Fisher designed all the quantitative genetic parameters in his model (Chapter 8) for a single population sharing a common gene pool and characterized by a single system of mating for the phenotype of interest. Moreover, in order to estimate some of these parameters when genotypes are unmeasured, Fisher also had to assume that the environment (actually residual factors) was constant across the generations in the sense that the probability distribution of the environmental deviations was unchanging with time and all individuals had an independent environmental deviation (this assumption can be relaxed in more complicated models). However, a frequent problem in quantitative and population genetics occurs when we want to compare two populations with distinct phenotypic distributions. Because two different populations may have distinct gene pools and may live in a different range of environmental conditions, the biological meaning of changes in the phenotype from one to the other are difficult to evaluate. Do the populations differ because they have different allele frequencies but have identical genotype-to-phenotype mappings and the same environment? Or do they differ because they have completely different allelic forms? Or do they differ because they have the same alleles and the same allele frequencies but differ in the probabilities of various environmental or other residual factors? Or do they differ because of a combination of different allele frequencies, different alleles, and different environments? The Fisherian model outlined in Chapter 8 does not address any of these questions because it was designed to be applicable only to a single population and does not contain one statistic related to the comparison of different populations.
The inability of the Fisherian model to compare populations is particularly evident when we want to understand a difference in mean phenotype between two populations. The first step in Fisher's model is to subtract off the mean of the population from all observations. All of Fisher's quantitative genetic parameters are defined in terms of deviations from the mean and hence are mathematically invariant to the overall mean value of the population. Consequently, none of Fisher's quantitative genetic measures such as broad-sense or narrow-sense heritability have anything to do with the mean phenotype of the population. Nevertheless, sometimes the argument is made that because a trait is heritable within two different populations that differ in their mean trait value, then the average trait differences between the populations are also influenced by genetic factors (e.g., Herrnstein and Murray 1994). Because heritability is a within-population concept that refers to variances and not to means, such an argument is without validity. Indeed, heritability is irrelevant to the biological causes of mean phenotypic differences between populations. To see this, we will consider four examples.
First, in Chapter 8 we examined the role of the amino acid replacement alleles at the ApoE locus in a Canadian population of men from the mid-1980s upon the phenotype of total serum cholesterol level. The mean phenotype in that population was 174.2 mg/dl. Hallman et al. (1991) studied the role of the same ApoE polymorphisms in nine different human populations, whose mean total serum cholesterol levels varied from 144.2 mg/dl (Sudanese) to 228.5 mg/dl (Icelanders). These mean differences in cholesterol levels span a range of great clinical significance, as values above 200 mg/dl are considered an indicator of increased risk for coronary artery disease. Hence, these nine populations are greatly different in their phenotypic distributions in a manner that is highly significant both statistically and biologically. Despite these large differences in mean total serum cholesterol levels, a Fisherian analysis of the ApoE polymorphism within each of these populations results in estimates of the average excesses and effects, heritabilities, and so on, that are statistically indistinguishable, as illustrated in Figure 9.3 for the average excesses for the Sudanese versus Icelanders. How can such large mean phenotypic differences between populations be totally invisible to the Fisherian model? It is known that the phenotype of total serum cholesterol is strongly influenced by many environmental factors, such as diet, exercise regimens, alcohol consumption, smoking, and so on. The populations examined by Hallman et al. (1991) differ greatly in these environmental variables, so it is not surprising that they also differ in their mean phenotypes. The homogeneity of the results of the Fisherian analyses of the ApoE polymorphism arises from two factors:
• The populations all have similar allele frequencies in their respective gene pools; for example, the frequencies of the e2, e3, and e4 alleles in the Sudanese with the lowest average total serum cholesterol are 0.081, 0.619, and 0.291, respectively, and in the Iceland population with the highest average total serum cholesterol they are 0.068, 0.768, and 0.165.
• The genotypes all map onto phenotypes in a similar manner relative to their population means; that is, a genotype that tends to be above the mean in one population tends to be above the mean and by the same relative amount in all populations.
The main impact of the environmental factors in this case is to shift the genotypic values (the mean phenotype of a genotype) up or down by approximately the same amount in all co 35
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