Quantitative Genetics, Heritability & Inbreeding

Overview

Every trait covered so far has been discrete — round or wrinkled, purple or white, affected or unaffected. Many real traits (height, skin color, crop yield, milk production) instead vary continuously across a population, because they are controlled by many genes of small individual effect rather than one or two genes of large effect. This page covers how such polygenic traits are analyzed genetically, how heritability partitions observed phenotypic variation into genetic and environmental sources, and — a related but distinct topic — how non-random mating, specifically inbreeding, changes the distribution of genotypes in a population without changing allele frequencies at all. This last topic deliberately does not re-derive the Hardy-Weinberg framework itself, covered fully in Evolution’s Population Genetics & Hardy-Weinberg Equilibrium — this page covers only the one HWE assumption (random mating) that classical population genetics treats as a simplifying assumption rather than analyzing in depth.

Key Concepts

Polygenic inheritance and continuous variation

A polygenic trait is controlled by multiple genes, each typically contributing a small, roughly additive effect to the phenotype — as the number of contributing genes increases, the number of possible genotypic classes increases, and their phenotypic distribution increasingly approximates a continuous normal (bell-curve) distribution rather than the small number of discrete classes seen in simple Mendelian crosses. A classic illustrative model: for a trait controlled by n additive gene pairs, a cross between two individuals heterozygous at all n loci produces 2n+1 phenotypic classes in a binomially-distributed ratio — with even a modest number of genes (3-4), the extreme classes become rare enough, and the intermediate classes numerous enough, that the population looks continuously distributed rather than showing distinct classes.

Source: mysciencesquad.weebly.com

Heritability: partitioning phenotypic variance

Heritability asks what fraction of the observed variation in a trait across a population is attributable to genetic differences, as opposed to environmental differences — a population-level statistic, not a property of any individual’s phenotype. Total phenotypic variance (V_P) is partitioned as:

$$ V_P = V_G + V_E $$

where V_G is genetic variance and V_E is environmental variance (interaction terms are set aside at this level of treatment). Broad-sense heritability (H²) is the proportion of phenotypic variance attributable to all genetic variance:

$$ H^2 = \frac{V_G}{V_P} $$

Narrow-sense heritability (h²), more useful for predicting a trait’s response to selective breeding, further partitions genetic variance into additive genetic variance (V_A — the component that breeds true and predicts offspring resemblance to parents) versus dominance and epistatic interaction variance (which do not transmit as reliably to offspring):

Twin studies are the standard method for estimating heritability in humans, comparing trait concordance between monozygotic twins (identical genotype) and dizygotic twins (share on average 50% of segregating alleles, like ordinary siblings) raised in comparably shared environments — a substantially higher concordance in monozygotic pairs is taken as evidence of a genetic contribution to the trait.

Source: slideshare.net

A critical caveat: heritability is population- and environment-specific

A heritability estimate is not a fixed biological constant for a trait — it depends entirely on the specific population and environmental range in which it was measured. In a population raised in a highly uniform environment, V_E shrinks toward zero and heritability rises toward 1, even for a trait that is substantially environmentally malleable in a more variable environment — heritability describes the current balance of variance sources in a specific study population, not an intrinsic, universal “how genetic is this trait” number. This is among the most commonly misapplied statistics in genetics and is worth stating explicitly as a caveat on any heritability claim.

QTL mapping

Quantitative trait locus (QTL) mapping identifies the approximate chromosomal locations of the individual genes contributing to a polygenic trait, by statistically associating trait values with genetic marker genotypes across a segregating population (e.g. an F2 population from a cross between two strains differing in the trait) — conceptually, this extends the linkage-mapping logic from Linkage, Recombination & Genetic Mapping to a continuously varying trait rather than a discrete phenotype, by looking for markers whose genotype correlates statistically with trait value rather than co-segregating with a single discrete phenotype class.

Non-random mating: inbreeding

The Hardy-Weinberg model assumes random mating; inbreeding — mating between relatives more often than random-mating expectation predicts — is the most genetically consequential violation of this assumption. Critically, inbreeding by itself does not change allele frequencies in a population; it changes the distribution of genotypes, systematically increasing the proportion of homozygotes and decreasing the proportion of heterozygotes relative to Hardy-Weinberg expectation, at every locus simultaneously (not just loci under selection).

The inbreeding coefficient (F) quantifies the probability that an individual’s two alleles at a given locus are identical by descent — both copies of the same specific ancestral allele, inherited via two different paths through a pedigree that eventually converge on a common ancestor. F is calculated from a pedigree by summing, over every path connecting the two parents through each common ancestor:

$$ F = \sum \left(\frac{1}{2}\right)^{n} (1 + F_A) $$

where n is the number of individuals in each specific path linking the two parents through one common ancestor (inclusive of both parents and the ancestor), and F_A is the inbreeding coefficient of that common ancestor itself (often approximated as 0 if the ancestor’s own ancestry is not inbred). For a full-sibling mating, F = 1/4; for a first-cousin mating, F = 1/16.

Source: ihh.kvl.dk

Inbreeding’s genotype-frequency consequence at a single locus: starting from allele frequencies p and q, the expected genotype frequencies become p² + Fpq (homozygote AA), 2pq(1−F) (heterozygote Aa), and q² + Fpq (homozygote aa) — visibly reducing to the standard Hardy-Weinberg p²:2pq:q² when F = 0.

Source: bioprinciples.biosci.gatech.edu

Inbreeding depression

Inbreeding depression is the reduced fitness (survival, fertility, growth) commonly observed in inbred offspring, arising because increased homozygosity raises the chance that rare, deleterious recessive alleles — normally masked in the heterozygous state in an outbred population — are exposed in the homozygous state. This is the practical, fitness-level consequence of the purely statistical genotype-frequency shift described above, and the reason natural and captive-breeding-program management alike actively work to avoid close inbreeding in small populations.

Assortative mating

Positive assortative mating (mating between phenotypically similar individuals more often than random) and negative (disassortative) mating (mating between dissimilar individuals) are non-random mating patterns based on phenotype similarity rather than relatedness — a distinct mechanism from inbreeding, worth not conflating on an exam. Positive assortative mating for a specific trait increases homozygosity at the loci controlling that trait specifically (similar to inbreeding’s effect, but locus-specific rather than genome-wide), without necessarily affecting other, unrelated loci the way true inbreeding (relatedness-based) does across the entire genome simultaneously.

Comparative Structures

Common Exam Questions

• “Inbreeding changes allele frequencies” is a frequent misstatement to correct explicitly — inbreeding changes genotype frequencies (more homozygotes, fewer heterozygotes) while leaving allele frequencies exactly unchanged; only processes like selection, drift, mutation, and gene flow change allele frequencies (see Genetic Drift, Gene Flow & Mutation).
• “A heritability of 0.8 means the trait is 80% genetically determined in every individual” is a near-universal misinterpretation to correct directly — heritability is a population-level statistic about the proportion of variance explained by genetic differences in one specific population/environment, not a per-individual “percent genetic” figure, and it can differ substantially between populations or environments for the identical trait.
• Distinguishing broad-sense (H², all genetic variance) from narrow-sense (h², additive variance only) heritability is a common conceptual question, particularly in the context of predicting a trait’s response to artificial selection, which depends specifically on the additive component.
• Calculating F from a simple pedigree (full-sib = 1/4, half-sib = 1/8, first-cousin = 1/16) is a frequent numeric question — always count every distinct path through every common ancestor, since a pedigree with more than one common ancestor requires summing multiple path contributions.

Visual Reference

Interactive

• A polygenic trait distribution simulator: adjust the number of contributing gene pairs and watch the F2 phenotype distribution shift from a few discrete classes toward a smooth, continuous bell curve.

(Static images are placed inline in Key Concepts above, next to the concept each one illustrates, rather than collected here. One item is only partially fulfilled: the Hardy-Weinberg-vs-inbred genotype-frequency comparison shows the HWE baseline only, not the inbred-population contrast — see the inline note above.)

Practice Problems

1. Two individuals are the offspring of a mating between full siblings. Using the inbreeding coefficient formula, confirm that F = 1/4 for full-sibling mating, and explain in words what this coefficient represents.

2. A dog breeder measures shoulder height in two different kennel populations of the same breed. In Population 1 (raised under highly standardized diet and exercise conditions), the estimated heritability of shoulder height is 0.85. In Population 2 (raised under widely varying diet and exercise conditions), the estimated heritability for the same trait is 0.40. Explain how the same trait, in the same breed, can show such different heritability estimates.

3. A conservation program managing a small captive population of an endangered species notices declining litter sizes and increased juvenile mortality across several generations, despite no new mutations or disease outbreaks being detected. Propose the most likely explanation, and describe one breeding-program strategy that would directly address it.