Estimates of Genetic Variance in an F2 Maize Population D. P. Wolf, L. A. Peternelli, and A. R. Hallauer
Maize (Zea mays L.) breeders have used several genetic-statistical models to study the inheritance of quantitative traits. These models provide information on the importance of additive, dominance, and epistatic genetic variance for a quantitative trait. Estimates of genetic variances are useful in understanding heterosis and determining the response to selection. The objectives of this study were to estimate additive and dominance genetic variances and the average level of dominance for an F2 population derived from the B73 ⴛ Mo17 hybrid and use weighted least squares to determine the importance of digenic epistatic variances relative to additive and dominance variances. Genetic variances were estimated using Design III and weighted least squares analyses. Both analyses determined that dominance variance was more important than additive variance for grain yield. For other traits, additive genetic variance was more important than dominance variance. The average level of dominance suggests either overdominant gene effects were present for grain yield or pseudo-overdominance because of linkage disequilibrium in the F2 population. Epistatic variances generally were not significantly different from zero and therefore were relatively less important than additive and dominance variances. For several traits estimates of additive by additive epistatic variance decreased estimates of additive genetic variance, but generally the decrease in additive genetic variance was not significant.
From Golden Harvest Research, North Platte, Nebraska (Wolf ), Universidade Federal De Vic¸osa, MG, Brazil (Peternelli), and the Department of Agronomy, Iowa State University, Ames, IA 50011 ( Hallauer). This is a contribution of the Department of Agronomy and journal paper no. J-18459 of the Iowa Agricultural and Home Economics Experiment Station (Ames), project 3495. This article is part of a dissertation submitted by D. P. Wolf in partial fulfillment of the requirements for a Ph.D. degree. Address correspondence to A. R. Hallauer at the address above or e-mail:
[email protected]. 2000 The American Genetic Association 91:384–391
384
Information on genetic variances, levels of dominance, and the importance of genetic effects have contributed to a greater understanding of the gene action involved in the expression of heterosis. The Design III mating design (Comstock and Robinson 1952) has primarily been used in maize F2 populations to determine the effects of linkage on estimates of additive and dominance genetic variances and on the average level of dominance ( Hallauer and Miranda Fo 1988). Design III has shown generally that genes controlling quantitative traits in maize F2 populations are in the partial to complete dominance range. There has been little evidence for genes with overdominance controlling quantitative traits. Pseudo-overdominance, when detected, has generally been due to linkage effects (Gardner et al. 1953; Gardner and Lonnquist 1959; Moll et al. 1964). Use of the Design III and other geneticstatistical models to estimate genetic variances usually assumes epistasis to be absent or of little importance. Several studies indicate that epistasis is not a significant component of genetic variability in maize populations (Chi et al. 1969; Eber-
hart et al. 1966; Silva and Hallauer 1975). Other studies have shown, however, that epistatic effects are important for specific combinations of inbred lines ( Bauman 1959; Gorsline 1961; Lamkey et al. 1995; Sprague et al. 1962). Specific crosses with epistatic effects likely have unique combinations of genes contributing to heterosis. These unique combinations are restricted to the specific cross and may be of little importance in a maize population, and if the frequency of genetic combinations that exhibit epistatic effects are low the variability due to epistasis may not be detected when effects are spread throughout the population ( Hallauer and Miranda Fo 1988). A previous study (Wolf and Hallauer 1997) determined that epistatic effects were significant for several traits in the B73 ⫻ Mo17 hybrid. In the present study it is possible to determine the importance of epistatic genetic variance relative to additive and dominance variance for this hybrid. The objectives of our study were to estimate additive and dominance genetic variances and the average level of dominance for the F2 population derived from
the B73 ⫻ Mo17 cross and use weighted least squares to determine the importance of digenic epistatic variances relative to additive and dominance variances.
Materials and Methods Genetic Materials The hybrid B73 ⫻ Mo17 was an important and widely grown hybrid in the central U.S. corn belt in the late 1970s and early 1980s. Inbred B73 was a selection from Iowa Stiff Stalk Synthetic after five cycles of half-sib recurrent selection for grain yield (Russell 1972). Inbred Mo17 was derived by selection from the single cross of inbred lines, CI187-2 ⫻ C103 ( Zuber 1973). In 1991 an F2 population derived from the B73 ⫻ Mo17 cross was grown at the Agronomy Research Farm near Ames, Iowa. Using the triple testcross ( TTC) mating design ( Kearsey and Jinks 1968), 100 random F2 plants (males) were crossed to both parents ( B73 and Mo17) and the F1. B73, Mo17, and the F1 were considered testers. Each F2 plant was selfed to form S1 progenies. For Design III analysis only data from B73 and Mo17 testcrosses are needed. Experimental Procedures Testcross and S1 progeny were evaluated in separate experiments. The 300 testcross entries were evaluated in a replications-within-sets, randomized incomplete block design with two replications per set. Ten sets were used, and each set included 30 entries comprised of three testcrosses from each of 10 different F2 plants. The S1 progeny were grown in a 10 ⫻ 10 lattice with two replications. Both experiments were grown at the Agronomy Research Center near Ames, the Atomic Energy Farm in Ames, and near Elkhart, Iowa, in 1992. In 1993, experiments were evaluated at the Agronomy Research Center and the Ankeny Research Farm. Each location by year combination was treated as a different environment. Each plot was a single row 5.49 m in length with 0.76 m between plots. Plots were overplanted and thinned to a stand of 57,520 plants/ha. Sixteen traits were measured in both experiments. Days from planting to 50% anthesis and silk emergence were recorded at the Agronomy Research Center in 1992 and 1993, and at the Atomic Energy Farm in 1992. Silk delay was calculated as the difference between anthesis and silk emergence. Plant and ear heights (cm) were calculated as the average measurement of
10 competitive plants within a plot at all environments, except Elkhart. Plant and ear height were measured from ground level to the collar of the flag leaf and uppermost ear node, respectively. Ten competitive plants within a plot were hand harvested (with gleaning for dropped ears) at all locations and ears were dried to a uniform moisture. Data for the following traits were measured as the average of 10 primary ears or plants, ear diameter (cm), cob diameter (cm), ear length (cm), kernel-row number, and ears per plant. Kernel depth was recorded as the difference between ear and cob diameter. Grain yield was determined from all primary and secondary ears and expressed in grams per plant. Barren plants were expressed as the percentage of 10 harvested plants that did not produce an ear. Root lodging (percentage of plants leaning more than 30 degrees from vertical), stalk lodging (percentage of plants broken at or below the primary ear node), and dropped ears (percentage of plants with dropped ears at harvest) were based on the total number of plants in a plot and recorded at five environments.
Statistical Analysis Genetic Variance Components The genetic-statistical model for Design III was followed to derive genetic variance components for the F2 reference population (Comstock and Robinson 1952). Analyses of variance (ANOVAs) combined across environments were used to estimate variance components. From the Design III analysis, additive genetic (A2 ), ad2 ditive by environment (AE ), dominance 2 genetic (D ), and dominance by environ2 ment (DE ) variance components were estimated. The necessary components were calculated as variation among males [M2 ⫽ covariance half-sibs ⫽ (1/4)A2 ]; envi2 2 ronment by male [EM ⫽ (1/4)AE ]; tester 2 by male [MT ⫽ D2 ]; and tester by male by 2 2 environment [TME ⫽ DE ]. Estimates of 2 2 A and D were used to estimate the aver2 age level of dominance as d¯ ⫽ MT /2M2 )1/2 ⫽ (2D2 /A2 )1/2. From the combined ANOVA for S1 progeny, genotypic (G2 ) and genotypic by en2 vironment (GE ) variance components were estimated. Because an F2 population was sampled, the expected gene frequencies of segregating loci are 0.5. At gene frequencies of 0.5 the G2 of S1 progeny can be expressed in genetic components as G2 ⫽ A2 ⫹ (1/4) D2 . Standard errors (SE) for all variance components were calculat-
ed using the method of Anderson and Bancroft (1952); SE ⫽ {2/C2 ⌺i [(MSi )2/(ni ⫹ 2)]}1/2, where MSi ⫽ i th mean square; ni ⫽ degrees of freedom associated with the i th mean square; and C ⫽ coefficient of the variance component in the expected mean square. Variance components were considered significantly different from zero if they were greater than twice their standard error. If estimates are distributed normally the 95% confidence interval will be bounded by ⫾2 standard errors of the estimate. Estimates were considered different from each other if their confidence intervals did not overlap. Because half-sib and S1 progeny were derived from the same F2 parents, the covariance between them can be translated into genetic variance components. Mean products were obtained from the combined analysis of covariance between halfsib and S1 means as discussed by Matzinger and Cockerham (1963). Mean products were multiplied by two to put them in the same magnitude as mean squares from the ANOVAs. Expectations for mean cross products have the same general form as for the mean squares (Mode and Robinson 1959), and therefore covariance components can be derived from mean products as MXY ⫽ rXYE ⫹ reXY; MXYE ⫽ rXYE; XY ⫽ (MXY ⫺ MXYE)/re; and XYE ⫽ MXYE/r, where MXY ⫽ mean product between halfsib ( X) and S1 ( Y) progeny; MXYE ⫽ interaction of environment by half-sib and S1 progeny mean product; XY ⫽ covariance of half-sib and S1 progeny; and XYE ⫽ covariance by environment interaction. The genetic covariance between half-sib and S1 progeny was derived by Bradshaw (1983) and rederived by Peternelli et al. (1999) for the special case of half-sib families obtained as the average of the three respective testcross families ( F2 ⫻ P1, F2 ⫻ P2, and F2 ⫻ F1) as used in the present study. For a genetic model that includes digenic epistatic variances this covariance can be 2 expressed as XY ⫽ (1/2)A2 ⫹ (1/4)AA and 2 2 XYE ⫽ (1/2)AE ⫹ (1/4) AAE. These components, however, may be biased by epistatic terms that include dominance effects (Peternelli et al. 1999). Standard errors of components of covariance were estimated by the following formula ( Dickerson 1969): SE ⫽ {1/C 2 ⌺i [(Mi XX)(Mi YY) ⫹ (Mi XY)2]/(nI ⫹ 2)}1/2, where C ⫽ coefficient of the component of covariance; Mi XX and Mi YY ⫽ mean squares for half-sib and S1 progeny; Mi XY ⫽ mean product for half-sib and S1 progeny; and ni ⫽ degrees of freedom of i th mean product. Wolf and Hal-
Wolf et al • Genetic Variance in Maize 385
lauer (1997) used the triple testcross analysis for the same type of population, and estimates of epistatic effects were significant for several traits. For the present study, however, the potential bias for the estimation of the different components of variance will be considered either absent or negligible to permit comparisons of the variance component estimates from the different populations. Weighted Least Squares From Design III, S1 progeny, and covariance combined analyses, there were 10 mean squares and mean products that were translated into genetic components of variance and error variances. Mean squares and products were expressed in terms of genetic components of variance through digenic epistatic components and error variances as follows: From the Design III, Males
2 2 ⫽ 2e1 ⫹ rt[(1/4) AE ⫹ (1/16) AAE ] 2 ⫹ rte[(1/4) A2 ⫹ (1/16) AA ];
⫹ rt[(1/4)
b
2AE
2D
2DE
2AA
2AAE
2DD
2DDE
2AD
2ADE
2e1
2e2
Design III Males (M) Males ⫻ environment ( E) Males ⫻ tester ( T ) M⫻T⫻E Error
5.00 0.00 0.00 0.00 0.00
1.00 1.00 0.00 0.00 0.00
0.00 0.00 10.00 0.00 0.00
0.00 0.00 2.00 2.00 0.00
1.25 0.00 0.00 0.00 0.00
0.25 0.25 0.00 0.00 0.00
0.00 0.00 10.00 0.00 0.00
0.00 0.00 2.00 2.00 0.00
0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
1.00 1.00 1.00 1.00 1.00
0.00 0.00 0.00 0.00 0.00
S1 progeny Genotypes (G) G⫻E Error
9.30 0.00 0.00
2.00 2.00 0.00
2.30 0.00 0.00
0.50 0.50 0.00
9.30 0.00 0.00
2.00 2.00 0.00
0.58 0.00 0.00
0.12 0.12 0.00
2.30 0.00 0.00
0.50 0.50 0.00
0.00 0.00 0.00
1.00 1.00 1.00
Mean products Half-sib/S1 Half-sib/S1 ⫻ E
4.70 0.00
1.00 1.00
0.00 0.00
0.00 0.00
2.40 0.00
0.50 0.50
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
All traits except plant and ear heights, anthesis, silk emergence, and silk delay. 2 2 Variance components: additive genetic ( 2A ); dominance (2D ); digenic epistasis of 2A and 2D ( AA , 2DD , AD ); interaction of these components by environment (2AE , 2DE , 2AAE , 2DDE , 2ADE ); experimental error of the design III ( 2e1); and experimental error of S1 progeny ( 2e2).
⫽ rXYE ⫹ reXY 2 ⫽ r[(1/2) 2AE ⫹ (1/4) AAE ]
Half-sib and S1 ⫻ environment
2 ⫽ 2e1 ⫹ rt EM
⫽
a
2A
2 ⫹ re[(1/2) A2 ⫹ (1/4) AA ]; and
Males ⫻ environment
2 AE
Variance componentsb
Half-sib and S1
2 2 ⫽ 2e1 ⫹ rt EM ⫹ rte M
2 e1
Table 1. Matrix of coefficients for mean squares and mean products in terms of genetic, genetic by environment, and error variances for combined analysis of traits measured in five environmentsa
⫹ (1/16)
2 AAE
];
Male ⫻ tester 2 ⫽ 2e1 ⫹ r ETM ⫹ re 2TM 2 2 ⫽ 2e1 ⫹ r( DE ⫹ DDE )
⫹ re( 2D ⫹ 2DD ); and Male ⫻ tester ⫻ environment 2 ⫽ 2e1 ⫹ r ETM 2 2 ⫽ 2e1 ⫹ r( DE ⫹ DDE ).
From the S1 progeny, Genotypes 2 ⫽ 2e2 ⫹ r GE ⫹ re G2
2 2 ⫽ rXYE ⫽ r[(1/2) AE ⫹ (1/4) AAE ]. 2 2 2 For the mean squares, AA , DD , and AD are the digenic epistatic variance compo2 2 2 nents; AAE , DDE , and ADE are the digenic 2 epistatic by environment variances; e1 is 2 the error variance of Design III; e2 is the error variance of S1 progeny; XY is the covariance of half-sibs and S1 progeny; and XYE is the covariance by environment. Translation matrices of mean squares and products into coefficients of genetic and error variance components for the complete model are presented in Table 1. Weighted least squares as discussed by Nelder (1960) were used to estimate genetic variance components. The weighted analysis can be expressed as
2 2 ⫽ 2e2 ⫹ r[ AE ⫹ (1/4) DE ⫹ 2AAE
⫹ (1/16) 2DDE ⫹ (1/4) 2ADE ] ⫹ re[ 2A ⫹ (1/4) 2D ⫹ 2AA 2 ⫹ (1/16) DD ⫹ (1/4) 2AD ]; and
Genotypes ⫻ environment 2 ⫽ 2e2 ⫹ r GE 2 2 ⫽ 2e2 ⫹ r[ AE ⫹ (1/4) DE ⫹ 2AAE
⫹ (1/16) 2DDE ⫹ (1/4) 2ADE ]. From the mean products,
386 The Journal of Heredity 2000:91(5)
ˆ ⫽ ( X⬘WX)⫺1 ( X⬘WY), B where ˆ ⫽ column vector of estimated genetic B and error variances; X ⫽ matrix of coefficients of the genetic and error variances; W ⫽ matrix with the inverse of the variances of mean squares and mean products on the diagonal and zero on the off diagonal; and Y ⫽ column vector of observed mean squares and products.
Standard errors of the parameter estimates were computed as the square root of the associated diagonal element of the ( X⬘WX)⫺1 matrix. Variances of mean squares and products were calculated by the methods of Mode and Robinson (1959). The following formula was used for the variance of a mean square: V(Mi ) ⫽ [2(Mi )2/dfi ⫹ 2], where Mi ⫽ ith mean square; and dfi ⫽ degrees of freedom of ith mean square. The following formula was used for the variance of a mean product: V(Mi XY) ⫽ [(MiXX )(MiYY ) ⫹ (MiXY ) 2 ] ⫼ (dfi ⫹ 2), where Mi XX and MiYY ⫽ ith mean squares for half-sib and S1 progeny; Mi XY ⫽ ith mean product of halfsib and S1 progeny; and dfi ⫽ degrees of freedom of the ith mean product. ˆ, To estimate the genetic parameters of B different models were tested. Not all genetic and error variances, however, could be estimated from a single model. A complete model included 10 genetic variances and two error variances. The adequacy of each model was tested using a chi-square test (Mather and Jinks 1982). X2 ⫽
冘 [(O ⫺ E) ·V], 2
where O ⫽ observed mean square or product,
Table 2. Estimates of variance componentsa (ⴞ standard error) and average level of dominance (d ) from the Design III ANOVA across five environmentsb for the B73 ⴛ Mo17 F 2 maize population Trait
2A
Yield (g/plant) Ear diameter (cm)d Cob diameter (cm)d Kernel depth (cm)d Ear length (cm) Kernel rows (no.) Ears/plant (no.)d Barren plants (%) Root lodging (%) Stalk lodging (%) Dropped ears (%) Plant height (cm) Ear height (cm) Anthesis (days) Silk emergence (days) Silk delay (days)
52.46 ⫾ 2.15 ⫾ 1.62 ⫾ 0.89 ⫾ 0.60 ⫾ 1.13 ⫾ 0.06 ⫾ 2.91 ⫾ 0.38 ⫾ 6.56 ⫾ 0.45 ⫾ 180.01 ⫾ 136.65 ⫾ 3.53 ⫾ 3.25 ⫾ 0.40 ⫾
2AE 18.25 0.43 0.28 0.23 0.14 0.18 0.02 1.50 0.25 1.78 0.24 28.15 21.68 0.63 0.58 0.13
2D
65.68 ⫾ 0.40 ⫾ 0.07 ⫾ 0.04 ⫾ 0.60 ⫾ ⫺0.02 ⫾ 0.09 ⫾ 6.55 ⫾ ⫺0.11 ⫾ 2.76 ⫾ 0.67 ⫾ 14.06 ⫾ 13.65 ⫾ 0.60 ⫾ 0.72 ⫾ 0.30 ⫾
27.65 0.31 0.13 0.28 0.13 0.03 0.04 2.77 0.53 2.21 0.46 4.01 3.79 0.25 0.22 0.14
155.63 ⫾ 1.16 ⫾ 0.24 ⫾ 0.50 ⫾ 0.40 ⫾ 0.12 ⫾ 0.00 ⫾ ⫺0.24 ⫾ 0.04 ⫾ 1.05 ⫾ 0.17 ⫾ 20.68 ⫾ 10.98 ⫾ 0.46 ⫾ 0.53 ⫾ 0.07 ⫾
2DE 27.45 0.23 0.06 0.13 0.08 0.02 0.01 0.50 0.11 0.49 0.11 3.78 2.32 0.12 0.12 0.05
15.96 ⫾ 0.14 ⫾ ⫺0.05 ⫾ 0.13 ⫾ 0.13 ⫾ ⫺0.02 ⫾ 0.02 ⫾ 2.70 ⫾ 0.08 ⫾ ⫺1.24 ⫾ 0.25 ⫾ 4.53 ⫾ 4.35 ⫾ 0.06 ⫾ 0.16 ⫾ 0.15 ⫾
2e 12.71 0.15 0.06 0.15 0.05 0.01 0.02 1.34 0.27 0.94 0.22 1.81 1.70 0.10 0.09 0.07
267.72 ⫾ 3.24 ⫾ 1.39 ⫾ 3.14 ⫾ 1.05 ⫾ 0.35 ⫾ 0.35 ⫾ 26.79 ⫾ 6.10 ⫾ 23.30 ⫾ 4.72 ⫾ 29.28 ⫾ 27.38 ⫾ 1.63 ⫾ 1.29 ⫾ 0.91 ⫾
12.27 0.15 0.06 0.14 0.05 0.02 0.02 1.23 0.28 1.07 0.22 1.50 1.40 0.10 0.08 0.05
d
2D/2A
2.44c 1.04 0.54c 1.06 1.16 0.46c 0.32 0.41 0.47 0.57 0.88 0.48c 0.40c 0.51c 0.57c 0.61
2.97 0.54 0.15 0.56 0.67 0.10 ⫺0.05 ⫺0.08 0.11 0.16 0.38 0.12 0.08 0.13 0.16 0.19
Variance components: additive genetic variance (2A ); additive genetic by environmental variance (2AE ); dominance genetic variance (2D ); dominance genetic by environmental variance ( 2DE ); and experimental error variance ( 2e ) b Plant and ear heights measured in four environments, and anthesis, silk emergence, and silk delay measured in three environments. c Average level of dominance deviated from complete dominance at 0.01 probability level. d Estimates and standard errors multiplied by 100.
a
E ⫽ expected mean square or product, and V ⫽ inverse of the variance of the mean square or product. Models that included the maximum number of parameters permitted by the number of independent equations often produced an X-matrix that was either singular or nearly singular. These models included two digenic epistatic terms and gave unrealistic estimates (very large or negative, with large standard errors). Chi et al. (1969), Silva and Hallauer (1975), and Wright et al. (1971) also obtained unrealistic and negative estimates as the number of epistatic terms in the model increased. Therefore models that included no more than one digenic epistatic term were used. The following six models were included to estimate genetic and error variances: Model
Parameters
1
2 2 2 2A , AE , e1 , e2
2
2 2 2 2 A2 , AE , 2AA , AAE , e1 , e2
3
2 2 2 2A , AE , D2 , 2DE , e1 , e2
4
2 2 2 2 2A , 2AE , 2D , 2DE , AA , AAE , e1 , e2
5
2 2 2A , 2AE , D2 , 2DE , DD , DDE , 2e1 , 2e2
6
2 2A , 2AE , 2D , 2DE , AD , 2ADE , 2e1 , 2e2 .
Heritabilities Heritability estimates (h2) were calculated on a progeny mean basis. Heritability of half-sib progeny means of Design III was calculated as 2 h2 ⫽ M2 /(2/rte ⫹ EM /te ⫹ M2 ).
Heritability of S1 progeny means was calculated as 2 /e ⫹ G2 ). h2 ⫽ G2 /(2/re ⫹ GE
Exact 90% confidence intervals for estimates of heritability were calculated, as defined by Knapp et al. (1985).
Results Design III Barren plants, root lodging, and dropped ears did not have estimates of A2 significantly different from zero ( Table 2). For other traits, estimates were generally two to five times greater than their standard 2 errors. Estimates of AE were not different from zero for ear and cob diameters, kernel depth, kernel-row number, root and stalk lodging, and dropped ears. Estimates 2 of AE were larger than estimates of A2 for yield, ears per plant, and barren plants. Estimates of dominance genetic variance (D2 ) were significantly different from zero for all traits except ears per plant, barren plants, root lodging, dropped ears, and silk delay ( Table 2). Significant estimates of D2 were generally greater than three times their standard errors. Ear length, barren plants, plant and ear heights, days-to-silk emergence, and silk 2 delay had significant estimates of DE . Estimates of A2 and D2 were not different from each other for ear diameter, kernel depth, and ear length, and both estimates were zero for barren plants, root lodging, and dropped ears. For grain yield, ˆ D2 was greater than ˆ A2 , while the opposite was true for the remaining traits. There-
fore the ˆ D2 ·ˆ A2 ratio was less than one for all traits except grain yield ( Table 2). Ratios were generally greater in magnitude for traits in this study compared with ratios reported by Han and Hallauer (1989), who also evaluated the F2 of the B73 ⫻ Mo17 cross. The average level of dominance deviated from complete dominance for yield, cob diameter, kernel-row number, plant height, ear height, anthesis, and silk emergence ( Table 2). Of these traits, grain yield had an average level of dominance in the overdominant range (2.44), while the remaining traits exhibited partial dominance. Han and Hallauer (1989) reported an average level of dominance for grain yield of 1.28, which did not deviate from complete dominance. The average levels of dominance for other traits in the present study were similar to those reported by Han and Hallauer (1989), who compared estimates in the F2 generation with those in the same F2 generation after five cycles of intermating. The average level of dominance for grain yield also was greater than estimates reported for F2 populations by Gardner et al. (1953), Gardner and Lonnquist (1959), Moll et al. (1964), and Robinson et al. (1949), which ranged from 1.03 to 2.14. For other traits the level of dominance was similar to estimates from these studies. S1 Progeny Genetic variance (G2 ) estimates were significantly different from zero for all traits except root lodging ( Table 3). Genetic by 2 environmental (GE ) variances were signif-
Wolf et al • Genetic Variance in Maize 387
Table 3. Estimates of variance componentsa (ⴞ standard error) from the S1 progeny ANOVA across five environmentsb for the B73 ⴛ Mo17 F 2 maize population Trait
2G
Yield (g/plant) Ear diameter (cm)c Cob diameter (cm)c Kernel depth (cm)c Ear length (cm) Kernel rows (no.) Ears/plant (no.)c Barren plants (%) Root lodging (%) Stalk lodging (%) Dropped ears (%) Plant height (cm) Ear height (cm) Anthesis (days) Silk emergence (days) Silk delay (days)
166.65 ⫾ 2.38 ⫾ 1.20 ⫾ 0.91 ⫾ 1.04 ⫾ 1.00 ⫾ 0.29 ⫾ 18.13 ⫾ ⫺0.09 ⫾ 4.30 ⫾ 0.26 ⫾ 210.67 ⫾ 163.22 ⫾ 4.95 ⫾ 4.07 ⫾ 0.47 ⫾
2GE 59.52 ⫾ 0.44 ⫾ 0.26 ⫾ ⫺0.02 ⫾ 0.07 ⫾ 0.05 ⫾ 0.35 ⫾ 32.42 ⫾ ⫺0.29 ⫾ 1.12 ⫾ 0.20 ⫾ 8.66 ⫾ 7.23 ⫾ 0.98 ⫾ 0.72 ⫾ 0.39 ⫾
29.20 0.40 0.21 0.18 0.17 0.15 0.06 4.39 0.13 1.07 0.10 30.96 23.93 0.80 0.67 0.12
2e 251.02 ⫾ 3.20 ⫾ 1.80 ⫾ 3.30 ⫾ 1.12 ⫾ 0.44 ⫾ 0.71 ⫾ 50.28 ⫾ 8.76 ⫾ 27.10 ⫾ 3.36 ⫾ 50.88 ⫾ 35.04 ⫾ 2.21 ⫾ 2.38 ⫾ 1.19 ⫾
15.89 0.18 0.10 0.15 0.06 0.02 0.06 4.53 0.42 1.40 0.18 3.47 2.48 0.23 0.22 0.11
16.55 0.21 0.12 0.20 0.07 0.03 0.05 3.32 0.58 1.79 0.22 3.78 2.60 0.19 0.20 0.10
Variance components: genetic variance ( 2G ); genetic by environmental variance ( 2GE ); and experimental error variance ( 2e ). b Plant and ear heights measured in four environments and anthesis, silk emergence, and silk delay measured in three environments. c Estimates and standard errors multiplied by 100.
a
icantly different from zero for all traits except kernel depth, ear length, root and stalk lodging, and dropped ears. The esti2 mate of GE for ears per plant and barren plants was larger but not significantly different from the estimate of G2 . For several traits the estimates of G2 were smaller than those reported by Han and Hallauer (1989). Covariance S1 and Half Sibs Covariance of S1 and half-sibs translated 2 into A2 and AE are presented in Table 4. Estimates of A2 were not different from zero for yield and dropped ears. Additive by environment variance was different from zero for yield, ear diameter, ears per plant, barren plants, days to anthesis, and 2 silk emergence. For yield, AE was signifi-
cantly greater than A2 , while for ears per plant and barren plants it was larger but not different from A2 . The estimates obtained from the covariance of S1 and half sibs generally were not significantly smaller than estimates from Design III. For ears per plant, barren plants, days to anthesis, and silk delay, estimates of A2 were larger than those obtained from Design III. Over2 all, estimates of A2 and AE from covariance analysis were generally within one standard error of estimates from Design III. Weighted Least Squares Across all traits the chi-square lack of fit was generally significant for models 1 and 2, while models 3 and 4 generally provided an adequate fit to the data ( Tables 5 and
Table 4. Estimates of additive genetic (2A ) and additive by environment (2AE ) variance components (ⴞ standard error) from analysis of covariance between S1 and half-sib progeny across five environmentsa for the B73 ⴛ Mo17 F 2 maize population Trait Yield (g/plant) Ear diameter (cm)b Cob diameter (cm)b Kernel depth (cm)b Ear length (cm) Kernel rows (no.) Ears/plant (no.)b Barren plants (%) Root lodging (%) Stalk lodging (%) Dropped ears (%) Plant height (cm) Ear height (cm) Anthesis (days) Silk emergence (days) Silk delay (days)
2A 28.09 ⫾ 1.46 ⫾ 1.26 ⫾ 0.50 ⫾ 0.62 ⫾ 0.98 ⫾ 0.10 ⫾ 5.51 ⫾ 0.40 ⫾ 4.98 ⫾ 0.03 ⫾ 172.87 ⫾ 138.15 ⫾ 4.01 ⫾ 3.44 ⫾ 0.50 ⫾
2AE 18.42 0.33 0.21 0.15 0.13 0.15 0.03 1.75 0.14 1.10 0.11 27.15 21.36 0.67 0.60 0.10
57.48 ⫾ 0.40 ⫾ ⫺0.06 ⫾ 0.05 ⫾ 0.08 ⫾ ⫺0.01 ⫾ 0.13 ⫾ 11.70 ⫾ ⫺0.17 ⫾ 1.02 ⫾ 0.31 ⫾ 0.36 ⫾ ⫺0.52 ⫾ 0.49 ⫾ 0.51 ⫾ ⫺0.05 ⫾
14.87 0.16 0.09 0.14 0.06 0.02 0.03 2.25 0.28 1.14 0.16 2.98 2.38 0.22 0.19 0.08
Plant and ear heights measured in four environments, and anthesis, silk emergence, and silk delay measured in three environments. b Estimates and standard errors were multiplied by 100.
a
388 The Journal of Heredity 2000:91(5)
6). Model 3 generally provided a good fit, with R 2 greater than 97% and smaller standard errors of the six models for the majority of traits. Silva and Hallauer (1975) and Wright et al. (1971) also obtained their best results from the same model. Esti2 2 mates of A2 , AE , D2 , and DE from model 3 were generally similar to estimates from Design III, and standard errors from weighted least squares were generally less than those from Design III. Only the esti2 mate of AE for ear length was significantly different between the two methods, and it was greater in Design III. Additive genetic variance (ˆ A2 ) from model 3 was not different from zero for root lodging, while ˆ D2 was not different from zero for ears per plant, barren plants, root lodging, dropped ears, and silk delay. Estimates of A2 and D2 were not significantly different from each other for ear and cob diameters, kernel depth, ear length, root lodging, and dropped ears. For the remaining traits, estimates of A2 were greater than D2 , except for yield. Inclusion of digenic epistatic variances in models 4, 5, and 6 generally improved the fit and increased the R 2 values compared with model 3, but the standard errors of ˆ A2 and ˆ D2 for models 4, 5, and 6 increased compared with model 3. The increase in standard errors as the number of epistatic components increased is likely unavoidable because of the high correlation between coefficients of the first-order variance components (A2 and D2 ) and coefficients of second-order components 2 2 2 (AA , AD , DD ) (Chi et al. 1969). With model 4, several traits had decreased estimates of A2 , while D2 was not affected compared with model 3 ( Tables 5 and 6). For example, the estimate of A2 for yield was decreased by 75% and for ear length decreased by 32%. Decreases in ˆ A2 resulted in nonsignificant estimates for yield, ears per plant, barren plants, and dropped ears in model 4. Dominance variance did not differ from zero for ears per plant, barren plants, root lodging, dropped ears, and silk delay. Additive genetic variance was significantly greater than ˆ D2 for cob diameter, kernel-row number, root and stalk lodging, plant and ear heights, days to anthesis and silk emergence, and silk delay in model 4. Dominance variance was significantly greater than ˆ A2 for yield. 2 When significant estimates of A2 , AE , D2 , 2 and DE were observed in model 4, they did not differ from corresponding estimates observed in model 3. 2 Inclusion of DD in model 5 generally gave unrealistically large estimates of D2
Table 5. Model 3a weighted least squares estimates (E) of variance components and their respective standard errors (SE) from the combined analysis across five environments for the B73 ⴛ Mo17 F2 maize population Variance componentsb and standard errors 2A
Trait Yield (g/plant) Ear diameter (cm) Cob diameter (cm) Kernel depth (cm) Ear length (cm) Kernel rows (no.) Ears/plant Barren plants (%) Root lodging (%) Stalk lodging (%) Dropped ears (%) Plant height (cm) Ear height (cm) Anthesis (days) Silk emergence (days) Silk delay (days)
E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE
54.78 11.85 1.83 0.22 1.30 0.13 0.68 0.10 0.72 0.08 1.02 0.09 0.09 0.02 4.48 1.10 0.15 0.09 4.85 0.71 0.16 0.07 185.94 16.63 145.23 12.92 4.00 0.40 3.50 0.35 0.46 0.06
2AE 57.46 10.08 0.40 0.11 0.06 0.06 0.02 0.09 0.11 0.04 0.002 0.01 0.14 0.02 12.62 1.62 ⫺0.13 0.21 1.43 0.81 0.26 0.11 6.02 1.96 4.60 1.56 0.68 0.13 0.62 0.12 0.13 0.06
2D
2DE
170.82 26.81 1.19 0.22 0.22 0.05 0.51 0.12 0.42 0.08 0.12 0.02 0.00 0.01 ⫺0.26 0.50 0.00 0.11 1.01 0.48 0.19 0.11 20.73 3.78 10.97 2.31 0.46 0.11 0.53 0.12 0.07 0.04
14.93 12.19 0.15 0.14 ⫺0.03 0.06 0.12 0.14 0.08 0.05 ⫺0.01 0.01 0.03 0.02 3.86 1.32 0.05 0.26 ⫺1.45 0.89 0.14 0.21 4.07 1.77 3.92 1.66 0.08 0.10 0.16 0.09 0.16 0.07
2e1
2e2
R2
2
269.15 10.99 3.20 0.13 1.38 0.06 3.14 0.12 1.12 0.04 0.34 0.01 0.33 0.01 25.13 1.13 6.17 0.24 23.73 0.94 4.87 0.19 30.37 1.41 28.56 1.31 1.61 0.09 1.30 0.07 0.93 0.05
251.87 15.20 3.21 0.19 1.93 0.10 2.96 0.17 1.07 0.07 0.47 0.02 0.79 0.04 56.29 3.11 8.24 0.45 27.00 1.53 3.30 0.19 51.87 3.39 35.96 2.38 2.31 0.18 2.41 0.18 1.28 0.09
99.4
8.2
99.8
2.2
99.2
7.7
99.8
2.8
98.8
17.5**
99.6
5.6
97.6
33.8**
97.1
41.8**
99.3
9.6*
99.8
2.3
99.6
5.1
99.3
8.5
99.0
11.4*
99.5
4.3
99.8
1.3
98.8
10.7*
*,** Chi-square lack of fit significant at the 0.05 and 0.01 probability levels, respectively. Model 3 included 2A, 2AE, 2D, 2DE, 2e1, and 2e2. b Variance components: additive genetic ( 2A ); dominance ( 2D ); interaction of these components by environment 2 2 2 ( 2AE, DE ); experimental error of Design III ( e1 ); and experimental error of S1 progeny ( e2 ).
a
2 and large negative estimates of DD , which may indicate the model was inadequate. 2 Estimates of AD generally had no effect on estimates of A2 and D2 in model 6. There2 fore AA had a greater effect on estimates 2 2 of A2 and D2 than did either DD or AD . If model 5 is considered inadequate, estimates of D2 were generally not biased by epistasis in the remaining models. Hallauer and Miranda Fo (1988) observed 2 that AA had the greatest bias on estimates 2 of A and D2 . Estimates of digenic epistatic components from models 2, 4, 5, and 6 often were negative, smaller than their standard errors, or unrealistically large compared with estimates of A2 and D2 . These results agree with those of Silva and Hallauer (1975) and Wright et al. (1971) who also observed unrealistic and negative estimates of digenic epistatic components.
Heritabilities Heritability estimates and their 90% confidence intervals of Design III and S1 progeny are presented in Table 7. Estimates were considered greater than zero if the
confidence interval did not overlap zero. Estimates in both analyses were significantly greater than zero for all traits, except for root lodging of S1 progeny, which was negative. In both analyses kernel-row number and plant and ear heights had the largest estimates (⬎0.91), which were significantly greater than estimates for other traits. Estimates for S1 progeny were larger than Design III estimates for several traits, particularly yield. Because the expected genetic variance of S1 progeny includes all the A2 and one-fourth of the D2 of the source population, heritabilities are expected to be larger compared with those based on half-sib progeny, which contain one-fourth of the A2 .
Discussion Variance Components In both the Design III and weighted least squares analyses, the estimate of D2 was significantly greater than the estimate of A2 for grain yield. For the remaining traits, ˆ A2 was greater than ˆ D2 . The average level of dominance from Design III was in the
overdominance range for grain yield and partial to complete dominance for the remaining traits, but the average level of dominance for grain yield may be biased upward due to linkage. In an F2 population, linkage disequilibrium will be at a maximum. If coupling phase linkages predominate, ˆ A2 and ˆ D2 will be biased upward. Repulsion phase linkages will cause a downward bias of ˆ A2 and upward bias of ˆ D2 . Both types of linkage may cause an upward bias in the average level of dominance. Han and Hallauer (1989) reported that the average level of dominance for grain yield decreased from 1.28 to 0.95 after five generations of random mating. Linkage did not bias estimates of A2 , but ˆ D2 decreased by 40% with random mating. However, the two estimates of average level of dominance did not differ from complete dominance, indicating linkage may have only a small bias on the average level of dominance. The average level of dominance for yield from the present study was 2.44. This is a distinct contrast to the estimate of 1.28. If ˆ D2 from the present study is reduced by 40%, the level of dominance is 1.89, which is still greater than the majority of estimates reported in previous studies (Gardner et al. 1953; Gardner and Lonnquist 1959; Moll et al. 1964). The estimated level of dominance (2.44) supports the presence of either important overdominant gene effects or pseudooverdominance because of linkage effects in the expression of yield. A significant difference in the estimate of the average level of dominance was observed by Gardner and Lonnquist (1959) for two samples of the single cross M14 ⫻ 187-2. Sample 1 had an average level of dominance estimate of 0.59, and sample 2 had an average level of dominance estimate of 1.59; both estimates deviated from complete dominance. Sample 1 had a larger estimate of A2 , and they suggested the environment in which sample 2 was grown may have suppressed the estimate of A2 , increasing the level of dominance. The sample 1 estimate of D2 was 67% of sample 2, and estimate of A2 sample 2 was 18% of that observed in sample 1. 2 The estimate of AE for yield was greater than observed by Han and Hallauer 2 (1989). The ˆ AE · ˆ A2 ratio was 1.25 compared with 0.16 in the study of Han and 2 Hallauer (1989), whereas the ˆ DE · ˆ D2 ratios were 0.10 and 0.16, respectively. The estimate of D2 for yield was less affected by environment than ˆ A2 , in the present study. The range of environments in which this study was conducted may have decreased
Wolf et al • Genetic Variance in Maize 389
Table 6. Model 4a weighted least squares estimates (E) of variance components and their respective standard errors (SE) from the combined analysis across five environments for the B73 ⴛ Mo17 F2 maize population Variance componentsb and standard errors 2A
Trait Yield (g/plant) Ear diameter (cm) Cob diameter (cm) Kernel depth (cm) Ear length (cm) Kernel rows (no.) Ears/plant Barren plants (%) Root lodging (%) Stalk lodging (%) Dropped ears (%) Plant height (cm) Ear height (cm) Anthesis (days) Silk emergence (days) Silk delay (days)
2AE
E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE E SE
13.31 24.98 1.73 0.50 1.65 0.31 0.62 0.25 0.49 0.18 1.12 0.20 ⫺0.01 0.04 ⫺1.75 2.46 0.74 0.24 6.78 1.80 0.10 0.20 164.97 34.66 128.98 26.81 3.10 0.81
62.40 26.21 0.38 0.29 ⫺0.20 0.14 0.13 0.25 0.37 0.11 ⫺0.06 0.03 ⫺0.03 0.05 ⫺3.38 3.76 ⫺0.01 0.53 1.71 2.10 0.54 0.32 7.67 4.54 ⫺3.90 3.88 0.33 0.30
E SE E SE
2.98 0.73 0.44 0.14
0.61 0.27 0.06 0.14
2D
2DE
161.32 14.88 27.28 12.61 1.18 0.14 0.22 0.15 0.23 ⫺0.06 0.05 0.06 0.51 0.14 0.13 0.14 0.40 0.11 0.08 0.05 0.12 ⫺0.02 0.02 0.01 0.00 0.02 0.01 0.02 ⫺0.24 2.54 0.50 1.34 0.04 0.10 0.11 0.27 1.06 ⫺1.37 0.49 0.92 0.18 0.20 0.11 0.22 20.65 4.20 3.78 1.80 10.90 3.66 2.32 1.69 0.46 0.05 0.11 0.10 0.53 0.12 0.07 0.04
2AA 87.38 46.23 0.17 0.76 ⫺0.54 0.43 0.09 0.36 0.41 0.29 ⫺0.17 0.30 0.27 0.08 16.79 5.51 ⫺0.82 0.31 ⫺2.83 2.43 0.09 0.26 38.02 55.22 30.99 42.63 1.77 1.36
0.16 0.09 0.13 0.07
0.97 1.18 0.03 0.22
2AAE
2e1
⫺7.44 270.00 36.48 11.56 0.03 3.21 0.41 0.14 0.43 1.43 0.21 0.06 ⫺0.17 3.12 0.35 0.13 ⫺0.36 1.09 0.15 0.04 0.10 0.35 0.05 0.01 0.35 0.35 0.09 0.02 33.08 27.18 7.10 1.19 ⫺0.29 6.06 0.84 0.26 ⫺0.49 23.59 3.06 1.00 ⫺0.40 4.82 0.45 0.20 ⫺2.74 30.22 6.83 1.46 6.78 29.44 5.45 1.36 0.57 1.64 0.45 0.09 0.02 0.41 0.32 0.22
1.30 0.07 0.95 0.05
2e2
R2
252.13 16.43 3.20 0.21 1.82 0.12 3.01 0.19 1.15 0.07 0.44 0.03 0.72 0.05 51.04 3.26 8.72 0.56 27.32 1.76 3.39 0.22 52.48 3.74 36.93 2.58 2.23 0.18
99.6
4.6
99.8
2.2
99.8
2.0
99.8
2.5
99.3
10.1**
99.9
0.6
99.8
2.0
99.8
2.8
99.9
0.9
99.9
0.8
99.6
4.3
99.3
7.9*
99.1
10.8**
99.9
0.7
99.9
0.6
99.0
8.4*
2.40 0.20 1.23 0.10
2
*,** Chi-square lack of fit significant at the 0.05 and 0.01 probability levels, respectively. 2 2 2 2 Model 4 included 2A, 2AE, 2D, 2DE, AA , AAE , e1 , and e2 . b Variance components: additive genetic ( 2A ); dominance (2D ); digenic epistasis of 2AA ; interaction of these com2 2 ponents by environment ( 2AE , 2DE , AAE ); experimental error of Design III ( e1 ); and experimental error of S1 progeny ( 2e2).
a
Table 7. Estimates of heritability (h2) with confidence intervals for half-sib progeny means from the Design III and S1 progeny means, based on the ANOVAs across five environmentsa for the B73 ⴛ Mo17 F 2 maize population Confidence intervalb
a
b
Confidence interval
Trait
Design III h2
Lower limit
Upper limit
S1 progeny h2
Lower limit
Upper limit
Yield (g/plant) Ear diameter (cm) Cob diameter (cm) Kernel depth (cm) Ear length (cm) Kernel rows (no.) Ears/plant (no.) Barren plants (%) Root lodging (%) Stalk lodging (%) Dropped ears (%) Plant height (cm) Ear height (cm) Anthesis (days) Silk emergence (days) Silk delay (days)
0.44 0.75 0.85 0.59 0.65 0.94 0.40 0.30 0.24 0.56 0.30 0.94 0.93 0.83 0.83 0.50
0.27 0.67 0.80 0.46 0.54 0.93 0.22 0.10 0.01 0.43 0.08 0.93 0.91 0.77 0.77 0.32
0.58 0.81 0.89 0.69 0.73 0.96 0.55 0.48 0.43 0.67 0.47 0.96 0.95 0.87 0.88 0.63
0.81 0.84 0.83 0.74 0.89 0.95 0.66 0.59 ⫺0.12 0.58 0.39 0.96 0.96 0.87 0.86 0.58
0.75 0.80 0.78 0.66 0.85 0.93 0.56 0.48 ⫺0.44 0.46 0.22 0.95 0.95 0.83 0.81 0.44
0.85 0.88 0.87 0.80 0.91 0.96 0.74 0.69 0.15 0.68 0.54 0.97 0.97 0.90 0.89 0.69
Plant and ear heights measured in four environments, and anthesis, silk emergence, and silk delay measured in three environments. Exact 90% confidence intervals as defined by Knapp et al. (1985).
390 The Journal of Heredity 2000:91(5)
the estimate of A2 , as suggested by Gardner and Lonnquist (1959). Estimates of A2 and D2 were 17% and 61%, respectively, of estimates reported by Han and Hallauer (1989), indicating both have decreased in the present study. Additive variance for grain yield may have been suppressed by a large interaction with environments. Other traits between the two studies were less affected by environments, and average level of dominance estimates were consistent between studies. Weighted Least Squares Weighted least squares analysis was conducted to determine the relative importance of epistatic variance compared with A2 and D2 . Triple testcross analysis indicated epistatic effects were important for several traits (Wolf and Hallauer 1997). 2 2 2 Generally, ˆ AA , ˆ DD , and ˆ AD were not greater than twice their standard errors, negative, or unrealistic. Models that did not include digenic epistatic components often provided an adequate fit and more precise 2 2 2 estimates. Therefore AA , DD , and AD were 2 2 less important than A and D for the majority of traits. Weighted least squares analysis indicat2 ed that inclusion of AA in model 4 de2 creased estimates of A for several traits. The decrease in ˆ A2 generally did not result in nonsignificant estimates or estimates different from those obtained with model 3. For yield, ˆ A2 was not significant in model 4, as were several other traits. Therefore, although ˆ A2 is biased upward if we assume epistasis is absent, the magnitude of bias is small. Generally ˆ D2 was not biased by epistasis in models 4 and 6. Bias observed in model 5 is likely the result of an inadequate model. Dominance variance was less important than A2 for most traits and may be less likely to be biased by epistasis. All traits had negative variance component estimates for various models, with model 5 generally having at least two negative estimates. By definition, a variance is always positive, but Searle (1971) indicated that there is nothing intrinsic about the ANOVA to prevent negative estimates from occurring. Negative estimates could arise from an inadequate model, inadequate sampling, or inadequate experimental techniques. Searle (1971) discussed possible solutions to negative estimates. The best solution would be to interpret them as zero and reestimate other components from a reduced model. Negative estimates in the present study were generally small and not greater than
their standard error. Negative estimates often occurred for variance components that were either nonsignificant or negative when estimated in Design III or S1 progeny experiments. Generally when a model gave negative estimates, another model for that trait had positive estimates with greater precision; hence negative estimates were not a serious problem. Dominance variance was important in the expression of heterosis for grain yield in the B73 ⫻ Mo17 cross. While epistasis was less important than dominance, the presence of significant positive epistatic effects may have contributed to the expression of heterosis and could explain why the B73 ⫻ Mo17 cross was an exceptional and widely grown hybrid. Epistatic variances were not important in the F2 population of the B73 ⫻ Mo17 cross, although epistatic effects have been previously detected.
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Wolf et al • Genetic Variance in Maize 391
