difference between frequencies in two generations, pt+1 - Pt. A difference is commonly symbolized with the Greek capital letter delta (A), so we could say that the change in the frequency of the A genotype is given by Ap = pt+1 - pt. To generate an expression for Ap we can compare the initial genotype frequency pt with its frequency a generation later after natural selection has acted via differential growth. We start with the basic expression for the difference in genotype frequency

If Ap is positive then the A genotype will increase in proportion in the population while it will decrease in proportion if Ap is negative. Substituting the expression for the expected frequency of the A genotype after natural selection (Table 6.1) gives pWi

pwa + qtwB

The ptwA + ptwB term in the denominator on the right side of equation 6.10 is the average relative fitness of the population (it is a frequency-weighted average and so depends on the sum of the product of the frequency and relative fitness for each genotype). Positive values of Ap occur when the frequency of the A genotype after natural selection is greater than the average fitness of both genotypes after natural selection. The computations in Table 6.1 show that the frequency of the A genotype multiplied by its relative fitness (ptwA) is greater than the average fitness so the A genotype will increase in proportion in the population over time. The average fitness will be covered in more detail when considering natural selection in sexual diploid populations.

One advantage of using the relative fitness is that the population growth rates of each genotype do not have to be known to model the proportions of the genotypes over time. Rather, the outcome of the growth process in terms of the relative frequencies of genotypes can be predicted strictly from the ratio of growth rates. This means that equation 6.8 potentially applies to organisms with very high absolute growth rates like bacteria as well as to species with absolute growth rates very near one such as elephants. Equation 6.8 even applies to cases where population sizes are declining through time. If a population is headed to extinction because it is composed of genotypes that all have growth rates less than one, the relative fitness will nonetheless accurately express the change in the proportion of genotypes in the population over time. In practice, the relative fitness can be estimated in competition experiments where two or more genotypes are placed in the same environment and their proportions estimated at a later point in time (see Problem box 6.1, below).

Although simplistic and requiring many assumptions, the model of natural selection among genotypes in organisms with clonal reproduction is nonetheless pertinent to a range of practical situations. One example is the evolution of drug resistance by natural selection in the human immunodeficiency virus (HIV). The genome of HIV (and other retroviruses) is single-stranded RNA. All the proteins inside a virus particle as well as the viral protein envelope itself are encoded by genes in this RNA genome. After infecting a host cell, HIV uses reverse transcriptase produced from its own gene to reverse-transcribe its genome into double-stranded DNA. This DNA version of the retrovirus genome is then integrated into the DNA of the host cell, where it is transcribed by the host cell into many new virus RNA genomes. These new viral RNA genomes are packaged into virus particles released from the host cell through a viral protease. One treatment strategy for HIV has utilized drugs that mimic nucleosides (nucleotides without phosphate groups) that interfere with the virus reverse transcriptase but do not interfere with host cell DNA polymerase. Another treatment uses protease inhibitors that interfere with polyprotein

Absolute fitness The genotype-specific rate of increase or population growth that predicts the absolute number of individuals of a given genotype in a population over time. Commonly symbolized as W or X. Average or mean fitness (T) The frequency-weighted sum of the relative fitness values of each genotype in the population. Relative fitness The growth rate of genotypes relative to one genotype picked as the standard of comparison (often the genotype with the highest absolute fitness). Called Darwinian fitness after Charles Darwin and symbolized as w in models where time is represented in discrete generations (also called Malthusian fitness after Thomas Malthus and symbolized as m in models where time is continuous).

cleavage necessary to produce new infectious virus particles. Unfortunately, HIV has shown rapid evolution of drug-resistant genotypes via natural selection. Figure 6.2 shows allele frequencies over time in the population of HIV particles infecting two patients who began protease inhibitor treatment at day 0. Individual HIV particles with protease alleles resistant to the drug have higher replication rates than HIV particles with wild-type protease alleles. This differential growth rate of HIV genotypes, or natural selection at the protease locus, rapidly changed the protease gene allele frequencies in the HIV population found within each patient. The combination of short generation time, high mutation rate, and large effective population size make natural selection a rapid process acting to change allele frequencies in HIV populations.

Problem box 6.1 Relative fitness of HIV genotypes

It is commonly thought that drug-resistant alleles have lower relative fitness than non-resistant alleles in the absence of drug exposure. To test this hypothesis for HIV-1, Goudsmit et al. (1996) monitored the frequency of alleles at codon 215 of the reverse transcriptase gene in an individual newly infected with HIV but who was not undergoing treatment with the nucleoside analog azidothymidine (AZT). Initially, the HIV alleles were all sequences (90% TAC and 10% TCC codons) known to confer AZT resistance. Over time, the non-resistant allele (a TCC codon) increased in frequency to 49% after 20 months.

Use this change in allele frequencies over 601 days and equation 6.8 to estimate the relative fitness of the non-resistant allele in the absence of AZT. Assume that the generation time of HIV is 2.6 days and that generations are discrete, and that the wild-type allele was initially present at a frequency of 1.0% and so was not initially detectable. Note that the exponent in an equation like a = y(xt) where a, y and x are constants can be removed by taking the log of both sides to get log(a) = log(y) + t log(x).

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