1
Lecture 7 - Evolution in Finite Populations, continued
III. Loss of genetic variation ("decay of heterozygosity") under genetic drift
Imagine a random mating population of one
self-fertile hermaphrodite, initially
heterozygous at the A locus, so fr(A1) = fr(A2) = 1/2.
What is the probability that the
single surviving offspring is heterozygous?
1/2.
What is the probability the grandchild is
heterozygous (1/2)(1/2).
How about the great grandchild (1/2)
3
.
What about after
t
generations? (1/2)
t
Now, let us introduce the concept of "identity by descent".
Two gene copies are
"identical by descent" if they are descended from the same gene copy
in a common
ancestor.
For example, if you and your brother got the same gene copy from your
mother, those gene copies are
identical by descent
.
See Figure
In a population of N diploid individuals, what is the probability that two randomly drawn
genes are identical by descent?
1/2N.
We call the probability that two genes are
identical by descent
f
.
The way to think about this is imagine the diploid N individuals produce an effectively
infinite number of gametes.
Draw one gamete.
The probability that any particular
individual is the parent is 1/N.
Now, what is the probability the gene copy in the gamete
is the one the individual inherited from (say) its mother?
1/2.
So,
f
= (1/N)(1/2) = 1/2N
Let F = the probability that two gene copies are the same allele, call this "identical in
state", i.e., both copies are A1 or both copies are A2.
You can think of F as the
probability of being homozygous, or "homozygosity".
There are two ways that two gene copies can be identical in state.
They can be I by D,
in which case they have to be I in S (barring mutation, which we will bar), with
probability 1/2N, or they can be not I by D, with probability 1 - 1/2N, and identical in
state, with probability F.
So, the probability that two gene copies are identical in state after one generation of
random mating is:
F' = 1/2N + (1 - 1/2N)F
Now, define heterozygosity H = 1 - F (i.e., the probability that two randomly drawn gene
copies are different in state, i.e., A1 and A2)
So, H' = 1 - F'