Population and Quantitative Genetics

Hardy-Weinberg equilibrium

Hardy-Weinberg equilibrium describes the genotype frequencies expected at a locus in a large, randomly mating population when mutation, migration, selection and drift are absent. For two alleles with frequencies p and q, where p+q=1, the zygote frequencies are p^2 (homozygous dominant), 2pq (heterozygote) and q^2 (homozygous recessive). Because only the recessive homozygote shows the recessive phenotype directly, the observed frequency of that phenotype is used to estimate q^2, from which q=\sqrt{q^2} and p=1-q follow. For example, if the recessive phenotype frequency is 0.16, then q=0.4 and the heterozygote frequency is 2pq=0.48. Hardy-Weinberg proportions are reached in one generation of random mating, so the same rule applies to any population that is randomly mating at the locus. If a subset of a population is randomly mated, the allele frequency must first be recalculated from the gene copies in that subset, and then p'^2 gives the offspring frequency of the homozygote for the new allele. The maximum expected heterozygosity for a locus with n alleles is H=1-sum(p_i^2), which is greatest when all alleles are equally frequent, giving H_{max}=1-1/n. Hardy-Weinberg expectations are useful for checking whether observed genotype counts agree with the expected numbers under random mating.

Genetic drift and bottleneck effects

Genetic drift is the random change in allele frequencies that results from sampling error when gametes are drawn from a finite population. Its effect is strongest in small populations, where alleles can be lost or fixed by chance within a few generations, and it reduces heterozygosity and allelic diversity within each population. A bottleneck is a sharp reduction in population size, and it intensifies drift because the surviving individuals carry only a sample of the original alleles. Expected heterozygosity then declines over time roughly as H_t=H_0(1-1/(2N_e))^t, so a smaller effective size produces a faster loss. Drift is non-directional, so the expected allele frequency stays at its starting value, but the variance among replicate populations grows. Because each isolated population drifts in its own random direction, drift tends to increase genetic differences between populations, whereas it decreases variation within them. Drift and mutation act in opposite directions for within-population variation. For a new neutral mutation in a diploid population of N individuals, the mutation begins at frequency 1/(2N) among 2N gene copies, and for a neutral allele the fixation probability equals this starting frequency, which is 1/(2N).

Effective population size and sex ratio

The effective population size N_e is the size of an idealised population that would experience the same rate of genetic drift or inbreeding as the actual population. Real populations usually have N_e smaller than the census size because of unequal sex ratios, fluctuating numbers, and variance in reproductive success. For a dioecious diploid species with N_m breeding males and N_f breeding females, the standard expression is N_e=4N_mN_f/(N_m+N_f). Because the product N_mN_f is largest when the two sexes are equal, a sex ratio (males to females) of 1 gives the maximum N_e for a fixed total number of breeding adults, with N_e equal to N in that case. Skewed sex ratios reduce N_e even when the census number is unchanged. For example, with 10 breeding males and 40 breeding females, N_e=4(10)(40)/50=32, which is less than the total of 50 breeding adults. Terms such as minimum viable population, carrying capacity and metapopulation size describe different conservation quantities and should not be confused with N_e.

Linkage disequilibrium

Linkage disequilibrium is a nonrandom association between alleles at two loci, meaning that certain allele combinations occur together more often than expected from their separate frequencies. It is generated and maintained when recombination between the loci is rare, because alleles that are physically close on a chromosome stay together across generations. Natural selection that favours particular combinations of alleles at linked loci is another cause, since it keeps the favoured haplotypes together in the population. High recombination between loci breaks up allele associations and drives the population toward linkage equilibrium, so it does not produce disequilibrium. Polyploidy alters chromosome number but does not by itself create nonrandom association between loci. Tight linkage is typically generated by structural changes that suppress recombination. A chromosomal inversion is the classic example, because crossing over inside the inverted segment in heterozygotes yields unbalanced or nonviable gametes, so the loci inside the inversion are inherited as a unit. Exon deletion, exon insertion and gene duplication change gene structure or copy number, but they do not in themselves suppress recombination between neighbouring loci.

Maintenance of genetic variation

Genetic variation at a locus can be maintained by mechanisms that keep more than one allele in the population for long periods. Heterozygote advantage (overdominance) favours individuals carrying two different alleles, so both alleles persist. Negative frequency-dependent selection favours alleles when they are rare, so that a set of alleles remains in balance. Mutation-selection balance maintains deleterious alleles at a stable equilibrium frequency, because recurrent mutation continually regenerates them while selection removes them. Genetic drift does not help maintain variation, because it randomly eliminates alleles in finite populations and tends to reduce diversity over time. This distinction is the key to answering questions that ask which process does not preserve polymorphism. Observed heterozygosity is a sample value that can differ from the random-mating expectation, and selection acting on genotypes can produce a heterozygote excess. Such a value can be consistent with balancing selection, while a neutral two-allele locus under random mating has expected heterozygosity 2pq, which cannot exceed 0.5. A neutral locus with three alleles can reach expected heterozygosity above 0.6. For example, allele frequencies of 0.5, 0.3 and 0.2 give H=1-(0.25+0.09+0.04)=0.62.

Mendelian disorders and diet

Most Mendelian disorders are caused by a single mutated gene and are inherited in a dominant or recessive pattern, but the clinical expression of some of them can be altered by environmental factors such as diet. Phenylketonuria is an autosomal recessive disorder in which the enzyme that converts phenylalanine to tyrosine is defective, so phenylalanine accumulates and damages the nervous system. Because the accumulation depends on dietary intake, restricting phenylalanine in the diet controls the toxic build-up, and this makes phenylketonuria the Mendelian disorder that is managed through diet. Cystic fibrosis, haemophilia and thalassaemia are also Mendelian disorders, but they are not controlled by dietary modification. Cystic fibrosis is caused by CFTR mutations, haemophilia by deficiency of clotting factors, and thalassaemia by defective globin chain synthesis, and none of these is treated by restricting food intake.