Mendel's laws and the classic ratios
Dominance, segregation and independent assortment with Mendel's peas: monohybrid, dihybrid and test crosses, the classic ratios, and the chi-square check.
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Gregor Mendel, working with garden peas in a monastery garden in Brno in the 1850s and 1860s, worked out that heredity is carried by discrete units, now called genes, that pass unchanged from generation to generation in predictable proportions. His paper of 1866 was largely ignored until 1900, when three botanists, Hugo de Vries, Carl Correns and Erich von Tschermak, independently reached the same results.
Every genetics and plant breeding syllabus starts here, and genetics problems built on Mendel's ratios appear in agriculture, veterinary and biology exams worldwide.
Why peas worked
Mendel's choice of organism was as important as his ideas.
- Peas self-pollinate naturally, so he could keep true-breeding lines, yet he could also cross them by hand, removing the anthers of one flower and dusting it with pollen from another.
- He studied seven traits, each with two clear-cut forms: seed shape (round or wrinkled), seed colour (yellow or green), flower colour (purple or white), pod shape (inflated or constricted), pod colour (green or yellow), flower position (axial or terminal) and stem length (tall or dwarf).
- He counted offspring in large numbers and treated the results statistically, which almost no biologist then did.
The vocabulary
| Term | Meaning |
|---|---|
| Gene and allele | A gene is a unit of heredity at a fixed place (locus) on a chromosome; alleles are its alternative forms, such as R (round) and r (wrinkled). |
| Homozygous / heterozygous | Two identical alleles (RR, rr) or two different ones (Rr). |
| Genotype / phenotype | The alleles an individual carries / the trait it shows. |
| Dominant / recessive | The allele expressed in a heterozygote / the one hidden by it. |
| P, F1, F2 | Parents, first filial generation, the generation from selfing or intercrossing the F1. |
| Test cross | Crossing an individual with the homozygous recessive to reveal its genotype. |
| Back cross | Crossing the F1 with either parent. |
The first law: segregation
Mendel crossed true-breeding round-seeded peas with wrinkled-seeded ones. All the F1 seeds were round. When he grew them and let them self-pollinate, the F2 had round and wrinkled seeds in a ratio close to 3 : 1 (he counted 5,474 round and 1,850 wrinkled, a ratio of 2.96 : 1).
His explanation, now the law of segregation:
- Each plant carries two factors (alleles) for each trait.
- The two separate (segregate) when gametes form, so each gamete carries only one.
- Gametes combine at random at fertilisation.
So an Rr plant makes R and r gametes in equal numbers, and selfing it gives:
| R | r | |
|---|---|---|
| R | RR | Rr |
| r | Rr | rr |
- Genotypic ratio in the F2: 1 RR : 2 Rr : 1 rr.
- Phenotypic ratio: 3 round : 1 wrinkled, because R is dominant.
Segregation happens because the two alleles sit on the two members of a chromosome pair, which separate in meiosis. Mendel knew nothing of chromosomes; the link was made by Walter Sutton and Theodor Boveri around 1902–1903.
A test cross of the round F1 (Rr) with wrinkled (rr) gives 1 round : 1 wrinkled, confirming that the F1 makes two kinds of gamete in equal numbers.
Dominance is sometimes called a law too, but it is not universal. With incomplete dominance the heterozygote is intermediate, as in red × white snapdragons or four-o'clocks giving pink F1 and a 1 red : 2 pink : 1 white F2. With codominance both alleles show, as in roan cattle (red and white hairs) and the human AB blood group. In both cases the phenotypic ratio equals the genotypic ratio, 1 : 2 : 1.
The second law: independent assortment
Mendel then followed two traits at once. He crossed round–yellow plants (RRYY) with wrinkled–green ones (rryy). The F1 were all round and yellow (RrYy). In the F2 he found four kinds of seed in a ratio close to 9 : 3 : 3 : 1.
The law of independent assortment: alleles of different genes separate into gametes independently of each other. An RrYy plant therefore makes RY, Ry, rY and ry gametes in equal proportions, a quarter each.
The 9 : 3 : 3 : 1 ratio is simply two independent 3 : 1 ratios multiplied: (3 round : 1 wrinkled) × (3 yellow : 1 green) = 9 round–yellow : 3 round–green : 3 wrinkled–yellow : 1 wrinkled–green. A dihybrid test cross (RrYy × rryy) gives 1 : 1 : 1 : 1.
Independent assortment holds for genes on different chromosomes, or far apart on the same one. Genes close together on a chromosome are linked and tend to be inherited together, which is why some crosses depart from 9 : 3 : 3 : 1. Mendel's seven traits happen to behave independently.
Counting combinations quickly
For a cross between two individuals heterozygous for n independently assorting genes with complete dominance:
| Formula | Monohybrid (n = 1) | Dihybrid (n = 2) | Trihybrid (n = 3) | |
|---|---|---|---|---|
| Kinds of gamete per parent | 2n | 2 | 4 | 8 |
| Squares in the Punnett square | 4n | 4 | 16 | 64 |
| Genotypes in the F2 | 3n | 3 | 9 | 27 |
| Phenotypes in the F2 | 2n | 2 | 4 | 8 |
| Fraction homozygous recessive for all | (1⁄4)n | 1⁄4 | 1⁄16 | 1⁄64 |
For many genes, multiplying probabilities is faster than drawing squares. In an AaBbCc × AaBbCc cross, the probability of an offspring showing all three dominant traits is 3⁄4 × 3⁄4 × 3⁄4 = 27⁄64.
Testing a ratio: the chi-square test
Real counts never match a ratio exactly. The chi-square (χ2) test asks whether the deviation is small enough to be due to chance.
where O is the observed and E the expected number in each class. Compare the result with the table value at (number of classes − 1) degrees of freedom, usually at the 5 per cent level.
Do Mendel's dihybrid counts fit 9 : 3 : 3 : 1?
Mendel's own F2 from the round–yellow × wrinkled–green cross: 315 round–yellow, 108 round–green, 101 wrinkled–yellow and 32 wrinkled–green, 556 seeds in all.
Expected numbers: 556 × 9⁄16 = 312.75; 556 × 3⁄16 = 104.25 (twice); 556 × 1⁄16 = 34.75.
| Class | O | E | (O − E)2 ⁄ E |
|---|---|---|---|
| Round–yellow | 315 | 312.75 | 0.016 |
| Round–green | 108 | 104.25 | 0.135 |
| Wrinkled–yellow | 101 | 104.25 | 0.101 |
| Wrinkled–green | 32 | 34.75 | 0.218 |
| Total | 556 | 556 | χ2 = 0.47 |
With 4 − 1 = 3 degrees of freedom, the 5 per cent table value is 7.815. Since 0.47 is far below it, the counts fit the 9 : 3 : 3 : 1 ratio. (The statistician Ronald Fisher noted in 1936 that Mendel's results as a whole fit his ratios more closely than chance would usually allow, a debate still discussed; his conclusions are not in doubt.)
When the ratios change: gene interaction
When two genes affect the same trait, the four dihybrid classes can merge, giving modified ratios that still add to 16. These are a favourite exam topic:
| F2 ratio | Name | Example usually cited |
|---|---|---|
| 9 : 7 | complementary genes | flower colour in sweet pea |
| 9 : 3 : 4 | recessive epistasis | coat colour in mice; bulb colour in onion |
| 12 : 3 : 1 | dominant epistasis | fruit colour in summer squash |
| 13 : 3 | dominant suppression (inhibitory gene) | plumage colour in poultry |
| 15 : 1 | duplicate dominant genes | seed capsule shape in shepherd's purse |
| 9 : 6 : 1 | duplicate genes with cumulative effect | fruit shape in summer squash |
Recognising the ratio from counts, and naming the interaction, is the usual question.
Key points
- Mendel (1866; rediscovered 1900) used true-breeding peas with seven contrasting traits and counted offspring.
- Law of segregation: two alleles per gene separate into gametes; monohybrid F2 = 3 : 1 phenotype, 1 : 2 : 1 genotype; test cross 1 : 1.
- Law of independent assortment: genes on different chromosomes assort independently; dihybrid F2 = 9 : 3 : 3 : 1; test cross 1 : 1 : 1 : 1.
- Incomplete dominance and codominance give 1 : 2 : 1 phenotypes; linkage breaks independent assortment.
- For n heterozygous genes: 2n gametes, 3n genotypes, 2n phenotypes, 4n combinations.
- Chi-square checks whether observed counts fit an expected ratio; modified ratios (9 : 7, 9 : 3 : 4, 12 : 3 : 1, 13 : 3, 15 : 1) reveal gene interaction.
What the exams ask
- School-leaving and entrance papers: Mendel's traits, definitions, the 3 : 1 and 9 : 3 : 3 : 1 ratios, test and back crosses, and simple Punnett squares.
- Degree and research-fellowship entrances: probability methods for multi-gene crosses, chi-square tests, incomplete and codominance, gene interaction ratios, and linkage as an exception.
- Animal and plant breeding examinations: applying Mendelian ratios to coat colour, polled and horned cattle, disease resistance genes, and recessive defects in breeding programmes.
Described in general terms: the kind of examination, not a past paper. Laisha never reproduces exam questions.
Questions people ask
Why is the genotypic ratio 1 : 2 : 1 but the phenotypic ratio 3 : 1?
Because RR and Rr look the same when R is dominant. The 2 heterozygotes and 1 homozygous dominant all show the dominant trait, giving 3; only rr shows the recessive one.
Do Mendel's laws apply to animals and humans?
Yes, to any trait controlled by single genes in sexually reproducing organisms: coat colours, many inherited diseases, blood groups. Most economically important traits, such as yield or milk production, are controlled by many genes and follow quantitative genetics instead.
How did Mendel get such clean results if linkage exists?
The traits he reported happen to be on different chromosomes or far apart on the same one, so they behave independently.
Sources
- Mendel, G. "Versuche über Pflanzen-Hybriden" (Experiments on Plant Hybridization). Verhandlungen des naturforschenden Vereines in Brünn 4, 3–47 (1866). English translation by the Mendel Web project.
- Griffiths, A. J. F. et al. Introduction to Genetic Analysis, 12th edition. W. H. Freeman, 2020. Chapters 2 and 6.
- Klug, W. S., Cummings, M. R., Spencer, C. A. and Palladino, M. A. Concepts of Genetics, 12th edition. Pearson, 2019.
- Fisher, R. A. "Has Mendel's work been rediscovered?" Annals of Science 1, 115–137 (1936).
- Singh, B. D. Fundamentals of Genetics. Kalyani Publishers, latest edition.
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