Quantitative traits and the response to selection

Many of the traits that matter most in evolution, from body size and flowering time to milk yield and disease risk, vary continuously and are influenced by many genes and by the environment. BIOL-201 introduced heritability and the breeder’s equation, R = h²S, which predicts how much a trait changes in one generation of selection. This lesson builds on that foundation. It looks at how heritability is estimated and what it does and does not mean, how selection on one trait changes others through genetic correlations, what long-term selection experiments reveal and how the polygenic architecture of traits, now visible through genome-wide studies, shapes their evolution.

Partitioning variation

The phenotypic variance of a trait, V_P, can be divided into a genetic part, V_G, and an environmental part, V_E. The genetic part is further split into additive variance, V_A, which comes from the average effects of alleles and makes offspring resemble their parents; dominance variance, V_D, from interactions between alleles at the same locus; and interaction (epistatic) variance, V_I, among loci. Broad-sense heritability is H² = V_G/V_P; narrow-sense heritability, h² = V_A/V_P, is the part that determines the response to selection, because only additive effects are reliably passed from parents to offspring.

Heritability is often misunderstood. It is a property of a population in a particular environment, not of a trait in general or of an individual. A heritability of 0.8 for height does not mean that 80 percent of a person’s height is due to genes. It means that, in that population and setting, 80 percent of the variation among people is due to additive genetic differences. Heritability can change if the environment changes; in a population where some people are malnourished, environmental variance increases and heritability falls. Crucially, high heritability within groups says nothing about the causes of average differences between groups, which may be entirely environmental.

Estimating heritability

The simplest estimate comes from the resemblance of offspring to their parents. If offspring trait values are regressed on the average of their two parents (the midparent), the slope estimates h². Regressing on a single parent gives h²/2. Twin and family studies in humans, animal pedigrees analyzed with mixed models (the animal model), and, increasingly, genome data from unrelated individuals all provide estimates.

import numpy as np
rng = np.random.default_rng(321)
# Simulate a trait with additive genetic variance VA = 0.6 and environmental variance VE = 0.4
n = 3000; VA, VE = 0.6, 0.4
a_mother = rng.normal(0, np.sqrt(VA), n); a_father = rng.normal(0, np.sqrt(VA), n)
z_mother = a_mother + rng.normal(0, np.sqrt(VE), n); z_father = a_father + rng.normal(0, np.sqrt(VE), n)
# Offspring breeding value: midparent plus Mendelian sampling (variance VA/2)
a_off = (a_mother + a_father)/2 + rng.normal(0, np.sqrt(VA/2), n)
z_off = a_off + rng.normal(0, np.sqrt(VE), n)

mid = (z_mother + z_father)/2
slope_mid = np.polyfit(mid, z_off, 1)[0]
slope_one = np.polyfit(z_mother, z_off, 1)[0]
print(f"true h2 = {VA/(VA+VE):.2f}")
print(f"offspring on midparent slope = {slope_mid:.2f} (estimates h2)")
print(f"offspring on one parent slope = {slope_one:.2f} (estimates h2/2 = {VA/(VA+VE)/2:.2f})")

true h2 = 0.60
offspring on midparent slope = 0.60 (estimates h2)
offspring on one parent slope = 0.27 (estimates h2/2 = 0.30)

Parent–offspring regression works because it relies only on additive genetic resemblance. In real data, shared environments, such as diet, culture or maternal effects, can inflate the estimate, which is why careful studies use cross-fostering in animals or compare relatives raised apart or with different degrees of relatedness.

The response to selection

The breeder’s equation predicts the response to one generation of selection. If parents chosen for breeding have a mean that differs from the population mean by the selection differential S, the offspring mean shifts by R = h²S. It is often more convenient to express selection as intensity, i, the selection differential in standard deviation units, which depends only on the proportion selected. Then R = i h² σ_P, and per year, dividing by the generation interval, L, gives the annual genetic gain: ΔG = i h² σ_P / L. This is the equation animal and plant breeders use to decide how many animals to select and how fast to turn over generations.

Selection in nature works the same way, although S is created by differential survival and reproduction rather than by a breeder. After the 1977 drought on Daphne Major, the Grants measured a selection differential on beak depth of about 0.5 mm and a heritability of about 0.8, predicting a response of about 0.4 mm in the next generation. The observed response was close to this, one of the first direct confirmations of quantitative genetic theory in the wild.

Genetic correlations and correlated responses

Genes often affect several traits at once (pleiotropy), and alleles at nearby loci may be inherited together. As a result, traits can be genetically correlated, and selection on one changes others. The additive genetic variances and covariances of a set of traits are summarized in the G matrix. Lande generalized the breeder’s equation to many traits: the response vector is Δz̄ = Gβ, where β is the vector of selection gradients from lesson 2. A trait can evolve in the opposite direction to its own selection if it is strongly correlated with another trait under stronger selection.

import numpy as np
# Two traits: wing length and body mass, both standardized; G matrix (additive variances and covariance)
G = np.array([[0.40, 0.30],
              [0.30, 0.50]])
beta = np.array([0.20, -0.10])     # selection prefers longer wings but lighter bodies
dz = G @ beta
print("selection gradients:", beta)
print("predicted change per generation (SD units):", np.round(dz, 3))
r = G[0, 1] / np.sqrt(G[0, 0]*G[1, 1])
print(f"genetic correlation = {r:.2f}")
# Which combination of traits evolves most easily? The leading eigenvector of G (g_max)
vals, vecs = np.linalg.eigh(G)
print("g_max direction:", np.round(vecs[:, -1], 3), f"explains {100*vals[-1]/vals.sum():.0f}% of genetic variance")

selection gradients: [ 0.2 -0.1]
predicted change per generation (SD units): [0.05 0.01]
genetic correlation = 0.67
g_max direction: [0.646 0.763] explains 84% of genetic variance

Although selection favors lighter bodies, body mass increases because it is positively correlated with wing length, which is under stronger selection. The direction in trait space with the most genetic variance, g_max, is the line of least resistance: populations tend to evolve along it, at least in the short term, because selection in that direction finds the most variation to work with. Dolph Schluter showed in 1996 that divergence among related species of sticklebacks, finches and other animals often follows g_max, a way in which genetic architecture constrains and channels evolution.

Long-term selection

How far can selection push a trait? The Illinois long-term selection experiment on maize began in 1896 and is the longest-running selection experiment in plants. Lines selected for high and low kernel oil and protein content have diverged dramatically over more than 120 generations: oil in the high line rose from about 5 percent to over 20 percent, while the low line fell close to zero, which is a physiological floor. Remarkably, the high line has continued to respond long after the original variation might have been expected to run out. Similar continued responses are seen in mice selected for body size or wheel running, and in poultry and dairy cattle.

import numpy as np
rng = np.random.default_rng(3215)
# Polygenic trait: 200 additive loci with small effects; truncation selection of the top 20%
L, N, gens = 200, 500, 100
effects = rng.exponential(0.1, L)
G = rng.binomial(2, 0.5, (N, L))                       # genotypes (0, 1, 2 copies of the '+' allele)
sd_e = (G @ effects).std()                             # environmental SD set so initial h2 is about 0.5
means, h2s = [], []
for t in range(gens):
    g = G @ effects
    z = g + rng.normal(0, sd_e, N)
    means.append(z.mean()); h2s.append(g.var()/z.var())
    parents = np.argsort(z)[-N//5:]                     # top 20%
    mums, dads = rng.choice(parents, N), rng.choice(parents, N)
    G = rng.binomial(1, G[mums]/2) + rng.binomial(1, G[dads]/2)   # each parent passes one allele per locus
    G = G + (rng.random((N, L)) < 1e-4) * (G < 2) * 1   # rare new '+' mutations
for t in (0, 10, 25, 50, 99):
    print(f"gen {t:3d}: mean {means[t]-means[0]:6.2f} above start, h2 = {h2s[t]:.2f}")

gen   0: mean   0.00 above start, h2 = 0.46
gen  10: mean  11.07 above start, h2 = 0.36
gen  25: mean  17.31 above start, h2 = 0.09
gen  50: mean  19.95 above start, h2 = 0.02
gen  99: mean  20.64 above start, h2 = 0.00

In the simulation, the trait mean rises rapidly at first and then more slowly as favorable alleles approach fixation, and heritability falls toward zero within about 50 generations; the rare new mutations in this small model add too little variation to keep the response going. In real long-term experiments, the response often lasts longer than simple models predict, because traits are influenced by very many loci, new mutations add variation each generation and some alleles have effects that only become visible once others are fixed. Eventually, responses usually slow or stop, sometimes because of opposing natural selection: chickens selected for very rapid growth develop leg and heart problems, and mice selected for large size become less fertile.

Polygenic architecture

Genome-wide association studies (GWAS), which scan hundreds of thousands to millions of genetic variants in large samples, have revealed that most complex traits are extremely polygenic. Height in humans is influenced by thousands of variants, each with a tiny effect. A 2022 study of about 5.4 million people identified over 12,000 independent variants associated with height, which together explained around 40 to 45 percent of variation in people of European ancestry but considerably less in other groups, largely because most study participants have been of European ancestry. Some researchers have even proposed an omnigenic model, in which nearly every gene expressed in relevant tissues has some small effect on a trait through interconnected regulatory networks.

Polygenic architecture changes how adaptation proceeds. When a trait is influenced by many loci of small effect, a shift in the optimum can be reached by small frequency changes at many loci rather than a sweep of one allele to fixation. Such polygenic adaptation leaves subtle signals in the genome. Early claims of strong polygenic selection on height across European populations were later found to be partly caused by subtle population structure in the GWAS data, a cautionary example showing how difficult these analyses are. Polygenic scores, which add up the estimated effects of many variants for an individual, are used in research and are being explored in medicine, but their predictive power is modest for most traits and is lower in groups underrepresented in the original studies.