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Introduction linkage analysis,
Genetic markers, mapping
functions
Lecture 3
Background Readings: Chapter 5 & 6 (190-193) of An introduction
to Genetics, Griffiths et al. 2000, Seventh Edition.
This class has been edited from several sources. Primarily from Terry Speed’s homepage at
Stanford and the Technion course “Introduction to Genetics” and several other courses as
specified on some slides. Changes made by Dan Geiger.
.
Purpose of human linkage analysis
To obtain a crude chromosomal location of the gene or genes
associated with a phenotype of interest, e.g. a genetic disease
or an important quantitative trait.
Examples: Cystic fibrosis (found), Diabetes, Alzheimer, and
Blood pressure.
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Linkage Strategies I
Traditional (from the 1980s or earlier)

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

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Linkage analysis on pedigrees
Association studies: candidate genes
Allele-sharing methods: Affected siblings
Animal models: identifying candidate genes
Cell – hybrids
Newer (from the 1990s)


Focus on special populations (Finland, Hutterites)
Haplotype-sharing (many variants)
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Linkage Strategies II
On the horizon (here)


Single-nucleotide polymorphism (SNPs)
Functional analyses: finding candidate genes
Needed (starting to happen)


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New multilocus analysis techniques, especially
Ways of dealing with large pedigrees
Better phenotypes: ones closer to gene products
Large collaborations
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Horses for courses
 Each
of these strategies has its domain of
applicability
 Each of them has a different theoretical basis
and method of analysis
 Which is appropriate for mapping genes for a
disease of interest depends on a number of
matters, most importantly the disease, and
the population from which the sample comes.
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The disease matters
Definition (phenotype), prevalence, features
such as age at onset
Genetics: nature of genes (Penetrance),
number of genes, nature of their contributions
(additive, interacting), size of effect
Other relevant variables: Sex, obesity, etc.
Genotype-by-environment interactions:
Exposure to sun.
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Example: Age at onset
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Example: Y-linked disease
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The population matters
History: pattern of growth, immigration
Composition: homogeneous or melting pot, or in
between
Mating patterns: family sizes, mate choice
Frequencies of disease-related alleles, and of
marker alleles
Ages of disease-related alleles
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Immigration
106 years
105 years
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Complex traits
Definition vague, but usually thought of as having multiple,
possibly interacting loci, with unknown penetrances; and
phenocopies.
Affected only methods are widely used. The jury is still out on
which, if any will succeed.
Few success stories so far.
Important: heart disease, cancer susceptibility, diabetes, …are
all “complex” traits.
We focus more on simple traits where success has been
demonstrated very often. About 6-8 percent of human
diseases are thought o be simple Mendelian diseases.
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Design of gene mapping studies
How good are your data implying a genetic
component to your trait? Can you estimate the size
of the genetic component?
Have you got, or will you eventually have enough
of the right sort of data to have a good chance of
getting a definitive result?
Power studies.
Simulations.
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Genotyping
A person is said to be typed if its markers have been genotyped.
Choice of markers: highly polymorphic preferred.
Heterozygosity and polymorphism information content
(PIC) value are measures commonly used.
Reliability of markers important too
Good quality data critical: errors can play a surprisingly
large role.
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Preparing genotype data for analysis
Data cleaning is the big issue
here.
Need much ancillary
data…how good is it?
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Analysis
A very large range of methods/programs are
available.
Effort to understand their theory will pay off
in leading to the right choice of analysis
tools.
Trying everything is not recommended, but
not uncommon.
Many opportunities for innovation.
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Interpretation of results of analysis
An important issue here is whether you have
established linkage. The standards seem to be
getting increasingly stringent.
What p-value or LOD should you use?
Dealing with multiple testing, especially in the
context of genome scans and the use of
multiple models and multiple phenotypes, is one
of the big issues. E.g., Bonferroni correction.
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References
Related topics (not covered in this course): Exclusion mapping,
homozygosity mapping, variance component methods, twin
studies, and much more.
Some of these topics plus others are covered in two books:
Handbook of Human Genetic Linkage by J.D. Terwilliger & J. Ott
(1994) Johns Hopkins University Press. Ordered, not available at
the library.
Analysis of Human Genetic Linkage by J. Ott, 3rd Edition (1999),
Johns Hopkins University Press.
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Problem with standard P-values
If a single test was to be employed to test a null hypothesis, using 0.05 as
the significance level and if the null hypothesis was actually true; the
probability of reaching the right conclusion (i.e., not significant) is 0.95.
If two such hypotheses were tested, then the probably of reaching the right
conclusion (i.e., not significant) on both occasions would be 0.95X0.95 =
0.90.
If more hypotheses (n) were tested and if all of them were in fact true, the
probability of being right on all occasions would decrease substantially
(0.95n).
In other words, the probability of being wrong at least once (or getting a
significant result erroneously) would increase drastically (1-0.95n).
Put simply, by running more tests on a given data set, there is an
increasing likelihood of getting a significant result by chance alone
Source: http://www.edu.rcsed.ac.uk/statistics/the%20bonferroni%20correction.htm
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The Bonferroni Correction for Non-statisticians
The Bonferroni correction for multiple significance testing is simply to
multiply the p value by the number of tests k carried out. The
corrected value kp is then compared against the level of 0.05 to decide if
it is significant. If the corrected value is still less than 0.05, only then is
the null hypothesis rejected.
Source: http://www.edu.rcsed.ac.uk/statistics/the%20bonferroni%20correction.htm
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Some Problems with the Bonferroni Correction [1]
1.
This test is for independent tests not for depended ones.
2.
If one carries out multiple tests on a single set of data, the interpretation of
a single relationship between two variables (or the p value) would actually
depend on how many other tests were performed.
3.
Perhaps too cautious. This means that significant results are lost and the
power of the study is reduced.
4.
If Bonferroni correction were to be made universal, to make results
significant, authors would not include many other tests they would have
done with non-significant results and thus would not apply Bonferroni to
same extent they should.
Also for tests published in other papers on the same set of patients or tests
done subsequently would need to be corrected taking into account the
number of previous tests.
Source (modified from): http://www.edu.rcsed.ac.uk/statistics/the%20bonferroni%20correction.htm
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When to use Bonferroni Correction ?
Because of the above problems due to the disagreements among statisticians
over its universal use, the use of the Bonferroni correction may best be
limited to instances like
•
a group of cases and controls subjected to a number of independent tests
of associations with different biological parameters
•
the same test being repeated in many subsamples, such as when stratified
by age, sex, income status, etc.
Even in these instances, if there is a biological explanation for the null
hypothesis to be rejected and only the non-corrected p value is significant,
but kp is not, one is allowed to conclude (with appropriate explanations,
of course!), the significant nature of the findings.
Source: http://www.edu.rcsed.ac.uk/statistics/the%20bonferroni%20correction.htm
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References to Bonferonni and other multiple test
1. Perneger, T.V. What’s wrong with Bonferroni
adjustments. BMJ, 1998. 316(7139):p. 1236-1238.
2. Bender, R. and S. Lange, Multiple test procedures other
than Bonferroni’s deserve wide use. BMJ, 1999.
318(7138):p.600-601.
3. Sankoh, A.J., M.F. Huque, and S.D. Dubey, Some comments
on frequently used multiple endpoint adjustment methods
in clinical trials. Stat Med, 1997. 16(22):p.2529-2542.
Source: http://www.edu.rcsed.ac.uk/statistics/the%20bonferroni%20correction.htm
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Replication of results
This has recently become a big issue with
complex diseases, especially in psychiatry.
Nature Genetics suggested in May 1998 that
they will require replication before publishing
results mapping complex traits.
Simulations by Suarez et al (1994) show that
sample sizes necessary for replication may be
substantially greater than that needed for first
detection.
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Chromosome Description Types
Our description of chromosomes has three distinct sources:
•the genetic description, derived from studies of the inheritance of
traits;
•the morphological description, derived from microscopic
examination of chromosomes; and
•the molecular description, derived from analysis of the DNA of
chromosomes.
Each description can be related experimentally to the others.
Source (modified from): http://opbs.okstate.edu/~melcher/MG/fMG01.html
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The Genetic Chromosome
The genetic chromosome is represented by a genetic map.
•Genetic maps are unbranched lines or circles with
marks indicating the relative positions of genetic
markers.
•Genetic markers are genetically determined traits or
characters that are polymorphic in the population being
studied. Polymorphic means that at least two forms of
the trait occur in the population.
•If two markers are genetically linked, they are on the
same genetic map, also called a linkage map. The set
of all markers on the same linkage map is called a
linkage group.
•If two markers are not genetically linked they are said
to be unlinked markers and belong to different linkage
groups.
Source (modified from): http://opbs.okstate.edu/~melcher/MG/fMG01.html
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A linkage map of tomatoes chromosomes from 1952
Picture from L.A. Butler.(Griffiths et al, pp.155).
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The Morphological Chromosome
Chromosome appearance varies with stage of the cell cycle and with cell type.
•Interphase nuclei have distinct regions discernable by staining.
•Metaphase chromosomes
•exhibit a condensed structure and
•can be distinguished by size and chromosome banding.
•Polytene chromosomes occur in insect salivary gland cells.
•Lampbrush chromosomes are observed during amphibian development.
Source (modified from): http://opbs.okstate.edu/~melcher/MG/fMG01.html
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‫)‪Fluorescent In Situ Hybridization (FISH‬‬
‫סימון כרומוזומים עם גלאי דנ"א פלורסנטי ספציפי לגן הנמצא בעותק יחיד בתא‪.‬‬
‫הכרומוזומים נראים בשלב לאחר הכפלת הדנ"א‪ ,‬בזמן המטפזה‪.‬‬
‫נראות ‪ -2‬כרומטידות אחיות בכל כרומוזום‪.‬‬
‫‪28‬‬
‫צביעת כרומוזומים ב‪ FISH -‬על ידי מספר רב של גלאים‪.‬‬
‫‪29‬‬
The Molecular Chromosome
Several kinds of maps are useful in understanding the molecular
description of a chromosome:
AAGATCCCGATCCGATTAGCTTAG
1. Restriction maps locate the relative positions of “specific
sequences” by selected restriction enzymes. Main examples
for “specific sequences” are RFLP (restriction fragment
length polymorphism), and VNTR (variable naumber
tandem repeats).
2. Conting maps locate the relative positions of cloned sequences
from a library.
3. Nucleotide sequences represent the ultimate molecular map,
being the linear order of nucleotides in the nucleic acid.
Source (modified from): http://opbs.okstate.edu/~melcher/MG/fMG01.html
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Linkage map of human chromosome 1,
correlated with chromosome banding
pattern.
Distances are given in centimorgans.
Total length is 356 cM – the longest
human chromosome.
Figure 5-16 in Griffiths et al, pp.155. Taken from B.R. Jasney et
al.,Science, September 30, 1994.
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Restriction Fragment Length
Polymorphism
Bacterial restriction enzymes cut DNA at specific target sequences
that exist by chance on other organisms (e.g. human).
Homolog 1
Homolog 2
3kb
2kb
1kb
Extend of probe
The probe (say AACCTT) cuts the second Homolog (say the
middle of TTGGAA) into two pieces. It does not cut the first
Homolog because the target sequence AACCTT is absent. These
represent two alleles at that locus. There are thousands of RFLP
markers.
Measuring the alleles uses electric field to separate the fragments
according to their molecular weights (Using Southern blotting).
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RFLPs in mapping
If an individual is heterozygous for presence and absence (+/-) of
that target sequence, then this locus can be used for mapping, like
any other genetic marker.
Consider the two individuals:
Homolog 1
Homolog 2
Homolog 1
Homolog 2
D
d
D
3kb
2kb
1kb
3kb
3kb
d
Half the progeny would show three fragments when probed and
half only one fragment, following Mendel’s first law of equal
segregation.
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Variable Number Tandem Repeats
(VNTRs)
Some locations have different number of repeats of the same
basic unit. Say AAAAA versus AAA. These can be regarded
as two alleles. A probe that cuts after the first three A’s can
distinguish long from short.
Homolog 1
Homolog 2
D
d
Probe
As before, if an individual is heterozygous for Long and short
(L/s) target sequences, then this locus can be used for mapping.
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
Measuring genetic distance:
Haldane’s mapping function
A natural measure of genetic distance is the expected number of
recombinants, denoted by m. Let  denote the expected number of
crossovers.
We assume that m = 0.5  because the expected number of
recombinants is believed to equal half the expected number of
crossovers . Can we measure m ?
The observed Recombination Fraction RF (just r for short) is
thus given by: r = 0.5 Prob(no crossover) = 0.5(1 - e-2m )
Inverting the formula yields Haldane’s mapping function:
m = -(1/2) ln(1-2r).
Recall that ln(1-x)=x for small x, hence m  r for small m. In practice
10 centi morgan (r =0.1) is considered small. So small m’s are additive. 35
The Poisson Distribution
Suppose a (rare) event of interest occurs with rate  (per length or time units).
For example number of dead birds along a highway. Number of births in one hour.
Or the number of crossovers along a chromosome.
If we assume that:
1. For an arbitrarily small unit  of distance (time) the probability of observing
an event is approximately equal to , and equals virtually zero for more than
one event.
2. The rate  is constant over the entire region.
3. The number of events occurring in one interval is independent of the number
of events occurring in a previous disjoint interval,
then, the probability for the number of events i occurring at an interval of length 1
is the Poisson distribution given by:
e   ( )i e 2 m (2m)i
f (i) 

i!
i!
In our case  =2 m.
;
e 2 m (2m) 0
f (0) 
 e 2m
0!
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Additivity for small regions
Consider three closely linked loci:
rdf = rde(1- ref) + ref(1-rde) If there is no interference.
rdf = rde+ ref – 2 rde ref = 0.06 + 0.08 –2(0.0048)  0.14
So in practice, for short chromosome segments,
map distance = observed recombination fraction,
i.e., 4% observed recombination = 4cm = 8% crossover events.
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Non-Additivity for longer regions
Consider three loci each separated by more than 10cm.
If there is no
interference
rac does not equal rab+rbc =40 but rather rac = rab+rbc – 2 rab rbc
Namely, rac = 0.2+0.2-2(0.04) = 0.32
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Chaismata Interference
Morgan’s data. Breed Drosophila so as to obtain female parental
gametes v+ cv ct and v cv + ct + and breed these females with triple
recessive males.
The female gametic genotypes are shown out of a sample of 1448
flies:
v
cv+
ct +
580
v+
cv
ct
592
v
cv
v+
cv +
ct
40
v
cv
ct
89
v+
cv +
ct +
94
v
v+
cv
+
cv
ct
+
45
rv,cv= (45+40+89+94)/1448= 18.5%
rv,ct= (89+94+3+5)/1448 = 13.2%
rct,cv= (45+40+3+5)/1448 = 6.4%
v
13.2
ct
3
ct +
5
cv
ct
6.4
Can we conclude the order
just by inspecting the table ?
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Interference
a
c
b
13.2
6.4
rac does not equal rab+rbc=0.196
but maybe rac = rab+rbc – 2 rab rbc, assuming no interference.
mac = 0.132+0.064 -2(0.132 * 0.064 ) = 0.1943 (Haldane’s mapping
function)
However, we observed recombination fraction rac between a and
c is 0.185 , namely, less recombinations then expected, even if
we take (independent) double crossovers into account.
Use Kosambi’s mapping function or other that take interference
into account.
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