Estimates of Genetic Variance in an F 2 Maize Population

Transcription

1 Estimates of Genetic Variance in an F Maize Population D. P. Wolf, L.. Peternelli, and. 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 F population derived from the B73 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 F 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 Viçosa, MG, Brazil (Peternelli), and the Department of gronomy, Iowa State University, mes, I (Hallauer). This is a contribution of the Department of gronomy and journal paper no. J of the Iowa gricultural and Home Economics Experiment Station (mes), project This article is part of a dissertation submitted by D. P. Wolf in partial fulfillment of the requirements for a Ph.D. degree. ddress correspondence to. R. Hallauer at the address above or hallauer@iastate.edu. 000 The merican Genetic ssociation 91: 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 195) has primarily been used in maize F 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 F 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; Eberhart 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. 196). 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). 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 F population derived from 384

2 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 197). Inbred Mo17 was derived by selection from the single cross of inbred lines, CI187- C103 (Zuber 1973). In 1991 an F population derived from the B73 Mo17 cross was grown at the gronomy Research Farm near mes, Iowa. Using the triple testcross (TTC) mating design (Kearsey and Jinks 1968), 100 random F plants (males) were crossed to both parents (B73 and Mo17) and the F 1. B73, Mo17, and the F 1 were considered testers. Each F plant was selfed to form S 1 progenies. For Design III analysis only data from B73 and Mo17 testcrosses are needed. Experimental Procedures Testcross and S 1 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 F plants. The S 1 progeny were grown in a lattice with two replications. Both experiments were grown at the gronomy Research Center near mes, the tomic Energy Farm in mes, and near Elkhart, Iowa, in 199. In 1993, experiments were evaluated at the gronomy Research Center and the nkeny 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 57,50 plants/ha. Sixteen traits were measured in both experiments. Days from planting to 50% anthesis and silk emergence were recorded at the gronomy Research Center in 199 and 1993, and at the tomic Energy Farm in 199. 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 nalysis Genetic Variance Components The genetic-statistical model for Design III was followed to derive genetic variance components for the F reference population (Comstock and Robinson 195). nalyses of variance (NOVs) combined across environments were used to estimate variance components. From the De- sign III analysis, additive genetic ( ), ad- ditive by environment ( ), dominance genetic ( D), and dominance by environ- ment ( DE) variance components were estimated. The necessary components were calculated as variation among males [ M covariance half-sibs (1/4) ]; envi- ronment by male [ (1/4) ]; tester EM by male [ MT D]; and tester by male by environment [ TME DE]. Estimates of and D were used to estimate the average level of dominance as d MT / M ) 1/ ( D/ ) 1/. From the combined NOV for S 1 progeny, genotypic ( G) and genotypic by en- vironment ( ) variance components GE were estimated. Because an F population was sampled, the expected gene frequencies of segregating loci are 0.5. t gene fre- E E quencies of 0.5 the G of S 1 progeny can be expressed in genetic components as G (1/4) D. Standard errors (SE) for all variance components were calculated using the method of nderson and Bancroft (195); SE {/C i [(MS i ) /(n i )]} 1/, where MS i i th mean square; n i 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 standard errors of the estimate. Estimates were considered different from each other if their confidence intervals did not overlap. Because half-sib and S 1 progeny were derived from the same F 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 S 1 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 NOVs. 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 M XY r XYE re XY ;M XYE r XYE ; XY (M XY M XYE )/re; and XYE M XYE /r, where M XY mean product between halfsib (X) and S 1 ( Y) progeny; M XYE interaction of environment by half-sib and S 1 progeny mean product; XY covariance of half-sib and S 1 progeny; and XYE covariance by environment interaction. The genetic covariance between half-sib and S 1 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 (F P 1,F P, and F F 1 ) as used in the present study. For a genetic model that includes digenic epistatic variances this covariance can be expressed as XY (1/) (1/4) and E E XYE (1/) (1/4). 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 i [(M i XX )(M i YY ) (M i XY ) ]/(n I )} 1/, where C coefficient of the component of covariance; M ixx and M i YY mean squares for half-sib and S 1 progeny; M i XY mean product for half-sib and S 1 progeny; and n i degrees of freedom of i th mean product. Wolf and Hal- Wolf et al Genetic Variance in Maize 385

3 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, S 1 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 e1 rt EM rte M e1 rt[(1/4) E (1/16) E] rte[(1/4) (1/16) ]; Males environment e1 rt EM e1 rt[(1/4) E (1/16) E]; Male tester e1 r ETM re TM e1 r( DE DDE) re( D DD); and Male tester environment e1 r ETM e1 r( DE DDE). From the S 1 progeny, Genotypes e r GE re G e r[ E (1/4) DE E (1/16) DDE (1/4) DE] re[ (1/4) D (1/16) DD (1/4) D]; and Genotypes environment e r GE e r[ E (1/4) DE E (1/16) DDE (1/4) DE]. From the mean products, 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 environments a Variance components b E D DE E DD DDE D DE e1 e Design III Males (M) Males environment (E) Males tester (T) M T E Error S 1 progeny Genotypes (G) G E Error Mean products Half-sib/S Half-sib/S 1 E a ll traits except plant and ear heights, anthesis, silk emergence, and silk delay. b Variance components: additive genetic ( ); dominance ( D ); digenic epistasis of and D (, DD, D); in- teraction of these components by environment ( E, DE, E, DDE, DE); experimental error of the design III ( ); and experimental error of S 1 progeny ( ). e1 Half-sib and S 1 r XYE re XY r[(1/) E (1/4) E] re[(1/) (1/4) ]; and Half-sib and S1 environment r XYE r[(1/) E (1/4) E]. For the mean squares,, DD, and D are the digenic epistatic variance components; E, DDE, and DE are the digenic epistatic by environment variances; e1 is the error variance of Design III; e is the error variance of S 1 progeny; XY is the covariance of half-sibs and S 1 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 where Bˆ (X WX) 1 (X WY), Bˆ column vector of estimated genetic 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. e 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(M i ) [(M i ) /df i ], where M i ith mean square; and df i degrees of freedom of ith mean square. The following formula was used for the variance of a mean product: V(MiXY) [(M ixx)(m iyy) (M ixy) ] (df ), i where M i XX and M iyy ith mean squares for half-sib and S 1 progeny; M i XY ith mean product of halfsib and S 1 progeny; and df i degrees of freedom of the ith mean product. To estimate the genetic parameters of Bˆ, different models were tested. Not all genetic and error variances, however, could be estimated from a single model. 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 198). X [(O E) V], where O observed mean square or product, 386 The Journal of Heredity 000:91(5)

4 Table. Estimates of variance components a ( standard error) and average level of dominance ( d) from the Design III NOV across five environments b for the B73 Mo17 F maize population Trait E D DE e d / Yield (g/plant) c.97 Ear diameter (cm) d Cob diameter (cm) d c 0.15 Kernel depth (cm) d Ear length (cm) Kernel rows (no.) c 0.10 Ears/plant (no.) d Barren plants (%) Root lodging (%) Stalk lodging (%) Dropped ears (%) Plant height (cm) c 0.1 Ear height (cm) c 0.08 nthesis (days) c 0.13 Silk emergence (days) c 0.16 Silk delay (days) a Variance components: additive genetic variance ( ); additive genetic by environmental variance ( E ); dominance genetic variance ( D ); dominance genetic by environmental variance ( DE); and experimental error variance ( e) b Plant and ear heights measured in four environments, and anthesis, silk emergence, and silk delay measured in three environments. c verage level of dominance deviated from complete dominance at 0.01 probability level. d Estimates and standard errors multiplied by 100. D 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,,, E e1 e, E,, E, e1, e, E, D, DE, e1, e, E, D, DE,, E, e1, e, E, D, DE, DD, DDE, e1, e, E, D, DE, D, DE, e1, e. Heritabilities Heritability estimates (h ) were calculated on a progeny mean basis. Heritability of half-sib progeny means of Design III was calculated as h /( M /rte EM/te M). Heritability of S 1 progeny means was calculated as h /( G /re GE/e G). 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 significantly different from zero (Table ). For other traits, estimates were generally two to five times greater than their standard errors. Estimates of E were not different from zero for ear and cob diameters, kernel depth, kernel-row number, root and stalk lodging, and dropped ears. Estimates of E were larger than estimates of for yield, ears per plant, and barren plants. Estimates of dominance genetic variance ( D) were significantly different from zero for all traits except ears per plant, barren plants, root lodging, dropped ears, and silk delay ( Table ). Significant esti- mates of D were generally greater than three times their standard errors. Ear length, barren plants, plant and ear heights, days-to-silk emergence, and silk delay had significant estimates of DE. Estimates of and were not differ- ent 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, D ˆ D was greater than ˆ, while the opposite was true for the remaining traits. There- fore the ˆ D ˆ ratio was less than one for all traits except grain yield (Table ). Ratios were generally greater in magnitude for traits in this study compared with ratios reported by Han and Hallauer (1989), who also evaluated the F 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 ). Of these traits, grain yield had an average level of dominance in the overdominant range (.44), while the remaining traits exhibited partial dominance. Han and Hallauer (1989) reported an average level of dominance for grain yield of 1.8, 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 F generation with those in the same F generation after five cycles of intermating. The average level of dominance for grain yield also was greater than estimates reported for F 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.14. For other traits the level of dominance was similar to estimates from these studies. S 1 Progeny Genetic variance ( G) estimates were significantly different from zero for all traits except root lodging (Table 3). Genetic by environmental ( ) variances were signif- GE Wolf et al Genetic Variance in Maize 387

5 Table 3. Estimates of variance components a ( standard error) from the S 1 progeny NOV across five environments b for the B73 Mo17 F maize population Trait G GE e 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) nthesis (days) Silk emergence (days) Silk delay (days) a Variance components: genetic variance ( ); genetic by environmental variance ( variance ( e). G GE ); and experimental error 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. icantly different from zero for all traits except kernel depth, ear length, root and stalk lodging, and dropped ears. The estimate of GE for ears per plant and barren plants was larger but not significantly different from the estimate of G. For several traits the estimates of G were smaller than those reported by Han and Hallauer (1989). Covariance S 1 and Half Sibs Covariance of S 1 and half-sibs translated into and E are presented in Table 4. Estimates of were not different from zero for yield and dropped ears. dditive by environment variance was different from zero for yield, ear diameter, ears per plant, barren plants, days to anthesis, and silk emergence. For yield, was signifi- E Table 4. Estimates of additive genetic ( ) and additive by environment ( E) variance components ( standard error) from analysis of covariance between S 1 and half-sib progeny across five environments a for the B73 Mo17 F maize population Trait E cantly greater than, while for ears per plant and barren plants it was larger but not different from. The estimates obtained from the covariance of S 1 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 were larger than those obtained from Design III. Overall, estimates of and from covari- 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) nthesis (days) Silk emergence (days) Silk delay (days) a 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. ance analysis were generally within one standard error of estimates from Design III. Weighted Least Squares cross all traits the chi-square lack of fit was generally significant for models 1 and, while models 3 and 4 generally provided an adequate fit to the data (Tables 5 and E 6). Model 3 generally provided a good fit, with R 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. Estimates of, E, D, 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 estimate of E for ear length was significantly different between the two methods, and it was greater in Design III. dditive genetic variance ( ˆ ) from model 3 was not differ- ent from zero for root lodging, while ˆ D was not different from zero for ears per plant, barren plants, root lodging, dropped ears, and silk delay. Estimates of and D 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 were greater than D, except for yield. Inclusion of digenic epistatic variances in models 4, 5, and 6 generally improved the fit and increased the R values compared with model 3, but the standard errors of ˆ and ˆ D 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 ( and D) and coefficients of second-order components (, D, DD) (Chi et al. 1969). With model 4, several traits had decreased estimates of, while D was not affected compared with model 3 (Tables 5 and 6). For example, the estimate of for yield was decreased by 75% and for ear length decreased by 3%. Decreases in ˆ 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. dditive genetic variance was significantly greater than ˆ D 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 for yield. When significant estimates of, E, D, and DE were observed in model 4, they did not differ from corresponding estimates observed in model 3. Inclusion of DD in model 5 generally gave unrealistically large estimates of D 388 The Journal of Heredity 000:91(5)

6 Table 5. Model 3 a weighted least squares estimates (E) of variance components and their respective standard errors (SE) from the combined analysis across five environments for the B73 Mo17 F maize population Trait Variance components b and standard errors E D DE e1 e R Yield (g/plant) E SE Ear diameter (cm) E SE Cob diameter (cm) E SE Kernel depth (cm) E SE Ear length (cm) E ** SE Kernel rows (no.) E SE Ears/plant E ** SE Barren plants (%) E ** SE Root lodging (%) E * SE Stalk lodging (%) E SE Dropped ears (%) E SE Plant height (cm) E SE Ear height (cm) E * SE nthesis (days) E SE Silk emergence (days) E SE Silk delay (days) E * SE *,** Chi-square lack of fit significant at the 0.05 and 0.01 probability levels, respectively. a Model 3 included, E, D, DE, e1, and e. b Variance components: additive genetic ( ); dominance ( D); interaction of these components by environment (, ); experimental error of Design III ( ); and experimental error of S 1 progeny ( ). E DE e1 e and large negative estimates of DD, which may indicate the model was inadequate. Estimates of D generally had no effect on estimates of and D in model 6. There- fore had a greater effect on estimates of and D than did either DD or D.If model 5 is considered inadequate, estimates of D were generally not biased by epistasis in the remaining models. Hallauer and Miranda Fo (1988) observed that had the greatest bias on estimates of and D. Estimates of digenic epistatic components from models, 4, 5, and 6 often were negative, smaller than their standard errors, or unrealistically large compared with estimates of and D. 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 S 1 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 S 1 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 S 1 progeny were larger than Design III estimates for several traits, particularly yield. Because the expected genetic variance of S 1 progeny includes all the and one-fourth of the of the source population, heritabilities are expected to be larger compared with those based on half-sib progeny, which contain one-fourth of the. Discussion Variance Components In both the Design III and weighted least squares analyses, the estimate of D was significantly greater than the estimate of for grain yield. For the remaining traits, ˆ was greater than ˆ D. The average level of dominance from Design III was in the D 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 F population, linkage disequilibrium will be at a maximum. If coupling phase linkages predominate, ˆ and ˆ D will be biased upward. Re- pulsion phase linkages will cause a downward bias of ˆ and upward bias of ˆ D. 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 1.8 to 0.95 after five generations of random mating. Linkage did not bias estimates of, but ˆ D 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.44. This is a distinct contrast to the es- timate of 1.8. If ˆ D 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 (.44) supports the presence of either important overdominant gene effects or pseudooverdominance because of linkage effects in the expression of yield. 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 M Sample 1 had an average level of dominance estimate of 0.59, and sample had an average level of dominance estimate of 1.59; both estimates deviated from complete dominance. Sample 1 had a larg- er estimate of, and they suggested the environment in which sample was grown may have suppressed the estimate of, increasing the level of dominance. The sample 1 estimate of D was 67% of sample, and estimate of sample was 18% of that observed in sample 1. The estimate of E for yield was greater than observed by Han and Hallauer (1989). The ˆ E ˆ ratio was 1.5 compared with 0.16 in the study of Han and Hallauer (1989), whereas the ˆ DE ˆ D ratios were 0.10 and 0.16, respectively. The estimate of D for yield was less affected by environment than ˆ, 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

7 Table 6. Model 4 a weighted least squares estimates (E) of variance components and their respective standard errors (SE) from the combined analysis across five environments for the B73 Mo17 F maize population Trait Variance components b and standard errors E D DE E e1 e R Yield (g/plant) E SE Ear diameter (cm) E SE Cob diameter (cm) E SE Kernel depth (cm) E SE Ear length (cm) E ** SE Kernel rows (no.) E SE Ears/plant E SE Barren plants (%) E SE Root lodging (%) E SE Stalk lodging (%) E SE Dropped ears (%) E SE Plant height (cm) E * SE Ear height (cm) E ** SE nthesis (days) E SE Silk emergence (days) E SE Silk delay (days) E * SE *,** Chi-square lack of fit significant at the 0.05 and 0.01 probability levels, respectively. a Model 4 included, E, D, DE,, E, e1, and e. b Variance components: additive genetic ( ); dominance ( D); digenic epistasis of ; interaction of these com- ponents by environment ( E, DE, E); experimental error of Design III ( e1); and experimental error of S 1 progeny ( e). Table 7. Estimates of heritability (h ) with confidence intervals for half-sib progeny means from the Design III and S 1 progeny means, based on the NOVs across five environments a for the B73 Mo17 F maize population Trait Design III h Confidence interval b Lower limit Upper limit S 1 progeny h Confidence interval Lower 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) nthesis (days) Silk emergence (days) Silk delay (days) Upper limit a Plant and ear heights measured in four environments, and anthesis, silk emergence, and silk delay measured in three environments. b Exact 90% confidence intervals as defined by Knapp et al. (1985). the estimate of, as suggested by Gardner and Lonnquist (1959). Estimates of and D were 17% and 61%, respectively, of estimates reported by Han and Hallauer (1989), indicating both have decreased in the present study. dditive 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 and D. Triple testcross analysis indicated epistatic effects were important for several traits (Wolf and Hallauer 1997). Generally, ˆ, ˆ DD, and ˆ D 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 estimates. Therefore, DD, and D were less important than and for the ma- jority of traits. Weighted least squares analysis indicated that inclusion of in model 4 de- creased estimates of for several traits. The decrease in ˆ generally did not result in nonsignificant estimates or estimates different from those obtained with model 3. For yield, ˆ was not significant in model 4, as were several other traits. Therefore, although ˆ is biased upward if we assume epistasis is absent, the magnitude of bias is small. Generally ˆ D 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 for most traits and may be less likely to be biased by epistasis. ll 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 NOV 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 D 390 The Journal of Heredity 000:91(5)

8 their standard error. Negative estimates often occurred for variance components that were either nonsignificant or negative when estimated in Design III or S 1 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 F population of the B73 Mo17 cross, although epistatic effects have been previously detected. References nderson RL and Bancroft T, 195. Statistical theory in research. New York: McGraw-Hill. Bauman LF, Evidence of non-allelic gene interaction in determining yield, ear height, and kernel row number in corn. gron J 51: Bradshaw JE, Estimating the predicted response to S 1 family selection. Heredity 51: Chi KR, Eberhart S, and Penny LH, Covariances among relatives in a maize (Zea mays L.) variety. Genetics 63: Comstock RE and Robinson HF, 195. Estimation of average dominance of genes. In: Heterosis (Gowen JW, ed). mes, I: Iowa State University Press; Dickerson GE, Techniques for research in quantitative animal genetics. In: Techniques and procedures in animal science research. lbany, NY: merican Society of nimal Science; Eberhart S, Moll RH, Robinson HF, and Cockerham CC, Epistatic and other genetic variances in two varieties of maize. Crop Sci 6: Gardner CO, Harvey PH, Comstock RE, and Robinson HF, Dominance of genes controlling quantitative characters in maize. gron J 45: Gardner CO and Lonnquist JH, Linkage and the degree of dominance of genes controlling quantitative characters in maize. gron J 51: Gorsline GW, Phenotypic epistasis for ten quantitative characters in maize. Crop Sci 1: Hallauer R and Miranda Fo JB, Quantitative genetics in maize breeding, nd ed. mes, I: Iowa State University Press. Han G-C and Hallauer R, Estimates of genetic variability in F maize populations. J Iowa cad Sci 96: Kearsey MJ and Jinks JL, general method of detecting additive, dominance, and epistatic variation for metrical traits. I. Theory. Heredity 3: Knapp SJ, Stroup WW, and Ross WM, Exact confidence intervals for heritability on a progeny mean basis. Crop Sci 5: Lamkey KR, Schnicker BS, and Melchinger E, Epistasis in an elite maize hybrid and choice of generation for inbred line development. Crop Sci 35: Mather K and Jinks JL, 198. Biometrical genetics. New York: Chapman & Hall. Matzinger DF and Cockerham CC, Simultaneous selfing and partial diallel test crossing. I. Estimation of genetic and environmental parameters. Crop Sci 3: Mode CJ and Robinson HF, Pleiotropism and the genetic variance and covariance. Biometrics 15: Moll RH, Lindsey MF, and Robinson HF, Estimates of genetic variances and level of dominance in maize. Genetics 49: Nelder J, The estimation of variance components in certain types of experiment on quantitative genetics. In: Biometrical genetics (Kempthorne O, ed). New York: Pergamon Press; Peternelli L, Hallauer R, and Bailey TB, Theoretical bias on the covariance of S 1 and half-sib families. In: bstracts of the North Central Corn Breeding Research Conference, mes, Iowa, 8 9 February Robinson HF, Comstock RE, and Harvey PH, Estimates of heritability and the degree of dominance in corn. gron J 41: Russell W, 197. Registration of B70 and B73 parental lines of maize. Crop Sci 1:71. Searle SR, Topics in variance component estimation. Biometrics 7:1 74. Silva JC and Hallauer R, Estimation of epistatic variance in Iowa stiff stalk synthetic maize. J Hered 66: Sprague GF, Russell W, Penny LH, Horner TW, and Hanson WD, 196. Effect of epistasis on grain yield in maize. Crop Sci : Wolf DP and Hallauer R, Triple testcross analysis to detect epistasis in maize. Crop Sci 37: Wright J, Hallauer R, Penny LH, and Eberhart S, Estimating genetic variance in maize by use of single and three-way crosses among unselected lines. Crop Sci 11: Zuber MS, Registration of 0 maize parental lines. Crop Sci 13: Received June 8, 1999 ccepted March 10, 000 Corresponding Editor: William F. Tracy Wolf et al Genetic Variance in Maize 391