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One way to illustrate the idea behind the fundamental theorem is to examine natural selection and the change in the average fitness of a population over time. For simplicity, assume that the organisms are entirely haploid and reproduce asexually or clonally and that generations are discrete (these assumptions are not required by the fundamental theorem itself but make the math much simpler). In the haploid case, the average fitness is fitness of each haplotype weighted by its frequency summed over all haplo-types in the population (recall equation 6.10). In an equation the mean fitness is w

an equation that can be rearranged by multiplying by — rather than dividing by w to give w

11 PW2i

117 1 1

It turns out that the term

I PiW2

in fitness (the variance is I(piwi - w)2 which is equivalent to I piw2 - w2 and when both terms are multiplied by the constant — it gives the term in w k brackets in equation 6.52). In addition, the relative fitness values of all the haplotypes can be scaled so that w = 1. This then leads to

and the conclusion that the change in mean fitness of the population after one generation of natural selection is equal to the variation in fitness. This variation in fitness is really genetic variation in the case of haploids, due to the frequencies of the different haplotypes in the population as well as to the different fitness values of each haplotype. Therefore, the change in fitness under natural selection is equal to the genetic variation in fitness. Further, since a variance can never be negative, the change in mean fitness by natural selection must then be greater than or equal to zero.

The point of Fisher's fundamental theorem can also be shown graphically for a diploid diallelic locus using a De Finetti diagram (introduced in Chapter 2) that also displays the mean fitness of the population (Fig. 6.13). To see this, let 20, P, and R represent the frequencies of the genotypes Aa, AA, and aa, respectively. The ratio of the genotype frequencies can be expressed as the square of half the heterozygote frequency divided by the product of the homozygote frequencies or X = 02/PR, where X is a measure of departure from Hardy-Weinberg genotype frequencies akin to the fixation index F. When genotype frequencies are in Hardy-Weinberg proportions, genotype frequencies are then 20 = 2pq, P = p2, R = q2, and X= 1. The two dashed lines in Fig. 6.13 have values of X less than one. Each point of the De Finetti diagram in Fig. 6.13 will also represent a value of the mean fitness of the population, depending on the specific values of the relative fitness values of the genotypes. Mean fitness on the De Finetti diagram is represented by the grayscale gradient with darker tones representing higher mean fitness.

The change in mean fitness under natural selection can be thought of as a two-step process on the De Finetti diagram (Edwards 2002). fn the first step, allele frequencies change from their current value to some new value while keeping the ratio of genotype frequencies constant. This is equivalent to moving from point z1 to point z? while remaining on the line that defines a constant value of X. in the second step, the population moves from point z1 to point z} by changing its genotype frequencies but not altering its allele frequencies. The first part of the change in mean fitness caused by selection is due to the change

Frequency of Aa 2 Q

Frequency of Aa 2 Q

Allele frequency before selection

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