Chapter 6 Equations

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Summary of significant equations

• Equation 6.1  Expectation that a small sample N will be homozygous.

  • chance of homozygosity = (p²)^N + (q²)^N

  • This equation is not especially useful by itself, but it does illustrate the point that small samples from a heterozygous population have a definite probability of being homozygous.

• Equation 6.2 and 6.3  Expectation of homozygosity or heterozygosity after one generation.
  • Expected homozygosity f ' = (1 / 2N) + (1 - [1 / 2N] ) f

  • Expected heterozygosity H' = (1 - [1 / 2N] ) H

  • These two equations are essentially identical, since homozygosity plus heterozygosity equals one (i.e., f + H = 1 and f ' + H' = 1; see Review and Study Question 6, p. 150).  Both equations illustrate the "march to homozygosity".
  • Note that these equations are for most likely expectations, and do not yield deterministic predictions.

• Equations 6.4, 6.5, 6.6 and 6.7  Polymorphism resulting from neutral mutation.
  • If mutation is occurring at rate u, then equation 6.2 for expected heterozygosity must be modified by the probability that neither allele will be a mutant.  If mutation occurs at rate u, then the probability that a particular allele is NOT a mutant will be (1 - u), so the chance neither allele in a genotype is mutant must be (1 - u)².  So,

  • Expected homozygosity f ' = {(1 / 2N) + (1 - [1 / 2N] ) f } (1 - u)²

  • We can find the equilibrium, when drift is eliminating alleles at the same rate that mutation is introducing them, by setting f = f '.  If we simplify the resulting equation 6.5 by approximating u² by 0 (the square of a very small fraction is very much smaller still), then

  • f* = 1 / (4Nu + 1) and

  • H* = 4Nu / (4Nu + 1)

  • The latter two equations are essentially identical, since homozygosity plus heterozygosity equals one (i.e., f* + H* = 1; see Review and Study Question 6, p. 150).
  • Again, note that these equations are for most likely expectations, and do not yield deterministic predictions.

• Equation 6.8  Effective population size N_e when the sex ratio of breeding individuals is not 1:1.
  • N_e = 4 (N_m N_f) / (N_m + N_f)

  • This equation is not derived in the text, but simply credited to Sewall Wright (one of the founders of the Modern Synthesis).

• Equation 6.9  Effective population size N_e when individuals vary in fertility.
  • We shall skip this equation, also credited to Sewall Wright.  But do note that variation in fertility does influence effective population size.

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