Introduction to SNP and Haplotype Analysis

1
Introduction to SNP and Haplotype Analysis
Yao-Ting Huang
Kun-Mao Chao
Algorithms and Computational Biology
Lab, Department of Computer Science Information
Engineering, National Taiwan University, Taiwan.
2
Genetic Variations

• The genetic variations in DNA sequences (e.g.,
  insertions, deletions, and mutations) have a
  major impact on genetic diseases and phenotypic
  differences.
• All humans share 99 the same DNA sequence.
• The genetic variations in the coding region may
  change the codon of an amino acid and alters the
  amino acid sequence.

3
Single Nucleotide Polymorphism

• A Single Nucleotide Polymorphisms (SNP),
  pronounced snip, is a genetic variation when a
  single nucleotide (i.e., A, T, C, or G) is
  altered and kept through heredity.
• SNP Single DNA base variation found gt1
• Mutation Single DNA base variation found lt1

C T T A G C T T
C T T A G C T T
99.9
94
C T T A G T T T
C T T A G T T T
0.1
6
SNP
Mutation
4
Mutations and SNPs
Observed genetic variations
Common Ancestor
5
Single Nucleotide Polymorphism

• SNPs are the most frequent form among various
  genetic variations.
• 90 of human genetic variations come from SNPs.
• SNPs occur about every 300600 base pairs.
• Millions of SNPs have been identified (e.g.,
  HapMap and Perlegen).
• SNPs have become the preferred markers for
  association studies because of their high
  abundance and high-throughput SNP genotyping
  technologies.

6
Single Nucleotide Polymorphism

• A SNP is usually assumed to be a binary variable.
• The probability of repeat mutation at the same
  SNP locus is quite small.
• The tri-allele cases are usually considered to be
  the effect of genotyping errors.
• The nucleotide on a SNP locus is called
• a major allele (if allele frequency gt 50), or
• a minor allele (if allele frequency lt 50).

A C T T A G C T T
T Major allele
94
A C T T A G C T C
C Minor allele
6
7
Haplotypes

• A haplotype stands for a set of linked SNPs on
  the same chromosome.
• A haplotype can be simply considered as a binary
  string since each SNP is binary.

8
Genotypes

• The use of haplotype information has been limited
  because the human genome is a diploid.
• In large sequencing projects, genotypes instead
  of haplotypes are collected due to cost
  consideration.

9
Problems of Genotypes

• Genotypes only tell us the alleles at each SNP
  locus.
• But we dont know the connection of alleles at
  different SNP loci.
• There could be several possible haplotypes for
  the same genotype.

or
We dont know which haplotype pair is real.
10
Research Directions of SNPs and Haplotypes in
Recent Years
SNPDatabase
HaplotypeInference
Tag SNPSelection

MaximumParsimony
Perfect Phylogeny
Statistical Methods
Haplotype block
LD bin
PredictionAccuracy
11
Haplotype Inference

• The problem of inferring the haplotypes from a
  set of genotypes is called haplotype inference.
• This problem is already known to be not only
  NP-hard but also APX-hard.
• Most combinatorial methods consider the maximum
  parsimony model to solve this problem.
• This model assumes that the real haplotypes in
  natural population is rare.
• The solution of this problem is a minimum set of
  haplotypes that can explain the given genotypes.

12
Maximum Parsimony
or

• Find a minimum set of haplotypes to explain the
  given genotypes.

13
Related Works

• Statistical methods
• Niu, et al. (2002) developed a PL-EM algorithm
  called HAPLOTYPER.
• Stephens and Donnelly (2003) designed a MCMC
  algorithm based on Gibbs sampling called PHASE.
• Combinatorial methods
• Gusfield (2003) proposed an integer linear
  programming algorithm.
• Wang and Xu (2003) developed a branching and
  bound algorithm called HAPAR to find the optimal
  solution.
• Brown and Harrower (2004) proposed a new integer
  linear formulation of this problem.

14
Our Results

• We formulated this problem as an integer
  quadratic programming (IQP) problem.
• We proposed an iterative semidefinite programming
  (SDP) relaxation algorithm to solve the IQP
  problem.
• This algorithm finds a solution of O(log n)
  approximation.
• We implemented this algorithm in MatLab and
  compared with existing methods.
• Huang, Y.-T., Chao, K.-M., and Chen, T., 2005,
  An Approximation Algorithm for Haplotype
  Inference by Maximum Parsimony, Journal of
  Computational Biology, 12 1261-1274.

15
Problem Formulation

• Input
• A set of n genotypes and m possible haplotypes.
• Output
• A minimum set of haplotypes that can explain the
  given genotypes.

16
Integer Quadratic Programming (IQP)

• Define xi as an integer variable with values 1 or
  -1.
• xi 1 if the i-th haplotype is selected.
• xi -1 if the i-th haplotype is not selected.
• Minimizing the number of selected haplotypes is
  to minimize the following integer quadratic
  function

17
Integer Quadratic Programming (IQP)

• Each genotype must be resolved by at least one
  pair of haplotypes.
• For genotype G1, the following integer quadratic
  function must be satisfied.

Suppose h1 and h2 are selected
or
18
Integer Quadratic Programming (IQP)
Find a minimum set of haplotypes

• Maximum parsimony
• We use the SDP-relaxation technique to solve this
  IQP problem.

to resolve all genotypes.
19
The Flow of the Iterative SDP Relaxation Algorithm
Integer Quadratic Programming
Semidefinite Programming
Vector Formulation
Vector Solution
SDP Solution
Integral Solution
20
Research Directions of SNPs and Haplotypes in
Recent Years
SNPDatabase
HaplotypeInference
Tag SNPSelection

MaximumParsimony
Perfect Phylogeny
Statistical Methods
Haplotype block
LD bin
PredictionAccuracy
21
Problems of Using SNPs for Association Studies

• The number of SNPs is still too large to be used
  for association studies.
• There are millions of SNPs in a human body.
• To reduce the SNP genotyping cost, we wish to use
  as few SNPs as possible for association studies.
• Tag SNPs are a small subset of SNPs that is
  sufficient for performing association studies
  without losing the power of using all SNPs.
• There are many definitions of tag SNPs.
• We will first study one definition of tag SNPs
  based on haplotype blocks model.

22
Haplotype Blocks and Tag SNPs

• Recent studies have shown that the chromosome can
  be partitioned into haplotype blocks interspersed
  by recombination hotspots (Daly et al, Patil et
  al.).
• Within a haplotype block, there is little or no
  recombination occurred.
• The SNPs within a haplotype block tend to be
  inherited together.
• Within a haplotype block, a small subset of SNPs
  (called tag SNPs) is sufficient to distinguish
  each pair of haplotype patterns in the block.
• We only need to genotype tag SNPs instead of all
  SNPs within a haplotype block.

23
Recombination Hotspots and Haplotype Blocks
24
A Haplotype Block Example

• The Chromosome 21 is partitioned into 4,135
  haplotype blocks over 24,047 SNPs by Patil et al.
  (Science, 2001).
• Blue box major allele
• Yellow box minor allele

25
Examples of Tag SNPs
Haplotype patterns
An unknown haplotype sample
P1
P2
P3
P4
S1

• Suppose we wish to distinguish an unknown
  haplotype sample.
• We can genotype all SNPs to identify the
  haplotype sample.

S2
S3
S4
S5
S6
SNP loci
S7
S8
S9
Major allele
S10
S11
Minor allele
S12
26
Examples of Tag SNPs
Haplotype pattern
P1
P2
P3
P4
S1

• In fact, it is not necessary to genotype all
  SNPs.
• SNPs S3, S4, and S5 can form a set of tag SNPs.

S2
S3
S4
S5
S6
SNP loci
P1
P2
P3
P4
S7
S8
S3
S9
S4
S10
S5
S11
S12
27
Examples of Wrong Tag SNPs
Haplotype pattern
P1
P2
P3
P4
S1

• SNPs S1, S2, and S3 can not form a set of tag
  SNPs because P1 and P4 will be ambiguous.

S2
S3
S4
S5
S6
SNP loci
P1
P2
P3
P4
S7
S1
S8
S2
S9
S3
S10
S11
S12
28
Examples of Tag SNPs
Haplotype pattern

• SNPs S1 and S12 can form a set of tag SNPs.
• This set of SNPs is the minimum solution in this
  example.

P1
P2
P3
P4
S1
S2
S3
S4
S5
S6
SNP loci
S7
S8
P1
P2
P3
P4
S9
S1
S10
S12
S11
S12
29
Problem Formulation
P1
P2
P3
P4

• The relation between SNPs and haplotypes can be
  formulated as a bipartite graph.
• S1 can distinguish (P1, P3), (P1, P4), (P2, P3),
  and (P2, P4).
• S2 can distinguish (P1, P4), (P2, P4), (P3, P4).

S1
S2
S3
S4
S1
(1,2)
(1,3)
(1,4)
(2,3)
(2,4)
(3,4)
30
Observation

• The SNPs can form a set of tag SNPs if each pair
  of patterns is connected by at least one edge.
• e.g., S1 and S3 can form a set of tag SNPs.
• e.g., S1 and S2 can not be tag SNPs.

(1,2)
(1,3)
(1,4)
(2,3)
(2,4)
(3,4)
Each pair of patterns is connected by at least
one edge.
31
Problems of Finding Tag SNPs

• The problem of finding the minimum set of tag
  SNPs is known to be NP-hard.
• This problem is the minimum test set problem.
• A number of methods have been proposed to find
  the minimum set of tag SNPs (Bafna et al., Zhang,
  et al.).
• In reality, we may fail to obtain some tag SNPs
  if they do not pass the threshold of data
  quality.
• In the current genotyping environment, the
  missing rate of SNPs is around 510.
• We proposed two greedy algorithms and one linear
  programming relaxation algorithm to solve this
  problem.

32
References

• Huang, Y.-T., Zhang, K., Chen, T. and Chao,
  K.-M., 2005, Selecting Additional Tag SNPs for
  Tolerating Missing Data in Genotyping, BMC
  Bioinformatics, 6 263.
• Chang, C.-J., Huang, Y.-T., and Chao, K.-M.,
  2006, A Greedier Approach for Finding Tag SNPs,
  Bioinformatics, 22 685-691.

33
Research Directions of SNPs and Haplotypes in
Recent Years
SNPDatabase
HaplotypeInference
Tag SNPSelection

MaximumParsimony
Perfect Phylogeny
Statistical Methods
Haplotype block
LD bin
PredictionAccuracy
34
Linkage Disequilibrium

• The problem of finding tag SNPs can be also
  solved from the statistical point of view.
• We can measure the correlation between SNPs and
  identify sets of highly correlated SNPs.
• For each set of correlated SNPs, only one SNP
  need to be genotyped and can be used to predict
  the values of other SNPs.
• Linkage Disequilibrium (LD) is a measure that
  estimates such correlation between two SNPs.
• We will formally introduce the detailed
  information of LD later.

35
Linkage Disequilibrium Bins

• The statistical methods for finding tag SNPs are
  based on the analysis of LD among all SNPs.
• An LD bin is a set of SNPs such that SNPs within
  the same bin are highly correlated with each
  other.
• The value of a single SNP in one LD bin can
  predict the values of other SNPs of the same bin.
  
• These methods try to identify the minimum set of
  LD bins.

36
An Example of LD Bins (1/3)

• SNP1 and SNP2 can not form an LD bin.
• e.g., A in SNP1 may imply either G or A in SNP2.

Individual SNP1 SNP2 SNP3 SNP4 SNP5 SNP6
1 A G A C G T
2 T G C C G C
3 A A A T A T
4 T G C T A C
5 T A C C G C
6 T G C T A C
7 A A A T A T
8 A A A T A T
37
An Example of LD Bins (2/3)

• SNP1, SNP2, and SNP3 can form an LD bin.
• Any SNP in this bin is sufficient to predict the
  values of others.

Individual SNP1 SNP2 SNP3 SNP4 SNP5 SNP6
1 A G A C G T
2 T G C C G C
3 A A A T A T
4 T G C T A C
5 T A C C G C
6 T G C T A C
7 A A A T A T
8 A A A T A T
38
An Example of LD Bins (3/3)

• There are three LD bins, and only three tag SNPs
  are required to be genotyped (e.g., SNP1, SNP2,
  and SNP4).

Individual SNP1 SNP2 SNP3 SNP4 SNP5 SNP6
1 A G A C G T
2 T G C C G C
3 A A A T A T
4 T G C T A C
5 T A C C G C
6 T G C T A C
7 A A A T A T
8 A A A T A T
39
Difference between Haplotype Blocks and LD bins

• Haplotype blocks are based on the assumption that
  SNPs in proximity region should tend to be
  correlated with each other.
• The probability of recombination occurs in
  between is less.
• LD bins can group correlated of SNPs distant from
  each other.
• A disease is usually affected by multiple genes
  instead of single one.
• The SNPs in one LD bin can be shared by other
  bins.
• The SNPs in a haplotype block do not appear in
  another block.

40
Introduction to Linkage Disequilibrium
A, B major alleles a, b minor alleles PA
probability for A alleles at SNP1 Pa probability
for a alleles at SNP1 PB probability for B
alleles at SNP2 PB probability for b alleles at
SNP2 PAB probability for AB haplotypes Pab
probability for ab haplotypes
A
b
SNP1
SNP2
B b Total
A PAB PaB PA
a PaB Pab Pa
Total PB Pb 1.0
41
Linkage Equilibrium

• PAB PAPB
• PAb PAPb PA(1-PB)
• PaB PaPB (1-PA) PB
• Pab PaPb (1-PA) (1-PB)

SNP2
B b Total
A PAB PaB PA
a PaB Pab Pa
Total PB Pb 1.0
SNP1
42
Linkage Disequilibrium

• PAB ? PAPB
• PAb ? PAPb PA(1-PB)
• PaB ? PaPB (1-PA) PB
• Pab ? PaPb (1-PA) (1-PB)

SNP2
B b Total
A PAB PaB PA
a PaB Pab Pa
Total PB Pb 1.0
SNP1
43
An Example of Linkage Disequilibrium
-- A -- -- -- G -- -- --
-- C -- -- -- G -- -- --
-- C -- -- -- C -- -- --
PA1/3PC2/3
PG2/3PC1/3

• Suppose we have three haplotypes AG, CG, and CC.
• There is no AC haplotype, i.e., PAC 0.
• Note that PAC 0, PAPC 1/9, and PAC ? PAPC.
• These two SNPs are linkage disequilibrium.

44
An Example of Linkage Equilibrium
Before recombination
After recombination
-- A -- -- -- G -- -- --
-- A -- -- -- G -- -- --
-- C -- -- -- G -- -- --
-- C -- -- -- G -- -- --
-- C -- -- -- C -- -- --
-- C -- -- -- C -- -- --
-- A -- -- -- C -- -- --
PA1/2PC1/2
PG1/2PC1/2

• After recombination,
• PAG PAPG 1/4,
• PCG PCPG 1/4,
• PCC PCPC 1/4, and
• PAC PAPC 1/4.
• These two SNPs are linkage equilibrium.

45
Linkage Disequilibrium

• There are many formulas to compute LD between two
  SNPs, and most of them are usually normalized
  between -11 or 01.
• LD 1 (perfect positive correlation)
• LD 0 (no correlation or linkage equilibrium)
• LD -1 (perfect negative correlation)
• LD 0.8 (strong positive correlation)
• LD 0.12 (weak positive correlation)

46
Linkage Disequilibrium Formulas

• Mathematical formulas for computing LD
• r2 or ?2
• D
• Chi-square Test.
• P value.

47
Correlation Coefficient

• The correlation between two random variables A
  and B can be measured by the correaltion
  coefficient

48
Examples of Computing LD
Individual SNP1 SNP2 SNP3 SNP4 SNP5 SNP6
1 A T A A G T
2 G T C C T T
3 G A C A G T
4 G A C C T T
5 G A C A G C
49
Minimum Clique Cover Problem

• This problem asks for a minimum set of LD bins.
• The minimum LD value required between two SNPs in
  one bin is usually set to 0.8.
• This problem is known to be the minimum clique
  cover problem (by Huang and Chao, 2005).
• Consider each SNP as nodes on the graph.
• There exists an edge between two nodes iff the LD
  of these two SNPs 0.8.

50
Relaxation of This Problem

• The minimum clique cover problem is not easy to
  be approximated.
• The relaxed problem asks for a minimum set of LD
  bins such that at least one SNP in an LD bin has
  r2 0.8 with other SNPs in the same bin.
• The relaxed problem is known to be the minimum
  dominating set problem.
• The minimum dominating set problem is still
  NP-hard but is easier to be approximated.

51
Minimum Dominating Set Problem

• Given a graph G(V, E), the minimum dominating set
  C is the minimum set of nodes, such that each
  node in V has at least one edge connecting to
  nodes in C.
• Consider each node as a SNP and each edge as
  strong LD (r2 0.8) between two SNPs.
• The minimum dominating set of this graph is the
  set of tag SNPs.
• We can only use this set of SNPs to predict other
  SNPs.

52
Experimental Data Sets

• Hinds et al. (2005) identified 1,586,383 SNPs
  across three human populations.
• African, Americans of European, and Asian.
• The database provides both genotype data and
  inferred haplotype data.