Genetic components of combining ability in a complete diallel

Abstract

Obtained and values were associated to theoretical concepts of the respective parameters in a complete
diallel with 28 parents and the simulation of five hypothetic variables with five different d/a relations (0, 0.5, 1.0, 1.5, and 2.0).
These were controlled by a single gene with two alleles whose parents were represented by different frequencies of the
favorable allele (1/28, 2/28,...,28/28). The conclusion was drawn that the existence of dominance deviations in the loci
regulating the trait influences the GCA estimates considerably and that there is a high correlation (0.96) between the absolute
and the respective values. The joint evaluation of and estimates provides information on the genetic quality of the
populations of the diallel.

Full text

338 Crop Breeding and Applied Biotechnology 4:338-343, 2004
FM Ferreira et al.
Crop Breeding and Applied Biotechnology 4:338-343, 2004
Brazilian Society of Plant Breeding. Printed in Brazil
Genetic components of combining ability in a complete
diallel
Fábio Medeiros Ferreira1*, José Ivo Ribeiro Júnior2, Cleso Antônio Patto Pacheco3, Carlos Henrique Osório Silva2, and
Sebastião Martins Filho4
Received 30 December 2003
Accepted 03 August 2004
ABSTRACT - Obtained and values were associated to theoretical concepts of the respective parameters in a complete
diallel with 28 parents and the simulation of five hypothetic variables with five different d/a relations (0, 0.5, 1.0, 1.5, and 2.0).
These were controlled by a single gene with two alleles whose parents were represented by different frequencies of the
favorable allele (1/28, 2/28,...,28/28). The conclusion was drawn that the existence of dominance deviations in the loci
regulating the trait influences the GCA estimates considerably and that there is a high correlation (0.96) between the absolute
and the respective values. The joint evaluation of and estimates provides information on the genetic quality of the
populations of the diallel.
Key words: combinatory ability, complete diallel, genetic components, genetic improvement.
1Departamento de Biologia Geral, Universidade Federal de Viçosa (UFV), 36571-000, MG, Brasil. *E-mail: ferreirafmf@yahoo.com.br
2Departamento de Informática, UFV
3Embrapa Milho e Sorgo, C. P. 151, 35701-970, Sete Lagoas, MG, Brasil
4Departamento de Engenharia Rural, Centro de Ciências Agrárias da Universidade Federal do Espírito Santo, C. P. 16, 29500-000, Alegre,
ES, Brasil
INTRODUCTION
The expressions ‘general combining ability’ (GCA) and
‘specific combining ability’ (SCA) are being used since the
past century (Sprague and Tatum 1942) to designate the
properties of the evaluated populations in hybridization
programs. One of the techniques that can verify the mean
performance of a parent in a series of hybrid combinations
(GCA) and make out certain hybrid combinations which are
relatively superior or inferior to the expected based on the
GCA of its parents (SCA), is the diallel analysis.
The complete diallel involves all crossing possibilities
within a parent group, and it is used to study polygenic
systems that determine quantitative traits since it offers
information regarding the predominant gene action. It can be
used at initial, intermediate, or final stages of an improvement
program. Depending on the nature of the parents, the
conclusions based on the results are also useful for the
formation of a basis population for intra or inter-populational
improvement (Viana et al. 1999).
The success of an improvement program is closely
linked to the selection of the best parents (families, lines,
etc). The estimates of the GCA, and effect SCA of the parent
with itself, provide information on the quality and genetic

Crop Breeding and Applied Biotechnology 4:338-343, 2004 339
Genetic components of combining ability in a complete diallel
diversity of the parents that compose the diallel. Although
inferences on the genotypic values of individuals are of great
interest, it is important to consider the genetic values of each
allelic form. It is the case of the allogamous species, whose
parents pass their alleles and not their genotypes on to the
descendents (Falconer 1987).
On these premises, the aim of our study was to
associate and estimates obtained from simulated data to
the theoretical concepts presented in literature and to identify
the factors that can alter the estimates of the same.
MATERIAL AND METHODS
In our study we considered a complete diallel with p = 28
populations (parents). Five hypothetic variables controlled by a
single gene with two alleles each were simulated, whose dominance
degree was represented by d/a relations of 0, 0.5, 1.0, 1.5, and 2.0,
expressing the respective situations of absence, partial, and complete
dominance, and two degrees of overdominance.
The kth parents for k = 1, 2,..., 28, was represented by the
favorable allele frequency equal to k/28. The analysis of the
data obtained from the 28 parents and the p(p-1)/2 = 378 F1
hybrids, a total of 406 genotypes, was realized by method 1,
model 1 (for fixed effects) of Griffing (1956). We assumed that
the maternal effects, effective population size, and inbreeding
depression effects were all negligible or absent in the seeds of
these 406 genotypes.
The statistical model is expressed by:
,
where is the observed average of the ijth hybrid for i j or
for the ith parent, when i = j; with i and j = 1, 2,..., p; m is the
overall mean; and are general combining ability effects
(GCA); sij is the specific combining ability effect (SCA), and
is the random non-observable experimental error term.
The averages values used for the complete diallel were
defined by the following equations:
and
Yij=µ+a(pi+pj-1)+(pi+pj-2pipj)d
where µ is the average genotypic value of the homozygote; a is
the deviation between the largest genotypic homozygote value
and µ; d is the deviation between the genotypic heterozygote
value and µ, called deviation due to dominance; and pi and pj are
frequencies of favorable alleles for the ith and jth parents,
respectively.
The genetic GCA and SCA values were determined by a
modified version of Vencovsky (1987)’s model, which considers
only one locus controlling a quantitative trait with two alleles
without epistatic effects in a diploid species with sexual
reproduction, given by:
(i)
and
(ii)
where is the average frequency of the favorable allele among
the parents of the analyzed diallel, and the equation
represents the mean effect of gene substitution ( ).
The study was based on the estimation of GCA ( )
effects and also SCA effects of the ith parent with itself ( ) for
the five hypothetical simulated variables, relating these estimates
to the equations (i) and (ii), respectively. The simulations and diallel
analyses were performed with software Genes (Cruz 1997).
RESULTS AND DISCUSSION
Table 1 shows the values of the 28 GCA estimates for
the different hypothetic variables. Oscillations from one
variable to the other were small for the same parent
(population) in spite of the different dominance degree. This
fact can be explained by the mean allelic frequency of 0.52.
When the mean estimated allelic frequency in a diallele is 0.5,
the dominance effects are cancelled. Consequently, the ’s
are predominantly function of the contribution of the additive
effects and the gene frequency in parent i (Cruz and Vencovsky
1989). Thus, if the parents that compose the diallel have a
broad genetic base (many segregant loci) and a gene frequency
mean of 0.5, the greatest effects of the GCA for a quantitative
trait will probably indicate the parents with the highest
concentration of predominantly additive genes. However, we
should bear in mind that despite the broad-based parents,
some loci might have allelic frequencies that differ from 0.5,
which does not necessarily cancel the expression ,
whose GCA is influenced by the gene dominance.
The additional effect ascribed to the dominance deviation
slightly reduced the estimates along with the increase of the d/a
relation. Although the dominance effects hampered the selection
of the parents, they allowed an exploration of the heterosis that,
together with the right of the intellectual property warranted by
the secret of the inbred line combinations, are the chief responsible
for the success in maize improvement.

340 Crop Breeding and Applied Biotechnology 4:338-343, 2004
FM Ferreira et al.
The GCA estimates of the parents 1 and 28 of smallest
and greatest frequency of the favorable allele were the highest
in absolute values for all presented dominance degrees. On
the contrary, the GCA estimates for the intermediate parents
tended to be zero, principally in parent 14, whose difference
was the smallest.
If the parents are considered open pollination
populations, the greatest GCA effect is related to the
population with greatest frequency of genes that increase the
trait expression and/or that presents the greatest difference
between its gene frequency and the mean frequency of the
populations involved in the diallel. If the parents are inbred
or pure lines, the greatest GCA effect is related to the line of
the largest number of genes that increase the expression of
the trait and, consequently, the greatest number of positive
differences between its gene frequency and the mean
frequency of the lines involved in the diallel (Viana 2000).
The GCA effect is therefore an indicator for the
superiority of a parent and/or of its greatest divergence
regarding the parents of the diallel, besides making information
on the parameters of the population effect (vj) and of the
varietal heterosis (Hj) of the Gardner and Eberhart (1966)
model available, as Pacheco et al. (2002) have pointed out.
Cruz and Vencovsky (1989), who studied the SCA
parameter of a parent with itself ( ), considered it an
indicator of varietal heterosis and the existence of
unidirectional dominance, which is negative for predominantly
positive dominance deviations and vice-versa. They further
showed that the sum of the is a linear function of the mean
Table 1. Estimates of GCA effects for the hypothetical variables controlled by a single gene with two alleles and different dominance
degrees (d/a ratios) for a complete diallel with 28 parents
Parents d/a = 2.0 d/a = 1.5 d/a = 1.0 d/a = 0.5 d/a = 0
1 -4.676 -4.712 -4.749 -4.785 -4.822
2 -4.300 -4.342 -4.383 -4.424 -4.464
3 -3.928 -3.973 -4.018 -4.062 -4.107
4 -3.559 -3.605 -3.654 -3.702 -3.750
5 -3.193 -3.243 -3.292 -3.343 -3.393
6 -2.831 -2.882 -2.933 -2.985 -3.036
7 -2.472 -2.521 -2.575 -2.626 -2.679
8 -2.117 -2.168 -2.217 -2.270 -2.322
9 -1.764 -1.815 -1.864 -1.914 -1.964
10 -1.415 -1.465 -1.512 -1.559 -1.607
11 -1.069 -1.116 -1.160 -1.205 -1.250
12 -0.729 -0.770 -0.811 -0.852 -0.893
13 -0.391 -0.427 -0.463 -0.502 -0.536
14 -0.055 -0.086 -0.117 -0.148 -0.179
15 0.276 0.252 0.228 0.203 0.178
16 0.604 0.587 0.570 0.553 0.536
17 0.929 0.920 0.911 0.902 0.893
18 1.251 1.250 1.249 1.250 1.250
19 1.569 1.578 1.588 1.598 1.607
20 1.883 1.904 1.924 1.945 1.964
21 2.195 2.227 2.258 2.290 2.321
22 2.503 2.547 2.590 2.635 2.678
23 2.807 2.864 2.921 2.979 3.036
24 3.108 3.179 3.251 3.322 3.393
25 3.405 3.492 3.578 3.664 3.750
26 3.700 3.802 3.903 4.005 4.107
27 3.991 4.109 4.227 4.346 4.464
28 4.278 4.414 4.550 4.686 4.821

Crop Breeding and Applied Biotechnology 4:338-343, 2004 341
Genetic components of combining ability in a complete diallel
heterosis, and that method 2 proposed by Griffing (1956), which
includes the parents, spawns basically the same information as
the methodology of Gardner and Eberhart (1966).
The effects, like any SCA effect, are strongly linked
to the dominance effects and the allelic frequencies (Table 2).
This is made evident by the variations in the d/a relations
that tend to increase the effect with the increase of d for
the same parent.
Since a single gene was considered in this study, the
estimates were negative because of the unidirectional
dominance. It was also verified that the parents with extreme
allelic frequencies, 1 and 28, had the highest absolute
values, which made them genetically the most divergent
in relation to the others.
The correlation established in this study between the
absolute gi and siivalues were 0.96 for any dominance degree
(d/a 0). Viana (2000) found similar results.
This correlation between populations is the
correlation between the inter-populational and intra-
populational combination ability. If the additive and dominant
effects are significant in the studied diallel, the populations
with high (positive) and have the highest heterozygous
Table 2. Estimates of sii effects for the hypothetical variables controlled by a single gene with two alleles and different dominance
degrees (d/a ratios) for a complete diallel with 28 parents
Parents d/a=2.0 d/a=1.5 d/a=1.0 d/a=0.5 d/a=0
1 -8.787 -6.587 -4.395 -2.195 -0.004
2 -7.550 -5.658 -3.768 -1.888 0.001
3 -6.404 -4.806 -3.198 -1.601 -0.003
4 -5.352 -4.021 -2.676 -1.341 0.003
5 -4.404 -3.306 -2.199 -1.100 -0.001
6 -3.548 -2.658 -1.778 -0.886 0.004
7 -2.786 -2.089 -1.393 -0.693 0.000
8 -2.116 -1.585 -1.059 -0.525 -0.004
9 -1.551 -1.162 -0.775 -0.387 0.001
10 -1.069 -0.801 -0.540 -0.267 -0.003
11 -0.691 -0.520 -0.343 -0.175 0.003
12 -0.402 -0.302 -0.202 -0.101 -0.001
13 -0.218 -0.158 -0.107 -0.051 0.004
14 -0.119 -0.089 -0.060 -0.030 0.000
15 -0.122 -0.085 -0.060 -0.031 -0.004
16 -0.218 -0.156 -0.104 -0.051 0.001
17 -0.408 -0.302 -0.206 -0.099 -0.003
18 -0.691 -0.522 -0.342 -0.176 0.003
19 -1.068 -0.808 -0.539 -0.272 -0.001
20 -1.546 -1.159 -0.772 -0.385 0.004
21 -2.119 -1.585 -1.060 -0.526 0.000
22 -2.785 -2.085 -1.394 -0.695 -0.004
23 -3.544 -2.660 -1.775 -0.883 0.001
24 -4.406 -3.300 -2.205 -1.100 -0.003
25 -5.360 -4.015 -2.679 -1.343 0.003
26 -6.410 -4.805 -3.200 -1.605 -0.001
27 -7.551 -5.659 -3.778 -1.887 0.004
28 -8.786 -6.589 -4.393 -2.196 0.000
Heterosis mean - 3.215 - 2.410 - 1.607 - 0.803 0.000
Varietal heterosisMean heterosis areaVarietal heterosis

342 Crop Breeding and Applied Biotechnology 4:338-343, 2004
FM Ferreira et al.
and homozygous loci frequencies. This condition ensures the
required variability for a successful selection process and good
performance, and indicates the most suitable populations for
intra-populational improvement. On the other hand, Pacheco et
al. (2002) claim that the population with the greatest together
with a large negative in a diallel is not appropriate for intra-
populational improvement due to a high frequency of homozygous
recessive loci. Likewise, a population with a very high and a
small , in spite of having a high frequency of favorable
homozygous alleles and, consequently, high yield means, may
not be the most indicated for intra-populational improvement
once the expected gains with selection tend to be small owing to
its probably smaller genetic variability.
The negative heterosis may be attributed to the
deviations of the negative dominance, indicating that the
dominance effect tended to diminish the value of the trait.
Since the dominance deviations in this simulation were
unidirectional and positive, the varietal heterosis was negative
for the parents 7 to 22, mainly for 14 and 15, since these
contributed to reduce the heterosis of the crosses in which
they participated and their heterozygous genotype
manifestation.
The more homozygous a parent is, the greater the
varietal heterosis, e.g., parents 1 and 28. Although the greatest
absolute value is related to parents with the highest
favorable or unfavorable allele concentration (1 and 28,
respectively), they have no variability. The best parent should
unite a high frequency of favorable alleles and high frequency
of loci in heterozygosis. If several regulating loci of any trait
were considered, instead of a single locus, the parents 24, 25,
and 26 would probably be the most indicated for intra-
populational selection; besides the high concentration of
favorable alleles, these have most individuals in the respective
populations with heterozygote genotypes.
CONCLUSIONS
1. The GCA estimates are influenced by the dominance
deviations in the regulating trait loci;
2. The high correlation between the absolute values to
the respective values in the evaluation of populations for
a particular polygenic trait makes it possible to identify and
select populations of good performance and a high genetic
variability. The joint interpretation of these two genetic
parameters provides complementary information on the
genetic quality of the populations included in the
improvement program.
ACKNOWLEDGEMENTS
This study was supported by a grant from Conselho
Nacional de Desenvolvimento Científico and Tecnológico
(CNPq) of Brazil.
Componentes genéticos das capacidades de combinação
do dialelo completo
RESUMO - Com o objetivo de associar os valores de e obtidos, com os conceitos teóricos dos respectivos parâmetros,
foi considerado um dialelo completo com 28 progenitores e a simulação de cinco variáveis hipotéticas com cinco diferentes
relações d/a (0, 0,5, 1,0, 1,5 e 2,0). Estas foram governadas por apenas um gene com dois alelos, cujos progenitores foram
representados por diferentes freqüências do alelo favorável (1/28, 2/28,..., 28/28). Concluiu-se que a existência de desvios da
dominância nos locos reguladores da característica influencia consideravelmente nas estimativas de CGC e que existe uma
alta correlação (0,96) entre os valores absolutos de com os respectivos valores de . A avaliação das estimativas
de e conjuntamente, traz informações sobre a qualidade genética das populações do dialelo.
Palavras-chave: capacidade combinatória, dialelo completo, componentes genéticos, melhoramento genético.
REFERENCES
Cruz CD (1997) Programa Genes versão Windows:
aplicativo computacional em genética e estatística.
Editora UFV, Viçosa, 648p.
Cruz CD and Vencovsky R (1989) Comparação de alguns
métodos de análise dialélica. Revista Brasileira de
Genética 12: 425-438.
Falconer DS (1987) Introdução à genética quantitativa. Editora
UFV, Viçosa, 279p.

Crop Breeding and Applied Biotechnology 4:338-343, 2004 343
Genetic components of combining ability in a complete diallel
Gardner CO and Eberhart SA (1966) Analysis and interpretation
of the variety cross diallel and related populations.
Biometrics 22: 439-452.
Griffing B (1956) Concept of general and specific combining
ability in relation to diallel crossing systems. Australian
Journal of Biological Sciences 9: 463-493.
Pacheco CAP, Santos MX, Cruz CD, Guimarães PEO, Parentoni
SN and Gama EEG (2002) Topics on the genetics of maize
yield stability. Crop Breeding and Applied Biotechnology
2: 345-354.
Sprague GF and Tatum LA (1942) General vs. specific combining
ability in single crosses of corn. Journal of American
Society of Agronomy 34: 923-932.
Vencovsky R (1987) Herança quantitativa. In: Paterniani E
and Viégas GP (eds.) Melhoramento e produção de milho.
2nd ed., Fundação Cargill, Campinas, p. 277-340.
Viana JMS, Cruz CD and Cardoso AA (1999) Theory and analysis
of partial diallel crosses. Genetic and Molecular Biology
22: 591-599.
Viana JMS (2000) The parametric restrictions of the Griffing
diallel analysis model: combining ability analysis. Genetic
and Molecular Biology 23: 877-881