Biology Reference
In-Depth Information
In our discussion so far, we have only considered the case where each
allele was equally likely. This isn't necessary. If we consider two alleles,
one (say R) occurring with probability p and the other (r) with probability
1
p, the probability that exactly k of the 2m alleles will be R is
p
k
2m
k
2m
k
ð
1
p
Þ
;
where k
¼
0
;
1
;
2
;
...
;
2m
:
(3-10)
1
2
;
Note that when p
¼
then :
k
2m
k
k
2m
k
1
2
1
2
1
2
1
2
1
2
2m
2m
k
p
k
ð
1
p
Þ
¼
1
¼
¼
and, thus, we obtain exactly the same results as in Eq. (3-9) for N
¼
2m.
Figure 3-10 presents the probability histograms for the binomial
probabilities with varying values for p and N. In contrast with Figure 3-9,
these histograms are not symmetric about the value k
m. The smooth
bell-shaped curve in Figure 3-10 depicts the graph of a function known
as a normal or Gaussian curve. The normal curve is a theoretical
construction and presents a special case of the Central Limit Theorem.
In this context, the central limit theorem guarantees that for sufficiently
large values of m, the histogram of the binomial probabilities is well
approximated by a certain Gaussian curve. In the symmetric case when
p
¼
1/2, the approximation is very good, even for small values of n.
When p
¼
1/2 the approximation gets better for larger values of m.
A widely used rule of thumb advises that Gaussian approximation for
p
6¼
6¼
1/2 is appropriate when 2mp
5 and 2m(1
p)
5.
The Gaussian approximation described above is useful, as it allows for a
natural transition from a qualitative to a quantitative differentiation
between the phenotypes. For example, if there are only three possible
colors of wheat kernels (e.g., white, pink, and red), there is no problem
referring to them as separate colors. With five different colors, the
qualitative description of the phenotypes becomes more challenging. In
our example above, we used ''white,'' ''light pink,'' ''pink,'' ''dark pink,''
and ''red'' to describe them. With seven, nine, or more different
phenotypes, coming up with appropriate names for all possible colors
becomes increasingly difficult. In such cases, it would be more
convenient to describe the trait quantitatively as a deviation from the
most commonly occurring characteristic.
The polygenic hypothesis, in combination with the central limit theorem,
provides what could be called a mechanistic insight to the observation
that certain quantitative traits exhibit approximately Gaussian
distribution, namely, that if m, the number of genes controlling the trait
is relatively large, the binomial histograms representing the actual
distribution of the trait is well-approximated by a Gaussian curve. More
details regarding this result will be presented in the next chapter. We
note that the converse may not be true. If a trait is assumed to be
