Question

In gerbils there is a recessive mutant gene that causes a lethal condition. For the purpose of this problem let the symbol A denote the normal allele and a the mutant. In heterozygote individuals who carry both versions of the allele (Aa), this causes a white spotting color pattern. Homozygotes for the mutant (aa) die as embryos and are never seen in live gerbils. Since the mutant allele is lethal in the homozygous form, natural selection will occur against the frequency of this allele in a population. The decrease is proportional to the frequency of individuals carrying the gene. Assume that in an isolated population of gerbils in generation F0 the frequency of the normal allele is p and the frequency of the mutant allele is q (with p + q = 1). (a) Assuming random mating between animals, compute the expected frequencies of offspring with genotypes AA, Aa, and aa, respectively. Use this information to compute the frequencies of the alleles A and a in the surviving F1 generation in terms of p and q. (b) Write down a general formula for the decrease in frequency of the recessive mutant allele after n generations. Hint: It may help, to write down the frequency formulas for the rst couple generations, simplify and hunt for a pattern. (c) After how many generations can we expect the allele frequency of the recessive mutant to have dropped under 1% of its value in generation F0?

Answer 1

For generation F1

P(AA) = p^2
P(Aa) = 2pq
P(aa) = q^2


(b) Note that in generation F1 all aa members are dead and cannot reproduce.

the proportion of A alleles in the population is:

(p^2 + pq) / (p^2 + 2pq)

the proportion of a alleles in the population is:

pq / (p^2 + 2pq)

to define the proportion for the nth generation we have:

p_i = proportion of allele A in the ith generation
p_(i-1) = proportion of allele A in the (i-1)th generation

q_i = proportion of allele a in the ith generation
q_(i-1) = proportion of allele a in the (i-1)th generation
p_i = ( p_(i-1)^2 + p_(i-1) * q_(i-1) ) / ( p_(i-1)^2 + 2 * p_(i-1) * q_(i-1) )
q_i = ( p_(i-1) * q_(i-1) ) / ( p_(i-1)^2 + 2 * p_(i-1) * q_(i-1) )

(c)  In general, it is observed that even mutants with relatively small heterozygous effects (say 0.1 phenotypic standard deviation) are practically ‘safe’ (i.e. their probability of loss from one generation to the next is smaller than, say, 10%) after just a few generations, typically less than 10. For selection with larger effective size, such as within-family selection, the mutant is ‘safe’ in the population somewhat earlier but eventual fixation takes a longer time.