1.7: Inbreeding and Heterosis (2024)

Inbreeding as a Probability

Inbreeding is characterized by a departure from random mating and involves preferential mating between relatives. The most significant effect of inbreeding is that replicates of a single allele in a common ancestor can come together through the mating of relatives to produce hom*ozygous progeny whose alleles are identical by descent. In contrast in a second type of hom*ozygote, two alleles that are said to be alike in state are those that have the same function, but did not derive from replication of a single ancestral gene. These two types of hom*ozygotes can be understood by analyzing pedigrees such as the ones shown in Figure3. The two copies of the A₁ allele in the A₁A₁ offspring individual at the bottom of Figure 3a descend from the same copy in a common ancestor; this type of genotype is termed autozygous. In contrast the two copies of the A₁ allele in the A₁A₁ offspring individual at the bottom in Figure 3b are descended from two different copies in ancestors; this type of genotype is allozygous.

Inbreeding can be estimated directly by studying pedigrees or lineages of ancestry showing genetic relationships.

Measurement of Inbreeding

The probability that two genes are identical by descent is called the coefficient of inbreeding (denoted as F) and will be the measure of relationship between mating pairs.

The coefficient of inbreeding (F) is defined as the probability that two alleles at the same locus are identical by descent. At the population level, F describes the average level of hom*ozygosity. The coefficient of inbreeding is always expressed relative to a specified base population. The base population is defined to be non-inbred (F=0). The range of F is 0 to 1, with 0 indicating random mating and no inbreeding, while 1 means prolonged selfing.

Consider a base population consisting of N individuals each shedding equal numbers of gametes uniting at random (although see Fehr, 1987, p.113, with respect to the concept of effective population size, or N_e). Because the base is defined to have F=0, each individual in this population carries genes that are non-identical. The only way a hom*ozygote that carries genes that are identical by descent can arise is by the mating of a male and female gamete from the same individual that carries a replication of the same gene. Because there are 2N gametes the probability that two mating gametes are identical by descent is 1 / 2N.

Second Generation

In the second generation, there are two ways genes that are identical by descent can be joined: 1) by a new replication of the same ancestral gene; and 2) by the previous replication that occurred in generation 1. The probability of a new replication event is . The remaining proportion of zygotes, , carry genes that are independent in origin from Generation 1, but may have been identical in their origin in Generation 0. The probability the genes are identical by descent from Generation 1 is the inbreeding coefficient of Generation 1 . Note that F₁, F₂, F_t, etc in this section (written in italics) denote inbreeding coefficients, and not generations as on pages 1 and 2 of this section and in earlier sections.

Therefore the probability of identical hom*ozygotes in Generation 2 is:

\[F_{2} = {1 \over 2N} + (1 - {1 \over 2N})F_{1}\]

Where F₁ and F₂ are the inbreeding coefficients of Generations 1 and 2. The same arguments apply to future generations, so we can write the recurrence equation:

\[F_{t} = {1 \over 2N} + (1 - {1 \over 2N})F_{t-1}\]

Generation Inbreeding

The inbreeding of any generation is composed of two components: New inbreeding, which arises from self-fertilization and the “old” that was already there.

Note that inbreeding is cumulative and the absence of inbreeding in Generation t does not change the fact that a population had old inbreeding from prior generations.

Through a series of algebraic steps, we can write the inbreeding coefficient as a function of the number of generations removed from the base populations:

\[\textrm{F}_t = 1 - (1 - \Delta F) ^t\]

where,

\[\Delta \textrm{F} = {1 \over 2N}\]

Genotype Frequencies

The genotype frequencies in a population can then be expressed as:

Table 1 Genotype frequencies for a locus with two alleles. Data from Falconer ad Mackay, 1996.

	Original Frequencies	Change due to inbreeding	Origin
			Independent	Identical
AA
Aa
aa
			Allozygous genes	Autozygous genes

We can examine inbreeding or the probability of identity of alleles by descent, by looking at the genotype frequencies in Table 1 under two extremes of the F statistic:

• if F = 0 (random mating; no inbreeding)
  • the equation for genotypic frequencies reduces to the familiar equation for genotypes in Hardy-Weinberg proportions: p² + 2pq + q²
• if F = 1 (alleles identical by descent)
  • the genotype equation reduces to a ratio of hom*ozygotes to heterozygotes as p:0:q

Inbreeding Coefficient

Thus inbreeding leads to hom*ozygosity (all or nearly all loci hom*ozygous), and almost a complete absence of heterozygosity (all or nearly all loci heterozygosity). As noted in Table 1, with inbreeding there is a deficit of heterozygotes equal to 2pqF and an excess of each hom*ozygous class equal to half the deficiency of the heterozygotes.

To illustrate one type of mating that promotes inbreeding, consider the case of a population that reproduces by self-fertilization so that F = 1.

Another way to express the inbreeding coefficient, F, is to compare the frequency of heterozygotes in the population to the frequency expected under random mating:

\[F = \frac{H_{0} - H}{H_{0}}\]

where,

• H = frequency of heterozygotes in the population
• H₀ = expected frequency under HWE, meaning 2pq

Therefore the inbreeding coefficient is the proportional reduction in heterozygosity relative to a random mating population with the same allele frequencies.

Inbreeding via Self-Pollination

In a population that reproduces by self-fertilization the inbreeding coefficient, F = 1. Let’s assume the population begins with genotypic frequencies in Hardy-Weinberg proportions (p² + 2pq + q²). With selfing, each hom*ozygote produces only progeny of the same genotype:

AA × AA ⇒ all AA

aa × aa ⇒ all aa

However only half of the progeny of a heterozygote will be like the parent

Aa × Aa ⇒ 1/4 AA, 1/2 Aa, 1/4 aa

Self-pollination therefore reduces the proportion of heterozygotes in the population by half with each generation until all genotypes in the population are hom*ozygous (Table 2).

Table 2 Increased over generations in frequency of hom*ozygous in a self-pollinated population starting with p = q = 0.05>>>frequency of hom*ozygotes.

	Genotype Frequencies
Generation	AA	Aa	aa
1	1/4	1/2	1/4
2	1/4 + 1/8 = 3/8	1/4	1/4 + 1/8 = 3/8
3	3/8 + 1/16 = 7/16	1/8	3/8 + 1/16 = 7/16
4	7/16 + 1/32 = 15/32	1/16	7/16 + 1/32 = 15/32
n
	1/2	0	1/2

Proportion of hom*ozygotes

With multiple segregating loci, the proportion of hom*ozygotes in various selfing generations can be estimated using the following formula: [(2^m – 1)/2^m]ⁿ: where m is the number of selfing generations(m = 1 for F₂; m = 2 for F₃ and so on) and n is the number of segregating loci. For example, an F₁ plant with four independent segregating loci will result in the following frequency of hom*ozygous plants in F₂, [(2¹ – 1)/2¹]⁴ = 1/16 = 6.25%. The expected proportion of completely hom*ozygous plants in F₂ and later generations of selfing for different numbers of segregating loci are indicated in Table 3. The proportion of hom*ozygotes decreases sharply with increasing heterozygosity (more segregating loci) in the F₁.

Increasing Inbreeding

Through increasing hom*ozygosity, inbreeding brings together identical alleles at a locus. hom*ozygosity permits the expression of recessive alleles that may have been previously masked by heterozygosity in the parent generation. If recessive alleles are less favorable than dominant ones, the overall fitness of the individual decreases. Inbreeding is often detrimental because it increases the appearance of lethal and deleterious recessive traits. The term inbreeding depression describes the decrease in fitness or performance that often accompanies inbreeding or random genetic drift. Recall that fitness is the relative ability of an individual to survive and reproduce to contribute its genes to the next generation. Inbreeding depression is further described in the next section.

Plant stature, vigor, yield and other traits decline with increasing inbreeding, although significant differences exist among species for the amount of inbreeding depression expressed—ranging from minimal among self-pollinated crops such as oat and wheat to severe in cross-pollinated polyploid species such as alfalfa, whereby hom*ozygous genotypes do not survive (Figure 4). Table 4 on the next screen summarizes some general differences between outbreeders and inbreeders.

Outbreeders vs. Inbreeders

Inbreeding Depression and hom*ozygosity

Inbreeding results in increased hom*ozygosity. In cross-pollinated species, increased hom*ozygosity results in inbreeding depression or reduced performance. Symptoms of inbreeding depression may include:

• Reduced plant vigor
• Smaller plant size
• Decline in fertility
• Suppressed seed production
• Decreased pollen production
• Inferior seed quality
• Greater susceptibility to insect or pathogen damage or
• Poorer standability.

Why? With increased hom*ozygosity, expression of deleterious recessive alleles may be revealed by fixation that had been masked by more favorable dominant alleles, or it can be caused by overdominance in which hom*ozygotes are less fit, resulting in poorer performance.

The severity of inbreeding depression varies with the species and genotype. If the inbreeding depression is too severe, it may be difficult or impractical to maintain or propagate inbred lines by seed. In such cases, some heterogeneity must be maintained or other propagation means used.

Mating Systems

The frequency of hom*ozygotes in a population can be increased using any of several mating systems:

• Self-pollination—Repeated self-pollination is routinely used to develop pure lines of self-pollinating crops. It can also be applied to cross-pollinating species to obtain inbred lines. Like pure-lines in self-pollinated species, inbred lines are hom*ozygous at nearly all loci.
• Half-sib mating—Plants having one parent in common are mated. The pollen source is random from the breeding population, but the female plants are identifiable.
• Full-sib mating—Plants having both parents in common are crossed.
• Backcrossing—A method in which a hybrid is mated to one of its parents, the recurrent parent, resulting in a backcross 1 (BC₁) population. The resulting backcross offspring are repeatedly crossed to the recurrent parent.

The inbreeding coefficient increases more rapidly with matings among more closely related individuals. The first three mating systems listed above are compared in Figure 5. In each case the initial value of the inbreeding coefficient is assumed to be F₀ = 0. Backcrossing—like self-fertilization—can be one of the most extreme forms of inbreeding.

Degree of Relatedness of Individuals

The coefficient of relatedness (r_IJ) is a measure of the degree of relatedness between individuals (Table 5). It is the proportion of genes shared between two individuals I and J due to common descent (e.g., identical by descent). It ranges from -1.0 (no genes in common, at least among the genetic markers assessed) to +1.0 (vegetative clones). For example, the value of r_IJ for a parent and each of their progeny is 0.5 for a sexually reproducing species because half of each offspring’s genes come from each parent. In the case of full-siblings the value of r_IJ is also 0.5, but on average because siblings receive half of their genes from each of the same pair of parents, but the haploid set of genes in each parental gamete is a random sample of half of the parental genome due to recombination (Conner and Hartl 2004).

*After selfing for many generations

Heterosis or Hybrid Vigor

When inbred lines of cross-pollinating species are mated, their progeny are often superior to either or both of the parents for one or more characters, a phenomenon referred to as hybrid vigor or heterosis. Recall that genetic drift is random, so that different inbred lines or subpopulations tend to be fixed for different alleles at various loci; when they are intermated, the F₁ population will be highly heterozygous. Heterosis is the opposite of inbreeding depression—heterosis is commonly expressed as

• Improved plant vigor,
• Greater plant size, and/or
• Increased productivity.

The performance of a hybrid relative to its parents can be described in two ways: mid-parent heterosis is the performance of a hybrid compared with the average of the performance of its parents; high-parent heterosis is a comparison of the performance of the hybrid with that of the best parent:

\[\text{mid-point heterosis (%)} = \frac{F_{1} - MP}{MP} \times 100\]

Where:

• F₁ = performance of the hybrid
• MP = average performance of the parents (Parent 1 + Parent 2)/2
• HP = performance of the best parent

Dominance vs. Overdominance Hypothesis

Heterosis tends to be greatest in the progeny of diverse genotypes. There are two common explanations for this phenomenon:

1. Dominance hypothesis—Heterosis results from the complete or partial dominance of favorable alleles at various loci.
2. Overdominance hypothesis—Heterosis reflects the superior performance of the heterozygote over either hom*ozygote.

The next few pages provide a summary of the dominance and overdominance hypotheses from Charlesworth and Willis (2009). Note the differences between models for a single locus and those describing multiple loci. They include a third model that involves deleterious alleles at closely linked loci and focuses on so-called “pseudo-overdominance”, which will not be further discussed here.

Main Genetic Hypothesis

Recessive deleterious mutations

Recessive deleterious mutations at closely linked loci

Single locus with heterozygous advantage

Heterozygote Advantage

Heterozygote advantage is a term that is frequently confused with heterosis. However, heterozygote advantage is a synonym of overdominance, and refers to a condition in which the heterozygous genotype has a higher phenotypic value (especially for fitness) than for either hom*ozygous genotype. Heterosis is usually due to a number of loci that control quantitatively inherited traits, although depending on the situation it could be due to phenotypes controlled by either single genes or polygenic traits. In contrast, heterozygote advantage describes effects at a single locus (Conner and Hartl 2004). Compare the circ*mstances for these two concepts in Table 6.

Genetics of Heterosis and Inbreeding Depression

Whatever genetic mechanisms can explain heterosis must also be able to explain inbreeding depression. Let’s take a look at hypothetical numeric examples of these two hypotheses to show how each can explain both.

Although conclusive evidence to support either of these hypotheses remains elusive, the dominance hypothesis has been more widely accepted. However, it is recognized that a gene’s effect is determined by its effect in combination with:

• Other alleles at the same locus,
• Itself (when hom*ozygous),
• Genes at other loci (epistasis), and
• Closely linked genes.

Thus, dominance, overdominance, and epistasis, most likely all contribute to heterosis.

Overdominance Theory

1. Heterosis: The genotype aa bb cc dd ee ff has a value of 40.The hom*ozygous dominant allele at any locus increases the value by 7 over the hom*ozygous recessive allele, i.e., NN = +7. Heterozygous at any locus increases the value by 10 over the hom*ozygous recessive, i.e. Nn = +10. The following hom*ozygous parents are crossed; each has a value of 61 because of having three loci with hom*ozygous alleles [40 + (3)(7) = 61]. The F₁ has a value of 100 because of having six heterozygous loci.
2. Inbreeding depression: Inbreeding depression can be explained with overdominance. If we self the Aa Bb Cc Dd Ee Ff individuals that have a value of 100, we will get a population of individuals with a mean value of 85 (see table above). As was the case for the “favorable dominant alleles theory”, this value is lower than the value of the Aa Bb Cc Dd Ee Ff parental individuals, thus showing depression upon inbreeding.

Genetic Analysis

Molecular techniques are now being used to try to shed more light on the mechanisms involved in heterosis. In a study published in 2006, Swanson-Wagner et al. assayed the gene expression of 13,999 maize genes from seedling plants of two inbred parents and their F₁ hybrid. When comparing the three genotypes they found 1,367 of the genes produced significantly different amounts of messenger ribonucleic acid (mRNA), which performs an essential role in protein synthesis.

Analysis of these genes showed that 78 percent of them had additive gene action, 15 percent showed either high-parent or low-parent dominance, and 3 percent showed either overdominance or underdominance. (The other 45 genes showed non-additive gene action but with the statistical analysis used these genes could not be classified as dominant, overdominant, or underdominant.) The experiment analyzed individual genes and was not designed to evaluate interactions among the genes, so epistasis could not be measured.

This study involved only two maize inbred lines and their hybrid offspring and the evaluation was based on tissue from seedling plants. Different genes will be active during different parts of the life cycle of the maize plant and further studies will most likely show differential gene action among different genes. Likewise, different parents and hybrids and different species may well show very different gene action patterns among the same genes that were tested in this study. With just this one experiment, where nearly 14,000 genes were evaluated, we can see that heterosis is complex and that multiple types of gene action are involved. Molecular methods in the context of hybrid breeding will be discussed in detail in the molecular plant breeding course.

When two genetically different parents are mated, heterosis is observed in their F₁ progeny.

• All seed and forage species express some level of heterosis.
• Most cross-pollinated species express greater heterosis than do self-pollinated species.

Because of the expense in producing hybrid cultivars, hybrid cultivars are generally developed only in those crops in which the level of heterosis results in significantly better performance.