Formalisations of Evolutionary Biology 511Sperm
fr(A 1 )=p fr(A 2 )=p
Ova fr(A 1 )=p fr(A 1 A 1 )=p^2 fr(A 2 A 1 )=pq
fr(A 2 )=P fr(A 1 A 2 )=pq fr(A 2 A 2 )=q^2The lower left box also yields apqfrequency for anA 1 A 2. Since the order doesn’t
matter,A 2 A 1 is the same asA 1 A 2 and hence the sum of frequencies is 2pq.
This proves that a population withA 1 :A 2 =p:qin an initial generation will
in the next generation have a frequency distribution: (p^2 )A 1 A 1 :(2pq)A 1 A 2 :
(q^2 )A 2 A 2. The second step is to prove that this distribution is an equilibrium in
the absence of perturbing factors. Given the frequency distribution (p^2 )A 1 A 1 :
(2pq)A 1 A 2 :(q^2 )A 2 A 2 ,p^2 of the alleles will beA 1 and half of theA 1 A 2 combi-
nation will beA 1 , that ispq. Hence, there will bep^2 +(pq)A 1 in this subse-
quent generation. Sinceq=(1−p), we can substitute (1−p)forq, yielding
p^2 +(p(1−p)) =p^2 +(p−p^2 )=p. Since the frequency ofA 1 in this generation is
the same as in the initial generation (i.e.,p), the same frequency distribution will
occur in the following generation (i.e., (p^2 )A 1 A 1 :(2pq)A 1 A 2 :(q^2 )A 2 A 2 ).
Consequently, if there are no perturbing factors, the frequency of alleles after the
first generation will remain constant. But, of course, there are always perturbing
factors. One central one for Darwinian evolution is selection. Selection can be
added to the dynamics by introducing a coefficient of selection. For each genotype
(combination of alleles at a locus^39 ) a fitness value can be assigned. Abstractly,
A 1 A 1 has a fitness ofW 11 ,A 1 A 2 has a fitness ofW 12 ,andA 2 A 2 has a fitness of
W 22. Hence, the ratios after selection will be:
W 11 (p^2 )A 1 A 1 :W 12 (2pq)A 1 A 2 :W 22 (q^2 )A 2 A 2.To calculate the ratiop:qafter selection this ratio has to be normalised to make
p+q= 1. To do this, the average fitness,, is calculated. The average fitness is
the sum of the individual fitnesses.
=w 11 (p^2 )+w 12 (2pq)+w 22 (q^2 ).Then each factor in the ratio is divided by, to yield:
((w 11 (p^2 ))/)A 1 A 1 :((w 12 (2pq)/)A 1 A 2 :((w 22 (q^2 )/)A 2 A 2.(^39) The term “genotype” is used in a number of different ways by different writers and in different
contexts. It can, as in this context, mean a pair of alleles at a locus. It is sometimes used to
designate an organism’s entire collection of loci and the alleles associated with them (see, for
example, [Hartl, 2000, p.2]). It is also used to designate a subset of loci that together determine
some characteristic of the organism. This variety of uses invites confusion but has now become
embedded in the language of evolutionary biologists, geneticists and ecologists. Hence, to avoid
problems that can arise from the equivocation on the term, clarity about the use in particular
contexts is essential.