Two Genes at Once: Independent Assortment and 9:3:3:1

From one gene to two

A dihybrid cross follows two genes at once. Mendel used seed shape (round R is dominant over wrinkled r) and seed color (yellow Y dominant over green y). He crossed true-breeding round-yellow plants (RRYY) with true-breeding wrinkled-green plants (rryy). As before, every F1 plant gets one allele of each gene from each parent, so all F1 are RrYy — round and yellow, because both dominants show.

The interesting question is what happens when these RrYy plants self-pollinate. Will round always travel with yellow, the way they were paired in the original parents? Or will the genes shuffle freely? The answer is Mendel's second law.

The law of independent assortment

The law of independent assortment says that the alleles of different genes sort into gametes independently of one another. Whether a gamete gets R or r has nothing to do with whether it gets Y or y. So an RrYy plant makes four kinds of gametes in equal proportions: RY, Ry, rY, and ry — each one-quarter of the total.

The 4×4 square and 9:3:3:1

With four gamete types from each parent, the Punnett square becomes a 4×4 grid with sixteen boxes. Sorting those sixteen by phenotype gives the classic phenotype ratio of the F2 generation.

Cross: RrYy x RrYy
Gametes from each parent: RY, Ry, rY, ry

        RY      Ry      rY      ry
      +-------+-------+-------+-------+
  RY  | RRYY  | RRYy  | RrYY  | RrYy  |
      +-------+-------+-------+-------+
  Ry  | RRYy  | RRyy  | RrYy  | Rryy  |
      +-------+-------+-------+-------+
  rY  | RrYY  | RrYy  | rrYY  | rrYy  |
      +-------+-------+-------+-------+
  ry  | RrYy  | Rryy  | rrYy  | rryy  |
      +-------+-------+-------+-------+

Count by phenotype (R_ = round, Y_ = yellow):
   9  round  yellow   (R_Y_)
   3  round  green    (R_yy)
   3  wrinkled yellow (rrY_)
   1  wrinkled green  (rryy)
                    ->  9 : 3 : 3 : 1

The dihybrid F2 square: sixteen equally likely boxes collapse to a 9:3:3:1 phenotype ratio.

Notice the two middle classes — round-green and wrinkled-yellow. These are new combinations that neither original parent had. They are the visible proof of independent assortment: if round had stayed locked to yellow, those recombinant types could never appear. Mendel's real F2 counts (315 : 101 : 108 : 32) sit close to 9:3:3:1, and the appearance of fresh trait combinations is exactly why this cross mattered so much.