Linkage Disequilibrium Bins

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Linkage Disequilibrium Bins
Bioinformatics and Computational Molecular
Biology (Fall 2005) Final Project
An Algorithm for Tag SNP Selection from SNP
database Working directory and program
fileshttp//bunny.idv.tw/sbb/work/bcc Detailed
experimental reporthttp//www.bunny.idv.tw/sbb/w
ork/bcc/report.html
R94922059 ???(sbb)
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Overview

• Basic concepts
• SNP
• Tag SNP
• Introduction to "Tag SNP Selection"
• Algorithm
• Linkage Disequilibrium
• Minimum Dominating Set Problem
• Applications
• Experiment result
• Summary

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Basic concept - SNP

• Single Nucleotide Polymorphisms
• A genetic variation when a single nucleotide
  (i.e., A, T, C, or G) is altered and kept through
  heredity.
• The most frequent form among various genetic
  variations
• The preferred markers for association studies
  because of their high abundance and
  high-throughput SNP genotyping technologies

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Basic concept - SNP
Observed genetic variations
Common Ancestor
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Basic concept - 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.

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Tag SNP Selection

• Tag SNP
• A small subset of SNPs that is sufficient for
  performing association studies without losing the
  power of using all SNPs
• Within a haplotype block, its tag SNPs is
  sufficient to distinguish each pair of haplotype
  patterns in the block

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Tag SNP
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
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Tag SNP - Example
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
S7
P1
P2
P3
P4
S8
S3
S9
S4
S10
S5
S11
S12
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Tag SNP - Wrong Example
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
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Tag SNP Selection

• 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
• Linkage Disequilibrium!

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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.
• Linkage Disequilibrium (LD) is a measure that
  estimates such correlation between two SNPs

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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.

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LD Bins Example (1/3)

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

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LD Bins Example (2/3)

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

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LD Bins Example (3/3)

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

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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
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Linkage Disequilibrium

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

SNP2
SNP1
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Linkage Equilibrium

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

SNP2
SNP1
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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)

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Linkage Disequilibrium Formulas

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

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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 Chao, K.-M., 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.

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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.

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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.

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MDS Problem

• Here we introduce 3 algorithms
• Greedy
• Approximative solution
• Iterative Greedy
• Approximative solution
• Block Prune Search
• Optimal solution

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MDS Greedy

• Approximative solution
• ????????(?????????)??????
• ???????dominating vertex,??????????????LD
  bin,????????????
• Time Complexity O(n)

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MDS Iterative Greedy

• ?Greedy??,??????dominating vertex?????????????????
  ??
• ???????????dominating vertex,???????????????LD
  bin,??????????????
• ???????????,???????
• ????dominating vertex???????????
• Time Complexity O(n E n log(n)) on
  worst caseO(E (E / n) log(E / n)) on
  average case.

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MDS Block Prune Search

• ????????connected-components,???block(??block?????
  ??????????????)
• ?????????????
• Prune Search ??????,??????????????????,??????????
  ????,????????????
• MDS????????????????????dominate??????,??prune?lo
  wer bound
• ?lower bound??????G'(V', E')??????,????????????
  ??????(a1,a2,a3,...,aV'),?????? k ?? k
  a1a2a3...ak gt V'
• ?? (???? k) gt ????????,????????????

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Experiment

• Program
• C STL
• Input
• Haplotype data from http//genome.perlegen.com/bro
  wser/download.html
• 24 Human chromosomes
• Output
• Tag SNPs
• Smallest of LD Bins found
• Some statistics

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