Population and Mathematical Genetics

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CHAPTER 8 Population and Mathematical Genetics

Do not worry about your difficulties in mathematics. I can assure you mine are still greater.

ALBERT EINSTEIN

In this chapter, some of the more mathematical aspects of gene inheritance are considered, together with how genes are distributed and maintained at particular frequencies in populations. This subject constitutes what is known as population genetics. Genetics lends itself to a numerical approach, with many of the most influential and pioneering figures in human genetics having come from a mathematical background. They were particularly attracted by the challenges of trying to determine the frequencies of genes in populations and the rates at which they mutate. Much of this early work impinges on the specialty of medical genetics, and in particular on genetic counseling, and by the end of this chapter it is hoped that the reader will have gained an understanding of the following.

1 Why a dominant trait does not increase in a population at the expense of a recessive one.
2 How the carrier frequency and mutation rate can be determined from the disease incidence.
3 Why a particular genetic disorder can be more common in one population or community than another.
4 How it can be confirmed that a genetic disorder shows a particular pattern of inheritance.
5 The concept of genetic linkage and how this differs from linkage disequilibrium.
6 The effects of medical intervention.

Allele Frequencies in Populations

On first reflection, it would be reasonable to predict that dominant genes and traits in a population would tend to increase at the expense of recessive ones. On average, three-quarters of the offspring of two heterozygotes will manifest the dominant trait, but only one-quarter will have the recessive trait. It might be thought, therefore, that eventually almost everyone in the population would have the dominant trait. However, it can be shown that in a large randomly mating population, in which there is no disturbance by outside influences, dominant traits do not increase at the expense of recessive ones. In fact, in such a population, the relative proportions of the different genotypes (and phenotypes) remain constant from one generation to another. This is known as the Hardy-Weinberg principle, as it was proposed, independently, by an English mathematician, G. H. Hardy, and a German physician, W. Weinberg, in 1908. This is a very important principle in human genetics.

The Hardy-Weinberg Principle

Consider an ‘ideal’ population in which there is an autosomal locus with two alleles, A and a, that have frequencies of p and q, respectively. These are the only alleles found at this locus, so that p + q = 100%, or 1. The frequency of each genotype in the population can be determined by construction of a Punnett square, which shows how the different genes can combine (Figure 8.1).

image

FIGURE 8.1 Punnett square showing allele frequencies and resulting genotype frequencies for a two-allele system in the first generation.

From Figure 8.1, it can be seen that the frequencies of the different genotypes are:

Genotype Phenotype Frequency
AA A p2
Aa A 2pq
Aa a q2

If there is random mating of sperm and ova, the frequencies of the different genotypes in the first generation will be as shown. If these individuals mate with one another to produce a second generation, Punnett square can again be used to show the different matings and their frequencies (Figure 8.2).

image

FIGURE 8.2 Punnett square showing frequencies of the different matings in the second generation.

From Figure 8.2 the total frequency for each genotype in the second generation can be derived (Table 8.1). This shows that the relative frequency or proportion of each genotype is the same in the second generation as in the first. In fact, no matter how many generations are studied, the relative frequencies will remain constant. The actual numbers of individuals with each genotype will change as the population size increases or decreases, but their relative frequencies or proportions remain constant. This is the fundamental tenet of the Hardy-Weinberg principle. When studies confirm that the relative proportions of each genotype remain constant with frequencies of p2, 2pq, and q2, then that population is said to be in Hardy-Weinberg equilibrium for that particular genotype.

Table 8.1 Frequency of the Various Types of Offspring from the Matings Shown in Figure 8.2

image

Factors that Can Disturb Hardy-Weinberg Equilibrium

So far, this relates to an ‘ideal’ population. By definition such a population is large and shows random mating with no new mutations and no selection for or against any particular genotype. For some human characteristics, such as neutral genes for blood groups or enzyme variants, these criteria can be fulfilled. However, several factors can disturb Hardy-Weinberg equilibrium, either by influencing the distribution of genes in the population or by altering the gene frequencies. These factors include:

1 Non-random mating
2 Mutation
3 Selection
4 Small population size
5 Gene flow (migration).

Non-Random Mating

Random mating, or panmixis, refers to the selection of a partner regardless of that partner’s genotype. Non-random mating can lead to an increase in the frequency of affected homozygotes by two mechanisms, either assortative mating or consanguinity.

Assortative Mating

This is the tendency for human beings to choose partners who share characteristics such as height, intelligence, and racial origin. If assortative mating extends to conditions such as autosomal recessive (AR) deafness, which accounts for a large proportion of all congenital hearing loss, this will lead to a small increase in the relative frequency of affected homozygotes.

Consanguinity

Consanguinity is the term used to describe marriage between blood relatives who have at least one common ancestor no more remote than a great-great-grandparent. Widespread consanguinity in a community will lead to a relative increase in the frequency of affected homozygotes but a relative decrease in the frequency of heterozygotes.

Mutation

The validity of the Hardy-Weinberg principle is based on the assumption that no new mutations occur. If a particular locus shows a high mutation rate, then there will be a steady increase in the proportion of mutant alleles in a population. In practice, mutations do occur at almost all loci, albeit at different rates, but the effect of their introduction is usually balanced by the loss of mutant alleles due to reduced fitness of affected individuals. If a population is found to be in Hardy-Weinberg equilibrium, it is generally assumed that these two opposing factors have roughly equal effects. This is discussed further in the section that follows on the estimation of mutation rates.

Selection

In the ‘ideal’ population there is no selection for or against any particular genotype. In reality, for deleterious characteristics there is likely to be negative selection, with affected individuals having reduced reproductive (= biological = ‘genetic’) fitness. This implies that they do not have as many offspring as unaffected members of the population. In the absence of new mutations, this reduction in fitness will lead to a gradual reduction in the frequency of the mutant gene, and hence disturbance of Hardy-Weinberg equilibrium.

Selection can act in the opposite direction by increasing fitness. For some autosomal recessive disorders there is evidence that heterozygotes show a slight increase in biological fitness compared with unaffected homozygotes—referred to as heterozygote advantage. The best understood example is sickle-cell disease, in which affected homozygotes have severe anemia and often show persistent ill-health (p. 159). However, heterozygotes are relatively immune to infection with Plasmodium falciparum malaria because their red blood cells undergo sickling and are rapidly destroyed when invaded by the parasite. In areas where this form of malaria is endemic, carriers of sickle-cell anemia (sickle cell trait), have a biological advantage compared with unaffected homozygotes. Therefore, in these regions the proportion of heterozygotes tends to increase relative to the proportions of normal and affected homozygotes, and Hardy-Weinberg equilibrium is disturbed.

Small Population Size

In a large population, the numbers of children produced by individuals with different genotypes, assuming no alteration in fitness for any particular genotype, will tend to balance out, so that gene frequencies remain stable. However, in a small population it is possible that by random statistical fluctuation one allele could be transmitted to a high proportion of offspring by chance, resulting in marked changes in allele frequency from one generation to the next, so that Hardy-Weinberg equilibrium is disturbed. This is known as random genetic drift. If one allele is lost altogether, it is said to be extinguished and the other allele is described as having become fixed (Figure 8.3).

image

FIGURE 8.3 Possible effects of random genetic drift in large and small populations.

Gene Flow (Migration)

If new alleles are introduced into a population as a consequence of migration, with later intermarriage, a change in the relevant allele frequencies will result. This slow diffusion of alleles across racial or geographical boundaries is known as gene flow. The most widely quoted example is the gradient shown by the incidence of the B blood group allele throughout the world (Figure 8.4). This allele is thought to have originated in Asia and spread slowly westward as a result of admixture through invasion.

image

FIGURE 8.4 Distribution of blood group B throughout the world.

(From Mourant AE, Kopéc AC, Domaniewska-Sobczak K 1976 The distribution of the human blood groups and other polymorphisms, 2nd ed. Oxford University Press, London, with permission.)

Validity of Hardy-Weinberg Equilibrium

It is relatively simple to establish whether a population is in Hardy-Weinberg equilibrium for a particular trait if all possible genotypes can be identified. Consider a system with two alleles, A and a, with three resulting genotypes, AA, Aa/aA, and aa. Among 1000 individuals selected at random, the following genotype distributions are observed:

AA 800
Aa/aA 185
aa 15

From these figures, the incidence of the A allele (p) equals [(2 × 800) + 185]/2000 = 0.8925 and the incidence of the a allele (q) equals [185 + (2 × 15)]/2000 = 0.1075.

Now consider what the expected genotype frequencies would be if the population were in Hardy-Weinberg equilibrium, and compare these with the observed values:

Genotype Observed Expected
AA 800 796.5 (p2 × 1000)
Aa/aA 185 192 (2pq × 1000)
aa 15 11.5 (q2 × 1000)

These observed and expected values correspond closely and formal statistical analysis with a χ2 test would confirm that the observed values do not differ significantly from those expected if the population is in equilibrium.

Next consider a different system with two alleles, B and b. Among 1000 randomly selected individuals the observed genotype distributions are:

BB 430
Bb/bB 540
bb 30

From these values, the incidence of the B allele (p) equals [(2 × 430) + 540]/2000 = 0.7 and the incidence of the b allele (q) equals [540 + (2 × 30)]/2000 = 0.3.

Using these values for p and q, the observed and expected genotype distributions can be compared:

Genotype Observed Expected
BB 430 490 (p2 × 1000)
Bb/bB 540 420 (2pq × 1000)
bb 30 90 (q2 × 1000)

These values differ considerably, with an increased number of heterozygotes at the expense of homozygotes. Such deviation from Hardy-Weinberg equilibrium should prompt a search for factors that could result in increased numbers of heterozygotes, such as heterozygote advantage or negative assortative mating—i.e., the attraction of opposites!

Despite the number of factors that can disturb Hardy-Weinberg equilibrium, most populations are in equilibrium for most genetic traits, and significant deviations from expected genotype frequencies are unusual.

Applications of Hardy-Weinberg Equilibrium

Estimation of Carrier Frequencies

If the incidence of an AR disorder is known, it is possible to calculate the carrier frequency using some relatively simple algebra. For example, if the disease incidence is 1 in 10,000, then q2 = image and q = image. Because p + q = 1, therefore p = image. The carrier frequency can then be calculated as 2 × image × image (i.e., 2pq), which approximates to 1 in 50. Thus, a rough approximation of the carrier frequency can be obtained by doubling the square root of the disease incidence. Approximate values for gene frequency and carrier frequency derived from the disease incidence can be extremely useful in genetic risk counseling (p. 266) (Table 8.2). However, if the disease incidence includes cases resulting from consanguineous relationships, then it is not valid to use the Hardy-Weinberg principle to calculate heterozygote frequencies because a high incidence of consanguinity disturbs the equilibrium by leading to a relative increase in the proportion of affected homozygotes.

Table 8.2 Approximate Values for Gene Frequency and Carrier Frequency Calculated from the Disease Incidence Assuming Hardy-Weinberg Equilibrium

Disease Incidence (q2) Gene Frequency (q) Carrier Frequency (2pq)
1/1000 1/32 1/16
1/2000 1/45 1/23
1/5000 1/71 1/36
1/10,000 1/100 1/50
1/50,000 1/224 1/112
1/100,000 1/316 1/158

For an X-linked recessive (XLR) disorder, the frequency of affected males equals the frequency of the mutant allele, q. Thus, for a trait such as red-green color blindness, which affects approximately 1 in 12 male western European whites, q = image and p = image. This means that the frequency of affected females (q2) and carrier females (2pq) is image and image, respectively.

Estimation of Mutation Rates

Direct Method

If an autosomal dominant (AD) disorder shows full penetrance, and is therefore always expressed in heterozygotes, an estimate of its mutation rate can be made relatively easily by counting the number of new cases in a defined number of births. Consider a sample of 100,000 children, 12 of whom have a particular AD disorder such as achondroplasia (p. 93). Only two of these children have an affected parent, so that the remaining 10 must have acquired their disorder as a result of new mutations. Therefore 10 new mutations have occurred among the 200,000 genes inherited by these children (because each child inherits two copies of each gene), giving a mutation rate of 1 per 20,000 gametes per generation. In fact, this example is unusual because all new mutations in achondroplasia occur on the paternally derived chromosome 4; therefore the mutation rate is 1 per 10,000 in spermatogenesis and, as far as we know, zero in oogenesis.

Indirect Method

For an AD disorder with reproductive fitness (f) equal to zero, all cases must result from new mutations. If the incidence of a disorder is denoted as I and the mutation rate as m, then as each child inherits two alleles, either of which can mutate to cause the disorder, the incidence equals twice the mutation rate (i.e., I = 2µ).

If fitness is greater than zero, and the disorder is in Hardy-Weinberg equilibrium, then genes lost through reduced fitness must be counterbalanced by new mutations. Therefore, 2µ = I(1 – f) or µ = [I(1 – f)]/2.

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