Genomics- The Science and Technology Behind the Human Genome Project. Charles R. Cantor, Cassandra L / genomics1-10 / 6

INTERVAL MAPPING

185

Thus the average contribution from one child to the ELOD is the

sum

of these two

cases weighted by their expected frequency.

Since

recombination

across

10

cM

occurs

only 10% of the time,

ELOD(0.1)

0.9 log

0.9

0.1 log

0.1

0.5

0.5

ELOD(0.1)

0.23 0.07 0.16

Thus observation of cosegregation of A and D adds to the probability of linkage, while

observation of separation of A and D subtracts from the evidence for linkage.

What we need to do is develop the tools to assess the statistical significance of a par-

ticular ELOD score. Since some markers

will appear to cosegregate by

chance

in

any

study with a relatively small number of affected individuals, there is always a chance of

seeing a significantly positive LOD score, simply because of the random fluctuations. A

near consensus in human genetics is that an observed LOD of 3.0 or higher is required

before the probability of purely accidental linkage can be reduced to the point where few

errors are made. For the example just described, the number of individuals segregating D

with unambiguous pedigrees that would have to be combined to generate a LOD score of

3.0 can be estimated as 3/0.16

18. For common inherited diseases this is not a problem,

but for very rare diseases it may be extremely difficult to find 18 genetically informative

individuals for a particular marker with an unambiguous diagnosis.

Note that several constraints apply to the linkage analysis described above. One must

have access to a parent with known phase between A

and D. The marker A to

be tested

for linkage must have useful heterozygosity. The diagnosis of D must be unambiguous in

all the individuals tested. Note that failing to diagnose an individual who is carrying D (a

false negative) does not hurt the analysis,

since in this case the individual and

the

parent

are not scored. However, misclassifying an individual as carrying D instead of d (a false

positive) causes serious problems because it will weaken the evidence about which alleles

at other loci are cosegregating with D.

INTERVAL

MAPPING

Once a

genetic map is available for a region of

interest,

the

process

of

linkage

analysis

can be made more powerful by examining several markers simultaneously. We will con-

sider the simplest possible case, illustrated in Figure 6.19

a . As in the previous discussion

of simple linkage analysis, we will calculate the average contribution of the LOD

score

from a single, informative individual inheriting a disease allele D. We wish to

test

a re-

gion of the genome containing two linked loci with markers A and B to see if the disease

allele D lies between them or is unlinked. (Here we ignore the case that it might be linked

to A and B but lie outside them rather than between them.)

Suppose that the loci containing A and B are 20 cM apart. This is a reasonable model

for how human genetic maps are used in average regions of the genome.

AB

0.2. First

we calculate the possible contributions from

a parent carrying D to a child,

also

carrying

D, if there is no linkage between A and B with D (Fig. 6.19

b ). Since A and B are on

the

same chromosome, D, if unlinked, they must

lie on a different chromosome. Assuming

that the parent is heterozygous and informative at all these loci, there are four possible

contributions from the parent to the child (Fig. 6.19

c ).

INTERVAL

MAPPING

187

If no recombination between A and B occurs (80% probability for markers 20 cM

apart), the child will either inherit ABD (0.4 odds) or abD (0.4 odds). If recombination

between A and B occurs (20% probability), the child will inherit either AbD (0.1 odds) or

aBD (0.1 odds).

If D is linked and

located between A and B,

assuming

the phase of the parent is

known, the two homologous chromosomes of the parent

carry alleles ADB and adb, as

shown in Figure 6.19

d . In principle, D may lie anywhere between A and B and the actual

position of D is a variable that must be included in the calculations. Here we will consider

the simple case where D lies midway between A and B. Assuming that the recombination

frequency is uniform in this

region of the chromosome, we can then place D 10 cM from

A and 10 cM from B (Fig. 6.19

e ). There are four possible sets of alleles that can be

passed from this parent to a child who inherits D (Fig. 6.19

f ). These are as follows:

ADB: resulting from no recombination between A and D, and no recombination be-

tween D and B (odds are 0.9

0.9).

ADb: resulting from no recombination between A and D but recombination has oc-

curred between D and B (odds are 0.9

0.1).

aDB: recombination

has occurred between A

and D, but

no recombination has oc-

curred between D abd B (odds are 0.1

0.9).

aDb (a double crossover event):

recombination has occurred both between A and D

and between D and B (odds are 0.1

0.1).

Thus the same four possible genotypes can

arise either with or without linkage.

However, the odds of particular genotypes vary

considerably in

the two

cases.

For

the

four possible offspring:

Alleles

ADB

ADb

aDB

aDb

Odds (linked/unlinked)

0.81/0.4

0.09/0.1

0.09/0.1 0.01/0.4

The ELOD for a single statistically representative child can be calculated from these re-

sults by realizing that if there is linkage, the probabilities of seeing the four patterns of al-

leles are 0.81, 0.09, 0.09, and 0.01, respectively. Thus the ELOD is given by

ELOD(0.2)

0.81 log

0.81

0.09 log

0.09

0.09 log

0.09

0.4

0.1

0.1

0.01 log

0.01

0.4

ELOD(0.2)

0.25 0.004 0.004 0.02 0.23

Figure 6.19

Interval mapping to test the hypothesis that a disease allele D is located equidistant

between two linked markers A and B, separated by 20 cM.

(a) Map of the test region.

(b) Parental

chromosomes if D is unlinked to A and B.

(c) Possible chromosomes inherited by an offspring car-

rying the disease allele in the absence of linkage.

(d) Parental chromosomes if D lies between A and

B. (e) Map location assumed for D for the example calculated in the text.

(f) Possible parental con-

tributions to an offspring inheriting the disease allele.

188

GENETIC

ANALYSIS

Note that this ELOD is larger in the case of interval mapping than in the simple case of

linkage analysis we considered earlier. The number of informative individuals that would

have to be examined to achieve a LOD score of 3 would be 3/0.23

14.

FINDING GENES BY GENETIC MAPPING

What is done, in practice, is to repeat the kinds of calculations previously described with

all

possible

values of

as

a variable using actual genotype data from real

families. For

simple linkage analysis the sorts of results obtained are shown schematically in Figure

6.20. These yield

the expected LOD score as a function of

. The critical results are the

maximum

LOD

value

and the confidence limits on possible values of

. With interval

mapping, the results are more complex, but the basic kind of information obtained is sim-

ilar, as shown by the example in Figure 6.21. For details, see Ott (1991) and Lalouel and

White (1966).

In a typical case, no a priori information exists about the putative location of a gene of

interest. To have a reasonable chance of finding it, one must test the hypothesis that it lies

near (or between) any of about 150 informative markers. This will subdivide the genome

into

intervals

spaced

about 20 cM apart. Each marker must be tested with a sufficient

number of informative individuals to achieve a LOD score of 3.0 or higher if that particu-

lar

marker is

linked

to the gene. With present technology this search is often carried out

one

marker

and

one

individual

at

a

time. It is easy to estimate that around 150

markers

40

to

60 individuals

(parents and offspring) must be tested in

ideal

cases

where parental phase is known and markers are very informative. If the analysis is carried

out by ordinary Southern blotting (Chapter 3) of DNA bands 6000 to 12,000 gel elec-

trophoresis lanes have to be examined by hybridization to afford a reasonable chance of

finding a gene, and this is an ideal case! Schemes have recently been described that can

reduce the workload by an order of magnitude through the use of pools of samples

(Churchill et al., 1993; see also Chapter 9 for examples of the power of pooling).

If a LOD score of 3.0 or greater is achieved, there is

a reasonable chance

that

the

correct

location

of

the

disease

gene

of

interest has been

found.

What

is usually done

is to celebrate, publish a preliminary report, and fend off overoptimistic members of the

press or families segregating the disease

of interest who confuse the first sighting of

a

gene location with the identification of the actual disease gene itself. Knowing the loca-

tion of a disease gene

does provide improved diagnostics for

the disease

but, initially,

only in those families where the phase of the disease allele and nearby markers is known.

FINDING GENES BY GENETIC MAPPING

189

Furthermore, at a 10 cM distance,

the amount of recombination between the marker and

the disease allele in each meiosis

is still 10%, so the

accuracy of any genetic testing is

quite limited. More accurate approaches are outlined in Box

6.3 and Box 6.4.

LINKAGE DISEQUILIBRIUM

191

This is a very dangerous practice,

since if one starts with a sufficient number of families,

it will almost always to possible to achieve an alluring LOD score by selectively choosing

among them. Clearly the appropriate statistical tests must be employed to discount the re-

sulting LOD score against such selective manipulation of the data. The real issue is not

whether one can increase a LOD score by dropping a family with a negative contribution.

The issue is whether the magnitude of the increase in LOD is sufficient to justify the ad-

ditional parameterization implicit in dropping this family. A much safer procedure is to

collect more families and try additional markers near the ones that have already shown a

hint of linkage if not yet compelling evidence for linkage. When this has been done, some

LODs of 2.0 eventually have produced the desired gene; others have faded into oblivion.

Eventually genetic linkage studies may narrow down the location of a gene to a 2 cM re-

gion. However, in such an interval of the genome, there may be a single gene or more than

80. It is very difficult to use conventional linkage analysis to narrow the location further. The

available families are likely to have only a limited number of recombination events in the re-

gion of interest because they represent just a few generations, which means any recombina-

tions seen must have occurred recently. A 2 cM localization means that already 50 informa-

tive meioses have been found. It is

usually not efficient to keep gathering more families at

this point, although it is efficient to keep trying to find additional informative markers, since

these can narrow down the location of any recombination events.

LINKAGE

DISEQUILIBRIUM

In fortunate cases, a variant on linkage analysis can be used to home in on the likely loca-

tion of a disease gene once it has

been assigned to a mapped

region of

a

chromosome.

Suppose

that most affected individuals

have the same disease allele

D. This

is

the case, for

Venezuela; it

is also the case with individuals affected with

sickle cell disease, and with

most individuals

of northern European descent afflicted with severe

cystic fibrosis. In such

If

a disease

allele

arose once

by mutation on

a single chromosome, it will be created

in the

context of

a particular

haplotype

(Fig.

6.24). The chromosome that first carries

the disease will have a particular set of polymorphic

markers. It will have a particular ge-

netic background. As

this

chromosome

is

passed

through

many generations of offspring, it

192

GENETIC ANALYSIS

will suffer frequent meiotic recombination events. These will tend to blur the memory of the

original haplotype of the chromosome; they will average out the original genetic background

with the general distribution of markers in the human population. However, those

markers

very close to the disease gene will tend, more likely than average, to retain the haplotype of

the original chromosome because, as the distance to the disease gene shrinks, it becomes less

likely that recombination events will have occurred in this particular location.

Humans are an outbred population.

Most alleles

were

established when

the

species

was established, and a sufficient number of generations

has

passed

since

then

that

fre-

quent recombination events have occurred between any pair

of

neighboring

loci resolved

on our genetic maps. For this reason the distribution of particular haplotypes in neighbor-

ing loci in the population (as opposed to particular families) should be close to random.

Consider the case shown in Figure 6.25, for two neighboring loci with two alleles each.

Within the population, the frequencies

X of

the

alleles

at

a

particular

locus must sum

to

1.0.

X a

X A

1.0

X b

X B

1.0

The frequencies of particular haplotypes,

f, should be given by simple binomial statistics:

fAB X A X B

fAb

X A X b

faB X a X B

fab X a X b

Deviations from these results, measured,

for example,

as

fAB

observed

fAB

calculated,

are evidence for linkage disequilibrium, and they indicate that

the

individuals

examined

are not a random sample of the population. Note, however, that deviation of allele fre-

quencies from those expected by binomial statistics may

have other causes besides ge-

netic linkage. Deviations can reflect improper sampling of the population, or they can re-

flect actual functional association between specific alleles. The latter process could occur,

for example, if the protein products of the two genes in question actually interacted bio-

chemically. (For further discussion see Ott, 1991.)

To search for a gene by linkage disequilibrium, one does not examine families segre-

gating a disease allele D. Instead, one looks across a broad spectrum of the population for

unrelated individuals who have the disease allele D. If evidence

for

linkage

disequilib-

rium is found, it reflects recombinations

along the chromosome all the way

back

in

time

to the original founder. Since this may extend back hundreds of years, more than ten gen-

erations may be involved, and thus the number of recombination

events

viewed

will

be

much greater than possible with any contemporary family. In the case of linkage disequi-

librium, we expect to see the general results shown in Figure 6.26. There will be a gradi-

ent of increasing deviation from equilibrium as the neighborhood of the disease gene is

reached because of the diminishing likelihood of recombination

events

occurring

in

an

ever-shrinking region.

COMPLICATIONS IN LINKAGE DISEQUILIBRIUM AND GENETIC MAPS IN GENERAL

193

Figure 6.27

complications

that can obscure evidence for linkage disequilibrium.

(a) Recombina-

tion hotspots near the disease gene.

(b) Mutation hot spots near the disease gene.