Although important, variation in the isoform of proteins may actually make up only a very small part of functional allelic variation. Much variation in physiology and behavior arises from a difference in expression of the gene. That is, much allelic variation may lead to more or less of a protein, rather than a different form of the protein. To illustrate the quantitative effects of allelic variation we use a modification of the example in the classical textbook on quantitative genetics by Falconer (Falconer and Mackay, 1996) and hypothesize a bi-allelic gene for systolic blood pressure (SBP) that has a decreaser allele "b" which causes lower blood pressure and an increaser allele "B" which causes higher blood pressure (see Fig. 28.2). For now, let us assume that the mean effect of all other factors translates into a SBP of 129 mm Hg and that none of these factors leads to variation between

1 What was left out here is large-scale genetic variation, including loss or gain of chromosomes or breakage and rejoining of chromatids. This variation is rare but often leads to profound developmental problems.

bb m Bb BB

Fig. 28.2 A hypothetical "SBP gene" with a bi-allelic polymorphism that influences systolic blood pressure level individuals. We observe three types of individuals, one with a SBP of 117 mm Hg and genotype "bb," one with a SBP of 133 mm Hg and genotypes "Bb," and one with a SBP of 141 mm Hg and genotype "BB." The allelic effects are typically defined with respect to the difference between the homozygotes (bb, BB). They can be formalized as "+a" and"-a," which is the difference between each of the two homozygotes and the homozygote midpoint (m), and d, which is the dominance deviation of the heterozygote from the homozygote midpoint. If the alleles were perfectly additive, the heterozygote (Bb) should have been equal to the homozygote midpoint (129 mm Hg). However, the increaser allele B of this SBP gene appears to "dominate" the decreaser allele d. The SBP gene is said to display dominance (d = 0).

Now suppose we would do the following breeding experiment (please note that this is a thought experiment!). We select pairs of heterozygous parents for the SBP locus (both have genotype Bb). Such parents will yield an offspring with genotypes BB, Bb, and bb and the expected frequencies and genotypic effects as in Table 28.1.

The contribution of this locus to the mean SBP in the offspring (mean genetic effect ^g) is the sum of the genotypic effects multiplied by their appropriate population frequencies, or

In the SBP gene example (a=12, d=4) the in the offspring is 1/4 x 12 + V2 x 4 +—V4 x 12 = 2 mm Hg. The mean population SBP will be 129 + 2 = 131 mm Hg. The genetic variance (Vg) around this mean is computed according to standard variance rules as the sum over all genotypes of the squared differences between the genetic effect of the genotype and the mean genetic effect weighted by the frequency of each genotype, or

In the example, the VG for SBP in the offspring is V4 x (12 - 2)2 + V2 x (+4 - 2)2 + V4 x (-12 - 2)2 = 76 mm Hg2.

Selective breeding and inbreeding are daily practice in animal and plant genetics, but of course completely unthinkable in human research, which must always be observational. In a human sample all possible matings (e.g., BBxBB, BBxBb, BBxbb) can occur and the three genotype frequencies in the offspring will not be V4,1/2, and 1/4 but something else. Common notation for the frequency of a bi-allelic gene in a population is p for the frequency of one allele and q for the frequency of the second allele, with p, q probabilities varying between 0 and 1, and summing to 1 (p = 1 - q). The genotype frequencies for BB, Bb, and bb are then p2,2pq, and q2. The mean genetic effect and genetic variance can still be derived exactly as before, but now p and q are additional unknowns that need to be estimated from a sample of the population (see Table 28.2).

Under the assumption of random mating, lack of selection according to genotype and absence of mutation or migration, the frequencies of the genotypes in the population are perfectly predicted by the frequencies of the alleles, which is referred to as Hardy-Weinberg equilibrium (HWE). As an example consider the SBP gene

Table 28.1 Expected frequency and genotypic effects of a heterozygous mating Bb x Bb

Genotype (i) |
BB |
Bb |
Bb |

Frequency (f) |
14 |
12 |
14 |

Genotypic effects (x;) |
a |
d |
-a |

Frequency x genotypic effect (f x x;) |
V4 a |
V2d |
-V4 a |

Table 28.2 Expected frequency and genotypic effects for a gene with frequency p for the B allele and frequency q for the b allele

Genotype (i)_BB_Bb_bb

Frequency fi) p2 2pq q2

Genotypic effects (xi) ad -a fi x Xi p2a 2pqd -q2a where we let the least frequent, or minor, allele B take up 40% of all alleles in the population (p=0.4). The expected frequencies for the three genotypes BB, Bb, bb are then p2 (0.16), 2pq (0.48), and q2 (0.36), respectively. We now genotype the 500 subjects in our sample and observe them to have genotypes BB, Bb, bb at frequencies 75, 235, 190, respectively (expected frequencies are 0.16x500, 0.48x500, and 0.36x500, or 80, 240, and 180). A x2 test for HWE simply compares the expected genotype frequencies to the observed genotype frequencies with a significant x2 value indicating that HWE does not hold. In the example the X2 = 0.97 with a p-value of 0.61, which means that the HWE assumption is not violated.

Under the HWE assumption, the mean genetic effect (^g) ofEq. (28.3) can now be more generally computed as ig = p2a + 2pqd - q2a

The genetic variance around this genetic mean will have two terms, reflecting additive genetic effects and dominance effects respectively or

VG = p2 x (2q(a - dp))2 + 2pq x (a(q - p) +d(1 - 2pq))2 + q2 x (-2p(a + dq))2 = 2pq(a + (q - p)d)2 + 4(pqd)2.

where 2pq(a + (q - p)d)2 is the additive genetic variance (VA) and 4(pqd)2 the dominance variance (VD). The contribution of the additive genetic term to the total trait variance is often referred to as the narrow sense heritability, whereas the total contribution of additive plus dominance effects is the broad sense heritability.

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