Extensions of Mendel: dominance, multiple alleles, lethal alleles and penetrance

Mendel's seven pea traits were unusually tidy: each was controlled by one gene with two alleles, one completely dominant, and the phenotypes were easy to classify. Most genes are messier. Heterozygotes may be intermediate or show both parental phenotypes; a gene may have dozens of alleles; some allele combinations kill the organism; and individuals with the same genotype may show a trait to different degrees, or not at all. None of these complications overturns Mendel's laws. Alleles still segregate and unlinked genes still assort independently. What changes is the relationship between genotype and phenotype, and therefore the ratios we observe.

Why dominance happens

Dominance is a property of the relationship between alleles at one gene, not of the alleles themselves, and it is easiest to understand at the level of the protein each allele encodes. Many recessive alleles are loss-of-function alleles: they produce a protein that does not work, or no protein at all. If one working copy produces enough protein for a normal phenotype, the heterozygote looks normal and the loss-of-function allele is recessive. This is the case for most enzymes, because cells usually make more enzyme than they need; a gene for which one copy is enough is called haplosufficient.

When one copy is not enough, a loss-of-function allele can be dominant; such genes are haploinsufficient. Other dominant alleles gain a new or excessive activity, or produce a faulty protein that interferes with the normal one. These dominant-negative alleles are common in proteins that work as parts of larger assemblies, such as collagen: a single defective chain can spoil a whole triple helix, which is why some forms of osteogenesis imperfecta (brittle bone disease) are dominant.

Incomplete dominance

In incomplete dominance the heterozygote has a phenotype intermediate between the two homozygotes. The classic example is flower color in snapdragons and four-o'clocks: crossing red-flowered (C^R C^R) with white-flowered (C^W C^W) plants gives pink F1 plants. In the F2 the red, pink and white phenotypes appear in a 1 : 2 : 1 ratio, the same as the genotypic ratio, because every genotype has its own phenotype. Pink is not blending: the red and white alleles reappear unchanged in the F2. The pink color arises because one copy of the functional allele makes about half the red pigment of two copies.

In humans, familial hypercholesterolemia behaves in a similar way. Heterozygotes, about 1 person in 250 in many populations, have blood cholesterol roughly twice normal and an increased risk of early heart disease; homozygotes, who are rare, have cholesterol four or more times normal and may have heart attacks in childhood without treatment.

Codominance and the ABO blood groups

In codominance the heterozygote shows the phenotypes of both alleles at once, rather than an intermediate. The human ABO blood group gene provides the best-known example and also illustrates multiple alleles. The gene encodes an enzyme that adds a sugar to a molecule on the surface of red blood cells. The I^A allele adds N-acetylgalactosamine, producing the A antigen; the I^B allele adds galactose, producing the B antigen; the i allele encodes an inactive enzyme and adds nothing, leaving the unmodified H antigen (type O).

I^A and I^B are codominant with each other, because an I^A I^B person makes both antigens, and both are completely dominant to i. One gene with three alleles thus gives six genotypes and four phenotypes. With k alleles, the number of possible genotypes is k(k + 1)/2: k homozygotes plus k(k − 1)/2 heterozygotes.

Whether a relationship looks dominant, incompletely dominant or codominant can depend on the level at which we look. Sickle-cell disease is a good illustration. At the level of the whole person, the sickle allele (HbS) is recessive: heterozygotes, said to have sickle-cell trait, are usually healthy. At the level of red blood cells under low oxygen, a small fraction of heterozygote cells sickle, so the allele is incompletely dominant. At the level of the hemoglobin protein, heterozygotes make both normal and sickle hemoglobin, so the alleles are codominant. Dominance is a description of a phenotype, and phenotypes can be measured at many levels.

Multiple alleles and allelic series

A population can carry many alleles of a gene even though each diploid individual carries at most two. The human leukocyte antigen (HLA) genes, which help the immune system recognize foreign cells, are the most variable genes known; thousands of alleles have been cataloged at some HLA loci, which is why finding a well-matched unrelated bone marrow donor can be so difficult and why registries such as Canadian Blood Services' Stem Cell Registry and the US National Marrow Donor Program list millions of potential donors.

Alleles can also form a dominance series. In rabbits, the C gene, which controls pigment production, has four common alleles: full color (C), chinchilla (c^ch), Himalayan (c^h) and albino (c), with the dominance order C > c^ch > c^h > c. The Himalayan allele encodes a temperature-sensitive enzyme that works only in the cooler extremities, giving dark ears, nose, feet and tail, the same pattern seen in Siamese cats. This is a striking example of the environment shaping the phenotype a genotype produces.

Lethal alleles

Some alleles are so damaging that individuals homozygous for them die, often before birth. Lethal alleles change the ratios of surviving offspring in characteristic ways. The first lethal allele was found in mice in 1905 by Lucien Cuénot. Yellow mice crossed with each other always produced yellow and non-yellow (agouti) offspring in a ratio of 2 : 1, never true-breeding yellow mice. The explanation is that the yellow allele A^Y is dominant for coat color but recessive for lethality: A^Y A^Y embryos die early in development.

The same pattern appears in Manx cats, where the tailless allele is lethal when homozygous, and in many crop and livestock breeding programs. In humans, homozygosity for the achondroplasia allele is usually lethal in infancy, so a couple who both have achondroplasia face a 1/4 risk of a lethal homozygous pregnancy, a 1/2 chance of a child with achondroplasia and a 1/4 chance of a child of average stature. Lethal alleles that act late in life, such as the Huntington disease allele, do not distort ratios at birth because they act after reproduction may already have occurred.

Penetrance and expressivity

Even a clear single-gene trait may not appear in everyone who has the relevant genotype. Penetrance is the percentage of individuals with a given genotype who show the expected phenotype at all. Expressivity is the degree to which the phenotype is expressed in those who show it. A dominant allele with 80 percent penetrance produces the trait in 80 percent of carriers; the other 20 percent are said to be non-penetrant, and they can still pass the allele on, which can make a dominant trait appear to skip a generation.

Incomplete penetrance and variable expressivity arise because genes do not act alone. Other genes (modifier genes), the environment and chance events during development all influence the outcome. For genetic counseling, penetrance changes risk calculations: the chance that a child of a carrier of a dominant allele with penetrance P will be affected is 1/2 × P.

from fractions import Fraction

def risk_affected_child(penetrance):
    """Child of a heterozygote for a dominant allele with given penetrance."""
    return Fraction(1, 2) * penetrance

def carrier_given_unaffected(penetrance):
    """Bayes: a child of a heterozygous parent who is unaffected (fully adult).
    prior carrier 1/2; P(unaffected | carrier) = 1 - penetrance."""
    prior = Fraction(1, 2)
    joint_carrier = prior * (1 - penetrance)
    joint_non = (1 - prior) * 1
    return joint_carrier / (joint_carrier + joint_non)

for pen in (Fraction(1), Fraction(9, 10), Fraction(7, 10), Fraction(1, 2)):
    print(f"penetrance {float(pen):.0%}: child affected {float(risk_affected_child(pen)):.3f}; "
          f"unaffected adult child is still a carrier with P = {float(carrier_given_unaffected(pen)):.3f}")

penetrance 100%: child affected 0.500; unaffected adult child is still a carrier with P = 0.000
penetrance 90%: child affected 0.450; unaffected adult child is still a carrier with P = 0.091
penetrance 70%: child affected 0.350; unaffected adult child is still a carrier with P = 0.231
penetrance 50%: child affected 0.250; unaffected adult child is still a carrier with P = 0.333

The second column shows a useful consequence. With full penetrance, an unaffected adult child of an affected parent cannot carry the allele. With 70 percent penetrance, that person still has about a 13 percent chance of carrying it and passing it on.

Pleiotropy and the environment

A single gene often affects several apparently unrelated traits, a phenomenon called pleiotropy. The sickle allele, for example, causes anemia, pain crises, organ damage and increased susceptibility to infection, all traceable to one change in hemoglobin. Marfan syndrome, caused by mutations in the fibrillin-1 gene, affects the skeleton (long limbs), the eyes (dislocated lenses) and the aorta (risk of dangerous dilation), because fibrillin is part of connective tissue throughout the body. Pleiotropy explains why a single gene can be under selection for several reasons at once.

Finally, phenotype is always the outcome of genotype and environment together. People with phenylketonuria have a recessive defect in the enzyme that breaks down the amino acid phenylalanine. On a normal diet, phenylalanine builds up and causes severe intellectual disability. Detected by newborn screening and treated with a low-phenylalanine diet, people with PKU develop normally. The genotype has not changed; the environment has. The range of phenotypes a genotype can produce across environments is called its norm of reaction.