2p2p3p30 1=211=2000 p1p20p1p3p2p3 1=2001 01=2 001=2011=2 0 B B B B B B @ 1 C C C C C C A G11 G12 G22 G13 G23 G33 0 B B B B B B @ 1 C C C C C C A ;ð10Þ G11 G12 G22 G13 G23 G33 0 B B B B B B @ 1 C C C C C C A¼ 12p22ð1p1Þp22p32ð1p1Þp32p2p3 112p22p1p2þp32p32p1p3p32p2p3p3 12ðp1þp3Þ2p1ð1p2Þ2p32p1p32ð1p2Þp3 12p22p1p2p212p32p1p3þp22p2p3p2 112p22p1p2p112p32p1p3p12p2p3þp1 12p22p1p22ðp1þp2Þ2p1ð1p3Þ2p2ð1p3Þ 0 B B B B B B @ 1 C C C C C C A l a12 d12 a13 d13 d23 0 B B B B B B @ 1 C C C C C C A :ð11Þ 1122 Genetica (2011) 139:1119–1134 123 it. Therefore, with only two alleles, His just the identity matrix (see ‘‘Appendix B’’ for details). The three columns of G dec give, respectively, the mean and the additive and dominance components of the genotypic values. This decomposition (14) is meant to be the one referred to in expression (3) and, thus, the second and third columns of the matrix G dec are the vectors of additive and dominance effects, a G =Na g and d G respectively. Using this information about expression (14) and expressions (12, 13), it is easy to see that the two components of the genetic variance can be obtained simultaneously, just as: V¼PT GGdec Gdec ðÞ:ð15Þ The first scalar of Vis actually the squared population mean, being the remaining two scalars the additive and the dominance variances, i.e. V=(l 2 ,V A ,V D ). The properties of the statistical models of genetic effects for multiple alleles are not exactly the same as the ones of the two-allele case. With only two alleles, all genetic effects are orghogonal, despite HWD (A ´lvarez-Castro and Carlborg 2007;Yang2004; Cockerham 1954). With three or more alleles there are several genetic effects accounting for additive (average) effects of allele substitutions. Since the effect of each allele substitution depends on the frequencies of all other alleles present in that locus, those additive parameters are dependent on each other. We want our parameters to reflect effects of substitutions and some of them will thus not be orthogonal to each other, even without HWD. We can explain the same fact mathematically. The multiallelic genetic effects come from a multiple regression (3)and some of them will necessarily be statistically dependent on each other. Indeed, the statistical formulation of NOIA for multiple alleles (5) is not fully orthogonal. It is nevertheless orthogonal by blocks that gather effects of the same type (additiveor dominant-related effects) for different pairs of alleles, as illustrated in ‘‘Appendix A’’ (26). Therefore, the decomposition of genotypic values given by (14) is not fully orthogonal, but again orthogonal by blocks of effect-types. Conveniently, though, the variance decomposition performed from expression (15) is fully orthogonal. This is so because the components of variance are computed using the effects within each of the orthogonal blocks of the statistical formulation of NOIA. Thus, the variance decomposition provided by the NOIA model (15) is orthogonal even under departures from the Hardy–Weinberg proportions. Furthermore, expression (15) can be straightforwardly extended to an arbitrary number of loci with arbitrary numbers of alleles and arbitrary HWD, under LE. Multiple loci The NOIA formulations can conveniently allow for straightforward extensions to multiple loci under LE. To distinguish the expressions of each of the locus, we implement the notation used so far with appropriate indicators for each locus name and number of alleles. We do this using subscripts and superscripts, respectively, in all matrices and vectors. For a locus Awith three alleles, for instance, we enunciate the statistical formulation as: Gð3Þ A¼Sð3Þ AEð3Þ A. We now consider a two-locus genetic system in which there is, in addition, a locus Bwith two alleles. Assuming LE between the two loci, we use the Kronecker product of the one-locus genetic-effect design matrices (as in A ´lvarez-Castro and Carlborg 2007), Sð3;2Þ AB ¼Sð2Þ BSð3Þ A, to obtain the two-locus statistical formulation as Gð3;2Þ AB ¼Sð3;2Þ AB Eð3;2Þ AB . The equivalent expression Eð3;2Þ AB ¼Sð3;2Þ AB  1Gð3;2Þ AB can be obtained by computing the inverse of the two-locus genetic-effect design matrix Sð3;2Þ AB , or equivalently using the inverses of the one-locus matrices as Sð3;2Þ AB  1¼Sð2Þ B  1Sð3Þ A  1. Either way, this entails the extension of the solution to Eq. (4) to multiple loci under LE. The general expression to obtain the statistical genetic-effect design matrix for an arbitrary number of loci, l, is:  1 k¼lSðrkÞ Lk  , where r k is the number of alleles at locus L k . Once the multiallelic Smatrix has been obtained, the decomposition of the genetic variance through expressions (14,15) naturally holds for multiple loci. The multilocus operator Hcan be built from the single-locus ones as  1 k¼lHðrkÞ Lk  , to account for the additional (due to interactions among loci) variance components. Applying expressions (14,15) to the system of loci Aand Bleads to a vector of variance components Vð3;2Þ AB in which the order of the variance components is the same as the one of the genetic effects in the vector Efor two alleles (A ´lvarez-Castro and Carlborg 2007). For the multilocus genetic systems, it is possible to test for orthogonality in the same way as shown for the onelocus case at the end of ‘‘Appendix A’’. By doing so at the system of loci Aand Bconsidered just above, we have obtained a matrix with nine independent non-zero blocks for the mean, additive effect of locus A, additive effects of locus B, dominance effect of locus A, dominance effects of locus B, and the additive-by-additive, additive-by-dominance, dominance-by-additive and dominance-by-dominance interactions—i.e. an analogous matrix to (26), although larger and having more independent blocks (not shown). These separate blocks reflect the orthogonality of all the variance components. Thus, expressions (14,15) comprise a straightforward routine to perform orthogonal decomposition of variance from the NOIA model for Genetica (2011) 139:1119–1134 1123 123 genetic systems of arbitrary numbers of alleles at multiple epistatic loci under LE. Applications Here we show a graphical interpretation of the decomposition of the genotypic values for a three-allele case under HWD and we make an analysis of real data on the human ACP1 polymorphism. Graphical interpretation A ´lvarez-Castro and Carlborg (2007, Figures 2 and 3A) have represented graphically the biallelic statistical formulation of NOIA, although here we point out a misprint in that representation—there lacks a factor 2 adjacent to every a i , which also applies to every a i in Figures 2 and 3B of that article. On the other hand, Kempthorne (1957) has provided a graphical interpretation of the statistical decomposition of the genotypic values for the one-locus three-allele case. Although Kempthorne (1957) illustrated his graphical interpretation under the Hardy–Weinberg proportions, an analogous interpretation holds when, like in the case we are dealing with here, there are departures from these proportions. In Fig. 1we actually provide a graphical interpretation that fits this case, based on the formulation of the NOIA model for non-equilibrium populations (6). Figure 1is produced using the one-locus three-allele example taken from Yang and A ´lvarez-Castro (2008) where the genotypic values G=(10, 30, 50, 36, 46, 42) T and the genotypic frequencies P=(0.12, 0.06, 0.195, 0.1, 0.15, 0.375). The genotypic values are represented by globes whose sizes are in accordance with their genotypic frequencies. The decomposition of the genotypic value G 11 =l?a 11 ?d 11 is marked by vertical grey arrows that represent the additive expectation a 11 (from the mean to the value predicted by the regression plane) and the dominance deviation d 11 (the departure between the multilinear prediction and the true genotypic value). The genotypic values are expressed as functions from the two-dimensional domain defined by the axis of the gene content of alleles A 2 and A 3 ,G(c 2 ,c 3 ), and so it is the regression plane, ^ Gðc2;c3Þ¼18:3 _c2þ15c3þ13. The intercept in this regression, 13, is the predicted value for the genotype A 1 A 1 , a 11 —for which the allele contents of the alleles A 2 and A 3 are zero. Under the Hardy–Weinberg proportions, the predictions of the regression plane are the breeding values. In Fig. 1it is possible to observe that, although only the genotypic value of the heterozygote A 1 A 3 lies outside the midpoint of the two flanking homozygotes, the regression plane does not meet, for instance, the genotypic value G 12 . In other words, the presence of functional dominance interaction at one only heterozygote suffices to cause non zero dominance deviations, d ij , for all genotypes—as already noted by Yang and A ´lvarez-Castro (2008). Analysis of the ACP1 polymorphism The acid phosphatase multiallelic polymorphism, ACP1, has been discovered in Europe almost half of a century ago (Hopkinson et al. 1963) and extensively studied ever since. Three alleles were found at different frequencies in northern European populations, ACP1*A,ACP1*B and ACP1*C (hereafter A, B and C, respectively), for which no significant deviations from an additive inheritance of enzyme activity have been found (Spencer et al. 1964; Greene et al. 2000; Eze et al. 1974). Indeed, taking a vector of genotypic values, G ac , from Greene et al. (2000, reproduced in our Table 1), and using the multiallelic functional formulation of NOIA (Yang and A ´lvarez-Castro 2008), we obtain functional genetic effects from the reference of the genotype AA, Eac AA, showing that dominance effects are very small compared to the additive effects Fig. 1 Graphical interpretation of the statistical decomposition of the genotypic values as a function of the content of alleles ‘‘2’’ and ‘‘3’’, G(c 2 ,c 3 ), for the one-locus three-allele example given by the genotypic values G=(10, 30, 50, 36, 46, 42) T and the genotypic frequencies P=(0.12, 0.06, 0.195, 0.1, 0.15, 0.375), which is one of the instances we have considered in a previous publication (Yang and A ´lvarez-Castro 2008). The genotypic values are represented by globes whose sizes are in accordance with their genotypic frequencies. The decomposition of the genotypic value G 11 =l?a 11 ?d 11 is marked by vertical grey arrows that represent the additive expectation a 11 (from the mean to the value predicted by the regression plane) and the dominance deviation d 11 (the departure between the multilinear prediction and the true genotypic value) 1124 Genetica (2011) 139:1119–1134 123 (Table 2). The transformation tool of NOIA (2) enables us to obtain a vector of statistical genetic effects from a functional one (A ´lvarez-Castro and Carlborg 2007). Any statistical genetic effects are associated to certain population frequencies. We take them from a study with a sample of 7059 individuals from a German population (Brinkmann et al. 1971, see our Table 1). Using these frequencies (referred to as the observed frequencies hereafter) we obtain the vector of genetic effects, Eac l, (Table 2). From this vector and (15) we obtain the following variance decomposition: V A =658.43, V D =0.97, V A /V G = 0.999, showing the very small contribution of the dominance effects of this trait to the genetic variance. The high proportion of additive variance indicates that directional selection on ACP1 activity would have a high response of the trait in this population, which would actually lead to the fixation of one of the alleles. Sensabaugh and Golden (1978) inspected whether the maintenance of the polymorphism could be explained by a different trait—the inhibition of ACP1 by folic acid. The genotypic values obtained for this trait, G in , and their standard deviations are shown in Table 1. From them we computed the functional and statistical genetic effects, Ein AA and Ein l, which are again largely additive (Table 2). The dominance effects are actually not significantly different from zero. From the observed frequencies, the decomposition of the genetic variance for this trait is: V A =44.54, V D =1.15, V A /V G =0.997. To determine the significance of the genetic effects, we have computed 95% confidence intervals of these estimates (Table 2). These just come from transforming the standard errors of the genotypic values into the standard errors of the genetic effects (Le Rouzic and A ´lvarez-Castro 2008). Then we have inspected Table 1 Genotypic values for the traits ‘‘ACP1 enzyme activity’’ and ‘‘ACP1 enzyme inhibition’’, observed frequencies used in our analyses and results obtained for the different geontypes—fixation indexes, fitness values minimizing V A /V G and equilibrium frequencies of these fitnesses ACP1 genotypes AA AB BB AC BC CC ACP1 activity a ,G ac 122.4 153.9 188.3 183.6 212.3 240.0 ACP1 inhibition b ,G in 41.2 37.9 34.4 58.7 53.1 76.0 SD of ACP1 inhibition b 4.3 5.2 5.2 8.3 3.8 15.5 Observed frequencies c ,p ij 0.1242 0.4139 0.3349 0.0445 0.0799 0.0025 Observed fixation indexes, F ij -0.003 -0.006 -0.015 0.028 -0.061 -0.027 Fitness values, x ij 0.9303 1 0.9667 1.0465 0.9658 0.5832 Equilibrium frequencies, p ij 0.1115 0.4323 0.3766 0.0295 0.0491 0.0001 a Greene et al. (2000). Expressed as lmol of p-niotrophenol liberated in 0.5 h at 37C per g hemoglobin b Sensabaugh and Golden (1978). Enzyme inhibition by 0.1 folic acid, expressed as lmol of p-niotrophenol liberated c Brinkmann et al. (1971) Table 2 Genetic effects for the traits ‘‘ACP1 enzyme activity’’ (ac), ‘‘ACP1 enzyme inhibition’’ (in, with confidence intervals, CI) and for the stabilizing (st) genotype-fitness (GF) map of fitnesses minimizing the ratio V A /V G . Genotypic values, fitness values and observed frequencies as in Table 1 Reference point and genetic effects a Ra AB a AC a BC d AB d AC d BC Enzyme activity, Eac AA 122.4 35.95 58.80 -23.85 -1.45 2.40 -1.85 Enzyme activity, Eac l167.7 33.00 59.19 26.19 -1.45 2.40 -1.85 Enzyme inhibition, Ein AA 41.2 -3.4 17.4 20.8 0.1 0.1 -2.1 CI for Ein AA ±1.99 ±0.85 ±5.87 ±5.72 ±2.30 ±10.56 ±7.58 Enzyme inhibition, Ein l39.4 -3.57 16.05 19.62 0.1 0.1 -2.1 Stabilizing GF map, Est AA 0.930 0.018 -0.174 -0.192 0.052 0.290 0.191 Stabilizing GF map, Est l0.978 0.000 b 0.000 b 0.000 b 0.052 0.290 0.191 a We use only Latin letters in the headings just for simplicity. It is understood that statistical genetic effects are named with the corresponding Greek letters, ainstead of a,dinstead of dand linstead of R b These values are zero to eight decimal places Genetica (2011) 139:1119–1134 1125 123 whether the proportion of additive variance varies significantly for parameter values within the confidence intervals of the estimates—we have used the theory provided in this communication (4–6,14,15) to plot the ratio V A /V G within those intervals. We have obtained that at least three estimates have to reach values at the edges of their intervals for the ratio V A /V G to be lower than 0.7 (Fig. 2). We can therefore conclude that this genotype-phenotype map is largely additive and that directional selection on the phenotype would lead to fixation of one of the alleles. Alternatively, we have computed the fixation indexes of the observed frequencies (Weir 1996), which are shown in Table 1. These fixation indexes reflect HWD with a deficiency of all homozygotes and heterozygote AC and an excess of heterozygotes AB and BC. According to the observations of Alvarez (2008), it is possible that the observed frequencies are equilibrium frequencies under viability selection acting on the ACP1 locus. However, this selection pressure cannot be directional selection acting either on enzyme activity or on inhibition by folic acid, which in the absence of significant dominance interactions would lead to fixation of one of the alleles—as illustrated above. In fact, Greene et al. (2000) suggested that this polymorphism could instead be maintained by stabilizing selection due to the balance of two forces. On the one hand, ACP1 genotypes with high enzyme-activity (particularly genotype CC) would not be well adapted to cold environments and therefore they could be selected against in BC 5.72 BC 19.62 BC 5.72 AC 5.87 0.7 0.8 0.8 0.9 0.9 0.9 0.95 0.99 10 5 0 5 10 6 4 2 0 2 4 6 0.9 0.95 0.95 0.99 10 5 0 5 10 6 4 2 0 2 4 6 0.95 0.99 0.99 10 5 0 5 10 6 4 2 0 2 4 6 AC 16.05 0.7 0.8 0.9 0.9 0.9 0.95 0.99 10 5 0 5 10 6 4 2 0 2 4 6 0.8 0.9 0.9 0.95 0.95 0.95 0.99 10 5 0 5 10 6 4 2 0 2 4 6 CI BC 0.9 0.95 0.95 0.99 10 5 0 5 10 6 4 2 0 2 4 6 CI BC AC 5.87 0.9 0.95 0.95 0.99 10 5 0 5 10 6 4 2 0 2 4 6 0.8 0.9 0.95 0.95 0.99 10 5 0 5 10 6 4 2 0 2 4 6 CI BC 0.9 0.95 0.95 0.99 10 5 0 5 10 6 4 2 0 2 4 6 CI AC CI BC CI AC CI BC CI AC CI BC CI BC CI BC CI AC CI AC CI AC CI AC CI AC CI AC CI BC Fig. 2 Contour plots of the ratio V A /V G for ACP1 activity (from Greene et al. 2000) with the observed frequencies (from Brinkmann et al. 1971) for the less accurate genetic effects estimates ranging within their confidence intervals 1126 Genetica (2011) 139:1119–1134 123 Northern Europe. On the other hand, phenotypes with the lowest enzyme activity levels (particularly genotype AA) were found to be associated with risk of macrosomia and adult obesity. Genotypes showing intermediate enzyme activities would thus in average perform better than the most extreme ones. Selection acts on the fitnesses of the different genotypes present in a population. With directional selection on the trait value, the fitnesses of the individuals with a particular genotype directly reflect their phenotypes—their genotypic values. This is why the decomposition of the genetic variance of a trait is informative about the response of that trait to directional selection (e.g. Falconer and MacKay 1996). This does however not hold for other selection regimes. With stabilizing selection the genotype-phenotype map is not a linear transformation of the genotype-fitness map and thus the former one cannot be used in substitution of the latest one, as it was the case for directional selection. Therefore, we hereafter address the variance decomposition of stabilizing genotype-fitness maps to analyze the effect of stabilizing selection on the ACP1 polymorphism. First, we have considered several ad-hoc genotype-fitness maps in accordance with the verbal model by Greene et al. (2000). In particular, we have fixed the fitness of the CC genotype at x CC =0.6 relative to the fitness of AB, x AB =1, and considered a variety of values for the fitnesses of the other genotypes. Then we have inspected whether they could explain the maintenance of the ACP1 equilibrium. To do so, we have plotted the ratio V A /V G for those genotype-fitness maps, using the observed Fig. 3 Contour plots of the ratio V A /V G for various fitness values in accordance with the verbal model by Greene et al. (2000) with the observed frequencies (from Brinkmann et al. 1971). 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