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Haplotype Analysis based on
Markov Chain Monte Carlo
By Konstantin Sinyuk
Overview
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Haplotype, Haplotype Analysis
Markov Chain Monte Carlo (MCMC)
The algorithm based on (MCMC)
Compare with other algorithms
result
Discussion on algorithm accuracy
What is haplotype ?
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A haplotype is a particular pattern of sequential
SNPs found on a single chromosome.
Haplotype has a block-wise structure separated by
hot spots.
Within each block, recombination is rare due to
tight linkage and only very few haplotypes really
occur
Haplotype analysis motivation
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Use of haplotypes in disease association studies
reduces the number of tests to be carried out, and
hence the penalty for multiple testing.
The genome can be partitioned onto 200,000 blocks
With haplotypes we can conduct evolutionary
studies.
International HapMap Project started in October
2002 and planned to be 3 years long.
Haplotype analysis algorithms
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Given a random sample of multilocus genotypes at
a set of SNPs the following actions can be taken:
 Estimate the frequencies of all possible
haplotypes.
 Infer the haplotypes of all individuals.
Haplotyping Algorithms:
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Clark algorithm
EM algorithm
Haplotyping programs:
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HAPINFEREX ( Clark Parsimony algorthm)
EM-Decoder ( EM algorithm)
PHASE
( Gibbs Sampler)
HAPLOTYPER
Motivation for MCMC method
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MCMC algorithm considers the underlying
configurations in proportion to their likelihood
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Estimates most probable haplotype configuration
Prof. Donnelly:
“If a statistician cannot solve a problem, s/he makes it more complicated”
Discrete-Time Markov Chain
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Discrete-time stochastic process {Xn: n = 0,1,2,…}
Takes values in {0,1,2,…}
Memoryless property:
P{ X n 1  j | X n  i, X n 1  in 1 ,..., X 0  i0 }  P{ X n 1  j | X n  i}
Pij  P{ X n 1  j | X n  i}
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Transition probabilities Pij
Pij  0,
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
P
j 0
ij
1
Transition probability matrix P=[Pij]
Chapman-Kolmogorov
Equations
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n step transition probabilities
Pijn  P{X nm  j | X m  i},

n, m  0, i, j  0
Chapman-Kolmogorov equations
nm
ij
P

  Pikn Pkjm ,
n, m  0, i , j  0
k 0
Pijn is element (i, j) in matrix Pn
Recursive computation of state probabilities
State Probabilities – Stationary
Distribution
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State probabilities (time-dependent)
πnj  P{X n  j},

πn  (π0n , π1n ,...)

P{ X n  j}   P{ X n 1  i}P{ X n  j | X n 1  i}  π   πin 1Pij
n
j
i 0

i 0
In matrix form:
π n  π n 1P  π n 2 P 2  ...  π 0 P n
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If time-dependent distribution converges to a limit
π  lim π n
n
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p is called the stationary distribution
π  πP
Existence depends on the structure of Markov chain
Classification of Markov Chains
Irreducible:
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States i and j communicate:
n, m : P  0, P  0
n
ij
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Aperiodic:
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m
ji
Irreducible Markov chain: all
states communicate
1
2
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State i is periodic:
 d  1: Piin  0  n   d
Aperiodic Markov chain: none
of the states is periodic
1
2
0
0
3
4
3
4
Existence of Stationary
Distribution
Theorem 1: Irreducible aperiodic Markov chain. There
are two possibilities for scalars:
π j  lim P{ X n  j | X 0  i}  lim Pijn
n 
1.
2.
pj = 0, for all states j
pj > 0, for all states j
n 
No stationary distribution
p is the unique stationary
distribution
Remark: If the number of states is finite, case 2 is the
only possibility
Ergodic Markov Chains
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Markov chain with a stationary distribution
π j  0, j  0,1, 2,...
States are positive recurrent: The process returns to
state j “infinitely often”
 A positive recurrent and aperiodic Markov chain is
called ergodic
 Ergodic chains have a unique stationary distribution
π j  lim Pijn
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n 
Ergodicity ⇒ Time Averages = Stochastic Averages
Balanced Markov Chain
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Global Balance Equations (GBE)


i 0
i 0
π j  Pji   πi Pij  π j  Pji   πi Pij ,
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i j
Detailed Balance Equations (DBE)
π j Pji  πi Pij
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i j
j0
i , j  0,1,...
π j Pji is the frequency of transitions from j to i
 Frequency of   Frequency of 
 transitions out of j    transitions into j 
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 
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Markov Chain Summary
Markov chain is a set of random processes
with stationary transition probabilities
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- matrix of transition probabilities,
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and
Markov chain is Ergodic if:
 Aperiodic –
 Irreducible Ergodic Markov chain has stationary distribution property:
pij(n) exists and is independent of i ( lim n   p ij( n )  p j )
 lim
n 
p
The vector
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between
 (p j ), p i  1 is stationary distribution of the chain
pP  p , p p  p
i
i
ij
j
Ergodic Markov chain is detailed balanced if:
i
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p p p p
i
ij
j
ji
Markov Chain Monte Carlo
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MCMC is used when we wish to simulate from a
distribution p known only up to a constant
(normalization) factor:p i  C  i (C is hard to
calculate)
Metropolis proposed to construct Markov chain { X S}
with stationary distribution p using onlyp i
pj
ratio
Define transition matrix P indirectly via Q = (qij)
matrix:
p  q
*
 q  Pr(
 j | X T  i ) - proposal probability
X
ij
  ij
- acceptance probability, selected such that

ij
ij
ij
Markov chain will be detailed balanced
Metropolis-Hastings algorithm
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Metropolis-Hasting (MH) algorithm steps:
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Start with X0 = any state
Given Xt-1 = i, choose j with probability qij
Accept this j (put Xt = j) with acceptance probability
1
  ij hij
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hij 
p q
j
ji
pq
i
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- Hastings ratio
ij
Otherwise accept i (put Xt = i)
Repeat step 2 through 4 a needed number of times
With such hij detailed balance is satisfied
With rejection steps the Markov chain is surely
irreducible
 ij
Metropolis-Hastings Graph
Example of
Metropolis-Hastings
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Suppose we want to simulate from
p ( x)  c * exp(  x ), x  R
4
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Metropolis algorithm steps:
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Start with X0 = 0
Generate Xt from the proposal distribution
N(Xt-1,1) p ( x* )
*4
4
t
 exp(  xt  xt )
Compute ht 
p ( xt 1)
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Repeat step 2 through 4 a needed number of times
Gibbs Sampler
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The Gibbs sampler is a special case of the MH
algorithm that considers states that can be
partitioned into coordinates i  (i1 , i2 ,..., in )
At each step, a single coordinate of the state is
updated.
*
*
Step from i  (i1 ,... i r 1 , i r , i r 1 ..., in ) to i  (i1 ,... ir1 , ir , ir1 ..., in )
given by
Gibbs sampler is used where each variable depends
on other variable within some neighborhood
The acceptance probabilities are all equal to 1
MCMC in haplotyping
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The Gibbs sampler is good for multilocus genotyping
of n persons.
Lets define:
d
is the
observed
phenotype of
individual i
at locus j
Ordered
genotype of
person i

i
j
The conditional distribution P(g|d) can be estimated
,
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The Markov chain obtained with Gibbs sampler may
not be ergodic.
The proposed algorithm
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Most algorithms search g maxthat maximize P(g|d)
The proposed algorithm seeks for
f (c)  {g : P( g | d )  c}
An ergodic Markov chain is constructed such that
stationary distribution is P(f(c)|d)
 The sampling is done with Gibbs sampler
 An Ergodic property of Markov chain is satisfied with
use of Metropolis jump kernels
 The Gibbs-Jumping name is assigned to algorithm
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Gibbs step of algorithm
i
For each individual i and locus j, alleles a j1 and
are sampled from the conditional distribution:
a

i
j2
children of
i and s
spouses of i
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The following assumption are commonly made in
order to compute transition probability P( g i | g f , g )
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Hardy-Weinberg Equilibrium
Linkage Equilibrium
No interference
i
mi
Jumping step of algorithm
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After Gibbs step the algorithm attempts to jump
from current state of multilocus genotype g to the
state g* in a different irreducible set.
 The Metropolis jumping kernel Q
is used
 {g , g ,..., g r }
G
Let
be the set of nonD
D
D
D
communicating genotypic configurations on locus j
|
j
1

j
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D
j
2
j
j
j
j
set of individuals who “characterize” irreducible set at j
A new state g* is formed by replacing the alleles
k
pair in g by those from g D for individuals in D j
P( g* | d )


min{
1
,
}
The g* is accepted with probability
j

P( g | d )
Gibbs-Jump trajectories
Results Comparison
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Gibbs-Jumping algorithm is estimating one.
So the algorithm should be tested on well-explored
genetic diseases.
Such explored diseases are:
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The original exploration was done by programs
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Krabbe disease (autosomal, recessive disorder)
Episodic Ataxia disease (autosomal, dominant disorder)
LINKAGE (Krabbe) – enumerating linkage analysis
SIMWALK (Ataxia) – using simulated annealing (MC)
The comparison of various haplotyping method was
carried out by Sobel.
So proposed algorithm results are compared to Sobel
work.
Krabbe disease
(Globoid-cell leukodystrophy)
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This autosomal recessively-inherited disease results
from a deficiency of the lysosomal enzyme
b-galactosylceramidase (GALC).
GALC enzyme plays a role in the normal turnover of
lipids that form a significant part of myelin, the
insulating material around certain neurons.
Affected individuals show progressive mental and
motor (movement) deterioration and usually die
within a year or two of birth.
Krabbe disease cont
Krabbe disease result compare
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The input data is genetic map of 8
polymorphic genetic markers on
chromosome 14.
The Gibbs-Jump algorithm assigned
the most likely haplotype
configuration with probability 0.69,
the same configuration as obtained
by Sobol enumerative approach.
By Sobol they enumerated 262,144
haplotype variations with CPU
time of couple of hours instead of
less than 1 minute run of 100
iterations of Gibbs-Jump.
Episodic Ataxia disease
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Episodic ataxia, a autosomal dominantlyinherited disease affecting the cerebellum.
Point mutations in the human voltage-gated
potassium channel (Kv1.1) gene on
chromosome 12p13
Affected individuals are normal between
attacks but become ataxic under stressful
conditions.
Episodic Ataxia result compare
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The input data is genetic map of 9 polymorphic
genetic markers on chromosome 12.
The Gibbs-Jump algorithm assigned the most likely
haplotype configuration with probability 0.41, that is
very similar to the obtained by Sobel with SIMWALK.
The second most probable haplotype configuration
obtained with 0.09 probability and is identical to the
one picked by Sobel.
Simulation data
To evaluate the performance of Gibbs-Jump on
large pedigrees (with loops) a haplotype
configuration was simulated.
 The genetic map of 10 co dominant markers
(5 alleles per marker) with  = 0.05 was taken.
 The founders haplotypes were sampled randomly
from population distribution of haplotypes.
 Haplotypes for nonfounders where then simulated
conditional on their parents’ haplotypes.
 Assuming HW equilibrium ,Linkage equilibrium and
Haldane’s no interference model for recombination.
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Simulation results
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100 iteration of Gibbs-Jump were performed.
The most probable configuration (with probability
0.41) is identical to the true (simulated ) one
There are 3 configurations with second largest
probability (0.07)
All 3 differ from the true configuration in one
person with one extra recombination event in each
The algorithm execution time took several minutes
Simulation accuracy
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Results of 10 runs of 100
realizations each.
In runs 1 and 3-10 the
most frequent configuration
was the true one .
The most frequent
configuration in run 2
differed from the true one
at one individual.
Simulation run-length
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Results of 5 runs of 10000
realizations each.
The figure shows that
there is a fair amount of
variability in the
estimates, but with very
little correlation between
consecutive estimates.
Autocorrelation = -0.02
Dot plot of the estimated
frequency of the
underlying true haplotype
configuration for 100
iterations.
Simulation run-length cont
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Estimates converges to the
true haplotype configuration
after 2000 steps.
The confidence bound is
95%
Four other runs also inferred
the true configuration with
probabilities:
34.54%,35.75%,37.08% and
35.27% respectively.
Cumulative frequency of
the most probable
configuration , plotted for
every 100 iterations and
the confidence bound.
Results of Sensitivity Analysis
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Computation of P(g) requires an assignment of
haplotype probabilities to the founders.
How inaccurate prediction of founder probabilities
affects the results?
The 4 sets for gene frequencies (different from
simulated) for one of 10 markers were used (other
markers were leaved unchanged)
For the above simulation set the resulting haplotype
configuration was as simulated one.
Conclusion
In this discussion was presented a new method,
Gibbs-Jump, for haplotype analysis, which explores the
whole distribution of haplotypes conditional on the
observed phenotypes.
 The method is very time-efficient.
 The result accuracy was compared to obtained by
other methods (described by Sobol).
 Method demonstrated the sensitivity tolerance to
founders probabilities sample.
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The End…
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