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It's like this: this equation links allele frequency to genotype frequency, assuming certain conditions are met. This means that:.
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p^2 + 2pq + q^2 = 1 Derivation: Take a gene with two alleles; call them A 1 and A 2. (Dominance doesn’t matter for our purposes; this works equally well with codominance or incomplete dominance.) In a population, some members will have the A 1 A 1 genotype, some will have the A 1 A 2 genotype, and some will have A 2 A 2. Now, imagine that you can somehow take all the gametes produced by the members of the population—for simplicity, we’ll assume that these are eggs and sperm. Some gametes, of course, carry A 1 , and some A 2. p = freq ( A 1 ) q = freq ( A 2 ) NOTICE that: the frequency of an allele is equal to the probability that a randomly chosen gamete will be carrying that allele. Also notice that p+q=. What’s the chance that an egg and sperm drawn randomly will both be carrying A 1? Obviously, it’s p × p , or p 2
. And the chance that both gametes will both bear A 2 is q × q , or q 2 . There are two other possibilities: sperm with A 1 and egg with A 2 , or sperm with A 2 and egg with A 1. The chance of either one happening is p × q , and the total probability of producing a zygote with the A 1 A 2 genotype is twice that: 2pq. All of these probabilities sum to 1. So p^2 + 2pq + q^2 = 1. [Since p+q=1 , (p+q)^2 = p^2 + 2pq + q^2 = 12 =1 .] WHO CARES? It’s like this: this equation links allele frequency to genotype frequency, assuming certain conditions are met. This means that:
Derivation: w in general means “relative fitness”: a measurement of the relative ability of individuals with a certain genotype to reproduce successfully. W 11 , for instance, means the relative ability of individuals with the A 1 A 1 genotype to reproduce successfully. w is always a number between 0 and 1. Adding fitness (w) to the Hardy-Weinberg equation as shown above allows you to predict the effect of selection on gene and allele frequencies in the next generation. Take the Hardy-Weinberg equation and multiply each term (the frequency of each genotype) by the fitness of that genotype. Add those up and you get the mean fitness, 𝑤 (“w-bar”). Divide through by 𝑤 and you get the second equation. Here, each term of the equation is multiplied by the fitness of a genotype divided by the mean fitness. If a genotype is fitter than average, this quotient is greater than 1, and that genotype will increase in frequency in the next generation. If a genotype is less fit than average, the quotient is less than 1, and that genotype will decrease in frequency in the next generation. A related term to fitness (w) that you may run across is the selection coefficient, s. The selection coefficient compares two phenotypes and provides a measure of the proportional amount that the phenotype under consideration is less fit. With no selection against a phenotype s=0 and if a phenotype is completely lethal s=1. The relation ship between relative fitness (w) and the selection coefficient (s) is s = 1-w.
𝒑 (^) 𝒕!𝟏 = ( 1 − μμ)𝑝! + ν𝑞! 𝛥𝑝 = 𝑝! 1 − 𝜇 − 𝜈 + 𝜈 − 𝑝! Derivation: Imagine that in each generation, allele A 1 mutates to allele A 2 with a frequency of μ, and that allele A 2 “back-mutates” to A 1 with a frequency of ν. Then in each generation, q , the frequency of the A 2 allele, increases by a factor of μ p (the rate of mutation of A 1 to A 2 times the frequency of A 1 ) and decreases by a factor of ν q. These will eventually balance each other out, so that 𝛥𝑝 = 0 (i.e. allele frequencies don’t change any further). When 𝛥𝑝 = 0 , it must be true that μ p = ν q. From this, with a little algebraic jugglery, you can derive the formula 𝑝 =
where 𝑝 (“p-hat”) is the equilibrium frequency. Similar equations let you derive 𝑞. This isn’t all that useful an equation, however. In humans for example, mutation rates are estimated at 1.2 x 10-^8 mutations/base pair/generation. Assuming an average gene size of
through migration or mutation, the population will eventually become fixed for only one of the alleles originally present.