Genetics and Molecular Biology, 28, 1, 67-74 (2005) Copyright by the Brazilian Society of Genetics. Printed in Brazil www.sbg.org.br Research Article Dominance, epistasis, heritabilities and expected genetic gains José Marcelo Soriano Viana Universidade Federal de Viçosa, Departamento de Biologia Geral, Viçosa, MG, Brazil. Abstract Although epistasis is common in gene systems that determine quantitative traits, it is usually not possible to estimate the epistatic components of genotypic variance because experiments in breeding programs include only one type of progeny. As the study of this phenomenon is complex, there is a lack of theoretical knowledge on the contribution of the epistatic variances when predicting gains from selection and on the bias in estimating genetic parameters when fitting the additive-dominant model. The objective of this paper is to discuss these aspects. Regarding a non-inbred population, the genetic value due to dominance and the epistatic components of the genotypic value are not indicators of the number of favorable genes present in an individual. Thus, the efficiency of a selection process should be based on the narrow-sense heritability, a function only of additive variance. If there is no epistasis, generally it is satisfactory to assess the selection efficiency and to predict gain based on the broad-sense heritability. Regardless of the selection unit or type of epistasis, the bias in the estimate of the additive variance when assuming the additive-dominant model is considerable. This implies overestimation of the heritabilities at half sib family mean, plant within family and plant levels, and underestimation if the selection units are full sib progenies. The predicted gains will have a bias proportional to that of the heritability. Key words: breeding, genetic parameters, gene interaction, selection. Received: October 24, 2003; Accepted: August 16, 2004. Introduction ported gene interaction in rapeseed plant resistance to The methodologies generally used to study quantita- Sclerotinia sclerotiorum. The predominant type of epistasis tive trait inheritance, including interaction between was additive x additive. With the objective of determining non-allelic genes, are generation mean analysis (Mather the importance of epistatic effects in maize hybrid produc- and Jinks, 1974), that permits estimation of linear compo- tion, Hinze and Lamkey (2003) observed significant nents of genotypic means, and triple test cross analysis epistatic interaction in five of the forty cases analyzed. Al- (Kearsey and Jinks, 1968), which permits testing the exis- though with a smaller contribution to soybean plant resis- tence of epistasis. Rebetzke et al. (2003) reported epistatic tance to Cercospora sojina, as compared to the mean gene action from crosses between a wheat variety with low effects of genes and allelic interactions, Martins et al. leaf conductance and three with high leaf conductance val- (2003) reported epistasis to be responsible for the degree of ues. Kumar et al. (2003) observed that wheat embryo resis- infection, number of lesions per foliole, mean lesion diame- tance to Noevossia indica also depended on epistatic ter and disease index. Khattak et al. (2003) used a triple test effects. In the presence of the pathogen, they identified cross to detect that epistasis was responsible for two sec- complementary epistasis for fresh weight of calli. Interac- ondary components of mungbean plant yield, only in the tion between gene combinations of two and three loci was spring/summer planting. Regarding number of branches observed by Sharma et al. (2003) in a study of spike length and biomass, partitioning of total epistasis revealed addi- inheritance in durum wheat plants. At normal and late sow- tive x additive, additive x dominant and dominant x domi- ing, digenic epistatic effects predominated compared to ad- nant epistatic interactions, with predominance of the first ditive and dominance effects. In a study of mungbean type. resistance to Erysiphe polygoni, Gawande and Patil (2003) The estimation of epistatic components of genotypic observed duplicate epistasis for disease incidence and area variance is unusual in genetic studies because of limitation under disease progress curve. Zhao and Meng (2003) re- of the methodology, as in the case of the triple test cross, the high number of generations to be produced and assessed Send correspondence to José Marcelo Soriano Viana. Universida- (Viana, 2000), and mainly because only one type of prog- de Federal de Viçosa, Departamento de Biologia Geral, 36.570- eny, half sib, full sib or inbred families, is commonly in- 000 Viçosa, MG, Brazil. E-mail: [email protected]. cluded in the experiments. Even with the phenotypic values 68 Viana of plants in the families and in one or more genetically ho- dominant x additive (DA) and dominant x dominant (DD) mogeneous populations, for estimation of the environmen- genetic values are nil. tal variance at plant level, the breeder has two equations, The mean of the genotypes containing gene Ai is: the mean squares between and within families, to estimate A K ==∑ ∑ ∑ α at least five genetic variances, assuming a non-inbred popu- GpppGM+i j m n ijmn i j m n lation and digenic epistasis (Hallauer and Miranda Filho, α A − The effect of gene A is, thus, =GK G .As- 1988). An exception is the study by Braga (1987). Com- i i i .... suming two allelic forms, ααA =q if the gene increases monly, the additive-dominant model is fitted, assuming i aa the trait expression, or ααA =-p if the gene decreases it, epistasis to be negligible or non-existent. As epistasis is i aa common and not occasional, the consequences of this nec- where pa is the frequency of the gene that increases the ex- pression of the characteristic and α =a+− (q p )d is essary simplification should be known. Furthermore, be- aa aaa cause of the complexity of theoretical studies on epistasis, the average effect of a gene substitution. Parameter aa is the there is a lack of information about the contribution of the deviation between the genotypic value of the homozygote epistatic components of genotypic variance when predict- of greatest expression and the mean of the homozygotes, ing gains from selection. The objective of this study was to and da is the deviation due to dominance (Falconer and produce theoretical knowledge on these problems. Mackay, 1996). The mean of the genotypes containing the Ai and Aj Methods, Results and Discussion alleles is: ==+ααδA A A GppGM++ij..∑ ∑ m n ijmn i j ij Genotypic variance in non-inbred populations m n The genotypic value of an individual in relation to The genetic value due to dominance of the genotype δ A =−−+ two loci (A and B) is (Kempthorne, 1955): AiAj is, therefore, ij GGGGij.. i... j.. .... Taking two δ A =− 2 alleles, ij 2qa d a if the individual is homozygous for =+()()ααααA +A +B +B + δδA +B + δ A = GMijmn i j m n ij mn the gene that increases the trait expression, ij 2paaa q d δ A =− 2 ()()ααA B ++++++ ααA B ααA B ααA B αδA B αδA B if it is heterozygous, or ij 2pa d a if it is homozygous i m i n j m j n i mn j mn for the gene that decreases the expression of the character- ()δαA B +=+++++δαA B +δδA B ij m ij n ij mn MADAAAD istic (Falconer and Mackay, 1996). The mean of the genotypes containing both the A and DA+ DD i Bm genes is: where α represents the effect of a gene, δ represents the ef- ==+ααA B () ααA B Gi. m.∑ ∑ ppG j n ijmn M+ i + m i m fect of interaction between alleles (due to the presence of a j n pair of alleles), αα represents the additive x additive ()ααA B −− + epistatic effect (due to the presence of a pair of non-allelic Thus, i m =Gi.m. G i... G ..m. G .... The mean genes), αδ represents the additive x dominant epistatic ef- of the genotypes containing genes A ,B and B is: fect (due to the presence of a gene of one locus and a pair of i m n alleles from another locus), and δδ represents the dominant ==αααδA B +++B B x dominant epistatic effect (due to the presence of two pairs GpGM++i. mn∑ j ijmn i m n mn of non-allelic genes). Under the restrictions j ()()()ααA B ++ ααA B αδA B ()ααααδA ===A ()B B =A i m i n i mn EpEppi ∑∑∑ii m mm jij i m j Thus, δααααB ==()A B ()A B = ∑ ppnmn ∑∑iim pmim n i m ()αδA B ++ +− − i mn =Gi.mn G i... G ..m. G ...n G i.m. ()αδA B = () αδA B == ()δαA B ∑ ppnimn ∑∑iimn p jijm −− GGi..n ..mn G .... n i j ()δαA B == () δδA B () δδA B = Finally, ∑∑pppmijm iijmn ∑ mijmn 0 m i m δδA B αααA A B −− αB δ−δA B − ij mn =G-M---ijmn i j m n ij mn where p represents the probability of a gene, M is the popu- ()()()()()ααA B −− ααA B ααA B −− ααA B αδA B − lation mean with i m i n j m j n i mn ()αδA B −()() δαA B −δαA B ( ) ===j mn ij m ij n EGijmn∑ ∑ ∑ ∑ ppp i j m pG n ijmn G.... M i j m n Assuming that the population is in linkage equilib- and the expectations of the additive (A), due to dominance rium, all the genetic effects and values are un-correlated (D), additive x additive (AA), additive x dominant (AD), variables because Dominance, epistasis, heritabilities and expected genetic gains 69 (ααA A )== ()() αA αA ()=++=+2 2 K 2 2 Covi , j ∑ ∑ pij p i j E N pa pb (4) qa qb (0) 2pab 2p i j σ 2 =++2 22K 2 22−+()2 = N pp(4)a b qa qb (0) 2pab 2p ()αA () αA ==K ( δαA B δδA B ) = ∑∑ppii jj Covij m , ij mn + 2paa q 2p bb q i j ()()δαA B δδA B = The covariances between the number of genes that in- ∑ ∑ ∑ ∑ p ijmnpp p ij m ij mn i j m n crease the trait expression and the components of the genotypic value are: ()δαA B ()δδA B = ∑ ∑ ppij∑ p m ij m ∑ p nijmn 0 i j m n =+2 2 ()2 () + Cov(N,A) pa .pb (4) A22 p a .2pbb q (3) A 21 p2 .q2 (2)() A++22 p q .p2 (3) () A p q .