# 9:3:3:1 Ratio

∞ generated and posted on 2016.12.04 ∞

The consequence in a dihybrid cross given complete dominance of one allele to the other, in both loci, and otherwise complete independence of both loci such that there is no epistasis nor linkage.

 A 9:3:3:1 Ratio is at ratio of phenotypes among offspring (progeny) that results when two dihybrids mate, e.g., AaBa × AaBa, where allele A is dominant to allele a, allele B is dominant to allele b, and the A and B loci otherwise have no impact on each other phenotypically (no epistasis) nor genotypically (no linkage).

Though 9:3:3:1 would appear to be too complex to easily appreciate, these ratios really are quite simple. In fact, they represent the default, easy to appreciate consequence of a dihybrid cross, or at least when such crosses involve complete dominance. In particular, a 9:3:3:1 ratio is simply a 3:1 ratio "squared"!

So what's the secret to understanding what is going on? To start, note that 9 is three times 3 while 3 is three times 1. That is, note that there are two 3:1 ratios in 9:3:3:1 (9:3 and 3:1). That there should be two is because there are two loci being followed in this cross and each locus gives rise to a 3:1 ratio of phenotypes.

Thus, there are 9 individuals that are dominant at both loci, 3 that are dominant at the first locus but recessive at the second (and, flipping that around, another 3 that are dominant at the second locus but recessive at the first), and, finally, one that is recessive at both loci, with 9 + 3 + 3 + 1 = 16 = 4 × 4.

Figure legend: Here black (B) is dominant to white (W) while squares (S) are dominant to circles (C). Note that there are 9 black squares, 3 black circles, 3 white squares, and 1 white circle. Note too that this result requires that the two loci involved are not interacting at all.

It is actually fairly easy to extrapolate this concept to trihybrid crosses or tetrahybrid crosses, though doing so using a Punnett square can be quite challenging. Instead, one employs probability theory. Thus, for the 9:3:3:1 ratio you have 3/4 × 3/4 = 9/16; 3/4 × 1/4 = 3/16; 1/4 × 3/4 = 3/16; and 1/4 × 1/4 = 1/16.

For a trihybrid cross (where D stands for dominant and r for recessive):

3/4 × 3/4 × 3/4 = 27/64 (DDD phenotype)
3/4 × 3/4 × 1/4 = 9/64 (DDr phenotype)
3/4 × 1/4 × 3/4 = 9/64 (DrD phenotype)
1/4 × 3/4 × 3/4 = 9/64 (rDD phenotype)
3/4 × 1/4 × 1/4 = 3/64 (Drr phenotype)
1/4 × 3/4 × 1/4 = 3/64 (rDr phenotype)
1/4 × 1/4 × 3/4 = 3/64 (rrD phenotype)
1/4 × 1/4 × 1/4 = 1/64 (rrr phenotype)

That is, with a trihybrid cross one ends up with a 27:9:9:9:3:3:3:1 ratio, but note that 27 is three times 9 and therefore that there are three 3:1 ratios found within this larger ratio, that is, 27:9, 9:3, and 3:1. Note also that 27 + 9 + 9 + 9 + 3 + 3 + 3 + 1 = 64 (as so too does 4 × 4 × 4).