Single Nucleotide Polymorphisms

1
Single Nucleotide Polymorphisms
Instructor Yao-Ting Huang
Bioinformatics Laboratory, Department of Computer
Science Information Engineering, National Chung
Cheng University.
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Genetic Variants

• We are distinguished from each other by genetic
  variants.
• Single Nucleotide Polymorphisms (SNP)
• Insertion/deletion
• Copy Number Polymorphism (CNP)
• Inversion

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Genetic Variants Over Time
Variants observed in a population
Mutations over time
Common Ancestor
time
present
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SNPs and Haplotypes

• A Single Nucleotide Polymorphism (SNP),
  pronounced snip, is a single DNA base variation
  observed in the human population.
• A haplotype stands for a set of linked SNPs on
  the same chromosome.

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Single Nucleotide Polymorphism

• We only consider SNPs observed with sufficient
  frequency in the population.
• SNP the minor allele frequency is at least 5.
• Mutation the minor allele frequency is less than
  5.

C T T A G C T T
C T T A G T T T
SNP
A C T T A G C T T
99.9
A C T T A G T T T
0.1
Mutation
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Single Nucleotide Polymorphism

• All humans share 99.9 the same DNA sequence
• SNPs occur about every 200600 base pairs.
• 90 of human genome variation comes SNPs.
• The human genome contains about four million
  SNPs.
• Because the probability of recurrent mutation at
  the same locus is quite low, we usually observe
  only two alleles at a SNP locus.

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Single Nucleotide Polymorphism

• The SNPs differ among members in the human
  population.

Black eye Brown eye Black eye Blue eye Brown
eye Brown eye
GATATTCGTACGGA-T GATGTTCGTACTGAAT GATATTCGTACGGA-T
GATATTCGTACGGAAT GATGTTCGTACTGAAT GATGTTCGTACTGAA
T
Haplotypes
AG- 2/6 GTA 3/6 AGA 1/6
DNASequences of 6 individuals
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Discovery of SNPs

• The DNA of two individuals differs in less than
  0.1.
• Hinds et al. identified 1,586,383 Single
  Nucleotide Polymorphisms across three human
  populations (Science, 2005).

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The HapMap Project

• The International HapMap project aims to provide
  a map of SNPs in the human genome (269
  individuals from 4 populations).
• Phase I 1,007,329 SNPs.
• Phase II (ongoing) 4.6 millions SNPs.

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Haplotype v.s. Genotype

• The collection of haplotypes has been limited
  because the human genome is a diploid.
• In above projects, genotypes instead of
  haplotypes are collected due to cost
  consideration.

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Haplotype v.s. Genotype

• 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 haplotype pairs
  for the same genotype.

or
We dont know which haplotype pair is real.
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Three Possible Genotypes at Each SNP Locus

• At SNP1, it is possible to observe three
  genotypes (A, C), (A, A), and (C, C) in the
  population.
• (A, C) Heterozygous (One major and one minor
  alleles).
• (A, A) Homozygous wild type (two major alleles).
• (C, C) Homozygous mutant (two minor alleles).

T
C
G3
C
T
SNP1
SNP2
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Haplotype Inference

• Inferring the haplotypes from a set of genotypes
  is called haplotype inference.
• Without further assumption, this problem can not
  be solved.
• Most combinatorial methods consider the maximum
  parsimony model to solve this problem.
• Methods based on this model search for a minimum
  set of haplotypes which can explain all
  genotypes.
• This problem is shown to be APX-hard (Lancia
  etal, 2005).

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Maximum Parsimony
or

• Find a minimum set of haplotypes that can explain
  all genotypes.

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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 for this problem.
• 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 programming for this problem.

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Our Results

• Huang et al. An approximation algorithm for
  haplotype inference by maximum parsimony, Journal
  of Computational Biology, 2005.

Yao-Ting Huang
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Approximation Approaches to NP-hard problems

• Formulate the problem to an integer linear
  problem
• Relax to a Linear Programming (LP) problem and
  solve it.
• Gusfield and Brown formulate the haplotype
  inference problem into integer programming.
• Formulate the problem to an integer quadratic
  programming (IQP) problem
• Relax to a Semi-Definite Programming (SDP)
  problem and solve it.
• We formulate the haplotype inference problem into
  an IQP problem.

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Integer Quadratic Programming

• 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.
• Finding a minimum set of haplotypes is to
  minimize the following function

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Integer Quadratic Programming

• Each genotype must be explained by at least one
  pair of haplotypes.
• For genotype G1, the following inequality must be
  satisfied.

Suppose h1 and h2 are selected
or
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Integer Quadratic Programming
Constraint Functions

• Maximum parsimony

Find a minimum set of haplotypes
which can explain all genotypes.
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An Iterative Semi-definite Programming Relaxation
Algorithm
Integer Quadratic Programming
Semi-definite Programming
Vector Formulation
Vector Solution
SDP Solution
Integral Solution
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Relaxation
Integer Quadratic Programming
Vector Formulation

• We relax xi into a (m1)-dimensional unit vector
  yi.
• Replace the integer constant 1 with another unit
  vector y0 (1, 0, , 0).

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SDP Formulation
Vector Formulation

• Let Y (y0 y1 ym)T(y0 y1 ym)

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Reformulation
Vector Formulation
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Solving SDP
Semidefinite Programming

• The SDP problem can be solved by algorithms such
  as the interior point method in polynomial time.
• We can obtain the SDP solution matrix Y.

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Decomposition
SDP Solution

• Recall that Y (y0 y1 ym)T(y0 y1 ym).
• Use the incomplete Choleskey decomposition method
  to obtain vector solutions y0, y1, , ym.

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Randomized Rounding
IntegralSolution
Vector Solution

• Randomly generate two unit vectors z1 and z2.
• Set xi 1 if
• ( z1 yi ) ( z1 y0 ) gt 0, and
• ( z2 yi ) ( z2 y0 ) gt 0.
• Set xi -1 otherwise.

We will discuss this later
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Iterative Process
Integer Quadratic Programming

• Check if all inequalities are satisfied.
• No, repeat this algorithm only for those
  unsatisfied inequalities.
• Yes, we are done.

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Analysis of the SDP-relaxation Algorithm

• Recall the randomized rounding
• Randomly generate two unit vectors z1 and z2.
• Set xi 1 if
• ( z1 yi ) ( z1 y0 ) gt 0, and
• ( z2 yi ) ( z2 y0 ) gt 0.
• Set xi -1 otherwise.
• We will show that the randomized rounding outputs
  a solution Ew at least as good as the optimal
  solution.

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Analysis of the SDP-relaxation Algorithm

• The randomized rounding method can output a
  solution Ew at least as good as the optimal
  solution.
• We will show OPT(IQP) OPT(SDP) Ew.
• The solution space of SDP includes that of IQP,
• We already have OPT(IQP) OPT(SDP).
• We can set yi (1,0,0,0, ) ? xi 1.
• We can set yi (-1,0,0,0, ) ? xi -1.

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Analysis of the SDP-relaxation Algorithm

• We still need to prove
• OPT(IQP) OPT(SDP) Ew.

gt lt?
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Analysis of the SDP-relaxation Algorithm

• Recall xi 1 if
• ( z1 yi ) ( z1 y0 ) gt 0, and
• ( z2 yi ) ( z2 y0 ) gt 0.
• Note that cos? vi vj
• Let the angle between vectors y0 and yi be ?.
• Recall that cos? gt 0 when ?0, p/2 or p, 3p/2.

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Analysis of the SDP-relaxation Algorithm

• Recall xi 1 if
• ( z1 yi ) ( z1 y0 ) gt 0, and
• ( z2 yi ) ( z2 y0 ) gt 0.
• Let the angle between vectors y0 and yi be ?.
• ( z1 yi ) ( z1 y0 ) gt 0 if z1 is within region
  (p-?) or the opposite region.
• ( z2 yi ) ( z2 y0 ) gt 0 if z2 is within region
  (p-?) or the opposite region..
• xi 1 with probability ((p-?) /p)2.

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Analysis of the SDP-relaxation Algorithm
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Analysis of the SDP-relaxation Algorithm

• We now complete the proof
• OPT(IQP) OPT(SDP) Ew.

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Simulation Methods

• The haplotypes are used to validate the result.
• We randomly pair two haplotypes to generate a
  genotype.

HaplotypeData
GenotypeData
Solution
h1 h2 hm
G1 h1h4 G2 h2hm Gn h1h2
G1 h1h4 G2 h1h2 Gn h1h2
SDPHapInferHAPARHAPLOTYPER PHASE
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Results

• We prove that SDPHapInfer gives a solution of
  O(log n)-approximation with a high probability,
  where n is the number of genotypes.
• We implement SDPHapInfer in MatLab.
• We compare the number of haplotypes found by
  different methods on simulated data sets.

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Experimental Results (1/2)
Error rate
Number of genotypes
100 simluated data sets of 10 haplotypes with 20
SNPs
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The Challenge

• The problem of inferring haplotypes for long
  genotypes is still a challenging problem.
• Existing methods are forced to
• partition the genotypes into small segments,
• infer haplotype in each segment,
• and concatenate inferred haplotypes to construct
  a final solution.

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The First Application of SDP on Approximation
Algorithms

• A 0.878 randomized approximation algorithm for
  the MAXCUT problem is developed by SDP relaxation
  technique.
• The LP-relaxation can only achieve 0.5
  approximation ratio.
• An upper bound has shown to be 0.941.
• Goemans, M. and Williamson, D. at ACM STOC 1994.

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The MAXCUT Problem

• Given an undirected graph with n nodes Gx1 , x2
  , , xn, find a cut to maximize the number of
  edges on the cut.
• Let xi be 1 if the vertex is at one side of the
  cut, and -1 if the vertex is at the other side of
  the cut.

-1
-1
-1
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Integer Quadratic Programming

• Define aij be 1 if the edge (xi , xj) exists and
  0 otherwise.

x2
x1
x3
x4

• Relax the integer constraint of xi to be the unit
  length vector in dimension m.

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Semidefinite Programming Formulation
x2
x1
x3
x4

• Let X be (v1 ,v2 , , vn)T ? (v1, v2 ,, vn).

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Randomized Rounding Method

• Once X is found, perform Cholesky decomposition
  to obtain the vector solutions v1, v2, , vn.
• Pick a random unit vector r and
• Set xi 1 if vi ? r 0
• Set xi -1 if vi ? r lt 0
• Note that cos? vi ? vj
• The edge (vi , vj) is on the cut iff (vi ? r )
  and (vj ? r) has different sign.

vi
r
?
vj
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Analysis

• Denote C as the size of the cut found by the
  above algorithm.
• The expectation that each edge (xi , xj) is the
  solution is

vi
r
?
vj
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Analysis

• The randomized rounding partition the nodes by a
  hyperplane.

r
1
1
1
-1
-1
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Linear Algebra Background

• A symmetric n?n matrix A is positive semidefinite
  iff xTAx ? 0 , for every x?Rn.
• ABTB , for some m?n matrix B.
• All the eigenvalues of A are non-negative.
• The inner product of symmetric matrices A and B is