Quantitative and population genetics

So far most of the traits in this course have fallen into clear categories: round or wrinkled, affected or unaffected, blood type A or B. Many of the traits that matter most, such as height, crop yield, milk production, blood pressure and the risk of common diseases, vary continuously. They are quantitative traits, influenced by many genes and by the environment. This lesson shows how Mendelian inheritance at many genes produces continuous variation, how heritability is defined and estimated, and how breeders predict the response to selection. It then extends the population genetics introduced in BIOL-108 to selection against harmful alleles, the balance between mutation and selection, and the effects of inbreeding. BIOL-322 develops population genetics in much more depth.

From discrete to continuous variation

In the early 1900s there was a bitter dispute between Mendelians, who studied discrete traits, and biometricians, who measured continuous traits such as height and doubted that Mendel's particles could explain them. The dispute was resolved by experiments such as those of the Swedish geneticist Herman Nilsson-Ehle on wheat kernel colour (1909) and by Ronald Fisher's mathematical demonstration in 1918 that many Mendelian genes, each with a small effect, would produce continuous, bell-shaped variation. This is the polygenic, or multiple-factor, model.

Nilsson-Ehle crossed wheat with dark red kernels and wheat with white kernels. The F1 were intermediate, and the F2 showed a range of shades from dark red to white, with only 1 in 64 as dark as the red parent and 1 in 64 as white as the white parent. He explained this with three genes, each with an additive allele that contributes one unit of colour and a non-additive allele that contributes none. The number of additive alleles an F2 plant carries ranges from 0 to 6, and their distribution follows the binomial coefficients 1 : 6 : 15 : 20 : 15 : 6 : 1.

from math import comb
import numpy as np
rng = np.random.default_rng(201)

def f2_distribution(genes):
    """Number of additive alleles in an F2 from AaBb... x AaBb...: binomial with 2n trials."""
    n = 2 * genes
    return [comb(n, k) for k in range(n + 1)], 4 ** genes

for g in (1, 2, 3):
    counts, total = f2_distribution(g)
    print(f"{g} gene(s): classes {' : '.join(map(str, counts))} (out of {total})")

# With many genes plus environmental noise, the distribution becomes smooth and bell-shaped
for g in (1, 3, 10, 40):
    alleles = rng.binomial(2 * g, 0.5, size=20_000)
    phenotype = alleles / (2 * g) * 100 + rng.normal(0, 6, size=20_000)
    hist, _ = np.histogram(phenotype, bins=10, range=(-20, 120))
    print(f"{g:2d} genes + environment: " + " ".join(f"{h:5d}" for h in hist))

1 gene(s): classes 1 : 2 : 1 (out of 4)
2 gene(s): classes 1 : 4 : 6 : 4 : 1 (out of 16)
3 gene(s): classes 1 : 6 : 15 : 20 : 15 : 6 : 1 (out of 64)
 1 genes + environment:   819  3776   440   111  4889  4897   104   469  3714   775
 3 genes + environment:    56   364  1462  3405  4599  4718  3435  1544   369    48
10 genes + environment:     0     8   280  2473  7253  7269  2455   255     7     0
40 genes + environment:     0     0     7   888  9226  8988   886     5     0     0

The general rule is that with n genes of equal additive effect, an F2 has 2n + 1 phenotypic classes and the fraction resembling either extreme parent is (1/4)ⁿ. If 1/256 of an F2 is as extreme as one parent, four genes are involved. Add environmental variation, which blurs the boundaries between classes, and the distribution becomes continuous.

Partitioning variation

Quantitative genetics describes variation with variances. The total phenotypic variance in a population, written V_P, can be divided into variance due to genetic differences (V_G) and variance due to environmental differences (V_E):

Genetic variance itself has components. Additive variance \(V_A\) comes from the average effects of alleles, which add up across loci. Dominance variance \(V_D\) comes from interactions between alleles at the same locus, and interaction variance \(V_I\) from interactions between loci (epistasis, lesson 5). The distinction matters because parents pass on alleles, not genotypes: a child receives one allele from each parent, so only the additive part of a parent's genetic value is reliably transmitted.

Heritability

Heritability measures how much of the variation in a trait, in a particular population and environment, is due to genetic differences. There are two versions:

Heritability is one of the most misunderstood ideas in biology. It describes variation in a population, not individuals: a heritability of 0.8 for height does not mean 80 per cent of a person's height is due to genes. It applies to a particular population in a particular environment and can change if either changes. And high heritability does not mean a trait cannot be changed by the environment. Height has a heritability of about 0.8 in well-nourished populations, yet average height rose by roughly 10 cm over the twentieth century in many countries because of better nutrition and health. The differences among individuals are largely genetic; the change in the average was environmental.

Estimating heritability

Heritability is estimated from the resemblance between relatives. The simplest method regresses offspring phenotypes on the average of their two parents (the midparent value). The slope of that regression estimates h². Twin studies compare identical (monozygotic) twins, who share all their genes, with fraternal (dizygotic) twins, who share half on average; the extra resemblance of identical twins indicates genetic influence. Today, genome-wide data on hundreds of thousands of people allow heritability to be estimated directly from DNA similarity among unrelated individuals.

import numpy as np
rng = np.random.default_rng(201)
# Simulate a trait with additive genes and environment, then estimate h2 from midparent regression
n_fam, loci = 2000, 100
h2_true = 0.6
def genotype(n):            # number of '+' alleles at each locus, frequency 0.5
    return rng.binomial(1, 0.5, size=(n, loci, 2))
mothers, fathers = genotype(n_fam), genotype(n_fam)
def breeding_value(g): return g.sum(axis=(1, 2)).astype(float)
va = breeding_value(mothers).var()
ve = va * (1 - h2_true) / h2_true
def phenotype(g): return breeding_value(g) + rng.normal(0, np.sqrt(ve), size=len(g))
# each child gets one random allele per locus from each parent
pick = lambda g: np.take_along_axis(g, rng.integers(0, 2, size=(n_fam, loci, 1)), axis=2)
child = np.concatenate([pick(mothers), pick(fathers)], axis=2)
mid = (phenotype(mothers) + phenotype(fathers)) / 2
kid = phenotype(child)
slope = np.polyfit(mid, kid, 1)[0]
print(f"true narrow-sense heritability: {h2_true}")
print(f"estimated from offspring-midparent regression slope: {slope:.2f}")

true narrow-sense heritability: 0.6
estimated from offspring-midparent regression slope: 0.62

Response to selection: the breeder's equation

Breeders select the best individuals as parents, hoping to improve the next generation. How much improvement can they expect? The answer is the breeder's equation:

The selection differential \(S\) is how much the selected parents exceed the population mean; the response \(R\) is how much the offspring mean exceeds the original mean. Only the heritable share of the parents' advantage is passed on, so when \(h^2\) is low, even strong selection produces little response.

Selection experiments show that most traits respond to selection, which demonstrates that they have additive genetic variance. The Illinois corn experiment, now running for more than 125 generations, has increased kernel oil content from about 5 per cent to over 20 per cent in the high line and reduced it to near zero in the low line. Dairy breeding provides another example: genomic selection, which predicts an animal's breeding value from DNA markers spread across the genome, has roughly doubled the rate of genetic improvement in milk yield in North American Holstein cattle since it was adopted around 2009.

Quantitative trait loci and polygenic scores

A quantitative trait locus (QTL) is a region of the genome containing one or more genes that affect a quantitative trait. QTLs are found by crossing lines that differ in a trait and testing which genetic markers (lesson 8) are associated with the trait in the offspring. In humans, genome-wide association studies (GWAS) test millions of variants in hundreds of thousands of people. Height, for example, has been linked to more than 12,000 variants, each with a tiny effect, which together explain a large share of its heritability. Combining these effects gives a polygenic score, an estimate of an individual's genetic predisposition. Polygenic scores are useful in research and are being tested in medicine, but they predict poorly in populations that differ in ancestry from the study participants, most of whom have been of European ancestry, raising important questions of equity. BIOL-410 returns to these methods.

Selection at a single locus

Population genetics asks how allele frequencies change. Consider selection against a recessive allele a with frequency q. If aa individuals do not reproduce at all (selection coefficient s = 1), the allele frequency in the next generation is

The second formula gives the frequency after t generations. Selection against a rare recessive is extremely slow, because almost all copies of a rare recessive allele are hidden in heterozygotes, where selection cannot see them. Reducing q from 0.01 to 0.005 takes 100 generations, about 2,500 years in humans. This is why proposals to eliminate recessive conditions by preventing affected people from reproducing, a goal of the eugenics movement (lesson 1), were biologically futile as well as unethical.

def generations_to_reach(q0, q_target, s=1.0):
    """Selection against a recessive (fitness of aa = 1 - s); iterate the exact recursion."""
    q, t = q0, 0
    while q > q_target:
        p = 1 - q
        mean_w = 1 - s * q * q
        q = (p * q + q * q * (1 - s)) / mean_w
        t += 1
    return t

for q0, qt in [(0.5, 0.25), (0.1, 0.05), (0.01, 0.005), (0.01, 0.001)]:
    t = generations_to_reach(q0, qt)
    print(f"q from {q0} to {qt}: {t:5d} generations (about {25 * t:,} years at 25 years per generation)")

q from 0.5 to 0.25:     3 generations (about 75 years at 25 years per generation)
q from 0.1 to 0.05:    11 generations (about 275 years at 25 years per generation)
q from 0.01 to 0.005:   100 generations (about 2,500 years at 25 years per generation)
q from 0.01 to 0.001:   901 generations (about 22,525 years at 25 years per generation)

Mutation–selection balance

If harmful alleles are constantly removed by selection, why do they persist? Because new mutations keep creating them. An equilibrium is reached when the rate at which mutation creates harmful alleles equals the rate at which selection removes them. For a recessive allele with mutation rate μ and selection coefficient s against homozygotes, the equilibrium frequency is approximately

For a lethal recessive (s = 1) with a mutation rate of \(10^{-5}\) per generation, \(\hat{q} \approx \sqrt{10^{-5}} \approx 0.003\). For a dominant harmful allele the equilibrium is much lower, about \(\mu / s\), because heterozygotes are exposed to selection. Some recessive disease alleles are much more common than mutation–selection balance predicts. The sickle allele, for example, reaches frequencies of 10 to 20 per cent in parts of Africa because heterozygotes are protected against severe malaria, an example of heterozygote advantage that maintains the allele by balancing selection.

Inbreeding

Inbreeding, mating between relatives, increases homozygosity without directly changing allele frequencies. Its effect is measured by the inbreeding coefficient F, the probability that the two alleles at a locus in an individual are identical by descent, meaning both copies came from the same ancestral allele. The child of first cousins has F = 1/16. Inbreeding raises the frequency of homozygous recessive genotypes from q² to q² + Fpq, which for rare alleles is a large relative increase.

Inbreeding depression, the reduced survival and fertility of inbred offspring, is a major concern in conservation, where small populations cannot avoid mating between relatives, and in animal breeding, where popular sires can father thousands of offspring. In many cultures marriage between cousins is common, and genetic counselling and carrier screening can help such couples understand and manage their risks.