38
M. Slatkin
is independent of the presence of an allele at the other locus. Yet the above results
show that independence between homologues is established in one generation, but
the independence of loci on different chromosomes is established more slowly.
These results can be related to human populations in a way that illustrates the
reason that linkage disequilibrium has become such an important part of human
population genetics. There are approximately 24,000 coding genes in the human
genome, which has a total recombination length of 30 Morgans or 3000 cM
(Lander 2001). Therefore, the average recombination distance between adjacent
coding genes is 1/8 cM or c = 0.00125. If D is initially nonzero between adjacent
coding genes, then D will decrease to 37% of its initial value in 1/0.00125 = 800
generations. The generation time in humans is about 25 years, so 800 generations
represent about 20,000 years. In other words, the extent of linkage disequilibrium
between adjacent coding genes in the human genome is expected to decay on
a timescale comparable to major events in the history of modern humans, i.e.,
the colonization of North America and the arrival of agriculture in Europe or the
domestication of horses, sheep, and cattle.
Another implication of the preceding theory gives us an additional reason for
being concerned with LD. We first recall that Eq. (2.5) tells us that D = 1 when
D takes either its maximum positive value or its minimum negative value. In that
case, the set of equations in (2.2) implies that at least one of the haplotypes has 0
frequency. The reverse is also true, namely, that if only three of four haplotypes
are present in a population, then necessarily D = 1. (Understanding this fact
allows a few students in population genetics courses to quickly answer examination
questions that occupy the rest of their classmates for a considerable time.)
Now consider what happens when a mutation occurs at a locus that was
previously monomorphic and is linked to another locus that is polymorphic. To be
specific, suppose the A/a locus is polymorphic and the B/b locus is initially fixed for
b (i.e., all individuals in the population carry the b allele). In a particular generation,
B appears in one copy as a new mutant. When B appears, it does so on a chromosome
that initially carries an A or an a, but it cannot appear on both types of chromosomes.
Suppose it appears on an A-bearing chromosome. In that generation, there are then
only three haplotypes, AB, Ab, and ab. The fourth haplotype (aB) is not present.
Therefore, when B appears as a new mutant allele, we know that D = 1 regardless
of the frequency of A and regardless of the recombination distance between A and B.
That is, when a new mutant allele appears, D = 1 between it and every polymorphic
locus on the same chromosome.
What happens to D after B appears depends on c. For loci sufficiently far apart on
the same chromosome that they are effectively unlinked (c = 1/2), D and hence D
decrease rapidly to zero. For more closely linked loci (c < 1/2), D and D decrease
more slowly, with the rate of decrease being slowest for very closely linked loci.
Thus, each new mutant will be expected to be in strong LD with alleles at closely
linked polymorphic loci for a long time, roughly 1/c generations. That prediction
is valid for every new mutant. Because mutants are appearing every generation,
the overall pattern of LD we expect is one in which D is large between closely
linked loci and then decreases with increasing recombination distance between loci.
M. Slatkin
is independent of the presence of an allele at the other locus. Yet the above results
show that independence between homologues is established in one generation, but
the independence of loci on different chromosomes is established more slowly.
These results can be related to human populations in a way that illustrates the
reason that linkage disequilibrium has become such an important part of human
population genetics. There are approximately 24,000 coding genes in the human
genome, which has a total recombination length of 30 Morgans or 3000 cM
(Lander 2001). Therefore, the average recombination distance between adjacent
coding genes is 1/8 cM or c = 0.00125. If D is initially nonzero between adjacent
coding genes, then D will decrease to 37% of its initial value in 1/0.00125 = 800
generations. The generation time in humans is about 25 years, so 800 generations
represent about 20,000 years. In other words, the extent of linkage disequilibrium
between adjacent coding genes in the human genome is expected to decay on
a timescale comparable to major events in the history of modern humans, i.e.,
the colonization of North America and the arrival of agriculture in Europe or the
domestication of horses, sheep, and cattle.
Another implication of the preceding theory gives us an additional reason for
being concerned with LD. We first recall that Eq. (2.5) tells us that D = 1 when
D takes either its maximum positive value or its minimum negative value. In that
case, the set of equations in (2.2) implies that at least one of the haplotypes has 0
frequency. The reverse is also true, namely, that if only three of four haplotypes
are present in a population, then necessarily D = 1. (Understanding this fact
allows a few students in population genetics courses to quickly answer examination
questions that occupy the rest of their classmates for a considerable time.)
Now consider what happens when a mutation occurs at a locus that was
previously monomorphic and is linked to another locus that is polymorphic. To be
specific, suppose the A/a locus is polymorphic and the B/b locus is initially fixed for
b (i.e., all individuals in the population carry the b allele). In a particular generation,
B appears in one copy as a new mutant. When B appears, it does so on a chromosome
that initially carries an A or an a, but it cannot appear on both types of chromosomes.
Suppose it appears on an A-bearing chromosome. In that generation, there are then
only three haplotypes, AB, Ab, and ab. The fourth haplotype (aB) is not present.
Therefore, when B appears as a new mutant allele, we know that D = 1 regardless
of the frequency of A and regardless of the recombination distance between A and B.
That is, when a new mutant allele appears, D = 1 between it and every polymorphic
locus on the same chromosome.
What happens to D after B appears depends on c. For loci sufficiently far apart on
the same chromosome that they are effectively unlinked (c = 1/2), D and hence D
decrease rapidly to zero. For more closely linked loci (c < 1/2), D and D decrease
more slowly, with the rate of decrease being slowest for very closely linked loci.
Thus, each new mutant will be expected to be in strong LD with alleles at closely
linked polymorphic loci for a long time, roughly 1/c generations. That prediction
is valid for every new mutant. Because mutants are appearing every generation,
the overall pattern of LD we expect is one in which D is large between closely
linked loci and then decreases with increasing recombination distance between loci.