What the cross is actually testing

A dihybrid cross is not just a bigger Punnett square. It is the experiment that decides whether two genes are inherited independently of one another. Mendel's law of independent assortment says that genes do not influence each other when alleles sort into gametes, so every possible allele combination is equally likely. If that holds, a heterozygote for two genes makes four gamete types in equal numbers, and the 9:3:3:1 ratio follows automatically. The physical basis sits in meiosis I, where different homologous pairs line up at the metaphase plate in random orientations, so a gamete can carry any combination of maternal and paternal chromosomes.

A worked example

In pea plants, round seed shape (R) is dominant to wrinkled (r) and yellow seed color (Y) is dominant to green (y). A true-breeding yellow round plant (YYRR) is crossed with a true-breeding green wrinkled plant (yyrr), and the F1 offspring are then self-crossed. What phenotypic ratio appears in the F2 generation, and how many of the 16 boxes are YyRr?

  1. Work out the F1. Each parent is homozygous, so the YYRR plant makes only YR gametes and the yyrr plant makes only yr gametes. Every F1 offspring is YyRr, and all of them are yellow and round.
  2. List the gametes of the F1. A YyRr plant segregates R or r independently of Y or y, so four equally likely gametes form: YR, Yr, yR and yr.
  3. Build the grid. Four gamete types along the top and four down the side give a 4 by 4 Punnett square with 16 equally likely boxes.
  4. Count the phenotypes across the 16 boxes: 9 are yellow and round, 3 are green and round, 3 are yellow and wrinkled, and 1 is green and wrinkled. That is 9:3:3:1, and 9 + 3 + 3 + 1 = 16, which confirms nothing was dropped.
  5. Check it with the product rule instead of the grid. Each characteristic on its own is a 3:1 monohybrid cross, so yellow and round is 3/4 times 3/4 = 9/16, yellow and wrinkled is 3/4 times 1/4 = 3/16, and green and wrinkled is 1/4 times 1/4 = 1/16. The two routes agree.
  6. Count the YyRr boxes. The genotype tally across the grid is 1 YYRR, 2 YYRr, 1 YYrr, 2 YyRR, 4 YyRr, 2 Yyrr, 1 yyRR, 2 yyRr, 1 yyrr, which also sums to 16. YyRr appears in 4 of the 16 boxes, a probability of 1/4.

= The F2 phenotypic ratio is 9 yellow round : 3 green round : 3 yellow wrinkled : 1 green wrinkled, and 4 of the 16 boxes are YyRr, which is 1/4 of the offspring.

Where students lose the marks

Two places. The first is the gamete list: a YyRr parent makes YR, Yr, yR and yr, not two gametes and not eight. Write those four down before drawing anything, and the grid builds itself. The second is reading the finished square. Sixteen boxes hold only nine distinct genotypes, because a genotype such as YyRr can be reached four different ways, so counting genotypes and counting phenotypes give different answers to different questions. If a problem asks for a probability rather than a ratio, the product rule is faster and far less error prone than drawing sixteen boxes by hand.

Where this leads next

The dihybrid cross only works once the monohybrid version is automatic, so the Punnett square page is the place to go back to if the 3:1 ratio is not yet second nature. Going forward, the ratio breaks in interesting ways: when neither allele is fully dominant the phenotype counts change even though the genotype counts do not, which is the incomplete dominance and codominance page. And because 9:3:3:1 is an expectation rather than a promise, a real data set is compared against it with a chi square test, the standard AP Biology follow up to this cross.

Common mistakes

  • Writing only two gamete types for a YyRr parent. Two genes segregating independently give four gamete types, and a 2 by 2 grid cannot produce a 9:3:3:1 ratio.
  • Reporting 9:3:3:1 as a genotypic ratio. It is the phenotypic ratio; the genotypes across the same 16 boxes fall as 1:2:1:2:4:2:1:2:1.
  • Assuming every dihybrid cross gives 9:3:3:1. That ratio is specific to a self-cross of double heterozygotes with simple dominance at both genes.
  • Treating the ratio as a guarantee for a small litter or pod. It is an expected proportion that only emerges over a large enough sample.

Related concepts

  • Incomplete dominance vs codominance: In incomplete dominance the heterozygote shows an intermediate phenotype, as a red and a white snapdragon parent give pink offspring.
  • Punnett square: A Punnett square applies the rules of probability to predict the outcomes of a genetic cross.
  • Genotype vs phenotype: A genotype is an organism's underlying genetic makeup, including alleles that are not expressed.

Sources