Biology Reference
In-Depth Information
These examples underscore the importance of maintaining crop
diversity to reduce the severity of infections and avoid massive crop
failures. Nature preserves genetic diversity, and we humans eliminate
it at our peril. We next create a mathematical model explaining why,
under certain conditions, genetic diversity is preserved.
Suppose we focus on a particular site of a particular chromosome.
We assume the gene at that locus has two possible alleles, A and a.How
does the proportion of alleles in the gene pool change over time? We
shall show that if certain idealized assumptions are met, then
equilibrium in the proportion of allele combinations (AA, Aa, or aa)is
reached after just one generation. We make the following assumptions:
1. The population is large (theoretically, infinite).
2. Mating is random.
3. All allele combinations have the same fitness (i.e., there is no
natural selection occurring).
4. There is no net mutation.
5. There is no immigration or emigration.
Although this set of assumptions may appear restrictive, they provide a
good approximation in many cases. We should not forget that whether
these assumptions are appropriate depends very much on the particular
genes being examined. For example, if one were looking at the genes for
''tall, dark, and handsome,'' the random mating assumption might not
apply. However, if one were considering a biochemical difference—one
not readily apparent—mating would almost certainly be random with
respect to that particular gene. In addition, mutation is always occurring,
but it occurs at a small rate that may be safely ignored for many
purposes. With these assumptions, we derive the result presented below,
called the Hardy-Weinberg Law of Genetic Equilibrium because of its almost
simultaneous publication in 1908 by British mathematician Godfrey H.
Hardy (1877-1947) (Hardy [1908]) and German physician Wilhelm
Weinberg (1862-1937) (for the English translation of Weinberg's paper,
see Boyer [1963], pp. 4-15).
Theorem (Hardy-Weinberg Law of Genetic Equilibrium). Let
assumptions 1-5 above be satisfied. Let P(aa), P(Aa), and P(AA)denotethe
proportions of the genotypes aa, Aa,andAA, respectively, in the parental
generation. Assume these proportions have the values P(aa)
¼
x,P(Aa)
¼
y,
and P(AA)
¼
1
x
y,where0
x
1, 0
y
1, and x
þ
y
¼
1. Then:
1. The proportion of the A and a alleles in the parental generation is
calculated, respectively, as
2
ð
1
x
y
Þþ
y
y
þ
2x
p
¼
and q
¼
:
(3-1)
2
2
