Linear Reduction Method for Tag SNPs Selection

1
Linear Reduction Method for Tag SNPs Selection

• Jingwu He
• Alex Zelikovsky

2
Outline

• SNPs , haplotypes and genotypes
• Haplotype tagging problem
• Linear reduction method for tagging
• Maximizing tagging separability
• Conclusions future work

3
Outline

• SNPs , haplotypes and genotypes
• Haplotype tagging problem
• Linear reduction method for tagging
• Maximizing tagging separability
• Conclusions future work

4
Human Genome and SNPs

• Length of Human Genome ? 3 ? 109 base pairs
• Difference b/w any people ? 0.1 of genome ? 3 ?
  106 SNPs
• Total single nucleotide polymorphisms (SNP) ?
  1 ? 107
• SNPs are mostly bi-allelic, e.g., alleles A and C
  
• Minor allele frequency should be considerable
  e.g. gt 1
• Diploid two different copies of each chromosome
  
• Haplotype description of single copy (0,1)
• Genotype description of mixed two copies
  (000, 111, 201)

0
1
1
1 0 0 1
1
0
0
1
1
1 0 0 1
1
0
Two
haplotypes
per individual
Two
haplotypes
per individual
?
1
1
0
1 0 0 1
0
0
1
1
0
1 0 0 1
0
0
Genotype for the individual
Genotype for the individual
2
1
2
1 0 0 1
2
0
2
1
2
1 0 0 1
2
0
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Haplotype and Disease Association

• Haplotypes/genotypes define our individuality
• Genetically engineered athletes might win at
  Beijing Olympics (Time (07/2004))
• Haplotypes contribute to risk factors of complex
  diseases (e.g., diabetes)
• International HapMap project http//www.hapmap.or
  g
• SNPs causing disease reason are hidden among 10
  million SNPs.
• Too expensive to search
• HapMap tries to identify 1 million tag SNPs
  providing almost as much mapping information as
  entire 10 million SNPs.

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Outline

• SNPs, haplotypes and genotypes
• Haplotype tagging problem
• Linear reduction method for tagging
• Maximizing tagging separability
• Conclusions future work

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Tagging Reduces Cost

• Decrease SNP haplotyping cost
• sequence only small amount of SNPs tag SNP
• infer rest of (certain) SNPs based on sequenced
  tag SNPs
• Cost-saving ratio m / k (infinite population)
• Traditional tagging linkage disequilibrium (LD)
  needs too many SNPs, cost-saving ratio is too
  small ( 2)
• Proposed linear reduction method cost-saving
  ratio 20

Number of SNPs m Number of Tags k
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Haplotype Tagging Problem

• Given the full pattern of all SNPs for sample
• Find minimum number of tag SNPs that will allow
  for reconstructing the complete haplotype for
  each individual

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Outline

• SNPs, haplotypes and genotypes
• Haplotype tagging problem
• Linear reduction method for tagging
• Maximizing tagging separability
• Conclusions future work

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Linear Rank of Recombinations

• Human Haplotype Evolution
• Mutations introduce SNPs
• Recombinations propagate SNPs over entire
  population
• Replace notations (0, 1) with (1, 1)
• Theorem Haplotype population generated from l
  haplotypes with recombinations at k spots has
  linear rank (l-1)(k2)
• It is much less than number of all haplotypes l
  k
• Conclusion use only linearly independent SNPs
  as tags

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Tag SNPs Selection

• Tag Selecting Algorithm
• Using Gauss-Jordan Elimination find Row Reduced
  Echelon Form (RREF) X of sample matrix S.
• Extract the basis T of sample S
• Factorize sample S T ? X
• Output set of tags T
• Fact In sample, each SNP is a linear
  combination of tag SNPs
• Conjecture In entire population, each SNP is
  same linear combination of tags as in sample

rref X
tags T
Sample S
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Haplotype Reconstruction

• Given tags t of unknown haplotype h
• and RREF X of sample matrix S
• Find unknown haplotype h
• Predict the h t ? X
• We may have errors, since predicted h may not
  equal to unknown haplotype h. we assign 1 if
  predicted values are negative and 1 otherwise.
  (RLRP)
• Variant randomly reshuffle SNPs before choosing
  tags (RLR)

Unknown haplotype h
rref X
Predicted haplotype h
tags set
?

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Results for Simulated Data

• Cost-saving ratio for 2 error for LR is 3.9 and
  for RLRP is 13

• P 1000 different haplotypes
• m 25000 sites
• Sample size k (number of tag SNPs)
  50,100,,750

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Results for Real Data

• Cost-saving ratio for 5 error for LR is 2.1 and
  for RLRP is 2.8

• P 158 different haplotypes (Daly el.,)
• m 103 sites
• Sample size k (number of tag SNPs)
  10,15,20,,90

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Outline

• SNPs, haplotypes and genotypes
• Haplotype tagging problem
• Linear reduction method for tagging
• Maximizing tagging separability
• Conclusions future work

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Tag Separability

• Correlation between number of zeros for SNPs in
  RREF X and number of errors in prediction column
• Greedy heuristic gives a more separable basis.
  For 5 error, cost-saving ratio 2.8 vs 3.3 for
  RLRP

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Conclusions and Future work

• Our contributions
• new SNP tagging problem formulation
• linear reduction method for SNP tagging
• enhancement of linear reduction using separable
  basis
• Future work
• application of tagging for genotype and haplotype
  disease association

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Thank you