Quantitative and complex traits: the polygenic synthesis, heritability, and gene-environment interaction

Short version. A quantitative trait is one whose phenotype varies continuously across a population (height, blood pressure, BMI). A complex trait is one whose presence-or-absence phenotype reflects an underlying continuous liability (schizophrenia, type 2 diabetes, cleft lip). Both are explained by the same polygenic synthesis: many small genetic effects, plus environment, summed under a roughly Gaussian distribution. The framework is for research and education; outputs computed from it on this platform are illustrative, not clinical.

Mendelism, biometrics, and Fisher's reconciliation

By 1900, two empirical traditions in biology contradicted one another. Mendelians, working with discrete phenotypes (yellow vs green peas, smooth vs wrinkled), described inheritance as the segregation of factors at definite ratios. Biometricians, following Galton's regression-to-the-mean and Pearson's correlation studies, described continuous traits (human stature) as smoothly inherited with a parent-offspring correlation around 0.5 and a sib-sib correlation also around 0.5. Each tradition treated the other's traits as exceptional.

Ronald Fisher's Transactions of the Royal Society of Edinburgh 52:399 paper of 1918, "The Correlation between Relatives on the Supposition of Mendelian Inheritance", is the reconciliation. Fisher showed mathematically that if a continuous phenotype is the sum of contributions from many independent Mendelian loci, each of small effect, then the population distribution converges on a Gaussian and the family correlations Galton observed fall out exactly — parent-offspring 0.5, full-sib 0.5, half-sib 0.25, monozygotic-twin 1.0 in the limit of additive genetic variance. The biometrical and Mendelian schools were describing the same biology at different limits of locus number and effect size. This is the polygenic synthesis, and the model has been the foundation of quantitative and complex-trait genetics ever since.

The detailed mathematical treatment is in Lynch & Walsh's textbook Genetics and Analysis of Quantitative Traits (Sinauer, 1998), still the authoritative reference for the population genetics underlying everything in this pillar. The book separates the additive genetic variance V_A (the contribution of allelic substitutions assuming no dominance or epistasis) from the dominance variance V_D, the epistatic variance V_I, and the environmental variance V_E, each of which has a defined contribution to the observed phenotypic variance V_P.

Three subtopics, three pages

This pillar splits into three thematic subtopics; each has its own page on this site.

Polygenic models — from infinitesimal model to PRS

Fisher's 1918 framework treats a quantitative phenotype as the sum of small contributions from a large (in the limit, infinite) number of additively acting loci, each with allele frequencies and effect sizes drawn from a population distribution. The model is called the infinitesimal model because every locus has, asymptotically, an infinitesimal effect, and the distribution of phenotypes is Gaussian by the central limit theorem. Two summary statistics dominate: the additive effect a at each locus (the half-difference between homozygote phenotypes) and the dominance deviation d (the deviation of the heterozygote from the midpoint between homozygotes). When d = 0 across loci, the model is purely additive; when d ≠ 0, dominance variance contributes.

Modern polygenic risk scores (PRS) are the operational descendants of the infinitesimal model. A PRS is the sum across many SNPs of the dose of each risk allele weighted by its effect size, where effect sizes come from a genome-wide association study. The construction is reasonably standardised — clumping-and-thresholding (P+T), LDpred, and PRS-CS are the dominant methods, and the PGS Catalog (Lambert et al. 2021, Nat Genet 53:420) curates published scores for reuse. Khera et al. 2018 (Nat Genet 50:1219) is the canonical demonstration that a coronary-artery-disease PRS can stratify population risk at scale; Martin et al. 2019 (Nat Genet 51:584) is the canonical demonstration that PRS performance degrades sharply when transferred between ancestry groups, the so-called portability problem. The PRS subtopic page covers both in detail. None of this is recommended as a basis for individual clinical decisions on Evagene; the framing on this platform is research and education.

Heritability and the liability-threshold model

Heritability h² is the proportion of phenotypic variance attributable to additive genetic variance (V_A / V_P, the narrow-sense heritability). The broad-sense heritability H² includes dominance and epistatic variance (V_G / V_P). Estimation is the empirical core of quantitative genetics. The classical method is the twin study: Falconer's formula h² = 2(r_MZ − r_DZ) extracts heritability from monozygotic and dizygotic twin correlations under the equal-environments assumption. Mayhew & Meyre 2017 review the modern landscape of human-genetics heritability methods.

For SNP-heritability, Yang et al. 2010 introduced GREML (genomic-relatedness-matrix REML), a method to estimate h² from genotyped SNPs in unrelated individuals. Bulik-Sullivan et al. 2015 introduced LD-score regression, which estimates h² directly from GWAS summary statistics. Both methods recovered substantial heritability for height and other complex traits but consistently fell short of twin-study estimates — the missing heritability problem framed by Manolio et al. 2009 (Nature 461:747).

For binary disease phenotypes, heritability is defined on an underlying continuous liability — the abstract trait that determines who crosses a threshold and is affected. The liability-threshold model is set out in Falconer 1965 (Ann Hum Genet 29:51): the population's liability follows a normal distribution, the prevalence K is the area in the right tail, and the recurrence risk to a relative is the integral of the conditional liability distribution above the same threshold given the relative's genotype. Smith / Carter / Harper empirical recurrence-risk tables, used by Evagene's complex-disease engine and described on complex-disease-pedigree-software, are the family-study realisation of this idea. The closely related multifactorial-threshold framework in Carter 1961 (Br Med Bull 17:251) extended the model with sex-differential thresholds — the well-known Carter effect, observed in pyloric stenosis and in cleft lip. Detail is on the heritability subtopic page.

Gene-environment interaction

A pure polygenic decomposition assumes genetic and environmental effects act additively, with no interaction (V_P = V_G + V_E). In reality, many traits show gene-environment (GxE) interaction: the genetic effect depends on environmental exposure, or vice versa. Phenylketonuria is the canonical illustration — the same homozygous PAH genotype has near-normal cognitive outcomes on a low-phenylalanine diet and severe impairment on an unrestricted diet. The G6PD-favism literature, FTO-physical-activity studies, and CYP2A6-smoking interactions are common contemporary examples.

Statistical detection of GxE is hard at scale. Genome-wide interaction studies (GWIS) require very large samples; Hunter 2005 set out the design issues. Mendelian randomisation, reviewed in Davey Smith & Hemani 2014, exploits genetic variants as instrumental variables to test environmental causality. Epigenetic mediation — DNA methylation as a G-E interface — is illustrated by the famous Heijmans et al. 2008 Dutch Hunger Winter cohort, which documented persistent IGF2 hypomethylation six decades after periconceptional famine. The GxE subtopic page covers all of these.

Polygenic vs Mendelian inheritance — how to tell them apart

For families, the distinction between Mendelian and polygenic inheritance is operational. A Mendelian disorder shows clear segregation ratios at single-locus expectations — ~50% of offspring of an affected autosomal-dominant heterozygote, ~25% of offspring of two heterozygous AR carriers, sex-restricted patterns for X-linked, etc. A polygenic / multifactorial condition shows familial clustering without those ratios: recurrence risks rise with the number of affected relatives, with the severity of the proband, with proband sex when one sex is more frequently affected (Carter effect), and with consanguinity, but no relative has a 50% risk and unaffected parents commonly produce affected offspring without obligate-carrier interpretation.

Evagene supports both kinds of inheritance modelling under explicitly research- and education-grade framing. The Mendelian inheritance calculator page covers single-locus models. The complex-disease pedigree software page covers the liability-threshold engine and the 20+ catalogued multifactorial conditions, with empirical Smith / Carter / Harper tables where available and a Falconer-1965 fallback when no table exists. The two engines are complementary, not interchangeable: forcing a Mendelian model onto a polygenic disease (or vice versa) is the most common modelling error in introductory teaching, and a cleanly drawn pedigree typically suggests which framework fits.

Why this matters for pedigree work

A research- or teaching-grade pedigree records the structure that lets a learner or a researcher distinguish between these models. Three generations from the proband, sex and affected status of every relative, age at onset, severity grade where the disease is severity-graded (cleft lip, congenital heart defects), and consanguinity all feed into the decision between a Mendelian segregation analysis and a Falconer / Carter liability-threshold analysis. A pedigree that captures only the proband loses the information that drives the decision; a pedigree that captures three full generations is usually enough.

Evagene's role here is research and education. The platform draws the pedigree, runs the chosen risk model, exports the output as a structured-English summary, and links each output to the canonical paper. It does not recommend actions, does not address output to named individuals as medical information, and is not a substitute for professional clinical judgement.

Canonical references for this pillar

• Fisher RA. 1918. The Correlation between Relatives on the Supposition of Mendelian Inheritance. Trans R Soc Edinb 52:399–433. The polygenic synthesis.
• Falconer DS. 1965. The inheritance of liability to certain diseases, estimated from the incidence among relatives. Ann Hum Genet 29:51–76. Liability-threshold model.
• Carter CO. 1961. The inheritance of congenital pyloric stenosis. Br Med Bull 17:251–254. Multifactorial-threshold framing and the sex-differential Carter effect.
• Lynch M, Walsh B. 1998. Genetics and Analysis of Quantitative Traits. Sinauer Associates, Sunderland MA. ISBN 0–87893–481–2. Authoritative textbook.
• Manolio TA et al. 2009. Finding the missing heritability of complex diseases. Nature 461:747–753.
• Khera AV et al. 2018. Genome-wide polygenic scores for common diseases identify individuals with risk equivalent to monogenic mutations. Nat Genet 50:1219–1224.
• Martin AR et al. 2019. Clinical use of current polygenic risk scores may exacerbate health disparities. Nat Genet 51:584–591.

Related reading

• Polygenic models — additive vs non-additive effects, polygenic risk scores
• Heritability and the liability-threshold model
• Gene-environment interaction
• Complex-disease pedigree software
• Mendelian inheritance calculator
• Hereditary cancer risk assessment