The Additive Genetic Model

Consider a single locus with two alleles, A₁ and A₂, affecting a continuous trait such as body weight. The three genotypes A₁A₁, A₁A₂, A₂A₂ have measurable average phenotypic values. By convention, we set the midpoint between the two homozygotes to zero and assign:

• A₁A₁ = +a
• A₁A₂ = d
• A₂A₂ = −a

Here a is the difference between a homozygote and the midpoint (a measure of the locus's overall effect size), and d is the dominance deviation — how far the heterozygote departs from the midpoint. If d = 0, the heterozygote is exactly intermediate and the locus is purely additive. If d = a, A₁ is completely dominant; if d = −a, A₂ is completely dominant; and |d| > a describes overdominance (heterozygote superiority), as famously invoked to explain hybrid vigor. These a and d values describe the genotype's actual effect — a fixed biological quantity. But an individual's genotype at that locus is not, by itself, what gets transmitted to offspring: only single alleles are transmitted, one from each parent, and which allele the offspring receives from the other parent is a matter of chance drawn from the population's allele frequencies. This is the conceptual leap the additive model makes: instead of asking "what does this genotype do?", it asks "what is the average effect of substituting one allele for the other, given the population's actual genetic background?" That average is the average effect of allele substitution (α), and the genotype's additive value (breeding value) is defined from α. The genotypic value is then decomposed as:

G = μ + A + D where μ is the population mean, A is the additive value (the sum of average allelic effects, transmissible), and D is the dominance deviation (a residual due to the specific pairing, not transmissible). When multiple loci are considered jointly, a further term I captures epistatic deviation — genotype effects that depend on specific combinations of alleles across loci, again not reliably transmitted because recombination and independent assortment reshuffle these combinations every generation. The full decomposition for a polygenic trait is:

G = μ + A + D + I

Step 1 — Assign genotypic values. For a locus with alleles A₁ (frequency p) and A₂ (frequency q = 1 − p), measure or define the mean phenotype of each genotype and rescale so the midpoint of the two homozygotes is zero: A₁A₁ = a, A₁A₂ = d, A₂A₂ = −a.

Step 2 — Compute the population mean. Assuming Hardy–Weinberg genotype frequencies (p², 2pq, q²), the population mean genotypic value is:

M = a(p − q) + 2pqd This already reveals something important: even a purely additive locus (d = 0) shifts the population mean whenever p ≠ q, and dominance (d ≠ 0) contributes an extra term weighted by heterozygote frequency 2pq — dominance affects the mean only through the frequency of heterozygotes, not through some intrinsic "extra effect" independent of allele frequency.

Step 3 — Derive the average effect of allele substitution (α). Imagine picking a random gamete from the population and substituting an A₁ for what would otherwise have been drawn; α is the resulting average change in offspring genotypic value, averaged over all the genetic backgrounds (i.e., over what the other allele in the offspring turns out to be, drawn at random from the population). Formally:

α = a + d(q − p) Notice α reduces to a (the simple homozygote effect) only when d = 0 or p = q; dominance distorts the substitution effect whenever allele frequencies are unequal.

Step 4 — Define the breeding value (additive value). The additive value of a genotype is the sum of the average effects of the two alleles it carries, expressed as a deviation from the population mean:

This is the value that predicts, on average, how much an individual's offspring will deviate from the population mean — because offspring receive single alleles, not intact genotypes, the breeding value (built from single-allele average effects) is exactly what is transmitted.

Step 5 — Compute the dominance deviation. Whatever is left over after removing the additive value from the genotypic value is the dominance deviation:

D depends on d and on the specific pairing of alleles present in that individual; it is real and affects the phenotype, but it cannot be "inherited" as such, because meiosis separates the two alleles and the offspring's own dominance deviation depends on which allele it happens to receive from the other parent and how that pairs up.

Step 6 — Extend to multiple loci: epistasis. When two or more loci jointly affect a trait, their combined effect need not equal the sum of single-locus effects. The residual after accounting for all single-locus additive and dominance effects is the epistatic (interaction) deviation, I, which itself has additive-by-additive, additive-by-dominance, and dominance-by-dominance sub-components. Like dominance, epistatic deviations depend on specific multi-locus combinations that recombination breaks apart, so they are largely non-transmissible in the short term (though they can matter enormously for hybrid breakdown and speciation).

Step 7 — Sum across the genome. For a polygenic trait, total genotypic value is approximately the sum of per-locus (G = μ + A + D + I) contributions across all contributing loci, plus environmental deviation E:

P = μ + ΣA + ΣD + ΣI + E This equation is the mechanistic backbone of the additive genetic model and feeds directly into variance partitioning (Partitioning Phenotypic Variance) and into narrow-sense versus broad-sense Heritability: Broad and Narrow Sense.

By the early 1900s, geneticists were split into two camps that seemed to be describing incompatible worlds. The biometricians, led by Karl Pearson and W. F. R. Weldon, studied continuously varying traits — human height, crab shell dimensions — using correlation and regression, and found that parent–offspring and sibling correlations for such traits were modest, smooth, and clearly did not follow neat Mendelian ratios. The Mendelians, following the rediscovery of Mendel's work around 1900, studied discrete traits (seed shape, flower color) that segregated in clean 3:1 or 9:3:3:1 ratios and argued that inheritance was fundamentally particulate. Each side considered the other's data irrelevant or wrong; Pearson and Weldon in particular were openly hostile to Mendelian claims about "unit characters."

Ronald A. Fisher's 1918 paper, "The Correlation between Relatives on the Supposition of Mendelian Inheritance" (Transactions of the Royal Society of Edinburgh), resolved the conflict mathematically rather than experimentally, but it made a testable, quantitative prediction that experimentalists then confirmed. Fisher's reasoning proceeded in three moves: 1. Assume a continuous trait is controlled by a large number of Mendelian loci, each segregating according to ordinary Mendelian rules, each with a small additive effect (plus possible dominance), summed together with environmental deviations. 2. Derive, from first principles, the algebraic decomposition of genotypic value into additive and dominance components described above (Fisher's paper is where the terms "average effect," "additive value," and "dominance deviation" were first defined rigorously), and use this to compute the expected correlation in trait value between relatives — parent–offspring, full sibs, half sibs, twins — purely from Mendelian segregation probabilities. 3. Show that these Mendelian-derived correlations matched the biometricians' own empirically observed correlations (e.g., parent–offspring correlations near 0.5 for height, sib correlations similarly in that neighborhood) far more closely than any non-Mendelian "blending inheritance" model could. Fisher's key quantitative result was that under additive inheritance with no dominance, the expected parent–offspring correlation is exactly 1/2 (each parent contributes half the offspring's alleles) and the full-sib correlation is also 1/2 in expectation (sharing, on average, half their alleles identically by descent), while dominance and shared-environment effects introduce small, calculable departures — for instance, dominance variance inflates the full-sib correlation above the parent–offspring correlation because full sibs can share both alleles at a locus (and hence the same dominance deviation) whereas a parent and offspring can never do so (they always differ by at least one allele-transmission event). This subtle asymmetry — sib correlation slightly exceeding parent–offspring correlation when dominance variance is present — was exactly the kind of pattern biometricians had measured in real data (e.g., in stature) but had been unable to explain within either pure blending or pure Mendelian-without-dominance frameworks. The empirical confirmation came from re-analysis of existing biometrical datasets (including Pearson's own human stature data and, in subsequent decades, extensive livestock and crop pedigree records) once Fisher's formulas gave biometricians a precise, falsifiable target correlation to check their numbers against, rather than a vague qualitative claim that "many genes are involved." Over the following decades, breeding experiments in Drosophila, maize, and livestock — most famously Sewall Wright's guinea pig work and large-scale selection experiments in poultry and dairy cattle — repeatedly verified that response to selection tracked the additive genetic variance predicted by Fisher's model far better than it tracked total (broad-sense) phenotypic variance, directly confirming that the additive component, and not dominance or epistatic deviations, is what determines a population's response to selection generation after generation. This is the core empirical payoff of the additive model: it does not merely organize algebra, it correctly predicts observable quantities — correlations between relatives and rates of response to artificial selection — that non-additive-blind models cannot.

Example 1 — A single locus with partial dominance and unequal allele frequencies. Suppose a locus affecting piglet birth weight has alleles B (frequency p = 0.2) and b (frequency q = 0.8). Measured genotypic values (kg, deviations from midpoint) are:

Here d = 0.6 while a = 1.2, so d/a = 0.5 — partial dominance of B. Population mean: M = a(p − q) + 2pqd M = 1.2(0.2 − 0.8) + 2(0.2)(0.8)(0.6) M = 1.2(−0.6) + (0.32)(0.6) M = −0.72 + 0.192 = −0.528 kg (relative to the homozygote midpoint) Average effect of allele substitution: α = a + d(q − p) = 1.2 + 0.6(0.8 − 0.2) = 1.2 + 0.6(0.6) = 1.2 + 0.36 = 1.56 kg Breeding (additive) values:

• BB: 2qα = 2(0.8)(1.56) = +2.496 kg
• Bb: (q − p)α = (0.6)(1.56) = +0.936 kg
• bb: −2pα = −2(0.2)(1.56) = −0.624 kg

Dominance deviations (D = G − A):

• BB: 1.2 − 2.496 = −1.296 kg
• Bb: 0.6 − 0.936 = −0.336 kg
• bb: −1.2 − (−0.624) = −0.576 kg

Check: weighted mean of A should equal weighted mean of G (both equal M relative to midpoint), and the weighted mean of D should be zero — confirming the algebra: 0.04(2.496) + 0.32(0.936) + 0.64(−0.624) = 0.0998 + 0.2995 − 0.3994 ≈ −0.0001 (rounding), consistent with the population mean of A being ≈ 0 relative to itself by construction, while G's mean (−0.528) is carried entirely by the (p−q) and dominance terms as derived above. The key lesson: because rare allele B (p = 0.2) is somewhat dominant, α (1.56) is inflated above a (1.2) — a minority favorable dominant allele has an outsized average effect of substitution, precisely because when it does appear, it is usually paired with the common recessive allele and its dominance is "wasted" pulling heterozygotes toward it — this asymmetric interaction between allele frequency and dominance is exactly why breeders must track allele frequency, not just genotype means, when predicting response to selection.

Example 2 — Predicting response to selection using only the additive variance. A poultry breeder selects egg-laying hens for number of eggs per year. The population has:

• Phenotypic variance, V_P = 900 (eggs²)
• Additive genetic variance, V_A = 270 (eggs²)
• Dominance + epistatic + environmental variance make up the remainder

Narrow-sense heritability (foreshadowing Heritability: Broad and Narrow Sense): h² = V_A / V_P = 270 / 900 = 0.30 The breeder selects the top 10% of hens by egg number; suppose this selected group's mean egg count exceeds the population mean by a selection differential S = 60 eggs/year. The breeder's equation, itself a direct application of the additive model (only the additive value is transmitted to offspring), predicts the response:

R = h² × S = 0.30 × 60 = 18 eggs/year improvement in the offspring generation's mean. Now contrast: if instead the same phenotypic gap of 60 eggs were driven mostly by dominance and environmental effects (say h² were only 0.05 rather than 0.30, all else equal), the predicted response would be R = 0.05 × 60 = 3 eggs/year — a six-fold weaker response despite an identical-looking selected group. This is the practical crux of the additive model: two populations can look phenotypically identical among the selected parents, yet respond to selection completely differently, because only the additive component of the variance predicts transmission to the next generation. This is exactly why livestock and plant breeders invest heavily in estimating h² and additive breeding values (via pedigree BLUP or genomic prediction) rather than simply selecting on raw phenotype.

Ronald Aylmer Fisher was a British statistician, evolutionary biologist, and geneticist who spent much of his early career (1919–1933) at Rothamsted Experimental Station, an agricultural research institute, where he had access to decades of crop-trial data and where he also developed the analysis of variance and modern experimental design alongside his genetics work. Fisher was mathematically formidable and notoriously terse in print, which meant his 1918 paper — though it settled a decade-long, often bitter dispute between biometricians and Mendelians — took years for many biologists to fully absorb. He went on to write The Genetical Theory of Natural Selection (1930), uniting Mendelian genetics with Darwinian natural selection and formalizing concepts like the fundamental theorem of natural selection, which states (in Fisher's own restrictive sense) that the rate of increase in mean fitness due to natural selection at any time is equal to the additive genetic variance in fitness at that time — a direct descendant of the additive/non-additive split introduced in 1918.

Livestock and crop breeding. Modern dairy cattle breeding routinely computes estimated breeding values (EBVs) for milk yield, fertility, and disease resistance directly from this additive framework, now combined with genome-wide marker data (genomic BLUP) rather than pedigree alone. A bull's EBV for milk yield is, mechanistically, an estimate of the additive value A summed across all loci affecting the trait — exactly the quantity Fisher defined in 1918 — and it is this quantity, not the bull's own phenotype (bulls do not lactate), that predicts his daughters' performance.

Genomic selection in agriculture. Genomic selection programs in maize, wheat, and salmon aquaculture estimate additive effects at thousands of SNP markers simultaneously and sum them into genomic estimated breeding values, allowing selection decisions within weeks of DNA extraction rather than after years of progeny testing — a direct industrial-scale application of G = μ + A + D + I where dominance and epistatic terms are typically modeled but the additive term drives most of the predicted genetic gain.

Human polygenic risk scores. Genome-wide association studies estimate the average effect of allele substitution (α) at each of many thousands of SNPs associated with traits like height, type 2 diabetes risk, or coronary artery disease, then sum these additive contributions into a polygenic score — the same mathematical object as a breeding value, applied to human genomics for research and, increasingly, for clinical risk stratification.

Conservation genetics. Estimating additive genetic variance in small, endangered populations helps conservation managers predict whether a population retains the capacity to adapt to environmental change (evolvability depends on V_A, not total phenotypic variance) versus whether inbreeding has eroded additive variance and increased the relative contribution of dominance-driven inbreeding depression.

Public health and epidemiology. Understanding that non-additive (dominance, gene-by-gene, and gene-by-environment) effects contribute to "missing heritability" in complex disease studies has directly shaped the design of modern GWAS and family-based heritability studies, motivating methods (e.g., variance-components models, twin studies) that explicitly try to separate additive from non-additive genetic contributions to disease risk.

Ask the tutor: How does adding dominance and epistatic variance change the breeder's equation when predicting multi-generation response to selection?