WITH increasing availability of fine-scale and highly frequencies between loci would be uncorrelated. On the

Transcription

1 Copyright 00 by the Genetics Society of America DOI: /genetics Epistasis of Quantitative Trait Loci Under Different Gene Action Models Rong-Cai Yang 1 Alberta Agriculture, Food and Rural Development, Edmonton, Alberta T6H 5T6, Canada and Department of Agricultural, Food and Nutritional Science, University of Alberta, Edmonton, Alberta T6G P5, Canada Manuscript received July 15, 003 Accepted for publication April 9, 00 ABSTRACT Modeling and detecting nonallelic (epistatic) effects at multiple quantitative trait loci (QTL) often assume that the study population is in zygotic equilibrium (i.e., genotypic frequencies at different loci are products of corresponding single-locus genotypic frequencies). However, zygotic associations can arise from physical linkages between different loci or from many evolutionary and demographic processes even for unlinked loci. We describe a new model that partitions the two-locus genotypic values in a zygotic disequilibrium population into equilibrium and residual portions. The residual portion is of course due to the presence of zygotic associations. The equilibrium portion has eight components including epistatic effects that can be defined under three commonly used equilibrium models, Cockerham s model, F -metric, and F -metric models. We evaluate our model along with these equilibrium models theoretically and empirically. While all the equilibrium models require zygotic equilibrium, Cockerham s model is the most general, allowing for Hardy-Weinberg disequilibrium and arbitrary gene frequencies at individual loci whereas F -metric and F -metric models require gene frequencies of one-half in a Hardy-Weinberg equilibrium population. In an F population with two unlinked loci, Cockerham s model is reduced to the F - metric model and thus both have a desirable property of orthogonality among the genic effects; the genic effects under the F -metric model are not orthogonal but they can be easily translated into those under the F -metric model through a simple relation. Our model is reduced to these equilibrium models in the absence of zygotic associations. The results from our empirical analysis suggest that the residual genetic variance arising from zygotic associations can be substantial and may be an important source of bias in QTL mapping studies. WITH increasing availability of fine-scale and highly frequencies between loci would be uncorrelated. On the dense genetic maps for different organisms, it is basis of this reference population, Cockerham (195) now possible to simultaneously identify and map several then developed a set of orthogonal scales (contrasts) so quantitative trait loci (QTL) controlling a trait of economic that the two-locus genotypic values could be partitioned and/or adaptive significance in plant and animal into independent components due to additive, domi- species. Unlike earlier efforts to map individual QTL one nance, and epistatic effects. Cockerham s model has been QTL at a time (see Lynch and Walsh 1998 for review), a standard for conventional quantitative genetic analysis this new approach requires that gene action models specify and recently for modeling multiple QTL (Kao and Zeng both allelic (additive and dominance) effects at individual 00). loci and nonallelic interactions (epistatic effects) between For populations derived from a cross between two loci. However, these genic effects must be defined in a inbred lines such as F and subsequent populations, two reference population because they depend not only on simpler models, F -metric and F -metric, have often been genotypic values but also on genotypic frequencies of the used. While translation of the genic effects defined under reference population. The most obvious choice of reference the F -metric and F -metric models can be easily population has been an ideal random mating popu- done (Van der Veen 1959), each model has its own lation where Hardy-Weinberg and linkage equilibria are characteristics. The F -metric model is actually a special assumed but the gene frequencies may be arbitrary at case of Cockerham s model where the gene frequencies different loci (Kempthorne 1957; Crow and Kimura are one-half and the genotypic frequencies are those 1970; Lynch and Walsh 1998). Cockerham (195) con- expected under Hardy-Weinberg equilibrium at each sidered a more general population where departures from of two uncorrelated loci. Thus the F -metric model pos- Hardy-Weinberg equilibrium at individual loci such as sesses the same property of orthogonality among the inbreeding could be accommodated but the genotypic genic effects as Cockerham s model and has been widely used (e.g., Hayman and Mather 1955; Kempthorne 1957; Cockerham and Zeng 1996; Goodnight 000; 1 Address for correspondence: Department of Agricultural, Food and Kao and Zeng 00). On the other hand, the genic effects Nutritional Science, 10 Agriculture/Forestry Centre, University of Alberta, Edmonton, AB T6G P5, Canada. defined under the F -metric model are not orthogonal, rong-cai.yang@ualberta.ca but this model has been equally popular (e.g., Crow Genetics 167: ( July 00)

2 19 R.-C. Yang TABLE 1 Joint frequencies of the nine genotypes for two loci, each with two alleles, A and a at locus A and B and b at locus B. These genotypic frequencies are expressed in terms of their single-locus genotypic frequencies and zygotic associations Locus B Locus A BB Bb bb Total AA P AB AB P A P B B AB AB P Ab AB P A P b B Ab AB P Ab Ab P A P b b Ab Ab P A A p A D A Aa P ab AB P a A P B B ab AB P AB ab P Ab ab P a A P b B ab AB P Ab ab P A a P b b Ab ab P A a p A p a D A Aa P ab ab P AaP a B B ab ab P ab ab P a P b B ab ab P ab ab P a P b b ab ab P a a p a D A Total P B B p B D B P B b p B p b D B P b b p b D B 1 The possible values of zygotic associations are constrained by the single-locus frequencies such that only four of the nine zygotic associations need to be defined and the remaining five are entirely expressed in terms of the four defined zygotic associations. For example, if the zygotic associations for four double homozygotes, AB, Ab, ab as follows: AB Ab ab, and ab, are defined, then the zygotic associations for the remaining five genotypes are expressed ( AB AB Ab), ab AB ( AB AB ab), ab ab ( ab ab ab), ab Ab ( Ab Ab ab), and ab AB AB AB Ab Ab ab ab ab. and Kimura 1970; Mather and Jinks 198; Yang and at the two loci using Cockerham s model and the F - Baker 1990; Haley and Knott 199; Carlborg et al. metric and F -metric models. 000; Yi and Xu 00) probably because of its easy and clear genetic interpretation as noted by Van der Veen COCKERHAM S MODEL WITH (1959) and Mather and Jinks (198). ZYGOTIC ASSOCIATIONS The key assumption with all these gene action models is that the genotypic frequencies at different loci are Two-locus genotypic frequencies and zygotic associa- products of the appropriate single-locus genotypic frelocus tions: For two loci, each with two alleles, A and a at quencies; in other words, the population is in zygotic A and B and b at locus B, there are 9 possible ge- equilibrium. However, zygotic associations can arise from notypes (10 if the coupling and repulsion double heterozy- physical linkages between different loci or from many gotes are distinguishable). Following Yang (000), we evolutionary and demographic processes even for unresult write frequencies of these genotypes as, P vz uy P vz uy, which linked loci (see Yang 00 for review). In the presence from union of gametes uy and vz with u, v A of zygotic associations, a large number of genic disequicies or a, and y, z B or b (Table 1). The genotypic frequen- libria including Hardy-Weinberg and linkage disequilibappropriate at individual loci are the marginal totals of the ria are required for a complete characterization of nonple, two-locus genotypic frequencies. For examria random associations at different loci (Cockerham and the frequency of genotype AA is Weir 1973; Weir 1996). These disequilibria have been considered in some general gene action models (e.g., P A A P AB AB P AB Ab P Ab. Gallais 197; Weir and Cockerham 1977; Kao and With the genotypic frequencies at locus A, the frequency Zeng 00) but a prohibitively large number of genic of allele A is effects arising from such models have prevented them p A P A A 1 P a from being estimated in experimental quantitative genetics A and QTL mapping studies. In this article, we first and that of allele a is p a 1 p A. A develop a new model that partitions the two-locus geno- Departures from Hardy-Weinberg equilibrium at locus typic values in a zygotic disequilibrium population into A are equilibrium and residual portions. Such partitioning is made possible by the decomposition of the two-locus D A P A A p A P a a p a [ 1 P a A p A p a ] (1a) genotypic frequencies into the expected frequencies and and those at locus B are deviations (zygotic associations) as given in Yang (000). The residual portion is of course due to the presence D B P B B p B P b b p b [ 1 P B b p B p b ]. (1b) of zygotic associations. We also examine the partitioning of the equilibrium portion into different genic effects In a Hardy-Weinberg equilibrium population, D A D B

3 Modeling Epistasis with Zygotic Associations In inbred populations, D A and D B depend on the level of inbreeding as measured by the inbreeding coefficients at loci A (f A ) and B (f B ). For example, the Hardy- Weinberg disequilibrium at locus A is D A f A p A p a, where f A P A A p A P a a p a 1 P a A. p A p a p A p a p A p a Yang (000) defined a zygotic association as a deviation of the two-locus genotypic frequency from the product of single-locus frequencies, which has the range of uy vz P uy vz P u v P y z, () P u v P y z vz uy min[p u v (1 P y z), (1 P u v )P y z]. Because these zygotic associations are constrained by the single-locus genotypic frequencies, u,v uy vz y,z uy vz 0, only of the 9 zygotic associations can be defined freely and the remaining 5 are expressed entirely in terms of the four defined zygotic associations (Table 1). For compact and clear presentation of subsequent developments, we describe the relations of two-locus genotypic frequencies with their expectations and zygotic associations (Table 1) in matrix form, where P, (3) P diag{p AB AB P AB Ab P Ab Ab P AB ab P AB ab P Ab ab P ab Ab P ab ab P ab Ab P ab}, diag{p A P B B P A P B b P A P b b P A a P B B P A a P B b P A a P b b P a P B B P a P B b P a P b} diag{ AB AB AB Ab Ab Ab AB ab AB ab Ab ab ab ab Ab ab ab} with diag{ } signaling a diagonal matrix. Cockerham s model: In the absence of zygotic associations ( 0 and P ), Cockerham (195) develand oped a regression model to partition the two-locus genotypic t is the coefficient of partial regression of G on w t, value into eight orthogonal scales, t (w t G)/(w t w t ). (5) G 1 It should be noted that the orthogonal scales given 8 w t t, () t 1 in the W matrix are slight modifications of those by Cockerham (195) to be consistent with more recent where G [G AB AB G AB Ab G Ab Ab G AB ab G AB ab G Ab ab G ab ab G ab ab G ab] is the uses of these scales (Cockerham and Zeng 1996; Kao vector of genotypic values corresponding to genotypes and Zeng 00). Like the usual orthogonal contrasts, [AABB AABb Aabb AaBB AaBb Aabb aabb aabb aabb], 1 the orthogonal scales satisfy the two basic requirements: is a vector of ones, (G 1) is the population (i) w t 1 0 and (ii) w t w t 0 for t t. In addition, mean, w t is the tth column vector of matrix W, these eight orthogonal scales correspond to the additive

4 196 R.-C. Yang and dominance effects at locus A (w 1 and w ), additive Departure from Cockerham s model: In the presence and dominance effects at locus B (w 3 and w ), and four of zygotic associations ( 0 and P ), howepistatic effects between the two loci, w 5 ( w 1 w 3 ), ever, a residual term (ε) needs to be added to Cockerw 6 ( w 1 w ), w 7 ( w w 3 ), and w 8 ( w w ). ham s regression model and Equation becomes The orthogonal scales in the W matrix can be used to partition the total genetic variance in a zygotic equi- G 1 8 w t t 1ε, librium population ( G) into eight independent components, t 1 (8) where is the mean of the zygotic disequilibrium population, G P1 G 1. Thus, the total genetic G G G 8 (w t w t ) t 8 t, (6a) variance in the zygotic disequilibrium population is where t 1 t 1 G G PG ( ) G G G G ( ) t (w t G) /(w t w t ). (6b) G ε ( ), (9) Cockerham (195) gave detailed expressions for t s where G is the total genetic variance under Cockerin an inbred population where Hardy-Weinberg disequiham s model as given in Equation 6a and ε is the librium is measured in terms of inbreeding coefficient. residual variance arising from zygotic associations. He also noted that in the absence of inbreeding (f A When the means and genetic variances in equilibrium f B 0 and D A D B 0), these variance components and nonequilibrium populations are known, ε can be became those in a Hardy-Weinberg equilibrium populaobtained from Equation 9, tion (Kempthorne 1957; Crow and Kimura 1970; Lynch and Walsh 1998). Because the different scales ε G G ( ) are orthogonal (i.e., w t w t 0 for all t t ), there are no covariances between them. G 8 t ( ). (10a) When p A p B 1 and D A D B 0(i.e., the gene and t 1 genotypic frequencies are those in an F population), Alternatively, it can be calculated directly from geno- Cockerham s model is reduced to the F -metric model. typic values and zygotic associations, In this particular case where the F population is in Hardy-Weinberg and zygotic equilibria [ 0 and P ( 1 16 )diag{1 111}], the expressions for ε G G (G). (10b) partial regression coefficients in Cockerham s model Using the relations among zygotic associations given in ( t s in Equation 5) are greatly simplified and are identi- Table 1, we have fied with specific genic effects, ε ABE 1 AbE abe 3 abe a A 1 (G A A G a a )/ d A (G A a G A A G a a )/ a B 3 (G B B G b b )/ d B (G B b G B B G b b )/ aa 5 (G AB AB G Ab Ab G ab ab G ab ab )/ ad 6 ( G AB AB G AB Ab G Ab Ab G ab ab G ab ab G ab ab )/ da 7 ( G AB AB G AB ab G ab ab G Ab Ab G Ab ab G ab ab )/ where [ ABe 1 Abe abe 3 abe ], E 1 (G AB) (G AB ab ) (G AB Ab ) (G AB ab ), E (G Ab) (G Ab ab ) (G AB Ab ) (G AB ab ), E 3 (G ab) (G AB ab ) (G ab ab ) (G AB ab ), E (G ab) (G Ab ab ) (G ab ab ) (G AB ab ), (10c) dd 8 (G AB AB G AB AB G Ab Ab G AB ab G AB ab G Ab ab G ab ab G ab ab G ab ab )/ (7) It should be noted that while ε is called the residual variance it is not necessarily positive because the zy- gotic associations in that are the deviations can be either positive or negative. (Cockerham and Zeng 1996; Cheverud 000; Kao and Zeng 00), where Gv u y,z P uy vz G uy vz and G z y u,v Pvz uy G uy vz are the average values of genotypes uv (AA, Aa, oraa) at locus A and yz (BB, Bb, orbb) at locus B, respectively. This set of genic effects along with the mean ( ) can be expressed in terms of another set under the F -metric model through a simple translation as given in Van der Veen (1959). A more detailed comparison of the F -metric model with the F -metric model is made later in F -metric vs. F -metric models. and e 1 G AB AB G AB ab G AB Ab G AB ab, e G Ab Ab G Ab ab G AB Ab G AB ab, e 3 G ab ab G AB ab G ab ab G AB ab, e G ab ab G Ab ab G ab ab G AB ab.

5 Modeling Epistasis with Zygotic Associations 197 Application: Doebley et al. (1995) identified two QTL, to those expected in CRP and the F population. Because UMC107 (designated as locus A) and BV30 (desigtypes of small numbers of individuals for some geno- nated as locus B), controlling differences in plant and (e.g., AAbb), we also carry out Monte Carlo exact inflorescence architecture between cultivated maize (Zea tests (Weir 1996). The exact P values are very similar mays ssp. mays) and teosinte (Z. mays ssp. parviglumis) in to the asymptotic P values from the chi-square tests. the BC 3 F population (Teosinte-M1L Teosinte-M3L) Thus, the assumption of F gene and genotypic frequen- derived from an original cross of Reventador maize cies used for analyzing Doebley et al. s data (Lynch and parviglumis teosinte. A total of nine morphological and Walsh 1998; Goodnight 000; Kao and Zeng 00) inflorescence traits were measured. Two of those traits, is probably justified. average length of vegetative internodes in the primary Despite insignificant departures from Hardy-Wein- lateral branch (LBIL) and percentage of cupules lackfor berg and zygotic equilibria, we retain these disequilibria ing the pedicellate spikelet (PEDS), have been reanaof illustrating the use of our new model. The mean values lyzed to examine the magnitude of epistasis between nine morphological and inflorescence traits for each loci A and B (Lynch and Walsh 1998; Goodnight of the nine genotypic classes as given in Doebley et al. s 000; Kao and Zeng 00). However, all these analyses (1995) Table 8 are used for our analysis. We partition have assumed p A p B 1 and D A D B 0asinan the genotypic values into eight genic effects defined F under Cockerham s model and the F population with two uncorrelated loci. We test to -metric model plus determine if such assumption holds for Doebley et al s a residual effect (Table 3). The estimated genic effects data. We then use the actual genotypic frequencies as under the F -metric model for LBIL are identical to given in Table 7 of Kao and Zeng (00) and the trait those in Lynch and Walsh (1998, pp ) and in values in Table 8 of Doebley et al. (1995) to illustrate Kao and Zeng s (00) Table 9 (under the Cocker- the application of our new model developed above. ham column), barring rounding errors. In most cases, The observed and expected genotypic frequencies the values under Cockerham s model and the F -metric and their differences (disequilibria) at loci UMC107 and model are similar as would be expected from insignifi- BV30 are given in Table. A sample of 183 individuals cant Hardy-Weinberg and zygotic disequilibria (cf. Table ). However, there are a few exceptions, particularly from the BC 3 F population was recorded in Doebley with the residual effects for LBIL and percentage of et al. (1995) but as Kao and Zeng (00) pointed out, staminate spikelets in primary lateral inflorescence 1 individuals had a missing value for one or more traits (STAM), where the two models differ most. These differmeasured. Thus, a sample size of 161 individuals is used ences are likely attributable to the deviations of gene for our analysis. The expected values are derived asfrequencies from one-half under Cockerham s model. suming Cockerham s reference population (CRP) and Table presents the partitioning of total genetic varithe F population. For a single locus, CRP may not ance ( G ) using (9) for each of the nine traits in the be in Hardy-Weinberg equilibrium with arbitrary gene presence of zygotic associations. Two different partitions frequencies whereas the F population is in Hardy-Weinare obtained, depending on whether the equilibrium berg equilibrium with the gene frequencies being conportion of total variance ( G) is calculated using Cockerstrained to the fixed value of one-half. Thus, the deviaham s model or the F -metric model. For example, for tions of observed from expected single-locus genotypic LBIL, G G ε ( ) frequencies in CRP are simply Hardy-Weinberg disequi (56.005)( ) under libria but such deviations in the F population are Hardy- Cockerham s model but G Weinberg disequilibria plus the differences between (56.888)( ) under the F -metric expected values in the two reference populations. For model. In other words, since the two models are based example, for genotype AA at locus A, the difference on different reference populations, they have different between observed and expected genotypic frequencies population means and genetic variances under zygotic in CRP is but the corre- equilibrium ( and G). Like the patterns of estimated sponding difference under the F population is genic effects in Table 3, contributions (percentages) , which equals to (0.8 due to individual variance components of G under 0.500). The expected two-locus genotypic frequencies Cockerham s model vs. the F -metric model are similar in CRP are the products of observed single-locus geno- in most cases; the residual variances are similar as well typic frequencies. For example, the expected frequency with the two notable exceptions for LBIL and STAM. of genotype AABb is ( ) In Further inspection on contributions due to individual contrast, the expected two-locus genotypic frequencies variance components of G under both models shows in the F population are the products of expected single- that additive effects are predominant components for locus genotypic frequencies. According to the chi-square all traits except for PEDS whereas dominance and epitests, neither Hardy-Weinberg disequilibria at individual static effects are minor. Epistasis is most important for loci nor zygotic associations between the two loci are PEDs with its effects accounting for nearly 0% of the significant. The observed genotypic frequencies fit well total equilibrium variance under both models.

6 198 R.-C. Yang TABLE One-locus and two-locus genotypic frequencies and chi-square tests in a population derived from a cross of Reventador maize parviglumis teosinte CRP F population Genotype Observed Expected Disequilibrium Expected Disequilibrium Locus A AA Aa aa Test for HWD 0.83; d.f. ; 1.76; d.f. ; at locus A: P P 0.15 Locus B BB Bb bb Test for HWD 0.50; d.f. ; 0.55; d.f. ; at locus B: P P Loci A B AABB AABb AAbb AaBB AaBb Aabb aabb aabb aabb Test for ZA 5.31; d.f. ;.19; d.f. ; for loci A B: P P Observed and expected genotypic frequencies and their differences (disequilibria) at two diallelic loci, UMC107 (designated as locus A) and BV30 (designated as locus B), in the BC 3 F population (Teosinte- M1L Teosinte-M3L) derived from an original cross of Reventador maize parviglumis teosinte as described by Doebley et al. (1995) and Kao and Zeng (00) are shown. The expected values are derived under Cockerham s reference population (CRP) and the F population. Chi-square tests are carried out to determine the presence of Hardy-Weinberg disequilibria (HWD) at individual loci and zygotic associations (ZA) between the two loci. Sample size is n 161. It is evident from Table that the residual portion maximum zygotic disequilibrium, compared to ε of the total genetic variance ( ε) is generally small compared with insignificant zygotic disequilibrium as to the equilibrium portion ( G). This is of course shown in Table ; in both cases, G Evi- due largely to insignificant zygotic disequilibrium between dently, the magnitude of zygotic association affects the the two interacting loci, UMC107 and BV30 partitioning of the total genetic variance. Further redently, (Table ). To gauge the impact of zygotic disequilibrium search is needed to examine detailed relationships beon the magnitude of ε, we let the Teosinte-M1L tween nonextreme values of zygotic disequilibrium and Teosinte-M3L population be at the maximum level of the residual variance. zygotic disequilibrium that is obtainable if double homozygotes and heterozygotes are equally frequent in the absence of single heterozygotes at both loci (Yang F -METRIC VS. F -METRIC MODELS 003). One of such possibilities is that the required P Conceptual distinction: As noted above, Cockerham s matrix would be estimated by (1/161)diag{ model is reduced to the F -metric model for F and }. With this particular genotypic distribution, other populations derived from a cross between two the magnitudes of ε are larger for all nine traits and inbred lines (cf. Equation 6). However, since the 1950s, for some traits are as much as or even greater than those of G (results not presented). For example, ε 5.8 for LBIL under the F -metric model with the the F -metric model has coexisted with another popular model of gene action, the F -metric model. In essence, genic effects under Cockerham s model or the F -metric

7 Modeling Epistasis with Zygotic Associations 199 TABLE 3 Partitioning the two-locus genotypic values in a population derived from a cross of Reventador maize parviglumis teosinte Genic effects Locus UMC107 Locus BV30 Epistasis between UMC107-BV30 Trait a ε CUPL CUPR DISA INNO LBIL LIBN PEDS STAM YOKE Partitioning the genotypic values for two interacting loci, UMC107 and BV30, affecting nine morphological traits in the BC 3 F population (Teosinte-M1L Teosinte-M3L) derived from an original cross of Reventador maize parviglumis teosinte as described by Doebley et al. (1995) and Kao and Zeng (00) is shown. The partitions for each trait include eight genic effects ( 1 8) under Cockerham s model (top) and the F -metric model (bottom) plus a residual (ε) arising from zygotic associations. a CUPL, average length of cupules (internodes) in the inflorescence (in millimeters); CUPR, number of cupules in a single rank of the ear; DISA, tendency of ear to shatter (1 5 scale), 1 100% disarticulating, 5 0% disarticulating; INNO, number of vegetative internodes in the lateral branch; LBIL, average length of vegetative internodes in the primary lateral branch (in millimeters); LIBN, number of branches in primary lateral inflorescence; PEDS, percentage of cupules lacking the pedicellate spikelet; STAM, percentage of staminate spikelets in primary lateral inflorescence; and YOKE, degree to which the fruitcases are in yoked pairs (1 5 scale). 1, 0% yoked fruitcases; 5, 100% yoked fruitcases. ficients of the genic effects defined under the F -metric model are uniquely associated with homozygosity and heterozygosity at different loci. Second, the expressions for describing heterosis and other genetic phenomena are simpler and more surveyable under the F -metric model. Finally the F -metric model leads to simpler and symmetrical conditions and descriptions of relationships identified for classical F segregation ratios with epistasis. In contrast, Kao and Zeng (00) recently argued against the use of the F -metric model particu- larly in QTL mapping studies because there is possible bias in estimating allelic effects in the presence of non- allelic (epistatic) effects due to the fact that the genic effects defined under the F -metric model in an F popu- lation are not orthogonal. In what follows, we clarify the relationships and differences between the two models through theoretical analysis and numerical evaluation. Parameter relations: With two uncorrelated loci in an F population (p A p B 1 and D A D B 0), the values of nine possible genotypes can be expressed as model are deviations from the mean of a noninbred equilibrium population whereas genic effects under the F -metric model are contrasts among genotypes without reference to any population. For example, the additive effect at locus A under Cockerham s model or the F -metric model ( 1 ) is a weighted average of the two differences (G A. A. G a.) A. and (G A. a. G a.) corresponding to the comparisons of genotypes, AA Aa and Aa aa, with each difference representing an effect of replacing a by A. The dominance effect ( ) would be present, i.e., (G A. A. G a.) A. (G a. A. G a.) 0, unless the two differences are exactly the same. On the other hand, the two genic effects at locus A under the F -metric model are one-half the difference between the two homozygotes and the deviation of the heterozygote from the homozygote mean, respectively (e.g., Mather and Jinks 198). The F -metric model is also referred to as the homozygote-based model (Wright 1987). Van der Veen (1959) gave three reasons why the F -metric model would be preferred. First, nonzero coef-

8 1500 R.-C. Yang TABLE Partitioning the two-locus genetic variance in a population derived from a cross of Reventador maize parviglumis teosinte Percentage of G Locus UMC107 Locus BV30 Epistasis between UMC107-BV30 Trait a G ε G CUPL CUPR DISA INNO LBIL LIBN PEDS STAM YOKE Partitioning the total genetic variance ( G ) for two interacting loci, UMC107 and BV30, affecting nine morphological traits in the BC 3 F population (Teosinte-M1L Teosinte-M3L) derived from an original cross of Reventador maize parviglumis teosinte as described by Doebley et al. (1995) and Kao and Zeng (00) is shown. The genetic variance with zygotic equilibrium ( G) for each trait is partitioned into eight independent components ( 1 8) under Cockerham s model (top) and the F - metric model (bottom). a Detailed descriptions of these traits are given in Table 3. G M E, (11) special case of Cockerham s model, columns 9 in the M 0 matrix are identical to the W matrix when p A p B where indicates whether the genic effects are defined 1 and D A D B 0. The translation of parameters from under the F -metric model ( 0) or F -metric model F -metric into F -metric can be made through the rela- (); E [ ( ) a A( ) d A( ) a B( ) d B( ) aa ( ) ad ( ) da ( ) dd ( ) ] tion E 0 TE 1 (Van der Veen 1959), where contains the parameters defined in (6) for 0 and M ( 1) 1 ( 1) 1 ( 1) 1 ( 1) 1 ( 1) 1 (1) Obviously, G M 0 E 0 and G M 1 E 1 describe the partitioning of genotypic values under F -metric and F -metric models, respectively. Since the F -metric model is a (13) Thus, the same genotypic values can be partitioned into the mean and eight genic effects defined under either the F -metric or the F -metric model so long as appropriate translations are identified, G M 0 E 0 M 0 TE 1 M 1 E 1 M 1 T 1 E 0. (1) In addition, it is quite obvious from (1) that M 0 M 1 T 1 and M 1 M 0 T.

9 Modeling Epistasis with Zygotic Associations 1501 TABLE 5 Coded genotypic values and F segregation ratios at two unlinked loci (A and B) for 10 nonepistatic models and 10 epistatic models Model Genotype F segregation no. AABB AABb AAbb AaBB AaBb Aabb aabb aabb aabb ratio 1 x 3x x 3x x x x x 0 1::6::1 x x x x x x x x 0 9:6:1 3 x x x x 0 0 x 0 0 1:6:9 x 5x x 5x 6x 3x x 3x 0 ::1:::1 5 x x x x x x x x 0 1:::1:: 6 6x x x 5x 3x x x x 0 1::3::3::1 7 6x 6x x 6x 6x x x x 0 9:3:3:1 8 6x x x x 0 0 x 0 0 1:3:3:9 9 6x 8x x 7x 9x 3x x 6x 0 :::3:1::1:1 10 6x 0 x 3x 3x x x x 0 1:1::1:3::: 11 x x 0 x x :7 1 x wx 0 wx wx :8:7 13 x x x x x x x x 0 15:1 1 x x 0 x x 0 x x x 13:3 15 x x 0 x x 0 x x x 9:3: 16 x x x x x x x x 0 1:3:1 17 x x 0 x x 0 x x x 7:6:3 18 x 3x 0 x 3x 0 x x 0 6:3:3: 19 x 3x x x 3x x 3x 0 x 7::3: 0 9x 9x 3x 9x 9x 3x 3x 3x x 9:1:6 With F gene and genotypic frequencies [i.e., 0 and P ( 1 16 )diag{1 111}]thepopulation mean is G (0), if F -metric (1) 1 d A(1) 1 d B(1) 1 dd (1), if F -metric. (15) The total equilibrium genetic variance can be partitioned under the F -metric model, G G G 0 E 0 M 0 M 0 E 0 (0) a A(0) d A(0) a B(0) d B(0) and under F -metric model, aa (0) ad (0) 8 G E 1 M 1 M 1 E 1 (1) 1 d A(1) 1 d B(1) 1 dd (1) [a a(1) 1 ad (1) ] aa (1) ad (1) 8 [d A(1) 1 dd (1) ] da (1) 8 dd (1) 16 da (0) 8 dd (0) 16, (16a) [a B(1) 1 da (1) ] [d B(1) 1 dd (1) ]. (16b) Numerical evaluation: We evaluate 0 genetic models exhibiting different segregating ratios for two independent loci in F populations (Table 5). These models are chosen to represent varying levels of additive, dominance, and epistatic effects under the F -metric and F -metric models. Yang and Baker (1990) also examined these genetic models, but only with the F -metric model. The first 10 models involve no epistasis and they are variants of the Mendelian F ratio of 9:3:3:1 for two indepen- dently segregating loci (model 7). These models vary from those of pure additive genic effects (models 1 and 6) to those of strong dominance (e.g., models 9 and 10). Models are those with classical epistatic ratios that can be found in F populations (Strickberger It is evident from comparing (16a) and (16b) that the two models differ in partitioning the total genetic variance G in an F population. Because, with F gene and genotypic frequencies, the genic effects under the F -metric model are not orthogonal the additive and dominance variances include covariances between allelic and nonallelic effects plus a portion of appropriate epistatic variances. It is also evident that the eight independent components of G as identified in Cockerham s model have simpler forms under the F -metric and F -metric models, (17)

10 150 R.-C. Yang TABLE 6 metric, thereby allowing for identification of relationships Coefficients required for equality among eight genic effects among the genic effects for each of the 0 genetic derived from the two-locus F -metric ( 0) and models (Table 6). The relationships identified under F -metric ( 1) for each of 0 genetic models F -metric and F -metric are the same for nonepistatic models (models 1 10) but different for epistatic models Model no. a a A( ) d A( ) a B( ) d B( ) aa ( ) ad ( ) da ( ) dd ( ) (models 11 0). For example, the genic relationships for model are a A( ) a B( ) d A( ) d B( ), aa ( ) ad ( ) da ( ) dd ( ) 0 under either F -metric ( 0) or F -metric ( 1), whereas the genic relationships for model 16 are aa (0) 10a B(0) d A(0) 10d B(0) aa (0) 5ad (0) 5da (0) 5dd (0) under F -metric ( 0) but a A(1) 3a B(1) d A(1) 3d B(1) 3aa (1) ad (1) 3da (1) 3dd (1) under F -metric ( 1) It is evident from Equation 13 that while the conditions of a A(0) a A(1) and a B(0) a B(1) always hold regardless of whether or not epistasis is present, those of d A(0) d A(1) and d B(0) d B(1) are true only in the absence of epistasis It should be emphasized that because these genic rela tionships are derived from the coded genotypic values with x being set to 1 (cf. Table 5), they will be multiplied by a constant if x is set to be any other nonzero integer The contributions due to individual genic effects are calculated using (17) (Table 7). The nonepistatic mod els (models 1 10) differ only in the sizes of additive genic effects and levels of dominance at each of the two loci. The 10 epistatic models (models 11 0) show varying levels of epistasis as evident from inspecting percentages of the contributions of individual epistatic effects to the total variance. Model 13 (duplicate epista sis) and model 19 (partial dominant epistasis) were the most epistatic of the models considered. Epistatic effects in model 19 account for almost 75% of the total variance whereas those in model 13 account for 61% of the total variance, although the constitutions of the four epistatic variance components in both F -metric and F -metric are quite different DISCUSSION The models of gene actions including epistasis be tween different loci developed for conventional quanti tative genetics and recent QTL mapping studies do not For example, the equality among the eight genic effects for have the same level of applicability because the genic model is a A( ) a B( ) d A( ) d B( ) ; aa ( ) ad ( ) da ( ) effects in these models are defined with reference to dd ( ) 0 under both F -metric ( 0) and F -metric ( 1). a different types of populations. These models range from Genetic models are described in Table 5. the ones for general experimental and natural populations where gene frequencies are arbitrary and Hardy- 1976). Model 1 is the partial complementary model, Weinberg disequilibrium is often present (e.g., Cockera modification of model 11 with 0 w 1. Model 0 ham s model by Cockerham 195) to the ones for is hypothetical with the relation of a A(0) a B(0) d A(0) special populations derived from a cross between two d B(0) aa (0) ad (0) da (0) dd (0) under the F -metric inbred lines with gene frequencies of one-half (e.g.,f -met- model. The multiplier x in each of the genotypic values ric model by Hayman and Mather 1955 or F -metric in Table 5 can be any nonzero integer. For convenience, model by Mather and Jinks 198). The key assumption we take x 1 and w 0.5 (for model 1) for subsequent involved in all the models is that the study population calculations. is in zygotic equilibrium (i.e., genotypic frequencies at We partition each genotypic value into the mean different loci are products of corresponding single-locus and eight genic effects under either F -metric or F - genotypic frequencies). However, zygotic associations

11 Modeling Epistasis with Zygotic Associations 1503 TABLE 7 Percentages of total variance in an equilibrium F population accounted for by eight individual variance components for 0 genetic models Percentage of total variance in F Locus A Locus B Epistasis between loci A and B Model no. a a Genetic models are described in Table 5. locus associations; in contrast, Kao and Zeng s model incorporated interlocus associations into Cockerham s model (F -metric model) by defining genetic and statisti- cal parameters for the same genic effects. Second, Kao and Zeng s model considered linkage disequilibrium in a Hardy-Weinberg equilibrium F population, which would be only one of several genic disequilibria required to completely characterize zygotic associations; the re- sidual variance ( ε) in our model essentially measures the relative importance of genetic variance due to zygotic associations (Equation 10c and see Table for examples). While our model does not expound the relationship between those genic disequilibria and zygotic associations, it can be easily achieved using Cockerham and Weir s (1973) disequilibrium functions (Yang 00). For example, the zygotic association for genotype AABB ( AB) can be expressed as can arise from physical linkages between different loci or from many evolutionary and demographic processes even for unlinked loci (Yang 00). Zygotic associations are considered in some general gene action models (e.g., Gallais 197; Weir and Cockerham 1977) but a prohibitively large number of genic effects arising from such models have prevented them from being estimated in experimental quantitative genetics and QTL mapping studies. This article develops a new model that partitions the two-locus genotypic values in a zygotic disequilib- rium population into equilibrium and residual portions. The equilibrium portion has eight genic effects that can be defined under those equilibrium models (Cocker- ham s model and the F -metric and F -metric models). The residual portion is of course due to the presence of zygotic associations and can be a substantial part of the total genetic variance (Table ). Thus, bias will occur when an equilibrium gene action model is used in modeling and detecting epistatic QTL for a zygotic disequilibrium population. Kao and Zeng (00) also considered associations of genotypic frequencies between different loci in modeling epistatic effects but our model differs from theirs in two ways. First our model separates the genotypic values and genetic variance into a portion arising from zero interlocus association (Cockerham s model) and the other portion arising from the presence of inter- AB AB P AB AB P A A P B B p A D AB.B p B D AB A. p A p B D AB.. p A p B D A..B (D AB.. ) (D A..B) D AB, where each genic disequilibrium (D) is the deviation of a frequency from that based on random association of genes and accounting for any lower-order disequilibria. For example, linkage disequilibrium (D AB.. ) is the deviation of frequency of gamete AB from the product of

12 150 R.-C. Yang frequencies of allele A at locus A and allele B at locus thus unbiased estimates of allelic effects can be obtained B, D AB.. P AB.. p A p B, with regardless of whether or not epistasis is present (Cheverud 000; Kao and Zeng 00). Since the translation P AB.. P AB AB P AB Ab P AB ab P AB ab. between the F -metric and F -metric models can be When zygotes result from random union of gametes readily done through the simple relation (13), we recas assumed in Kao and Zeng (00), all nongametic ommend that in QTL mapping studies the F -metric disequilibria including Hardy-Weinberg disequilibrium model should be used to first estimate the genic effects disappear (e.g., D A. A. D.B A. D AB.B D AB AB 0). In and then the estimated genic effects should be con- this case, the zygotic association for genotype AABB verted into those under the F -metric model to capture ( AB) reduces to its interpretation advantages. AB AB p A p B D AB.. (D AB.. ), Our comparative assessment of different gene action models suggests the need for caution when reading the which agrees with the frequency of the same genotype standard labels of additive genetic variance, domi- given in Table 6 of Kao and Zeng (00). The same ar- nance variance, etc., in the literature. As shown in this gument is true for the remaining eight genotypes. article and elsewhere (e.g., Cockerham 1963; Cocker- Our theoretical and empirical (Table ) analyses clearly ham and Tachida 1988), genic effects and their variances show substantial differences between Cockerham s model change with the population of reference. For exshow and the F -metric model. Kao and Zeng (00) consid- ample, in the CRP, the additive variance at locus A can ered the F population where gene frequencies are onehalf be obtained from (6b) and (7), and genotypic frequencies are those expected un- der Hardy-Weinberg equilibrium at each of the two loci. 1(CRP) p A p a P A. A. (G A. A. G a.) A. P a. a. (G a. A. Ga.) p A p a P A. A. P a. a. p A p a. In this case, Cockerham s model is the same as the F - metric. In all other cases, however, the two models would If CRP is an inbred population (IP), this additive varibe different. In analyzing the PEDS data of Doebley et al. ance becomes (1995), Goodnight (000) used Cockerham s model to calculate eight variance components of the total genetic variance using three gene frequencies at both UMC107 1(IP) p A p a (1 f A ) a A(0) 1 f A (p a p A )d A(0) 1 f A, and BV30: p A p B 0.5, 0.5, and 0.75, but no Hardywhich is reduced to the more familiar form in a random Weinberg disequilibrium was considered at either locus. His case of p mating population (f A A p B 0.5 corresponds to the F -metric 0), as evident from our analysis of the same trait (Table ). Nevertheless, it is evident from his Table that changes 1(f A 0) p A p a [a A(0) (p a p A )d A(0) ]. in gene frequencies greatly affect the distributions of Clearly, the so-called additive variance is contaminated variance components due to the genic effects, which is with the dominance effect unless the gene frequencies certainly consistent with our finding. are one-half as obtained under F -metric model (Equa- We distinguish the genic effects defined under the tion 17) or the dominance effect is absent. On the other F -metric model ( 0) from those under the F -metric hand, as shown in Equation 17, the additive variance model ( 1). Such distinction should help to reduce under the F -metric model is contaminated with the the confusion arising from the use of the same notations epistatic effect. With zygotic associations (ZA) as in most for both models. Such distinction also enables us to natural populations, the additive variance takes an even clearly identify the differences in parameter relations more complicated form, between the two sets of parameters for different genetic models describing F segregation patterns (Table 6). A(ZA) (CRP) 3(CRP) part of ε, Until recently, there was a lack of appreciation that dif- whose amount can be quantified in terms of various ferent gene action models would lead to different partitions genic disequilibria. For example, part of the dominance of genic effects and subsequently different vari- and additive additive variances within finite popula- ance components due to the genic effects. Because the tions undergoing bottleneck, drift, or population sub- F -metric model has clear interpretation advantages as division has been found to behave as additive variance noted by Van der Veen (1959) and Mather and Jinks (Cockerham and Tachida 1988; Goodnight 1988; (198), it has often been used for modeling multiple Whitlock et al. 1993). Such nonadditive genetic variances QTL (e.g., Haley and Knott 199; Carlborg et al. contribute to immediate and permanent response 000; Yi and Xu 00). However, because the genic to selection in finite populations in an intricate fashion. effects defined under the F -metric model with the F These discussions serve to emphasize that the additive gene and genotypic frequencies are not orthogonal, bias variance is not free from the influence of nonadditive will occur in estimating allelic effects when nonallelic genic effects and thus does not have simplicity and (epistatic) effects are present; in contrast, the genic clean meaning unless they arise from the F -metric effects under the F -metric model are orthogonal and model. Fortunately, most of the current QTL mapping

13 Modeling Epistasis with Zygotic Associations 1505 activities particularly with plant and laboratory species J. B. Wolf, E.D.Brodie III and M. J. Wade. Oxford University Press, New York. have focused on the use of planned mating designs cor- Cockerham, C. C., 195 An extension of the concept of partitioning responding to the F -metric model (Lynch and Walsh hereditary variance for analysis of covariances among relatives 1998). when epistasis is present. Genetics 39: Cockerham, C. C., 1963 Estimation of genetic variances. Statistical The extension of our two-locus model for nonequilib- genetics and plant breeding. NAS-NRC 98: rium populations to three or more loci is straightforto selection for quantitative characters in finite populations. Proc. Cockerham, C. C., and H. Tachida, 1988 Permanency of response ward. For example, for three uncorrelated loci each Natl. Acad. Sci. USA 85: with two alleles, the frequencies of 7 possible genotypes Cockerham, C. C., and B. S. Weir, 1973 Descent measures for two are the appropriate products of three single-locus geno- loci with some applications. Theor. Popul. Biol. : Cockerham, C. C., and Z-B. Zeng, 1996 Design III with marker loci. typic frequencies. These three-locus genotypic frequen- Genetics 13: cies are required for partitioning the three-locus geno- Crow, J. F., and M. Kimura, 1970 An Introduction to Population Genetic typic values into 6 genic effects as expected under Theory. Harper & Row, New York. Doebley, J., A. Stec and C. Gustus, 1995 teosinte branched1 and Cockerham s model or the F -metric model: 3 additive, origin of maize: evidence for epistasis and the evolution of dominance. Genetics 11: dominance, 1 two-locus epistatic, and 8 three-locus epistatic effects. In the presence of two-locus and/or Gallais, A., 197 Covariances between arbitrary relatives with link- age and epistasis in the case of linkage disequilibrium. Biometrics three-locus zygotic associations, an additional residual 30: 9 6. genic effect arises from nonzero deviations of the observed Goodnight, C. J., 1988 Epistasis and the effect of founder events on the additive genetic variance. Evolution : 1 5. from expected three-locus genotypic frequencies. Such Goodnight, C. J., 000 Quantitative trait loci and gene interaction: deviations are actually functions of 1 two-locus zygotic the quantitative genetics of metapopulation. Heredity 8: associations and 8 three-locus zygotic associations. It Haley, C. S., and S. A. Knott, 199 A simple regression method for mapping quantitative trait loci in line crosses using flanking remains to be investigated what the exact forms of these markers. Heredity 69: functions are. Perhaps the functions of multilocus heter- Hayman, B. I., and K. Mather, 1955 The description of genetic ozygosity developed by Yang (00) may be modified interactions in continuous variation. Biometrics 11: Kao, C.-H., and Z-B. Zeng, 00 Modeling epistasis of quantitative to accommodate individual genotypes as required here. trait loci using Cockerham s model. Genetics 160: Our model can also be extended to the case of multiple Kempthorne, O., 1957 An Introduction to Genetic Statistics. John Wiley & Sons, New York. alleles per locus. For example, with three alleles at each Lynch, M., and B. Walsh, 1998 Genetics and Analysis of Quantitative of two independent loci, there are 6 possible genotypes Traits. Sinauer Associates, Sunderland, MA. at each locus and 36 genotypes at the two loci. The Mather, K., andj. L. Jinks, 198 Biometrical Genetics, Ed. 3. Chapman & Hall, London. genotypic values are partitioned into the mean and 35 Strickberger, M. W., 1976 Genetics, Ed.. Macmillan, New York. genic effects under Cockerham s model or the F -metric Van der Veen, J. H., 1959 Tests of non-allelic interaction and linkage model: 10 (5 at each locus) allelic effects and 5 nonalleloid pure lines. Genetica 30: for quantitative characters in generations derived from two dip- lic (epistatic) effects. With zygotic associations between Weir, B. S., 1996 Genetic Data Analysis II. Sinauer Associates, Sunderland, MA. the two loci, the residual genic effect is due to the presence of 5 zygotic associations that can be freely Weir, B. S., and C. C. Cockerham, 1977 Two-locus theory in quanti- tative genetics, pp in Proceedings of the International Conferdefined. ence on Quantitative Genetics, edited by E. Pollak, O. Kempthorne and T. B. Bailey, Jr. Iowa State University Press, Ames, IA. I am grateful to Marjorie Asmussen for her help and regret her re- Whitlock, M. C., P. C. Phillips and M. J. Wade, 1993 Gene interaccent and sudden death while this manuscript was still under review. tion affects the additive genetic variance in subdivided popula- I thank Yun-Xin Fu for stepping in to complete the review and two tions with migration and extinction. Evolution 7: reviewers for valuable comments. This research was supported in part Wright, A. J., 1987 Covariance of inbred relatives with special reference to selfing. Genetics 115: by the Natural Sciences and Engineering Research Council of Canada grant OGP Yang, R.-C., 000 Zygotic associations and multilocus statistics in a nonequilibrium diploid population. Genetics 155: Yang, R.-C., 00 Analysis of multilocus zygotic associations. Genetics 161: Yang, R.-C., 003 Gametic and zygotic associations. Genetics 165: LITERATURE CITED Yang, R.-C., and R. J. Baker, 1990 Effects of epistasis on tests for linkage Carlborg, O., L. Andersson and B. Kinghorn, 000 The use of in self-pollinated species. Theor. Appl. Genet. 79: 19. a genetic algorithm for simultaneous mapping of multiple inter- Yi, N., and S. Xu, 00 Mapping quantitative trait loci with epistatic active quantitative trait loci. Genetics 155: effects. Genet. Res. 79: Cheverud, J. M., 000 Detecting epistasis among quantitative trait loci, pp in Epistasis and the Evolutionary Process, edited by Communicating editor: Y.-X. Fu