We will first explore the consequences of natural selection through a simple extension of the one-locus, two-allele model introduced in Chapter 2. As before, we need to go through a complete generation transition, ending up in the next generation at a comparable point to the starting generation. In the models used in Chapter 2 we went from parental genotypes to gametes via Mendelian probabilities, then to offspring genotypes through the mechanisms of uniting gametes (population structure). To include natural selection, we need to expand this model by explicitly modeling the life stages measured by the fitness components. In this chapter, we will assume each of the fitness components can be measured as a constant probability that is assigned as a genotypic value to a particular group of individuals sharing a common genotype.
Figure 11.2 presents the basic one-locus, two-allele model that allows us to incorporate natural selection. As we did in Chapter 2, we assume the life stages are discrete with no overlapping of generations. We start with a gene pool at the initial generation. Our genetic model considers only variation at a single autosomal locus with two alleles, A and a, so the initial gene pool is completely characterized by specifying the frequency of the A allele to be p (the frequency of a is q). We initially consider only a single, isolated local population. This means that we can ignore gene flow as a component of population structure. We also assume that the population is infinite in size, so we can equate probabilities to their frequencies. This means that we can ignore genetic drift as a component of population structure. However, we make no assumptions at this point about the deme's system of mating. Accordingly, the results to be given apply both to random-mating and non-random-mating populations. Under these assumptions, the deme's system of mating will determine the probabilities of the various combinations of gamete pairs to yield the zygotic genotype frequencies, z. These zygotic genotype frequencies represent the frequencies of the possible genotypes at the moment of fertilization. In the models given in Chapter 2 where there was no natural selection, the zygotic genotypic frequencies were equated to the adult genotype frequencies, reflecting the assumption that all zygotes were equally viable.
We now deviate from the models of Chapter 2 by assuming that the individuals develop phenotypes in an environment that is constant through time. The first phenotypic attribute that we focus upon is viability. We assume that each genotype has a genotypic value of lij that measures the probability of a zygote with genotype ij living to adulthood in the environment in which it grows and develops. The frequency of an adult with genotype ij is proportional to the product of its initial zygotic frequency, zij, times the probability that it lives to adulthood, lij (Figure 11.2). Because some or all of the lij can be less than 1 (not every zygote lives to be an adult), these products do not define a probability distribution. Hence, it is necessary to divide the products zijlij by the average viability, l = zAAlAA + zAal Aa + zaataa, to obtain the adult genotype frequencies, zijlij /l, which sum to 1.
In Chapter 2 we also assumed that all adults are mated, or at least that the same proportion of adults are mated in each genotype category. We now assume that each genotype has a genotypic value of mij that measures the probability of an adult with genotype ij successfully finding a mate in the environment (Figure 11.2). The frequency of a mated adult with genotype ij is proportional to the product of its adult genotypic frequency times the probability that it finds a mate, mij. These products are proportional to zijlijmij. Because some or all of the mij can be less than 1 (not every adult finds a mate), these products do not define a probability distribution. Hence, it is necessary to divide the products zylymy by the average product of viability with mating success, ml = Z AAm AAlAA + ZAamAal Aa + Zaamaalaa, to obtain the mated adult genotype frequencies, zylymy/ml, which sum to 1.
In Chapter 2 we assumed that all mated adults contributed equally to the next generation's gene pool. However, we now assume that mated adults interact with their environment to produce the phenotype of the number of gametes passed on to the next generation (that is,
FUNDAMENTAL EQUATION OF NATURAL SELECTION: MEASURED GENOTYPES
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