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Griffiths AJF, Gelbart WM, Miller JH, et al. Modern Genetic Analysis. New York: W. H. Freeman; 1999.

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# Genotypes and Phenotypic Distribution

Using the concepts of distribution, mean, and variance, we can understand the difference between quantitative and Mendelian genetic traits. Suppose that a population of plants contains three genotypes, each of which has some differential effect on growth rate. Furthermore, assume that there is some environmental variation from plant to plant because of inhomogeneity in the soil in which the population is growing and that there is some developmental noise (see page 14). For each genotype, there will be a separate distribution of phenotypes with a mean and a standard deviation that depend on the genotype and the set of environments. Suppose that these distributions look like the three height distributions in Figure 18-3a. Finally, assume that the population consists of a mixture of the three genotypes but in the unequal proportions 1:2:3 (a / a: A / a:A / A). Then the phenotypic distribution of individuals in the population as a whole will look like the black line in Figure 18-3b, which is the result of summing the three underlying separate genotypic distributions, weighted by their frequencies in the population. This weighting by frequency is indicated in Figure 18-3b by the different heights of the component distributions that add up to the total distribution. The mean of this total distribution is the average of the three genotypic means, again weighted by the frequencies of the genotypes in the population. The variance of the total distribution is produced partly by the environmental variation within each genotype and partly by the slightly different means of the three genotypes.

#### Figure 18-3

(a) Phenotypic distribu-tions of three genotypes. (b) A population phenotypic distribution results from mixing individuals of the three genotypes in a proportion 1:2:3 (a / a:A / a:A / A).

Two features of the total distribution are noteworthy. First, there is only a single mode. Despite the existence of three separate genotypic distributions underlying it, the population distribution as a whole does not reveal the separate modes. Second, any individual whose height lies between the two arrows could have come from any one of the three genotypes, because they overlap so much. The result is that we cannot carry out any simple Mendelian analysis to determine the genotype of an individual organism. For example, suppose that the three genotypes are the two homozygotes and the heterozygote for a pair of alleles at a locus. Let a / a be the short homozygote and A / A be the tall one, with the heterozygote being of intermediate height. Because there is so much overlap of the phenotypic distributions, we cannot know to which genotype a given individual belongs. Conversely, if we cross a homozygote a / a and a heterozygote A / a, the offspring will not fall into two discrete classes in a 1:1 ratio but will cover almost the entire range of phenotypes smoothly. Thus, we cannot know that the cross is in fact a / a × A / a and not a / a × A / A or A / a × A / a.

Suppose we grew the hypothetical plants in Figure 18-3 in an environment that exaggerated the differences between genotypes—for example, by doubling the growth rate of all genotypes. At the same time, we were very careful to provide all plants with exactly the same environment. Then, the phenotypic variance of each separate genotype would be reduced because all the plants were grown under identical conditions; at the same time, the differences between genotypes would be exaggerated by the more rapid growth. The result (Figure 18-4b) would be a separation of the population as a whole into three nonoverlapping phenotypic distributions, each characteristic of one genotype. We could now carry out a perfectly conventional Mendelian analysis of plant height. A “quantitative” character has been converted into a “qualitative” one. This conversion has been accomplished by finding a way to make the differences between the means of the genotypes large compared with the variation within genotypes.

#### Figure 18-4

When the same genotypes as those in Figure 18-3 are grown in carefully controlled stress environments, the result is a smaller phenotypic variation in each genotype and a greater difference between genotypes. The heights of the individual distributions (more...)

### MESSAGE

A quantitative character is one for which the average phenotypic differences between genotypes are small compared with the variation between individuals within genotypes.

It is sometimes assumed that continuous variation in a character is necessarily caused by a large number of segregating genes, so continuous variation is taken as evidence for control of the character by many genes. But, as we have just shown, this is not necessarily true. If the difference between genotypic means is small compared with the environmental variance, then even a simple one-gene–two-allele case can result in continuous phenotypic variation.

If the range of a character is limited and if many segregating loci influence it, then we expect the character to show continuous variation, because each allelic substitution must account for only a small difference in the trait. This multiple-factor hypothesis (that large numbers of genes, each with a small effect, are segregating to produce quantitative variation) has long been the basic model of quantitative genetics, although there is no convincing evidence that such groups of genes really exist. A special name, polygenes, has been coined for these hypothetical factors of small-but-equal effect, in contrast with the genes of simple Mendelian analysis.

It is important to remember, however, that the number of segregating loci that influence a trait is not what separates quantitative and qualitative characters. Even in the absence of large environmental variation, it takes only a few genetically varying loci to produce variation that is indistinguishable from the effect of many loci of small effect. As an example, we can consider one of the earliest experiments in quantitative genetics, that of Wilhelm Johannsen on pure lines. By inbreeding (mating close relatives), Johannsen produced 19 homozygous lines of bean plants from an originally genetically heterogeneous population. Each line had a characteristic average seed weight ranging from 0.64 g per seed for the heaviest line to 0.35 g per seed for the lightest line. It is by no means clear that all these lines were genetically different (for example, five of the lines had seed weights of 0.450, 0.453, 0.454, 0.454, and 0.455 g), but let’s take the most extreme position—that the lines were all different. Obviously, these observations would be incompatible with a simple one-locus–two-allele model of gene action. In that case, if the original population were segregating for the two alleles A and a, all inbred lines derived from that population would have to fall into one of two classes: A / A or a / a. If, in contrast, there were, say, 100 loci, each of small effect, segregating in the original population, then a vast number of different inbred lines could be produced, each with a different combination of homozygotes at different loci.

However, we do not need such a large number of loci to obtain the result observed by Johannsen. If there were only five loci, each with three alleles, then 35 = 243 different kinds of homozygotes could be produced from the inbreeding process. If we make 19 inbred lines at random, there is a good chance (about 50 percent) that each of the 19 lines will belong to a different one of the 243 classes. So Johannsen’s experimental results can be easily explained by a relatively small number of genes. Thus, there is no real dividing line between polygenic traits and other traits. It is safe to say that no phenotypic trait above the level of the amino acid sequence in a polypeptide is influenced by only one gene. Moreover, traits influenced by many genes are not equally influenced by all of them. Some genes will have major effects on a trait; others, minor effects.

### MESSAGE

The critical difference between Mendelian and quantitative traits is not the number of segregating loci but the size of phenotypic differences between genotypes compared with the individual variation within genotypic classes.

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