AGRICULTURE AND BIOLOGY JOURNAL OF NORTH AMERICA ISSN Print: 2151-7517, ISSN Online: 2151-7525, doi:10.5251/abjna.2016.7.2.95.106

© 2016, ScienceHuβ, http://www.scihub.org/ABJNA

Application of diallel analyses in crop improvement

1Chukwu, S.C., 1Okporie, E.O., 2Onyishi, G.C., 1Ekwu, L.G., 1Nwogbaga, A.C. and 3Ede, N. V.

1Department of Crop Production and Landscape Management, Ebonyi State University, Abakaliki, Nigeria

2Department of Crop Science and Technology, Federal University of Technology, Owerri, Nigeria

3Department of Agronomy and Ecological Management, Enugu State University of Science

and Technology, Enugu.

ABSTRACT

For proper recommendation of the most reliable method of effective crop improvement, the knowledge and practice of diallel analysis is paramount. A diallel cross is a mating scheme used by plant and animal breeders, as well as geneticists, to investigate the genetic underpinnings of qualitative and quantitative traits. There are four main types of diallel mating design: i. Full diallel in which parents and reciprocal crosses are involved along with F1 ii. Half diallel with parent and without reciprocal crosses iii. Full diallel without inclusion of parents and iv. Half diallel without parents or reciprocal crosses. Several breeding methods are available for improvement of both quantitative and qualitative traits of different crops. Diallel crosses represent the best strategy for determining the general (GCA) and specific (SCA) combining abilities between putative parents and subsequently, the choice of breeding method. In this way, the objective of the present review is to elucidate the import of diallel analyses in crop improvement.

Key words: Crop improvement, diallel crosses, diallel analyses.

INTRODUCTION

Crop improvement refers to the genetic alteration of plants to satisfy human needs (Sansern et al., 2010; Falconer and Mackay, 1996). In prehistory, human beings in various parts of the world brought into cultivation a few hundred species from the hundreds of thousands available. In the process they transformed elements of these species into crops though genetic alterations that involved conscious and unconscious selection, the differential reproduction of variants (Allard, 1999). Through a long history of trial and error, a relatively few plant species have become the mainstay of agriculture and thus the world's food supply (Mackay, 2001). This process of domestication involved the identification of certain useful wild species combined with a process of selection that brought about changes in appearance, quality, and productivity. The exact details of the process that altered the major crops are not fully understood, but it is clear that the genetic changes were enormous in many cases. In fact some

crop plants have been so changed that for many of them, maize, for example, their origins are obscure, with no extant close wild relatives (Obi, 1991).

The selection process was unconscious in many cases. For example, in wild wheats, the grains scatter by disarticulation, separation of the seed from the seed head (Balzarini et al., 2002; Barbosa et al., 1996). When these grains were harvested by cutting the heads with a sickle, an unconscious selection occurred for "nonshattering" types that would then be continually replanted. For some crops a clear conscious selection occurred, especially when the variation was obvious and would be maintained by vegetative propagation (Balzarini et al., 2002). Something so clearly useful as a seedless banana must have been immediately seized upon and maintained ("fixed") by planting offshoots of the plant. The changes wrought in domestication included alteration in organ size and shape; loss of many survival characters, such as bitter or toxic substances; disarticulation of seeds in grains;

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protective structures, such as spines and thorns; seed dormancy; and change in life span—increased in crops grown for roots or tubers and decreased in crops grown for seed or fruit. Selection by bulking desirable types (mass selection) is a powerful technique for making rapid changes easily while maintaining genetic diversity in the population (Okporie and Obi, 2002).

According to Barton and Keightley (2002), the selection of naturally occurring variants is the basis of crop improvement. Over thousands of years, this technique resulted in the development of modern basic crops. The discovery of techniques for asexual (vegetative) propagation, such as by using natural offshoots, rooting stem cuttings, or various grafting techniques, made it possible to "fix" genetic variants. This was the technique used for many tree fruits, enabling identical plants to be cultivated in orchards. Naturally produced seedlings derived from intercrosses of these selected plants were then available for selection again. Many present-day fruit crops are similar to those cultivated in antiquity, and some ancient selections are still cultivated—dates, for example. As humans carried improved crops to new locations, opportunities opened to increase genetic variation from natural intercrosses with new wild populations (Boppenmaier et al., 1992).

The changes that occur can be dramatic over time, as seen in the proliferation of breeds of animals and especially the wide range of changes brought about by fanciers of dogs, chickens, and pigeons. The observation of these changes influenced the thinking of Charles Darwin to suggest that natural selection, the survival of the fittest, could lead to enormous genetic changes if carried out over a long enough time, and could lead to the origin of new species (Allard, 1999).

In the eighteenth and nineteenth centuries a conscious attempt was made to predict the performance of plants that could be expected from one seed generation to the next (Cheres and Knapp, 1998; Chloupek and Hrstkova, 2005). The concept that ancestry was important in crop improvement led to refinement in the selection process, brought about by keeping records and the assessment of lineage. Furthermore, it became obvious that variation could be managed by controlling the mating process, an extension of what had long been known in animal breeding. This new type of selection, termed pedigree selection, was found to increase the

efficiency of the process (Corbellini et al., 2002). Progeny testing (evaluating the genetic worth by progeny performance) increased efficiency of this process (Okporie et al., 2014). The origins of commercial plant breeding began in the second half of the nineteenth century among seed producers. It involved controlled crosses (hybridization) between selections to control genetic recombination, followed by selection of improved types (Okporie and Obi, 2004). This is still the basis of traditional plant breeding. Interestingly much of this early type of plant breeding was carried out without a clear understanding of the genetic mechanism involved in inheritance.

Until the famous experiments with the garden pea by Gregor Mendel within 1822–1884, a Catholic priest in Brünn, a Moravian town then in the Austro-Hungarian Empire, the basic theory of inheritance involved the concept of blending (Allard, 1999). Mendel unraveled the basic concept of inheritance and clearly showed that characters in the pea were due to elements, later called genes, that remained unaltered as they were inherited. Many characters in peas, such as tallness and dwarfness, were shown to be controlled by a pair of genes, of which one member was not always expressed (the concept of dominance and recessiveness) (Allard, 1999). Mendel demonstrated that the gametes of the pea contained one member of the gene pairs that controlled characters and that recombined randomly at fertilization. Mendel's paper was published in 1866, but it had no impact until the paper was "rediscovered" in 1900, when it created a sensation. It was soon obvious that the differences in appearance among plants (phenotypes) could be explained by the interaction of various genes (genotypes) as well as interaction with the environment (Obi, 2013).

In the twentieth century plant breeding developed a scientific basis, and crop improvement was understood to be brought about by achieving favorable accumulations and combinations of genes (Allard, 1999; Flachenecker et al., 2006). Taking advantage of known genetic diversity could facilitate this, and appropriate combinations were achieved through recombinations brought about by the sexual process (hybridization) (Carvalho et al., 2003b). Furthermore it was possible to move useful genes by special breeding strategies. Thus a gene discovered in a wild plant could be transferred to a suitable adapted type by a technique known as the backcross method. Carvalho et al. (2003a), made a sexual

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hybrid, followed by a series of backcrosses to the desirable (recurrent) parent, while selecting for the new gene in each generation. After about five or six back-crosses, the offspring resembled the recurrent parent but contained the selected gene.

In the early twentieth century it was demonstrated that the extra vigor long associated with wide crosses (called hybrid vigor or heterosis), particularly in naturally cross-pollinated crops, could be exploited in plant breeding. For maize, a new system of hybrid breeding was developed, using a combination of inbreeding and outbreeding (Lefebvre et al., 2001; Lanza et al., 1997). Inbreeding was accomplished by crossing the plant with itself. This led to a decline in vigor as the step was repeated over several generations. Outbreeding was achieved by intercrossing the inbred lines to restore vigor. The hybrid between inbreds derived from divergent inbreds (called a single cross or F1 hybrid) was uniform (homogeneous), and some were superior to the original populations before inbreeding. During the process of inbreeding, the inbreds became weak, but vigor was restored in the F1. To increase seed set from weak inbreds, two hybrids were crossed; this was known as the double cross method (Sawazaki et al., 2000).

Hybrid breeding technique in a sense is similar to arranging a Rubic's cube, where contradictory steps need to be taken to achieve the appropriate reformulation. In hybrid breeding, the first step produces a series of weak inbreds, followed by a series of specific combination, to produce a series of new hybrids. Hybrid maize breeding led to enormous increases in productivity, which were soon exploited in a wide variety of seed-propagated crops, including naturally self-pollinated ones, such as tomato and rice (Obi, 1991; Stuber et al., 1992).A number of genetic techniques were developed and refined in twentieth-century breeding, such as improved techniques to search for and store increased genetic variability, different techniques to develop variable populations for selection, and improved methods of testing to separate genetic from environmental effects. The exact details of the process for crops necessarily differed among naturally cross-pollinated plants (such as maize) and naturally self-pollinated plants (such as soybean or tomato) as well as those plants in which vegetative propagation (usually cross-pollinated) permitted the fixing of improved types directly (Ubi, 2013).

Conventional plant breeding can be defined as systems for selection of superior genotypes from genetically variable populations derived from sexual recombination (Allard, 1999). The system is powerful because it is evolutionary; progress can be cumulative, with improved individuals continually serving as parents for subsequent cycles of breeding. Genetic improvement by conventional breeding has made substantial changes when the efforts have been long-term. Characters improved include productivity, quality, and resistance to diseases, insects, and stress (Okporie et al., 2013; Obi, 1991). There are, however, limits to the progress of conventional breeding. These are due to limitations of the sexual system, because it is usually not possible to incorporate genes from nonrelated species or to incorporate small changes without disturbing the particular combination of genes that make a particular type unique (Ubi, 2013). Thus a useful gene in cabbage cannot be transferred to wheat. Limitations of conventional breeding are particularly apparent when a needed character (such as disease or insect resistance) is unavailable in populations that can be incorporated by sexual crosses. Mutations may be induced, but they are often deleterious or connected with undesirable effects (Ubi, 2013; Dankert et al., 2009).

With conventional breeding, it is also not possible to improve a unique genotype, such as "Bartlett" pear, by adding a single character, since the recombination that results from hybridization makes it impossible to recon-figure this cultivar exactly. Finally, conventional breeding has technical or economic limitations to detect infrequent or rare recombinants, the lack of sufficient time to generate cycles of recombination, space to grow necessary populations to recover superior recombinants, or resources to be able to select, identify, evaluate, and fix desired recombinants (Ubi, 2013). Therefore, the present review will discuss the import of diallel analyses in crop improvement.

Developments Using DNA: It has been suggested that many of the limitations of conventional breeding can be overcome with advances in molecular biology that rely on DNA, the genetic material (Sawazaki et al., 2000).Recombinant DNA technology, called transgene technology or genetic engineering, is the most powerful and revolutionary of the new genetics developed in the last half of the twentieth century (Ubi, 2013; DAS, 2013; Sawazaki et al., 2000;). It is possible to isolate stretches of DNA from one

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organism, store it in a bacterial host, select unique combinations, and then incorporate them into the DNA of another species, where it can be expressed. This technique, which relies on cell and tissue culture, is truly a marvelous process. Refinements in the technique make it possible to concentrate mutations in desired genes, further increasing variability. Other uses of molecular biology known as genomics involve the detailed mapping of the DNA and the identification of useful stretches of the molecule (Clemente and Cahoon, 2009). This makes it possible to improve the efficiency of selection, because it is based directly on the genes rather than the organism, where the effects may be confounded by environment and genetic interactions (Ubi, 2008).

The limitations of the new breeding methods include technical problems, such as the difficulty of transformation, problems of gene expression, or the lack of knowledge concerning suitable genes to transfer. There are also nontechnical issues, such as legal problems, since the techniques and the genes are usually patented. However, in the short run the greatest impediment has been problems of consumer acceptability and fear of the unknown (Plant Breeding News, 2012a). The term "Frankenfood" has been coined to refer to food altered by the process of using exotic genes incorporated by transgene technology. No convincing evidence shows that genetic engineering has produced harmful products, and an abundance of evidence shows that many foods derived from traditional systems have inherent problems (consider the allergic reactions of many people to peanuts). Nevertheless, molecular techniques have incited fear of this new technology in many people. Moreover, the surplus of food in the West has reduced the imperative to make the case for the need for new technology to consumers (Plant Breeding News, 2012b).

The biotechnology industry has sold the technology to farmers (who have accepted it) and ignored consumers. They have not been sophisticated in exploiting the environmental virtues inherent in the new technology, such as reducing pesticides, or in making the case that increased yield could free up the agricultural use of fragile environments. Because of the benefits that could accrue from this new technology, especially in the problem areas of the world, it seems certain that future progress in plant breeding will involve both conventional and unconventional techniques, but the immediate course of events is fraught with uncertainty (Rennie and

Heffer, 2010). However, given the present situation on ground, i.e., apathy of people to the general acceptability of genetically modified crops (GMOs) and farmers‘ unwillingness to buy planting material each cropping season, they rather prefer saving planting materials from the previously planted materials. Therefore, conventional breeding techniques such as diallel crosses still remain the best available option for crop improvement.

Genetic crop improvement: Genetic crop improvement can be achieved by introducing improved alleles at existing loci through conventional crossing, aided by marker and other technology, and by adding new loci by transformation (Schaart et al., 2004). At present it is not possible to use transformation to replace alleles at existing loci, although the expression of genes at existing loci can be blocked by introducing antisense or co-suppression constructs (Ruiz and Daniell, 2005).

The ability to transform crop plants has developed remarkably since the first transformed plants were reported in 1983. However, there is still a need to develop improved methodology, for example, to improve the frequency of transformation, to increase the number of genes that can be transferred, to allow better control of the expression of the transferred genes, and to enable the genes to be inserted at defined positions. Given the pace of past progress, there is every reason to believe that many of the necessary goals will be achieved (Okporie et al., 2013; Okporie and Obi, 2004).

Agronomic crop improvement: Effort has concentrated on genetic improvement because that is where the greatest potential lies to improve the yield of crops. This is in part because of the great success that has already been achieved in crop production through agronomy and crop protection. There will be tremendous advances in these areas in the future, but these are more likely to address improving the efficiency in the use of fertilizers and crop protection agents, and in minimizing the side-effects of these on the environment (Mendel Biotechnology, 2012). The developments of combinatorial chemistry and the identification of new target sites from genomics research are likely to enhance the quality of agrochemicals at the farmer's disposal. Sophisticated systems to support decision making, allied machinery capable of implementing those decisions precisely, particularly in respect to the use of water, fertilizers and crop protection agents, will undoubtably improve the quality of agriculture (Funke et al., 2006).

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Diallel crosses: A diallel cross is a mating scheme used by plant and animal breeders, as well as geneticists, to investigate the genetic underpinnings of qualitative and quantitative traits (Hallauer and Miranda, 1988). In a full diallel, all parents are crossed to make hybrids in all possible combinations. Variations include half diallels with and without parents, omitting reciprocal crosses (Obi, 2013). Full diallels require twice as many crosses and entries in experiments, but allow for testing for maternal and paternal effects (Obi, 2013). If such "reciprocal" effects are assumed to be negligible, then a half diallel without reciprocals can be effective.

Common analysis methods utilize general linear models to identify heterotic groups, (Griffing, 1956) estimate general or specific combining ability, (Gardner and Eberhart, 1966) interactions with testing environments and years, or estimates of additive, dominant, and epistatic genetic effects (Hayman, 1954) and genetic correlations (Okporie, 2013).

Types of diallel mating design: According to Obi (2013), there are four main types of diallel mating design:

(i) Full diallel in which parents and reciprocal crosses are involved along with F1

(ii) Half diallel with parent and without reciprocal crosses

(iii) Full diallel without inclusion of parents

(iv) Half diallel without parents or reciprocal crosses

Illustrations of the various types of diallel mating designs

♂ ♀

1 2 3 4 5 6 7

1 (X) X X X X X X

2 X (X) X X X X X

3 X X (X) X X X X

4 X X X (X) X X X

5 X X X X (X) X X

6 X X X X X (X) X

7 X X X X X X (X)

Fig. 1: Full diallel in which parents and reciprocal crosses are involved along with F1

Fig. 2: Half diallel with parent and without reciprocal crosses

Fig. 3: Full diallel without inclusion of parents

Fig. 4: Half diallel without parents or reciprocal crosses

The diallel cross involves all possible way crosses among several parents, and provides a method for evaluating parental entries by comparing their performance in combination with each other parent in the diallel. The average performance of a line in hybrid combinations is termed ―general combining ability,‖ (GCA) and relates to additive gene action.

♂ ♀

1 2 3 4 5 6 7

1 (X) X X X X X X

2 (X) X X X X X

3 (X) X X X X

4 (X) X X X

5 (X) X X

6 (X) X

7 (X)

♂ ♀

1 2 3 4 5 6 7

1 X X X X X X

2 X X X X X X

3 X X X X X X

4 X X X X X X

5 X X X X X X

6 X X X X X X

7 X X X X X X

♂ ♀

1 2 3 4 5 6 7

1 X X X X X X

2 X X X X X

3 X X X X

4 X X X

5 X X

6 X

7

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The performance of specific hybrid crosses, after taking into account the GCA of the two parents, is termed ‗specific combining ability‖ (SCA) and measures the dominance deviation from the additive model. Information of GCA and SCA – or the types of gene action influencing various traits – enables the plant breeder both to evaluate parental entries, and to select the best breeding system for maximum character improvement (Obi, 2013).

Diallel crosses represent the best strategy for determining the general (GCA) and specific (SCA) combining ability between putative parents. However, the major barrier for their use is the need of a large number of crosses for evaluation. The interpretation can be affected by the number and quality of data needed to obtain a precise estimate (Burow and Coors, 1993). Another point is that an increase in the number of genotypes used in the crosses can preclude the experiment feasibility and increase the difficulty in the analysis (Okporie, 2013).

According to this technique, it is necessary to cross all the selected genotypes (complete diallel) and evaluate their progenies or one can opt for the loss of some genetic information and perform part of the crosses (incomplete diallel). Another limitation is the difficulty in obtaining hybrids due to occurrences of species incompatibility or specific environmental requirements.

Despite these limitations, this type of analysis provides detailed information regarding the genotypes involved, estimates for parameters useful for the selection of the best parental combinations and an understanding of the genetic effects involved in the targeted characters. The most commonly used techniques are those proposed by: i) Griffing (1956), in that the effects for the general and specific combining ability between parents are estimated; ii) Gardner and Eberhart (1966), in that the variety and heterosis are evaluated; and iii) Hayman (1954), that provides information regarding the character's basic mechanism of inheritance on the genetic values of the parents used and the selection limit. The Gardner and Eberhart Analyses II and III have recently been revisited with some interesting interpretations (Murray et al., 2003). Furthermore, some software such as DIALLELSAS05 (Zhang et al., 2005) is available for helping breeders better design their diallel matings. Some examples of diallel analyses used for the selection of parents are available for wheat (Barbieri et al., 2001), oats (Lorenzeti et al., 2005), common beans (Machado et al., 2002b), maize (Melani and

Carena, 2005), soybeans (Cruz et al., 1987), and green pepper (Miranda et al.,1998).

In oats, the work of Lorencetti et al. (2005), employing five parents combined with each other in a complete diallel scheme without the reciprocals, adds to the understanding of how this analysis contributes to determining the best parents. In their study, the high individual performance of genotype UPF 16(168.83 g) was decisive for the increase in the progeny means in those crosses where it appears as one of the parents. This resulted in a higher estimate of GCA (35.79), and indicates that the parent was useful for crosses aiming to improve grain yield in oats. The SCA estimates are useful to breeders as a way to promote the selection of hybrid combinations for direct use by farmers in species where heterosis is exploited or for the recommendation of promising specific combinations for the selection of superior recombinants such as those in the present example.

Therefore, the major SCA effects observed for the crosses UPF 16×UPF 18, UFRGS 7×UFRGS 17, and UFRGS 17×URPel 95-015 reveal that the use of these combinations in breeding programs will produce promising progenies from which superior lines could originate. In general, for plant breeding, hybrid combinations with high SCA and those with at least one parent with high GCA are the most sought after (Paini et al.1996). Diallel crossing programs have been applied to provide a systematic approach for the detection of suitable parents and crosses for investigated characters. In addition, diallel analysis gives plant breeders the opportunity to choose the efficient selection method by allowing them to estimate several genetic parameters (Unay et al., 2004). Combining ability describes the breeding values of parental lines to produce hybrids.

In many studies, GCA effects for parents and SCA effects for crosses were estimated in maize (Araujo and Miranda, 2001). Non-additive gene effects for grain yield were found to be significant in maize (Dehanghpour et al., 1996 and Kalla et al., 2001) which suggested that several combinations among parental lines by their mean performance and genetic nature had the potential for the development of more yielding and earlier genotypes.

The diallel cross mating schemes have been extensively used in breeding programs for the evaluation of the genetic potential of populations or genotypes (Obi, 2013). When two distinct groups of varieties or populations (e.g., local vs exotic, dent type vs. flint type, late flowering vs. early flowering,

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etc) are available the intergroup partial diallel cross can be used (Geraldi & Miranda, 1988). Those models allow the analysis of variance and estimation of effects for varieties within groups and heterosis and its components between groups. In this work, two groups of populations (adapted and exotic) were used and their genetic potentials were studied through the intergroup partial diallel scheme (Miranda & Geraldi, 1984), aiming at the identification of new sources of genes for tolerance to the acid soil conditions.

A diallel cross was established (Valdemar et al., 1981) to determine the general and specific combining ability for yield in a diallel cross among 18 maize populations. General combining ability (GCA) was found to be significant but specific combining ability (SCA) was not. Both location x GCA and location x SCA was found to be significant. The results showed that CMS 06 and CMS 05 have the highest GCA effects while the cross (CMS 07 x CMS 10) gave the highest SCA effect.

Diallel analyses: Diallel crosses as mating designs are used to study the genetic properties of inbred lines in plant breeding experiments. Plant breeders frequently need overall information on average performance of individual inbred lines in crosses known as general combining ability, for subsequent choosing the best among them for further breeding. For this purpose, diallel crossing techniques are employed. Consider a hypothetical population involving large number of lines and crosses so that all means are estimated without error. Crossing a line to several others provides the mean performance of the line in all its crosses. This mean performance, when expressed as a deviation from the mean of all crosses, is called the general combining ability of the line. Any particular cross, then, has an expected value which is the sum of the general combining ability of the two lines in combination. In statistical terms, the general combining abilities are main effects and the specific combining ability is an interaction. Griffing (1956) defines diallel crosses in terms of genotypic values where the sum of general combining abilities for the two gametes is the breeding value of the cross (i, j). Similarly, specific combining ability represents the dominance deviation value in the simplest case ignoring epistatic deviation (Mayo, 1980).

From the population consisting of a large number of lines the plant breeder carries out a diallel cross experiment after drawing a random sample of p lines. Since we are observing a sample from a large

hypothetical population of lines and crosses, the expected value of an observation Yij (conditional on the realized value of the general combining ability and specific combining ability) arising out of cross (i, j) involving lines i and j, i< j; i, j = i, …, p is modeled as

E(Yij) = µ + ɡi* + ɡj

*+ sij

*,

(1.1)

Where µ is the general mean, ɡi*(ɡj

*) is the realized

value of ɡi(ɡj), the general combining ability effect of sampled i-th (j-th) line and sij

*is the realized value of

sij, the specific combining combining ability effect of cross (i, j).

Accordingly, in experimental mating design, the analysis of the observations arising out of n crosses involving p lines will be carried out based on a model

Yijl = µ + ɡi + ɡj+ eijl;i< j, (1.2)

Where Yijlis the observation out of the l-th replication of the cross (i, j), ɡiis the i-th line effect with E(ɡi) = 0, Var (ɡi) = σ

2ɡ, Cov (ɡi, ɡj) = 0, µ is the general mean

and eijl is the random error component with expectation zero and variance σ

2e, i< j; i, j = 1,…,p.

Here the specific combining ability effects are assumed to be negligible and have been absorbed in the error component. In the model, as given in (1.2), µ is the fixed effect while ɡi, ɡj(i< j) and eijlare random effects. This is characteristic of what is called a random effects model, named Model II by Eisenhart (1947) and Griffing (1956).

The basic idea in the study of variation among observations arising out of crosses is its partitioning into components attributed to different causes like additive value, dominance deviation and epistatic deviation (Falconer, 1991). The relative magnitude of these components determines the genetic properties of the population. One of such properties is heritability which depends on the breeding policies. The relative importance of heredity in determining phenotypic values is called the heritability of a character in broad sense. Thus the ratio σ

2ɡ/σ

2p gives

a measure of heritability, where σ2p = σ

2ɡ+ σ

2e is the

phenotypic variance and σ2

ɡis the genotypic variance. Such a measure expresses the extent to which individual‘s phenotypes are determined by the genotypes.

Our primary interest is thus in h2=σ

2ɡ/(σ

2ɡ+ σ

2e). It

may also be mentioned here that the best linear unbiased predictor (BLUP) of the unobserved line effects µ + ɡi depends on good estimates of σ

2ɡand

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σ2e. The BLUP (ɡ) of ɡ is used to rank the values of

inbred lines, which are unobservable, so that the predictors of ɡiandɡjhave the same pairwise ranking as ɡiand ɡjwith maximum probability. Thus in order to get a good estimate of h

2and an efficient BLUP, we

propose optimal designs for estimation of σ2ɡ and σ

2e.

To find the estimates of the variance components σ

2ɡand σ

2e, we adopt the Henderson Method III

(Casella and McCulloch, 1992). Such a method has been used here since it gives an unbiased estimate of the variance components.

An experiment is carried out using a diallel cross design with p lines and n crosses. A diallel cross design is said to be complete if each of the (

p2)

crosses appear equally often in the design otherwise it is said to be a partial diallel cross design. Customarily, diallel cross experiments have been carried out using a completely randomized design or a randomized complete block design. Several methods of obtaining such diallel cross designs have been given, together with their efficiency factors, by Kempthorne and Curnow (1961), Curnow (1963) and others. However, with the increase in the number of lines p, the number of crosses in the experiment increases rapidly, and in such a situation, adoption of a complete block design is not appropriate. Singh and Hinkelmann (1995) used conventional partially balanced incomplete block designs both to select the diallel crosses to be observed and to arrange them into blocks. Some orthogonal blocking schemes had also been advocated by Gupta et al. (1995). Most of the theory of optimal diallel cross designs is based on standard linear model assumptions where the general combining ability effects are taken as fixed and the primary interest lies in comparing the lines with respect to their combining ability effects. Under such a model, Dash and Ghosh (1999) have characterized and obtained optimal completely randomized designs and incomplete block designs for diallel crosses. In many practical situations, the fixed effects assumption may not be tenable since we are studying only a sample of inbred lines, from a possibly large hypothetical population. A random effects model is proposed that allows us to first estimate the variance components and then obtain the variances of the estimates. We address the issue of optimal designs in this context by considering the A-optimality criteria. We obtain designs that are A-optimal for the estimation of heritability in the sense that the designs minimize the sum of the variances of the estimates of the variance components.

In section 2 we first obtain the estimate of σ2ɡ and

σ2e, under an unblocked model, and then obtain the

variances of these estimates. In section 3 similar thing is done under a block model. Finally, in section 4 we characterize A-optimal designs.

Assumptions of diallel cross analysis: The following are the assumptions of diallel cross analysis as described by Obi (2013)

1. Normal diploid segregation 2. Lack of maternal effects 3. Absence of multiple alleles 4. Homozygosity of parents 5. Absence of linkage among genes affecting

the character 6. Lack of epistasis 7. Random mating

Approaches to Diallel Analysis: There are two approaches to diallel analysis

a) Hayman‘s graphical approach b) Griffings numerical approach

Hayman’s graphical approach: The following six components of variation are estimated:

D = additive genetic variance

H1 = dominance variance

H2 = H1 [(1 – (U – V)2]

Where u and v are proportions of positive and negative genes, respectively, in the parents.

E = expected environmental component of variance

F = mean of FV over the arrays, where FV is the covariance of additive and dominance effects in a single array.

h2 = dominance effect, as algebraic sum over all the

loci in heterozygous phase, in all the crosses.

Genetic ratios

1. Average degree of dominance = √ H1/D = (H1/D)

½

0 = there is no dominance >0<1 = parental dominance 1, 0 = complete dominance

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>1 = over dominance 2. The ratio of dominant and recessive genes in the

parents is estimated as follows: [(4DH1)

½ + F]/[(4DH1)

½ - F]

Of this ratio: 1 = dominant and recessive genes in the parent are in equal proportions <1 = excess of recessive genes >1 = an excess of dominant genes is indicated ii. The number of gene groups, which control the character and exhibit dominance is given by the following formula: Number of gene groups = h

2/H

2

iii. The proportion of genes with positive and negative effects in the parents is estimated using the following formula: Genes with – and + effects = H2/4H1

If the positive and negative alleles are symmetrically distributed, this ratio equals 0.25. The above components are estimated from the data on F1 generation of a diallel cross.

However, in 1956, Jinks gave the procedure for estimating these components, from F2 data. The co-efficient of H1 and H2 are ¼ in F2, while they are 1 in F1. The h2 and F have a co-efficient of ½ because of the one generation of inbreeding to attain F2. Therefore, the genetic ratios in F2 are worked out as follows:

(i) Degree of dominance = [¼(H1/D)]½ (ii) Proportion of dominant and recessive genes in

the parents [¼{(4DH1)]½ + ½ F}/¼{(4DH1)½ - ½F}]

But the formula for proportions of genes with positive and negative effects in the parents = (H2/4H1), and the number of groups of genes which control the character and exhibit dominance are the same in F2 as those in F1.

(2) Vr – Wr Graph Vr = the variance of the rth array, and Wr = the covariance between the parents and the offspring on the rth array Inferences from the Vr – Wr graph

(a) When the regression line passes through the origin it depicts complete dominance (D = H1)

(b) When the regression line passes above the origin cutting Wr axis, it suggests the existence of partial dominance

(D > H1) (c) When it passes above the origin, cutting the

Wr axis and touching the limiting parabola, it suggests absence of dominance.

(d) But when the regression line passes below the origin, cutting the Vr axis, it denotes the presence of over dominance.

(e) The position of parental points on the regression line indicating the dominance order of the parents. The parents with more dominant genes are located nearer to the origin, while those with more recessive genes fall farther from the origin. The parents with equal frequencies of dominant and recessive genes occupy an intermediate position.

Griffing’s Numerical Approach: This is based on the estimation of general combining ability (GCA) and specific combining ability (SCA) variances and effects. Griffing (1956) gave four different methods of diallel analysis depending on the material (s) included in the experimentation.

Materials included in the experiment number ofentries in the experiment (F1‘s + parents)

Parents, F1‘s and Reciprocals n2

Parents and F1‘s (w/o Reciprocals) n (n + 1)/2

F1‘s and Reciprocals (w/o Parents) n (n – 1)

F1‘s (w/o parents & w/o Reciprocals) n (n – 1)/2

In their approach, gene action is deduced through the estimates of GCA and SCA variances and effects. The GCA component is primarily a function of additive genetic variance.

However, if epistasis is present, the GCA will include the additive x additive interaction as well. On the other hand, the SCA variance is mainly a function of dominance variance but it would include all the three types of epistatic interaction component; if epistasis were present.

CONCLUSION

With diallel analyses, it is easy to separate variation component of different varieties, whether genotypic, phenotypic or environmental variance. It also helps to sought out varieties that have general combining ability or specific combining ability. This will also help

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the plant breeder to estimate the type of gene action (additive gene action, dominant gene action or additive-additive gene action, etc). With these, the plant breeder will be able to know the type of breeding method to adopt in his crop improvement.

ACKNOWLEDGEMENT

The authors wish to appreciate Professor I.U. Obi for providing resources especially through his research grant foundation (Prof. I.U. Obi Research Grant Foundation).

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