Polygenic Inheritance and the Normal Distribution

Before considering the impact of recent research in detail, it is necessary to outline briefly the scientific basis of what is known as polygenic or quantitative inheritance. This involves the inheritance and expression of a phenotype being determined by many genes at different loci, with each gene exerting a small additive effect. Additive implies that the effects of the genes are cumulative, i.e. no one gene is dominant or recessive to another.

Several human characteristics (Box 9.2) show a continuous distribution in the general population, which closely resembles a normal distribution. This takes the form of a symmetrical bell-shaped curve distributed evenly about a mean (Figure 9.1). The spread of the distribution about the mean is determined by the standard deviation. Approximately 68%, 95%, and 99.7% of observations fall within the mean plus or minus one, two, or three standard deviations, respectively.

FIGURE 9.1 The normal (Gaussian) distribution.

It is possible to show that a phenotype with a normal distribution in the general population can be generated by polygenic inheritance involving the action of many genes at different loci, each of which exerts an equal additive effect. This can be illustrated by considering a trait such as height. If height were to be determined by two equally frequent alleles, a (tall) and b (short), at a single locus, then this would result in a discontinuous phenotype with three groups in a ratio of 1 (tall-aa) to 2 (average-ab/ba) to 1 (short-bb). If the same trait were to be determined by two alleles at each of two loci interacting in a simple additive way, this would lead to a phenotypic distribution of five groups in a ratio of 1 (4 tall genes) to 4 (3 tall + 1 short) to 6 (2 tall + 2 short) to 4 (1 tall + 3 short) to 1 (4 short). For a system with three loci each with two alleles the phenotypic ratio would be 1-6-15-20-15-6-1 (Figure 9.2).

FIGURE 9.2 Distribution of genotypes for a characteristic such as height with 1, 2, and 3 loci each with two alleles of equal frequency. The values for each genotype can be obtained from the binomial expansion (p + q)⁽²ⁿ⁾, where p = q = 1/2 and n equals the number of loci.

It can be seen that as the number of loci increases, the distribution increasingly comes to resemble a normal curve, thereby supporting the concept that characteristics such as height are determined by the additive effects of many genes at different loci. Further support for this concept comes from the study of familial correlations for characteristics such as height. Correlation is a statistical measure of the degree of resemblance or relationship between two parameters. First-degree relatives share, on average, 50% of their genes (Table 9.1). Therefore, if height is polygenic, the correlation between first-degree relatives should be 0.5. Several studies have shown that the sib–sib correlation for height is indeed close to 0.5.

Table 9.1 Degrees of Relationship

Relationship	Proportion of Genes Shared
First degree
Parents
Siblings
Children
Second degree
Uncles and aunts
Nephews and nieces
Grandparents
Grandchildren
Half-siblings
Third degree
First cousins
Great-grandparents
Great-grandchildren

In reality, human characteristics such as height and intelligence are also influenced by environment, and possibly also by genes that are not additive in that they exert a dominant effect. These factors probably account for the observed tendency of offspring to show what is known as regression to the mean. This is demonstrated by tall or intelligent parents (the two are not mutually exclusive!) having children whose average height or intelligence is slightly lower than the average or mid-parental value. Similarly, parents who are very short or of low intelligence tend to have children whose average height or intelligence is lower than the general population average, but higher than the average value of the parents. If a trait were to show true polygenic inheritance with no external influences, then the measurements in offspring would be distributed evenly around the mean of their parents’ values.

Multifactorial Inheritance—The Liability/Threshold Model

Efforts have been made to extend the polygenic theory for the inheritance of quantitative or continuous traits to try to account for discontinuous multifactorial disorders. According to the liability/threshold model, all of the factors which influence the development of a multifactorial disorder, whether genetic or environmental, can be considered as a single entity known as liability. The liabilities of all individuals in a population form a continuous variable, which has a normal distribution in both the general population and in relatives of affected individuals. However, the curves for these relatives will be shifted to the right, with the extent to which they are shifted being directly related to the closeness of their relationship to the affected index case (Figure 9.3).

FIGURE 9.3 Hypothetical liability curves in the general population and in relatives for a hereditary disorder in which the genetic predisposition is multifactorial.

To account for a discontinuous phenotype (i.e., affected or not affected) with an underlying continuous distribution, it is proposed that a threshold exists above which the abnormal phenotype is expressed. In the general population, the proportion beyond the threshold is the population incidence, and among relatives the proportion beyond the threshold is the familial incidence.

It is important to emphasize again that liability includes all factors that contribute to the cause of the condition. Looked at very simply, a deleterious liability can be viewed as consisting of a combination of several ‘bad’ genes and adverse environmental factors. Liability cannot be measured but the mean liability of a group can be determined from the incidence of the disease in that group using statistics of the normal distribution. The units of measurement are standard deviations and these can be used to estimate the correlation between relatives.