Gene interaction and epistatic ratios
Why two genes acting on one trait change Mendel's 9:3:3:1, each modified ratio explained by pathways, test-cross ratios, and a chi-square sum.
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Mendel chose traits each controlled by a single gene, and his dihybrid crosses gave the famous 9 : 3 : 3 : 1 ratio. Soon after his work was rediscovered in 1900, geneticists found crosses that gave 9 : 7, 9 : 3 : 4, 12 : 3 : 1 or 15 : 1 instead. These were not failures of Mendel's laws. The genes were still segregating and assorting independently; they were simply acting on the same trait, so their effects combined. This is gene interaction, and when one gene masks the effect of another it is called epistasis, a term introduced by William Bateson in 1909.
Recognising modified ratios and explaining them is a favourite exam question in genetics and plant breeding, and the ideas matter in breeding, where hidden gene interactions can make crosses behave unexpectedly.
Allelic and non-allelic interaction
- Allelic (intra-allelic) interaction: between alleles of the same gene, as in complete dominance, incomplete dominance and codominance.
- Non-allelic (inter-allelic) interaction: between different genes affecting the same trait. Epistasis is the main kind.
The gene that masks is epistatic; the gene that is masked is hypostatic. This is different from dominance, which is about alleles of one gene.
Why ratios change: pathways
Most traits are the result of biochemical pathways, chains of enzymes each encoded by a gene. If two genes control two steps in the same pathway, the outcome depends on both, and some of the nine genotype classes of a dihybrid cross look the same.
The modified dihybrid ratios
All of these come from an F1 dihybrid (AaBb) selfed or intercrossed, with the 16 combinations grouped as 9 A_B_ : 3 A_bb : 3 aaB_ : 1 aabb. The underscore means either allele.
9 : 3 : 3 : 1 with interaction (comb shape in poultry)
Two genes give four phenotypes, but the double-dominant type is new. W. Bateson and R. C. Punnett showed that rose comb (R_pp) × pea comb (rrP_) gives walnut (R_P_) in the F1, and the F2 shows walnut 9 : rose 3 : pea 3 : single 1 (rrpp).
9 : 7, complementary genes (duplicate recessive epistasis)
Both dominant alleles are needed for the trait, as two enzymes act one after the other. Bateson and Punnett's cross of two white-flowered sweet pea lines gave all purple F1 and an F2 of 9 purple : 7 white. Recessive homozygosity at either gene blocks the pathway.
9 : 3 : 4, recessive epistasis (supplementary genes)
The recessive genotype of one gene masks the other. In Labrador retrievers, gene B gives black (B_) or brown (bb) pigment, while gene E controls whether pigment is deposited in the hair: ee dogs are yellow whatever their B genotype. The F2 gives black 9 : chocolate 3 : yellow 4. Similar patterns occur in onion bulb colour and in maize aleurone colour.
12 : 3 : 1, dominant epistasis
A dominant allele of one gene masks the other. In summer squash, W_ gives white fruit regardless of the other gene; among ww plants, Y_ gives yellow and yy gives green. F2: white 12 : yellow 3 : green 1.
13 : 3, dominant suppression (inhibitory gene)
A dominant allele of one gene suppresses the expression of the other. In poultry, C gives coloured feathers, but a dominant inhibitor I prevents colour. Crossing White Leghorns (IICC) with White Wyandottes (iicc) gives white F1, and an F2 of 13 white : 3 coloured; only iiC_ birds are coloured.
15 : 1, duplicate dominant genes
Either dominant allele alone produces the trait; only the double recessive differs. G. H. Shull found this for seed capsule shape in shepherd's purse (Capsella): 15 triangular : 1 ovoid. H. Nilsson-Ehle's work on red and white wheat grain colour gave 15 : 1 and 63 : 1 ratios from two and three duplicate genes, foreshadowing polygenic inheritance.
9 : 6 : 1, duplicate genes with cumulative effect
Each dominant allele adds to the trait. In summer squash fruit shape, A_B_ gives disc, A_bb or aaB_ gives spherical, and aabb gives elongated: 9 : 6 : 1.
Summary table
| F2 ratio | Interaction | Test-cross ratio (AaBb × aabb) |
|---|---|---|
| 9 : 3 : 3 : 1 | none, or new phenotype from interaction | 1 : 1 : 1 : 1 |
| 9 : 7 | complementary (duplicate recessive) | 1 : 3 |
| 9 : 3 : 4 | recessive epistasis | 1 : 1 : 2 |
| 12 : 3 : 1 | dominant epistasis | 2 : 1 : 1 |
| 13 : 3 | dominant suppression | 3 : 1 (or 1 : 3) |
| 15 : 1 | duplicate dominant | 3 : 1 |
| 9 : 6 : 1 | duplicate with cumulative effect | 1 : 2 : 1 |
The test-cross ratios come from grouping the four equal test-cross classes (AaBb, Aabb, aaBb, aabb) in the same way as the F2 classes.
Identifying a ratio from data
Which ratio fits?
An F2 of 320 plants has 182 with coloured grain, 61 with brown grain and 77 colourless. Try 9 : 3 : 4:
- Expected: 320 × 9⁄16 = 180, 320 × 3⁄16 = 60, 320 × 4⁄16 = 80.
- χ2 = (182 − 180)2 ⁄ 180 + (61 − 60)2 ⁄ 60 + (77 − 80)2 ⁄ 80 = 0.022 + 0.017 + 0.113 = 0.15.
- With 3 classes, degrees of freedom = 2; the 5 per cent critical value is 5.99.
χ2 is far below 5.99, so the data fit 9 : 3 : 4 well, suggesting recessive epistasis.
A two-class example. An F2 of 160 sweet pea plants has 93 purple and 67 white. For 9 : 7, expected values are 90 and 70: χ2 = 9 ⁄ 90 + 9 ⁄ 70 = 0.10 + 0.13 = 0.23, below the critical value of 3.84 for 1 degree of freedom, so 9 : 7 fits.
A quick first step is to divide each count by the total and multiply by 16: 182, 61 and 77 out of 320 become about 9.1, 3.1 and 3.9 sixteenths.
Why it matters in breeding
- Unexpected segregation: crossing two lines with the same phenotype, such as two white-flowered parents, can give coloured progeny if they carry different blocked genes (complementation).
- Hidden variation: epistatic genes can conceal useful alleles that appear only in certain combinations.
- Quantitative traits: yield and quality involve many genes whose interactions contribute to epistatic variance, which complicates prediction and is part of why hybrid performance is hard to predict.
- Complementation tests: crossing two recessive mutants with the same phenotype tells whether the mutations are in the same gene (no complementation) or different genes (complementation, wild-type F1).
Key points
- Gene interaction occurs when two or more genes affect one trait; epistasis is masking of one gene by another.
- Dominance is between alleles of a gene; epistasis is between genes.
- Ratios: 9 : 7 complementary, 9 : 3 : 4 recessive epistasis, 12 : 3 : 1 dominant epistasis, 13 : 3 inhibitory, 15 : 1 duplicate dominant, 9 : 6 : 1 cumulative, and 9 : 3 : 3 : 1 with new phenotypes (poultry combs).
- All modified ratios add to 16 and come from the same 16 dihybrid combinations.
- Use the chi-square test to check which ratio fits.
What the exams ask
- School-leaving and entrance papers: what epistasis is, and examples such as comb shape, sweet pea flower colour and Labrador coat colour.
- Degree and research-fellowship entrances: naming modified ratios from data, test-cross ratios, chi-square tests, and pathway explanations.
- Research and breeding examinations: epistatic variance, complementation tests, and implications for selection.
Described in general terms: the kind of examination, not a past paper. Laisha never reproduces exam questions.
Questions people ask
Is epistasis the same as dominance?
No. Dominance is the masking of one allele by another allele of the same gene. Epistasis is the masking of one gene's effect by a different gene.
Do modified ratios mean Mendel was wrong?
No. The genes still segregate and assort independently. Only the way their genotypes show up as phenotypes differs, because they affect the same trait.
How can I remember which ratio is which?
Think of the pathway. If both steps are needed, it is 9 : 7. If one gene must work before the other can show, it is 9 : 3 : 4. If one dominant allele hides everything, it is 12 : 3 : 1. If either dominant allele is enough, it is 15 : 1.
Sources
- Bateson, W. Mendel's Principles of Heredity. Cambridge University Press, 1909.
- Bateson, W., Saunders, E. R. and Punnett, R. C. Reports to the Evolution Committee of the Royal Society, II and III. Harrison and Sons, London, 1905–1906.
- Griffiths, A. J. F. et al. Introduction to Genetic Analysis, 12th edition. W. H. Freeman, 2020. Chapter 6.
- Phillips, P. C. "Epistasis: the essential role of gene interactions in the structure and evolution of genetic systems." Nature Reviews Genetics 9, 855–867 (2008).
- Singh, B. D. Genetics. Kalyani Publishers, latest edition.
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