In 1908, G. H. Hardy (a British mathematician) and Wilhelm Weinberg (a German doctor) independently formulated a principle, showing how under specific circumstances the genetic environment of a population will not change. This treatment of Mendel's first law resolved a divergence of opinions (and outright misunderstandings) regarding its expected outcome. The dissolving of this riddle ultimately led to the acceptance of evolution as a shift of an allele frequency in a population over a period of time.
The "Hardy-Weinberg equilibrium" theorem posits that evolution will not take place - if certain conditions obtain. It shows how allele and genotype frequencies remain stable (in a state of equilibrium) when undisturbed by intervening "forces of nature", and it draws a picture of an idealized, null evolutionary scenario.
Again Mendel's first law, "The Law of Segregation" states:
(1) A gene can exist in more than one form.(2) Individual organisms inherit two alleles for each trait.
(3) When gametes are produced by meiosis, allele pairs separate leaving each cell with a single allele for each trait.
(4) After combining when the two alleles of a pair are different, one is dominant and the other is recessive.
Each individual carries two, unblending copies of each gene. During meiosis, each copy of the gene pair separates from the other, and after combining, a gamete receives one copy of each gene selected at random from the pair.
Mendel's law permits the following outcome: in a large, sexually reproducing population, allele and genotype frequencies will remain constant across generations unless disrupted by evolutionary processes acting on them; That is, no evolution will occur, if the following conditions hold:
(1) There is no selection pressure operating on individuals; All individuals have equal rates of survival success;
(2) There are no mutations (affecting the genes of individuals) which convert one allele to another;
(3) There is no migration of individuals (gene flow) into or out of the population;
(4) The population is sufficiently large that there are no random events that cause some individuals to pass on more of their genes than other individuals;
(5) Individuals choose their mates at random, i.e., "panmictic" mating is universal; All individuals have equal rates of reproductive success.
Only if ALL of the above conditions are satisfied, then the relative frequencies of alleles and genotypes will not shift from generation to generation; Evolution will not take place. If ANY one of the above five constraints is violated, then alleles shift and evolution will occur. Given that in nature such an equilibrium does not exist, the observable violation of the Hardy-Weinberg relation in natural populations set the scene for population genetics (the ideas of Fisher, Haldane and Wright) to emerge and dominate evolutionary biology during the early Synthesis, allowing evolution to be unpacked through a mathematical analysis of alleles.
By two, simple algebraic equations, calculations from them allow us to determine whether evolution has taken place in a population:
(1) p^2 + 2pq + q^2 = 1
(2) p + q = 1
p = frequency of the dominant allele in the population
q = frequency of the recessive allele in the population
p^2 = percentage of homozygous dominant (XX) individuals
q^2 = percentage of homozygous recessive (xx) individuals
2pq = percentage of heterozygous (Xx) individuals
An example problem, below, regarding a population that is assumed to be in Hardy-Weinberg equilibrium. If the population were not in equilibrium, then the following calculations would be erroneous.
Suppose that there are 16 pigs in poke, 12 pigs have pink coats and 4 pigs have black coats.
1. Calculate for the percent of the population that are homozygous recessive, or q^2.
Since four of the sixteen individuals show the recessive phenotype (the black coat), the answer is 25% or 0.25.
2. Find the frequency of the recessive allele, or q.
Since the square root of q^2 yields q (the frequency of the recessive allele), the answer is 0.5.
3. Find the frequency of the dominant allele, or p.
Since the sum of the frequencies of both alleles = 100%, p + q = 1 and q = 0.5, the answer is (p = 1 - 0.5) = p = 0.5
4. Find the frequency of the heterozygotes, or 2pq.
Since the frequency of the heterozygotes is 2pq (the percent of the population that is heterozygous for the pink coat), the answer is 2pq = 2(0.5) (0.5) = 0.5. Or, 50% of the population is heterozygous.
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