1. Linkage Disequilibrium
Joe Mychaleckyj
Center for Public Health Genomics

2. Today we’ll cover… Haplotypes Linkage Disequilibrium Visualizing LD
HapMap

3. References Principles of Population Genetics,
Fourth Edition (Hardcover) by Daniel L. Hartl,
Andrew G. Clark (Author)
Genetic Data Analysis II Bruce S Weir x x x

4. References Statistical Genetics: Gene Mapping Through
Linkage and Association Eds Benjamin M.
Neale, Manuel A.R. Ferreira,
Sarah E. Medland, Danielle Posthuma

5. 2N (ie very large diversity possible)
SNP1 SNP2 SNP3
[A / T] [C / G] [A / G]
A C G
A C A
T G G
Haplotype: specific combination of alleles occurring (cis) on the same chromosome (segment of chromosome) N SNPs - How many Haplotypes are possible ?
2N (ie very large diversity possible)

6. Terminology
Haplotype: Specific combination (phasing) of alleles occurring (cis) on the same chromosomal segment
Linkage/Linked Markers: Physical co-location of markers on the same chromosome
Diplotype: Haplogenotype ie pair of phased haplotypes one maternally, one paternally inherited

7. Major Allele Freq: p(A) p(B) Minor Allele Freq: p(a) p(b)
SNP1 [ A / a ]
SNP2 [ B / b ]
Major Allele Freq: p(A) p(B)
Minor Allele Freq: p(a) p(b)
Independently segregating SNPs:
Haplotype Frequency p(ab) = p(a) x p(b)
LINKAGE EQUILIBRIUM
(How many haplotypes in total ?)
LINKAGE DISEQUILIBRIUM
Haplotype Frequency p(ab)≠ p(a) x p(b)

8. Linkage Disequilibrium
Non-random assortment of alleles at 2 (or more) loci
The closer the markers, the stronger the LD since recombination will have occurred at a low rate
Markers co-segregate within and between families

9. p(A)p(B)+p(a)p(B)=p(B){ p(A)+p(a)} = p(B)
* LINKAGE EQUILIBRIUM *
Not a Punnett
Square!
SNP2 Allele
B b
SNP1 Allele A a
p(A)p(B)
p(a)p(B)
p(A)p(b) p(A)
p(a)p(b) p(a)
p(B) p(b)
Example:
p(A)p(B)+p(a)p(B)=p(B){ p(A)+p(a)} = p(B)

10. Major Allele Freq: p(A) p(B) Minor Allele Freq: p(a) p(b)
SNP1 [ A / a ]
SNP2 [ B / b ]
Major Allele Freq: p(A) p(B)
Minor Allele Freq: p(a) p(b)
LINKAGE DISEQUILIBRIUM
Haplotype Frequency p(ab) = p(a) p(b) + D
(sign of D is generally arbitrary, unless comparing D values between populations or studies)
D: Lewontin’s LD Parameter (Lewontin 1960)

11. p(A)p(B)+D + p(a)p(B)-D =p(B){ p(A)+p(a)} = p(B)
* LINKAGE DISEQUILIBRIUM *
SNP2 Allele
B b
SNP1 Allele A a
p(A)p(B)+D
p(a)p(B)-D
p(A)p(b)-D p(A)
p(a)p(b)+D p(a)
p(B) p(b)
p(A)p(B)+D + p(a)p(B)-D =p(B){ p(A)+p(a)} = p(B)

12. Since p(ab) = p(a)p(b)+ D +D was used and D is +ve here, but arbitrary
b B
What is the LD ?
≠ 0
p(ab) ≠ p(a) p(b) a A
0.16
0.04
p(a)=0.20
0.14
0.66
p(B)=0.80
p(b)=0.30 p(B)=0.70
p(ab) = p(a) p(b) + D
0.16 = 0.2 x D
D = 0.1
Since p(ab) = p(a)p(b)+ D
+D was used and D is +ve here, but arbitrary
eg can relabel alleles A,B as minor

13. Range of D values (-ve to +ve)
D has a minimum and maximum value that depends on the allele frequencies of the markers
Since haplotype frequencies cannot be -ve
p(aB) = p(a)p(B) - D ≥ 0 D ≤ p(a)p(B)
p(Ab) = p(A)p(b) - D ≥ 0 D ≤ p(A)p(b)
These cannot both be true, so D ≤ min( p(a)p(B), p(A)p(b) )
p(ab) = p(a)p(b) + D ≥ 0 D ≥ -p(a)p(b)
p(AB) = p(A)p(B) + D ≥ 0 D ≥ -p(A)p(B)
These cannot both be true, so D ≥ max( -p(a)p(b), -p(A)p(B) )
* Similar equations if we had defined p(ab) = p(a)p(b) - D

14. Limits of D LD Parameter
Limits of D are a function of allele frequencies
Standardize D by rescaling to a proportion of its maximal value for the given allele frequencies (D')
D’ = D
Dmax

15. D’ (Lewontin, 1964) D’ = D / Dmax
Dmax = min (p(A)p(B), p(a)p(b)) D < 0
Dmax = min (p(A)p(b), p(a)p(B)) D > 0
Again, sign of D’ depends on definition
D’ = 1 or -1 if one of p(A)p(B), p(A)p(b), p(a)p(B), p(a)p(b) = 0
= Complete LD (ie only 3 haplotypes seen)
D’=1 or -1 suggests that no recombination has taken place between markers
Beware rare markers - may not have enough power/sample size to detect 4th haplotype

16. D’ Interpretation D=0 ; Dmax undefined D=Dmax =0.14 ; D’ = +1
b B
b B
0.06
0.14
p(a)=0.20
0.2
p(a)=0.20 a A a A
p(A)=0.80
0.1
0.7
P(A)=0.80
0.24
0.56
p(b)=0.30 p(B)=0.70
p(b)=0.30 p(B)=0.70
D=0 ; Dmax undefined
D=Dmax =0.14 ; D’ = +1
p(a) = 0.2
p(b)= 0.3
D’=1 (perfect LD using D’ measure
- No recombination between marker
- Only 3 haplotypes are seen

17. Creation of LD
Easiest to understand when markers are physically linked
Creation of LD
Mutation
Founder effect
Admixture
Inbreeding / non-random mating
Selection
Population bottleneck or stratification
Epistatic interaction
LD can occur between unlinked markers
Gametic phase disequilibrium is a more general term

18. A B A A b a B a A B A b a B a b SNP1 SNP1 SNP2 n=3 haplotypes
Recombination
n=2 haplotypes
A b a
a B
SNP1
SNP2
A B
A b
a B
a b
n=4 haplotypes

19. Destruction of LD Main force is recombination
Gene conversion may also act at short distances (~ 100-1,000 bases)
LD decays over time (generations of interbreeding)

20. Probability Recombination occurs = θ
SNP1
SNP2
Probability Recombination occurs = θ
Probability Recombination does not occur = 1-θ
Initial LD between SNP1 - SNP2: D0
After 1 generation
Preservation of LD:
D1 = D0(1-θ)
After t generations:
Dt = D0 (1- θ)t
NB: Overly simple model - does not account for allele frequency drift over time

21. Dt = D0 (1-θ)t

22. r2 LD Parameter (Hill & Robertson, 1968)
r2 = D2
p(a)p(b)p(A)p(B)
Squared correlation coefficient varies
Frequency dependent
Better LD measure for allele correlation between markers - predictive power of SNP1 alleles for those at SNP2
Used extensively in disease gene or phenotype mapping through association testing

23. r2 Interpretation D=0 ; Dmax undefined D=Dmax =0.14 ; D’ = +1 r2 = 0
b B
b B
0.06
0.14
p(a)=0.20
0.2
p(a)=0.20 a A a A
p(A)=0.80
0.1
0.7
p(A)=0.80
0.24
0.56
p(b)=0.30 p(B)=0.70
p(b)=0.30 p(B)=0.70
D=0 ; Dmax undefined
D=Dmax =0.14 ; D’ = +1
r2 = 0
r2 = 0.14/0.24 = 0.58
p(a) = 0.2
p(b) = 0.3
r2 ≠ 1 Correlation is not perfect, even though D’ = 1
r2 = 1 if D’ = 1 and p(a) = p(b) = 0.3

24. r2 Interpretation
p(a) = 0.3
p(b) = 0.3
Only 2 haplotypes:
r2 = 1 Correlation is perfect
D’ =1 (less than 4 haplotypes)
p(a) = p(b) (= 0.3 in this example)
r2=1 when there is perfect correlation between markers and one genotype predicts the other exactly
Only 2 haplotypes present
D’ = 1 ≠> r2 = 1
No recombination AND markers must have identical allele frequency
SNPs are of similar age
Corollary
Low r2 values do not necessarily = high recombination
Discrepant allele frequencies

25. Common Measures of Linkage Disequilibrium
Recombination
Correlation
Other LD Measures exist, less common usage

26. Visualizing LD metrics

27. SNP
| D’ |
SNP1
SNP2
SNP3
SNP4
SNP5
SNP6
1.0
0.8
0.6
0.2
Not usually worried about sign of D’

29. Haploview: TCN2 (r2)

30.
Launched October 2002

31. International HapMap Project
Initiated Oct 2002
Collaboration of scientists worldwide
Goal: describe common patterns of human DNA sequence variation
Identify LD and haplotype distributions
Populations of different ancestry (European, African, Asian)
Identify common haplotypes and population-specific differences
Has had major impact on:
Understanding of human popualtion history as reflected in genetic diversity and similarity
Design and analysis of genetic association studies

32. HapMap samples
90 Yoruba individuals (30 parent-parent-offspring trios) from Ibadan, Nigeria (YRI)
90 individuals (30 trios) of European descent from Utah (CEU)
45 Han Chinese individuals from Beijing (CHB)
44 Japanese individuals from Tokyo (JPT)

33. Project feasible because of:
The availability of the human genome sequence
Databases of common SNPs (subsequently enriched by HapMap) from which genotyping assays could be designed
Development of inexpensive, accurate technologies for highthroughput SNP genotyping
Web-based tools for storing and sharing data
Frameworks to address associated ethical and cultural issues

34. HapMap goals Define patterns of genetic variation across human genome
Guide selection of SNPs efficiently to “tag” common variants
Public release of all data (assays, genotypes)
Phase I: M markers in 269 people
1 SNP/5kb (1.3M markers)
Minor allele frequency (MAF) >5%
Phase II: M markers in 270 people

38. HapMap publications
The International HapMap Consortium. A Haplotype Map of the Human Genome. Nature 437,
The International HapMap Consortium. The International HapMap Project. Nature 426,
The International HapMap Consortium. Integrating Ethics and Science in the International HapMap Project. Nature Reviews Genetics 5,
Thorisson, G.A., Smith, A.V., Krishnan, L., and Stein, L.D. The International HapMap Project Web site. Genome Research,15:

39. ENCODE project
Aim: To compare the genome-wide resource to a more complete database of common variation—one in which all common SNPs and many rarer ones have been discovered and tested
Selected a representative collection of ten regions, each 500 kb in length
Each 500-kb region was sequenced in 48 individuals, and all SNPs in these regions (discovered or in dbSNP) were genotyped in the complete set of 269 DNA samples

40. Comparison of linkage disequilibrium and recombination for two ENCODE regions
Nature 437,

41. LD in Human Populations

42. Haplotype Blocks
N SNPs = 2N Haplotypes possible, ie very large diversity possible
But: we do not see the full extent of haplotype diversity in human populations
Extensive LD especially at short distances eg ~20kbases.
Haplotypes are broken into blocks of markers with high mutual LD separated by recombination hotspots
Non-uniform LD across genome

43. Haplotype Blocks
Haplotype blocks: at least 80% of observed haplotypes with frequency >= 5% could be grouped into common patterns
Whole Genome Patterns of Common DNA Variation in Three Human Populations, Science 2005, Hinds et al.

44. Length of LD spans r2
We fitted a simple model for the decay of linkage disequilibrium to windows of 1 million bases distributed throughout the genome. The results of model fitting are summarized for the CHB+JPT analysis panel, by plotting the fitted r2 value for SNPs separated by 30 kb. The overall pattern of variation was very similar in the other analysis panels.

45. Example: Large block of LD on chromosome 17
Cluster of common (frequent SNPs In high LD)
518 SNPs, spanning 800 kb
25% in EUR, 9% in AFR, missing in CHN
Genes:
Microtubule-associated protein tau
Mutations associated with a variety of neurodegeneartive disorders
Gene coding for a protease similar to presenilins
Mutations result in Alzheimer’s disease
Gene for corticotropin-releasing hormone receptor
Immune, endocrine, autonomic, behavioral response to stress

46. Chromosome 17 LD Region Prevalent inversion in EUR human population
~25%