Biology Reference
In-Depth Information
controlled by a number of genes, each of which could contribute
a unit of height, weight, or other measurable characteristics. In the
discussion that follows, we use grain color and the original work by
Nilsson-Ehle as a reference, but the ideas adapt easily to other
quantitative traits.
To determine whether the polygenic hypothesis holds promise, the
assumptions just made should result in a model that could explain the
following:
1. The increased variability of F
2
phenotypes, and
2. The F
2
phenotypic ratios arising from the experiments.
The effect of the polygenic hypothesis can be explained by the binomial
theorem and the related binomial distribution, which we now review.
Assuming that the parents are true-breeding lines, the F
1
generation has
a genetic make-up formed of exactly 50% contributing and 50%
noncontributing alleles. To determine the make-up of the F
2
, the
mathematical question we want to answer is: If we are filling N slots
with one of two alleles R or r that are equally likely, what is the
probability that there are k slots filled with R's and N
k slots filled with
r's? The answer is
:
N
;
¼
1
2
N
N
k
N
k
!
1
1
if k
¼
0
where
and k
!
¼
2
k
if k
>
0
:
k
!
ð
N
k
Þ
!
(3-9)
N
1
2
The factor
comes from the fact that we have N positions to fill
independently with two alleles that are equally likely to be selected. This
means that any string of length N composed of R's and r's has a
N
. The
term comes from the fact that there are
1
2
N
k
probability of
different arrangements of k R's and N
N
k
k r's.
The Binomial Theorem states that for a positive integer N,
a
k
b
N
k
X
N
N
k
n
ð
a
þ
b
Þ
¼
:
K
¼
0
¼
¼
If we set a
b
½ , we see that
1
2
k
N
k
1
2
N
N
X
X
N
N
1
2
1
2
þ
1
2
N
k
N
k
¼
¼
¼
1
;
k
¼
0
k
¼
0
so that we indeed have a probability density.
We now show how the polygenic hypothesis gives rise to a characteristic
having several manifestations and why the ''central manifestations'' are
