∞ 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).

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