A dihybrid cross tracks inheritance at two gene loci simultaneously, combining Mendel’s law of segregation (each parent forms gametes carrying one allele per gene) with independent assortment (unlinked genes assort independently during meiosis). A standard Mendelian model uses two autosomal genes, complete dominance at each locus, and no linkage between the loci.
Genetic setup and assumptions
Two loci are denoted by A/a and B/b, with A dominant over a and B dominant over b. The canonical dihybrid cross uses heterozygous parents: \[ \text{AaBb} \times \text{AaBb}. \]
Unlinked loci (independent assortment), complete dominance, random fertilization, and equal gamete viability are assumed. Deviations in real organisms commonly arise from linkage (recombination fraction \(r\)), epistasis, lethality, or incomplete dominance.
Gametes and independent assortment
Under independent assortment, a heterozygote AaBb produces four gamete types in equal proportions. Each gamete receives one allele from the A/a locus and one allele from the B/b locus.
| Parent genotype | Gamete types | Gamete probabilities |
|---|---|---|
| AaBb | AB, Ab, aB, ab | \(P(AB)=P(Ab)=P(aB)=P(ab)=\frac{1}{4}\) |
Punnett square representation
A 4×4 Punnett square displays the 16 equally likely zygote combinations for AaBb × AaBb when all gametes occur with probability \( \frac{1}{4} \). Cell colors indicate phenotype classes under complete dominance at both loci.
Probability structure and phenotype ratio
Independent assortment allows multiplication of probabilities across loci. The marginal probabilities for a heterozygous monohybrid cross are: \[ P(A\_) = \frac{3}{4},\quad P(aa)=\frac{1}{4},\quad P(B\_) = \frac{3}{4},\quad P(bb)=\frac{1}{4}. \]
Phenotype probabilities in AaBb × AaBb follow directly: \[ P(A\_B\_) = P(A\_)\cdot P(B\_) = \frac{3}{4}\cdot\frac{3}{4}=\frac{9}{16}, \] \[ P(A\_bb) = P(A\_)\cdot P(bb)=\frac{3}{4}\cdot\frac{1}{4}=\frac{3}{16}, \quad P(aaB\_) = P(aa)\cdot P(B\_)=\frac{1}{4}\cdot\frac{3}{4}=\frac{3}{16}, \] \[ P(aabb)=P(aa)\cdot P(bb)=\frac{1}{4}\cdot\frac{1}{4}=\frac{1}{16}. \]
| Phenotype class | Meaning | Probability | Count out of 16 |
|---|---|---|---|
| A_B_ | At least one A and at least one B | \(\frac{9}{16}\) | 9 |
| A_bb | At least one A and bb | \(\frac{3}{16}\) | 3 |
| aaB_ | aa and at least one B | \(\frac{3}{16}\) | 3 |
| aabb | aa and bb | \(\frac{1}{16}\) | 1 |
The phenotype ratio for the dihybrid cross AaBb × AaBb is therefore 9:3:3:1 under the stated Mendelian assumptions.
Genotype frequencies as a product of two 1:2:1 distributions
Each locus in Aa × Aa follows \( \text{AA}:\text{Aa}:\text{aa} = 1:2:1 \). Independence across loci gives a 3×3 genotype grid whose probabilities multiply: \[ P(\text{genotype at A locus})\cdot P(\text{genotype at B locus}). \]
| Offspring genotype | Probability | Count out of 16 |
|---|---|---|
| AABB | \(\frac{1}{16}\) | 1 |
| AABb | \(\frac{2}{16}\) | 2 |
| AAbb | \(\frac{1}{16}\) | 1 |
| AaBB | \(\frac{2}{16}\) | 2 |
| AaBb | \(\frac{4}{16}\) | 4 |
| Aabb | \(\frac{2}{16}\) | 2 |
| aaBB | \(\frac{1}{16}\) | 1 |
| aaBb | \(\frac{2}{16}\) | 2 |
| aabb | \(\frac{1}{16}\) | 1 |
Representative probability statements
Probability of aaBb in AaBb × AaBb factors into locus-wise events: \[ P(\text{aaBb}) = P(\text{aa})\cdot P(\text{Bb})=\frac{1}{4}\cdot\frac{1}{2}=\frac{1}{8}. \]
Probability of the double recessive phenotype aabb is: \[ P(\text{aabb}) = P(\text{aa})\cdot P(\text{bb})=\frac{1}{4}\cdot\frac{1}{4}=\frac{1}{16}. \]
Common extensions and mismatches with the 9:3:3:1 model
Linkage between loci produces gamete frequencies that depend on recombination fraction \(r\), shifting outcomes away from equal \( \frac{1}{4} \) gametes and away from the 9:3:3:1 phenotype ratio. Epistasis alters phenotype mapping so that genotype classes collapse into different phenotype groupings, even when Mendelian segregation and assortment remain intact at the gene level.
Common pitfalls
Phenotype counting errors frequently arise from mixing genotype categories (nine genotypes) with phenotype categories (four classes under complete dominance), or from treating linked loci as independent. The dihybrid cross assumptions should match the biology before a Punnett square or probability product is interpreted as a prediction.