In medical genetics, a population is a group of individuals that live and reproduce together in a defined area. Population genetics studies genetic structure and the change in genetic structure of populations. Genetic structure is the frequencies of alleles and genotypes. Gene frequency, or allele frequency, is the incidence of an allele among all the alleles of the same gene or locus. Genotype frequency is the incidence of a certain genotype among all the genotypes of the same gene or locus.

For a gene with one dominant allele, A, and one recessive allele, a, in a population, there is three genotypes, AA, Aa and aa. The frequency of genotype AA is calculated as f_AA = N_AA/(N_AA+N_Aa+N_aa), where N represents the counts of the individuals with subscript genotype. The frequency of other genotypes are calculated similarly. The frequency of allele A and allele a are commonly denoted by letters p and q respectively: p = N_A/(N_A+N_a) = (2N_AA+ N_Aa)/(2N_AA+2N_Aa+2N_aa) and q = N_a/(N_A+N_a) = (2N_aa+ N_Aa)/(2N_AA+2N_Aa+2N_aa) = 1-p.

In a large, random-mating population, if there is no disruptive circumstances (e.g., gene mutation, migration, natural selection), both genotype and allele frequencies will remain constant between generations. Such a population is called equilibrium population, or stated as being on genetic equilibrium or Hardy-Weinberg equilibrium. In the simplest case, for a locus with two alleles A and a with frequencies p and q, f_AA=p², f_Aa=2pq, and f_aa = q². Such a principle are termed Hardy–Weinberg principle after Godfrey Hardy and Wilhelm Weinberg who established it independently.

In a big enough population, according to Hardy–Weinberg principle, the gene frequency of the recessive allele (q) of an AR disease could be estimated as the square root of the prevalence (f) of the disease in the population: q=f^1/2. The carrier frequency could be estimated as 2(1-q)q. And the carrier frequency should not be ignored when estimating the recurrence of the AR disease. For example, the prevalence of the AR disorder phenylketonuria (PKU) in a population is 1/10000. We can estimate the gene frequency of the pathogenic allele as q = (1/10000)^1/2 = 1/100. The carrier frequency is calculated as f_Aa = 2(1-q)q = 2×(1-1/100)×(1/100) ≈ 1/50. Thus, when a carrier mated with a random individual, the recurrence risk of their children could be estimated as 1 × 1/50 × 1/4 = 1/200.

In the case of X-linked dominant inheritance, the gene frequency of the dominant allele is equal to the prevalence in males and about half of the prevalence in females (i.e., p²+2pq ≈ 2p, as q is almost 1).

In the case of X-linked recessive inheritance, the gene frequency of the recessive allele is equal to the prevalence in males and about the square root of the prevalence in females. For example, the prevalence of red-green color blindness in males in England is 1/12. We can estimate the gene frequency of the pathogenic allele as q = 1/12, and the gene frequency of the normal allele as p = 1-q = 11/12. The incidence of carrier is 2pq = 2 × 1/12 × 11/12 = 22/144 = 0.15. The prevalence in females is q² = (1/12)² = 1/144 = 0.007.

Hardy-Weinberg equilibrium population is an ideal population that is not exist. The genetic equilibrium of a real population is affected by non-random mating, gene mutation, selection, genetic drift and gene flow.

Non-random mating includes assortative mating, Hierarchical marriage, and consanguineous mating. It can lead to an increases in the genotype frequency of affected homozygotes. Assortative mating is the tendency for individuals to choose partners with similar characteristics including height, intelligence and racial origin. Consanguineous mating is the mating between close relatives. The chance for an individual being homozygous of an identical allele from the common ancestor(s) of his/her parents is termed inbreeding coefficient (typically denoted by the letter F) of the individual or the marriage of his/her parents. In clinical genetics, a consanguineous marriage is defined as that with an F equal or higher than 1/64 (that between second cousins or closer relatives).

For an autosomal locus, the inbreeding coefficient (F) is equal to a half of the coefficient of relationship (R) of the parents when the common ancestors are not inbred. A marriage between first cousins (R=1/8) will give birth of children with inbreeding coefficient of 1/16, supposing all the common ancestors are not inbred.

The chain-counting method could be employed to calculate the inbreeding coefficient from a pedigree. A chain for a certain common ancestor starts from one parent of the inbred individual, goes up, generation by generation, to the common ancestor, and then goes down to the other parent. The contribution of this common ancestor is calculated as (1/2)N × (1+F_AC), where N is the count of the individuals in the chain and F_AC is the breeding coefficient of the common ancestor. Then the contributions of all possible chains through all common ancestors are summed up to get the inbreeding coefficient of the inbred individual.

Figure 6.1 A Hypothesized Pedigree with Inbred Individuals

In the pedigree, to calculate the inbreeding coefficient of individual VIII1, all chains and common ancestors of the parents are listed (Table 6.1). The inbreeding coefficient of the inbred common ancestor should be calculated beforehand (Table 6.2). For IV2, there are two common ancestors, I1 and I2, which have two five-link chains III3-II1-I1-II3-III4 and III3-II1-I2-II3-III4 respectively, thus F_IV2 = (1/2)⁵ + (1/2)⁵ = 1/16 = 6.25%. For VIII1, there is four common ancestors with 6 chains in total, each of I1 and I2 has two eleven-link chains, II2 has one nine-link chain and IV2 has one five-link chain. F_VIII1 = (1/2)¹¹ + (1/2)¹¹ + (1/2)¹¹ + (1/2)¹¹ + (1/2)⁹ + (1/2)⁵ × (1+1/16) = 19/512 = 4.10%.

Table 6.1 Calculation of Breeding Coefficient for VIII1 in Figure 6.1

Table 6.2 Calculation of Breeding Coefficient for IV2 in Figure 6.1

For the X-linked inheritance, there is no father-to-son transmission and thus all the chains with father-to-son transmission should be excluded. Then the count of females is used for calculation of the contribution of the chain.

Table 6.3 Calculation of Breeding Coefficient of X-linked inheritance for VIII1 in Figure 6.1

Table 6.4 Calculation of Breeding Coefficient of X-linked inheritance for IV2 in Figure 6.1

Average inbreeding coefficient, the average across all individuals, is employed for the level of inbreeding in a population. In practice, the average inbreeding coefficient is simply calculated as the average inbreeding coefficient of marriages. One study presented 1000 marriages in a population. The types of marriage and calculation the average inbreeding coefficient are listed in Table 6.3. In this example, the inbreeding coefficient of each marriage type (F) is multiplied by the count of marriages in this type (n), sum the results, and divide by the total count of marriages: ΣnF/Σn.

Table 6.5 Computation of the Average Inbreeding Coefficient for a Population

If a population starts with all individuals not inbred, the average inbreeding coefficient will have become larger than zero after a few generations. We need to consider the average inbreeding coefficient when dealing with the genetic impact of inbreeding on the population. Inbreeding affects the Genotype frequencies but not the allele frequencies. In a population with average inbreeding coefficient F, f_AA = p² + pqF, f_Aa = 2pq(1 - F), f_aa = q²+pqF, where f is the genotype frequency of the subscript genotype, p and q are the allele frequency of A and a. In an inbreeding population, the genotype frequency of homozygotes will increase generation by generation. The average inbreeding coefficient is also the probability that an individual gets two identical alleles from an unknown ancestor under purely random mating.

There is no selection for or against any genotype in an ideal Hardy-Weinberg equilibrium population, however, in a real population some characteristics are negatively selected, as the affected individuals have reduced fitness. The fitness (typically denoted by the letter w) refers to the probability of survival and reproduction for a genotype in certain environment. The fitness is expressed as the relative fertility and fertility is expressed as average offspring count. Let Na affected individuals have Ca children in total, and Ns siblings with Cs children, the fitness could be calculated as w = (Ca/Na)/(Cs/Ns). w = 1 means normal fertility. Under selection, the change in fitness is termed selection coefficient (typically denoted by the letter s). s = 1 - w.

A survey in Denmark shown that 108 individuals affected by achondroplasia have 27 children in total while their 457 unaffected siblings have 582 children in total. Thus the fitness w = (27/108)/(582/457) = 0.20, and the selection coefficient s = 1 - w = 1 - 0.20 = 0.80 = 80%.

In reality, mutations occur at almost all loci at different rates. Mutation rate (typically denoted by the letter µ) refers to the frequency of mutation at a certain locus in a population. If a particular locus has a high mutation rate, the gene frequency of the mutant alleles will increase. Usually, the introduction and loss of mutant alleles are balanced by reduced fitness, termed mutation-selection balance. If a population is found to be in Hardy-Weinberg equilibrium, μ = sp for AD and XD disorders, μ= sq² for AR disorders, and μ = sq/3 for XR disorders.

In a population, 12 individuals are affected by an AD disorder in 100000 newborns, and only 2 of these 12 newborns have affected parents. Thus, 10 of these are new mutants. The mutation rate could be calculated directly as μ = 10/(2×100000) = 1/20000 = 5 × 10^-5.

In a Hardy-Weinberg equilibrium population, 10 individuals are affected by an AD disorder (s = 0.80) in 94075 newborns. The dominant homozygotes and heterozygotes are affected, f_AA + f_Aa = 10/94075. The mutation rate could be calculated as μ = sp = s(1-q) = s(1-f_aa^1/2) = s[1 – (1- f_AA - f_Aa)^1/2] = 0.80 × [1 - (1 -10/94075)^1/2] = 4.25×10^-5. In a Hardy-Weinberg equilibrium population, the prevalence of phenylketonuria (AR) is 6.0 × 10^-5. Known s = 0.85, the mutation rate could be calculated as μ = sq² = 0.85 ×6.0 × 10^-5 = 5.1 × 10^-5.

In a Hardy-Weinberg equilibrium population, the prevalence of haemophilia A (XR) in males is 8.0 × 10^-5. Known s = 0.75, the mutation rate could be calculated as μ = sq/3 = 0.75 ×8.0 × 10^-5 /3 = 2.0 × 10^-5.

In small isolated population, one allele may be transmitted to the next generation with higher proportion by chance, which changes the gene frequency and results in Hardy-Weinberg disequilibrium. This is termed genetic drift. In extreme cases, one allele gets lost and the other gets fixed.

Migration and intermarriage introduces new or more of one alleles in to a population. Such a diffusion of alleles across racial or geographical boundaries is termed gene flow.

The harmful or lethal alleles in a population lead to decrease of the fitness. The overall effect of all deleterious alleles in a population is termed genetic load (typically denoted by letter L). Genetic load is defined as L = (w_max-) , where w_max is the maximum fitness and is the average fitness in a population at a certain time point. = Σf_iw_i, where f_i and w_i denote the genotype frequency and corresponding fitness of the ith genotype respectively. The factors that affects Hardy-Weinberg equilibrium would affect the genetic load as well.

Deleterious mutation is the main contributor to genetic load. In some cases, genetic load also be simplified as the average harmful mutated genes per individual in a population. In average, human carries hundreds of loss-of-function mutations and hundreds of less severely deleterious amino acid variants.

Segregation and recombination may also rise homozygous individuals (aa) with lower fitness from the heterozygotes (Aa). Inbreeding in short term may increase genetic load as more aa individuals are generated. But these individuals will be selected against and get purged.