Principles of Plant Genetics and Breeding. George Acquaah

there are very few homozygotes.

      Source: Adapted from Falconer (1981).

      Most of the important variation displayed by nearly all plant characters affecting growth, development, and reproduction is quantitative (continuous or polygenic variation; controlled by many genes). Polygenes demonstrate the same properties in terms of dominance, epistasis, and linkage as classical Mendelian genes. The Hardy‐Weinberg equilibrium is applicable to these characters. However, it is more complex to demonstrate.

      Another state of variability is observed when more than one gene affects the same polygenic trait. Consider two independent loci with two alleles each: A, a and B, b. Assume also the absence of dominance or epistasis. It can be shown that nine genotypes (AABB, AABb, AaBb, Aabb, AaBB, AAbb, aaBb, aaBB, aabb) and five phenotypes ([AABB, 2AaBB] + [2AABb; AAbb, aaBB] + [4AaBb, 2aaBb] + [2Aabb] + [aabb]) in a frequency of 1 : 4 : 6 : 4 : 1, will be produced, following random mating. Again, the extreme genotypes (AABB, aabb) are the source of completely free variability. But, also AAbb and aaBB, phenotypically similar but contrasting genotypes, contain latent variability. Termed homozygotic potential variability, it will be expressed in free state only when, through crossing, a heterozyote (AaBb) is produced, followed by segregation in the F₂. In other words, two generations will be required to release this potential variability in free state. Further, unlike the 50% : 50% ratio in the single locus example, only 1/8 of the variability is available for selection in the free state, the remainder existing as hidden in the heterozygotic or homozygotic potential states. A general mathematical relationship may be derived for any number (n) of genes as 1 : n : n − 1 of free:heterozygotic potential:homozygotic potential.

Another level of complexity may be factored in by considering dominance and non‐allelic interactions (AA = Aa = BB = Bb). If this is so, the nine genotypes previously observed will produce only three phenotypic classes ([AABB, 4AaBb, 2AaBB, 2AABb] + [2Aabb, 2aaBb, AAbb, aaBB] + [aabb]), in a frequency of 9 : 6 : 1. A key difference is that 50% of the visible variability is now in the heterozygous potential state that cannot be fixed by selection. The heterozygotes now contribute to the visible instead of the cryptic variability. From the plant breeding standpoint, its effect is to reduce the rate of response to phenotypic selection at least in the same direction as the dominance effect. This is because the fixable homozygotes are indistinguishable from the heterozygotes without a further breeding test (e.g. progeny row). Also, the classifications are skewed (9 : 6 : 1) in the positive (or negative) direction.

      A key plant breeding information to be gained from the above discussion is that in outbreeding populations, polygenic systems are capable of storing large amounts of cryptic variability. This can be gradually released for selection to act on through crossing, segregation, and recombination. The flow of this cryptic variability to the free state depends on the rate of recombination (which also depends on the linkage of genes on the chromosomes and the breeding system).

      Given a recombination value of r between two linked genes, the segregation in the second generation depends on the initial cross, as M.D. Haywood and E.L. Breese demonstrated as follows:

Initial cross	Free	Homozygous potential
1. AABB × aabb	1(1 − r)	2r
2. AAbb × aaBB	2r	2(r − 1)

      The second cross shows genes linked in the repulsion phase. The flow of variability from the homozygous potential to the free state depends on how tight a linkage exists between the genes. It will be at its maximum when r = 0.5 and recombination is free, and diminish with diminishing r. This illustration shows that with more than two closely linked loci on the same chromosome, the flow of variability would be greatly restricted. In species where selfing is the norm (or when a breeder enforces complete inbreeding), the proportion of heterozygotes will be reduced by 50% in each generation, dwindling to near zero by the 8th or 9th generation.

      The open system of pollination in cross‐pollinated species allows each plant in the gene pool to have both homozygous and heterozygous loci. Plant breeders exploit this heterozygous genetic structure of individuals in population improvement programs. In a natural environment, the four factors of genetic change mentioned previously are operational. Fitness or adaptive genes will be favored over nonadaptive ones. Plant breeders impose additional selection pressure to hasten the shift in the population genetic structure toward adaptiveness as well as increase the frequencies of other desirable genes.

       An example of a breeding application of Hardy‐Weinberg equilibrium

      In disease resistance breeding, plant breeders cross an elite susceptible cultivar with one that has disease resistance. Consider a cross between two populations, susceptible × resistant. If the gene frequencies of an allele A in the two populations are represented by P ₁ and P ₂ , the gene frequency in the F ₁ = (P ₁ + P ₂ )/2 = p. Assuming the frequency of the resistance gene in the resistant cultivar is P ₁ = 0.7 and that in the susceptible elite cultivar is P ₂ = 0.05, the gene frequency in the progeny of the cross p would be obtained as follows;

      Consequently, the gene frequency for the resistant trait is reduced by about 50% (from 0.7 to 0.375).

3.2 Issues arising from Hardy‐Weinberg equilibrium

      In order for Hardy‐Weinberg equilibrium to be true, several conditions must be met. However, some situations provide approximate conditions to satisfy the requirements.

      3.2.1 Issue of population size

      The Hardy‐Weinberg equilibrium requires a large random mating population (among other factors as previously indicated) to be true. However, in practice, the law has been found to be approximately true for most of the genes in most cross‐pollinated species, except when non‐random mating (e.g. inbreeding and assortative mating – discussed next) occur. Whereas inbreeding is a natural feature of self‐pollinated species, assortative mating can occur when cross‐pollinated species are closely spaced in the field.

      3.2.2 Issue of multiple loci

Research has shown that it is possible for alleles at two loci to be in random mating frequencies and yet not in equilibrium with respect to each other. Further, equilibrium between two loci is not attained after one generation of random mating as Hardy‐Weinberg law concluded, but is attained slowly over many generations. Also, the presence of genetic linkage will further slow the rate of attainment of equilibrium (Figure 3.2). If there is no linkage (c = 0.5), the differential between actual frequency and the equilibrium frequency is reduced by 50% in each generation. At this rate, it would take about seven generations to reach approximate equilibrium. However, at c = 0.01 and c = 0.001, it would take about 69 and 693 generations, respectively, to reach equilibrium. A composite gene frequency can be calculated for genes at the two loci. For example, if the frequency at locus Aa = 0.2 and that for locus bb = 0.7, the composite frequency of a genotype Aabb = 0.2 × 0.7 = 0.14.

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