Genetic aspects of populations, Hardy-Weinberg equilibrium

Population

What is a population from genetic point-of-view? It is a group of inbreeding individuals of the same species that inhabit the same space (prescribed geographical area) and time. The key to a population is that they must be able to interbreed.

Within any given population there is variation (differencies) of/in:

• phenotypes; the proportion of individuals within a population that are of a particular phenotype is phenotype frequency
• genotypes; the proportion of individuals within a population that are of a particular genotype is genotype frequency
• alleles; the proportion of all copies of a gene in a population that are of a particular allele type is allelic frequency

The gene pool is the sum total of all of the alleles (of one locus) present and carried by the population.

• Gametic gene pool – sum of all alleles in gametes.
• Zygotic gene pool – sum of all alleles in zygotes.

For a gene with 2 alleles, A and a:

• N_AA is the number of AA homozygotes
• N_Aa is the number of heterozygotes Aa
• N_aa is the number of aa homozygotes
• N_AA + N_Aa + N_aa = N, number of individuals in population

Estimating/calculating of allele frequencies in a population

• with three distinct phenotypes for a trait (e.g. in MN blood group system)

Let p = frequency of allele A, and q = frequency of a. Then:

pA = (2N_AA + N_Aa) / 2N

qa = (2N_aa + N_Aa) / 2N

Variant procedure: Calculating allele frequencies from (known) frequencies of genotypes AA, Aa, aa

pA = f_AA + ½ f_Aa

qa = f_aa + ½ f_Aa

p + q = 1

• where (only) two distinct phenotypes exist (i.e., in allelic relation of full dominance/recessivity); estimate of frequency of unfavourable (mutant, deleterious, recessive) allele (Rh blood group system, tasting of PTC or AR diseases)

Genotype frequencies

If frequency of allele A in a population is p, frequency of allele a in a population is q:

• the probability that both the egg and the sperm contain the A allele is p x p = p²
• the probability that both the egg and the sperm contain the a allele is q x q = q²
• the probability that the egg and the sperm contain different alleles is (p x q) + (q x p) = 2pq

Hardy-Weinberg Equation

Hardy-Weinberg law

A population that is not changing genetically is in Hardy-Weinberg equilibrium (1908). It comes if these 5 assumptions are correct:

• Random mating (panmixia)
• Large population size (N approaching infinity)
• No migration between populations
• No (or negligible) mutations
• Natural selection does not affect alleles being considered

If these assumptions are true, it follows that:

• Allele frequencies remain constant from one generation to the next
• After one (or more) generations of random mating (breeding), the genotype frequencies (for a 2-allele gene with allele frequencies p, q) are in the proportions: p²_(AA) : 2pq_(Aa) : q²_(aa), and population will be in H-W equilibrium. Var.: H-W equilibrium in a large population will be reached after one generation of (random) breeding.
• For a population to be in Hardy Weinberg equilibrium, the observed genotype frequencies must match those predicted by the equation p² + 2pq + q².

Graphic demonstration of H-W equilibrium

(relation between frequencies of alleles and frequencies of genotypes)

Multiple alleles

Multinomial expansion for two alleles a and b with frequencies p and q p² + 2pq + q² is a binomial expansion of (p + q)²

p²(AA) + 2pq(Aa) + q²(aa) = (p + q)² = (1)² = 1

For three alleles a, b and c with frequencies p, q and r, the multinomial expansion is (p + q + r)² which expands into: p²+ q² + r² + 2pq + 2pr +2qr , where the first 3 terms being homozygotes and the remaining three heterozygotes.

p + q + r = 1 p²+ q² + r² + 2pq + 2pr +2qr = 1