Evolution__3rd_Edition

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Linkage disequilibrium ( D)
r = 0.5
0 1 2 3 4
Generation t
5 6 7 8
Linkage disequilibrium tends to
decay over time
r = 0.2
r = 0.1
1
a′ = a − r ad + r bc
2 2
1
2 2
a′ =a − r (ad − bc)
Now, the expression (ad − bc) is simply equal to the linkage disequilibrium D. (This is
easy to confirm by multiplying out ad − bc from the definitions above of a, b, c, and d.)
If D = 0, i.e., if the genes are randomly associated, the haplotype frequencies are
constant: a′ =a. But if there is an excess of A 1 B 1 haplotypes, the excess decreases by
an amount rD per generation. The same relation holds true for any successive pair
of generations. We can see what is happening graphically if we substitute for a in the
equation:
a′=p 1 q 1 + D − rD
a′ −p 1 q 1 = (1 − r)D
The difference between a and p 1 q 1 is the amount of “excess” of the A 1 B 1 haplotype (i.e.,
the amount by which the frequency exceeds the random frequency). It is also equal
to the linkage disequilibrium (D = a − p 1 q 1 ). Therefore:
D′=(1 − r)D
CHAPTER 8 / Two-locus and Multilocus Population Genetics 201
Figure 8.2
Non-random associations between genes at different loci
are measured by the degree of linkage disequilibrium (D).
Recombination between the loci breaks down the linkage
disequilibrium, which decays at an exponential rate equal
to the recombination rate between the loci.
In the absence of selection and in an infinite random mating population, the amount
of linkage disequilibrium undergoes exponential decay at a rate equal to the recombination
rate between the two loci (Figure 8.2). In other words, the difference between
the actual frequency of a haplotype such as A 1 B 1 (a) and the random proportion (p 1 q 1 )
decreases each generation by a factor equal to the recombination rate between the loci.
Over time, any non-random genic associations will disappear; recombination will
destroy the association. The higher the rate of recombination, the more rapid the
destruction. The highest possible value of r is 1 /2, which is true when the two loci are on
different chromosomes. Genic associations persist longer for tightly linked loci on the
same chromosome, as we would intuitively expect.