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Review of Population Genetics Equations, Study notes of Genetics

It's like this: this equation links allele frequency to genotype frequency, assuming certain conditions are met. This means that:.

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Review
of
Popu l a tion
Gen e tics
Equation s
1.
Hardy-Weinberg Equation:
p2 + 2pq + q2 = 1
Derivation: Take a gene with two alleles; call them A1 and A2. (Dominance doesn’t
matter for our purposes; this works equally well with codominance or incomplete
dominance.) In a population, some members will have the A1 A1 genotype, some will
have the A1 A2 genotype, and some will have A2 A2.
Now, imagine that you can somehow take all the gametes produced by the members of
the populationfor simplicity, we’ll assume that these are eggs and sperm. Some
gametes, of course, carry A1, and some A2.
p = freq (A1)
q = freq (A2)
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=1.
What’s the chance that an egg and sperm drawn randomly will both be carrying A1?
Obviously, it’s p × p, or p2. And the chance that both gametes will both bear A2 is q × q,
or q2. There are two other possibilities: sperm with A1 and egg with A2, or sperm with A2
and egg with A1. The chance of either one happening is p × q, and the total probability of
producing a zygote with the A1 A2 genotype is twice that: 2pq. All of these probabilities
sum to 1. So p2 + 2pq + q2= 1. [Since p+q=1, (p+q)2 = p2 + 2pq + q2 = 12 =1.]
WHO CARES? It’s like this: this equation links allele frequency to genotype frequency,
assuming certain conditions are met. This means that:
1) If you know allele frequencies, you can predict the genotype frequencies, and compare
them with the actual frequencies. If they don’t match, then one of your assumptions is
violatedmaybe there is natural selection going on, or immigration, or non-random
mating. . .
2) If you know genotype frequencies, you can predict allele frequencies, and compare
them with the actual frequencies. Again, if they don’t match, then one of your
assumptions is violated.
3) If you know phenotype frequencies, then you can estimate genotype and allele
frequenciesbut you can’t test the underlying assumptions.
pf3
pf4

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Review of Population

Genetics Equations

1. Hardy-Weinberg Equation:

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:

  1. If you know allele frequencies, you can predict the genotype frequencies, and compare them with the actual frequencies. If they don’t match, then one of your assumptions is violated—maybe there is natural selection going on, or immigration, or non-random mating...
  2. If you know genotype frequencies, you can predict allele frequencies, and compare them with the actual frequencies. Again, if they don’t match, then one of your assumptions is violated.
  3. If you know phenotype frequencies, then you can estimate genotype and allele frequencies—but you can’t test the underlying assumptions.

2. Fitness:

𝒑𝟐^

+ 𝒒𝟐^

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.

3. Mutation:

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.

  1. The time to fixation on a single allele is directly proportional to population size, and the amount of uncertainty associated with allele frequencies from one generation to the next is inversely related to population size.