Principles of Plant Genetics and Breeding. George Acquaah

biological boundaries to be made. Breeding of cross‐pollinated species tends to focus on improving populations rather than individual plants, as is the case in breeding self‐pollinated species. To understand population structure and its importance to plant breeding, it is important to understand the type of variability present, and its underlying genetic control, in addition to the mode of selection for changing the genetic structure.

      3.1.2 Mathematical model of a gene pool

      As previously stated, gene frequency is the basic concept in population genetics. Population genetics is concerned with both the genetic composition of the population as well as the transmission of genetic material to the next generation. The genetic constitution of a population is described by an array of gene frequencies. The genetic properties of a population are influenced in the process of transmission of genes from one generation to the next by four major factors: population size, differences in fertility and viability, migration and mutation, and the mating system. Genetic frequencies are subject to sample variation between successive generations. A plant breeder directs the evolution of the breeding population through the kinds of parents used to start the base population in a breeding program, how the parents are mated, and artificial selection.

      The genetic constitution of individuals in a population is reconstituted for each subsequent generation. Whereas the genes carried by the population have continuity from one generation to the next, there is no such continuity in the genotypes in which these genes occur. Plant breeders often work with genetic phenomena in populations that exhibit no apparent Mendelian segregation, even though in actuality, they obey Mendelian laws. Mendel worked with genes whose effects were categorical (kinds) and were readily classifiable (ratios) into kinds in the progeny of crosses. Breeders, on the other hand, are usually concerned about differences in populations measured in degrees rather than kinds. Population genetics uses mathematical models to attempt to describe population phenomena. To accomplish this, it is necessary to make assumptions about the population and its environment.

       Calculating gene frequency

      To understand the genetic structure of a population, consider a large population in which random mating occurs, with no mutation or gene flow between this population and others, no selective advantage for any genotype, and normal meiosis. Consider also one locus, A, with two alleles, A, and a. The frequency of allele A ₁ in the gene pool is p, while the frequency of allele A ₂ is q. Also, p + q = 1 (or 100% of the gene pool). Assume a population of N diploids (have two alleles at each locus) in which two alleles (A, a) occur at one locus. Assuming dominance at the locus, three genotypes – AA, Aa, and aa – are possible in an F₂ segregating population. Assume the genotypic frequencies are D (for AA), H (for Aa), and Q (for aa). Since the population is diploid, there will be 2N alleles in it. The genotype AA has two A alleles. Hence, the total number of A alleles in the population is calculated as 2D + H. The proportion or frequency of A alleles (designated as p) in the population is obtained as follows:

      The same can be done for allele a, and designated q. Further, p + q = 1 and hence p = 1 – q. If N = 80, D = 4, and H = 24,

      Since p + q = 1, q = 1 − p, and hence q = 1 – 0.2 = 0.8.

       Hardy‐Weinberg equilibrium

      Consider a random mating population (each male gamete has an equal chance of mating with any female gamete). Random mating involving the previous locus (A/a) will yield the following genotypes: AA, Aa, and aa, with the corresponding frequencies of p ² , 2pq, and q ² , respectively. The gene frequencies must add up to unity. Consequently, p ² + 2pq + q ² = 1. This mathematical relationship is called the Hardy‐Weinberg equilibrium. Hardy of England and Weinberg of Germany discovered that equilibrium between genes and genotypes is achieved in large populations. They showed that the frequency of genotypes in a population depends on the frequency of genes in the preceding generation, not on the frequency of the genotypes.

      Considering the previous example, the genotypic frequencies for the next generation following random mating can be calculated as follows:

      The Hardy‐Weinberg equilibrium is hence summarized as:

      This means that in a population of 80 plants as before, about 3 plants will have a genotype of AA, 26 will be Aa, and 51, aa. Using the previous formula, the frequencies of the genes in the next generation may be calculated as:

      and q = 1 – p = 0.8.

      The allele frequencies have remained unchanged, while the genotypic frequencies have changed from 4, 24, and 52, to 3, 26, and 51, for AA, Aa, and aa, respectively. However, in subsequent generations, both the genotype and gene frequencies will remain unchanged, provided:

      1 Random mating occurs in a very large diploid population.

      2 Allele A and allele a are equally fit (one does not confer a superior trait than the other).

      3 There is no differential migration of one allele into or out of the population.

      4 The mutation rate of allele A is equal to that of allele a.

In other words, the variability does not change from one generation to another in a random mating population. The maximum frequency of the heterozygote (H) cannot exceed 0.5 (Figure 3.1). The Hardy‐Weinberg law states that equilibrium is established at any locus after one generation of random mating. From the standpoint of plant breeding, two states of variability are present: two homozygotes (AA, aa), called “free variability” that can be fixed by selection; and the intermediate heterozygous (Aa), called “hidden or potential variability” that can generate new variability through segregation. In outcrossing species, the homozygotes can hybridize to generate more heterozygotic variability. Under random mating and no selection, the rate of crossing and segregation will be balanced to maintain the proportion of free and potential variability at 50% : 50%. In other words, the population structure is maintained as a dynamic flow of crossing and segregation. However, with two loci under consideration, equilibrium will be attained slowly over many generations. If genetic linkage is strong, the rate of attainment of equilibrium will even be much slower.

Figure 3.1 The relationship between gene frequencies and allele frequencies in a population in Hardy‐Weinberg equilibrium for two alleles is depicted in the figure. The frequency of the heterozygotes cannot be more than 50%, and this maximum occurs when the gene frequencies are p = q = 0.5. Further, when the frequency of an allele is low, the rare allele occurs