Algebraic derivation of Mendelian ratios
as binomial traits
Mendelian ratios may also be obtained from the calculation of the binomial
distribution of two independent traits. For two alleles at one
locus, the genotype ratios are
1A + 1a
1A + 1a
_________________
1Aa + 1aa
1AA + 1 Aa
_________________
1AA + 2Aa + 1aa gives a 1 : 2 : 1 ratio
and the phenotype ratios are therefore (1AA + 2Aa) + 1 aa =
(1+2) A + 1 a = 3:1
For two alleles at two loci, the genotypic and phenotypic
ratios above give:
1AA + 2Aa + 1aa
1BB + 2Bb +
1bb
____________________________________________________________________
+ 1AAbb + 2Aabb +
1aabb
+
2AABb + 4AaBb + 2aaBb
1AABB + 2AaBB + 1aaBB
= 1 : 2 :
1 : 2 : 4 : 2 : 1 : 2 : 1
for which the phenotypes are
1 AB + 2
AB + 1 aB + 2 AB
+ 4 AB + 2 aB + 1
Ab + 2 Ab + 1 ab
rearranging
(1 + 2 + 2 + 4) AB + (1 + 2) aB + (1 + 2) Ab + 1 ab gives a 9
: 3 : 3 : 1 ratio
Alternatively,
3A + 1a
3B + 1b
____________________
3Ab + 1ab
9AB + 3aB
____________________
9AB + 3aB + 3Ab + 1ab
HOMEWORK: Show the algebraic
derivation of the Mendelian genotypic and phenotypic ratios
for a trinomial trait, generated by three independent
characters. Hint: Look at the pattern
in the binomial genotypic ratios. Can you expand this pattern of nine
terms for one with 27 ?