FINE SCALE MAPPING

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FINE SCALE MAPPING

• ANDREW MORRIS
• Wellcome Trust Centre for Human Genetics
• March 7, 2003

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Outline

• Introduction fine scale mapping using
  high-density SNP haplotype data.
• Bayesian framework.
• Gene trees and the coalescent process.
• Genetic heterogeneity and shattered gene trees.
• Markov chain Monte Carlo (MCMC) algorithm.
• SNP genotype data.
• Example cystic fibrosis.

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Introduction

• Candidate region of the order of 1Mb in length.
• Refine location of putative disease locus within
  region.
• Make use of high-density maps of single
  nucleotide polymorphisms (SNPs).
• Type sample of affected cases and unaffected
  controls.

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Once upon a time

• Disease predisposition determined by single locus
  in candidate region.
• Each case chromosome carries a copy of a disease
  allele, resulting from a single recent mutation
  event at disease locus.
• Each control chromosome carries a copy of the
  ancient normal allele at the disease locus.

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In an ideal world

• Excess sharing of SNP haplotypes in the vicinity
  of the disease locus, among cases and not among
  controls.
• Decreased probability of sharing as distance from
  disease locus increases.
• Approximate location of disease locus inferred.

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Problems

• Gene tree and ancestral haplotypes are unknown.
• Marker mutations lead to mismatch of alleles
  within preserved regions.
• Multiple disease genes, multiple mutations, and
  dominance.

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Example Cystic fibrosis (CF)

• Fully penetrant recessive disorder, incidence
  1/2500 live births in white populations, less
  common in other populations.
• Preliminary linkage analysis suggested 1.8Mb
  candidate region for a single CF gene on
  chromosome 7q31.
• More recently, a 3bp deletion, ?F508, has been
  identified in the CFTR gene at 0.88Mb into the
  candidate region.
• Now known that ?F508 accounts for 66 of all
  chromosomal mutations in individuals with CF.
• Remainder of CF chromosomes carry copies of many
  other rare mutations in the same gene.
• 23 RFLPs used to identify haplotypes in 92
  control chromosomes and 94 case chromosomes, 62
  of which have been confirmed to carry ?F508.

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Challenges

• The ?F508 locus does not lie at the centre of the
  region of high LD.
• Non-?F508 case chromosomes are not expected to
  share the same founder marker haplotype.
• Useful test-data set for fine-scale mapping
  methods

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Challenges

• The ?F508 locus does not lie at the centre of the
  region of high LD.
• Non-?F508 case chromosomes are not expected to
  share the same founder marker haplotype.
• Useful test-data set for fine-scale mapping
  methods

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Published methods
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Bayesian framework (1)

• Assume disease locus exists in candidate region
  aim is then to estimate its location.
• Approximate the posterior distribution of
  location.
• Allows assignment of probabilities that disease
  locus lies in any particular area of the
  candidate region.

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Bayesian framework (2)

• Aim is to approximate the posterior density of
  location of the disease locus, given SNP
  haplotypes in cases A and controls U, denoted
  f(xA,U).
• Depends on other model parameters M, including
  gene tree, population haplotype frequencies, etc
• Recover marginal posterior density by integration
  over these nuisance parameters,
• f(xA,U) ?f(x,MA,U)dM

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Bayesian framework (3)

• By Bayes Theorem
• f(x,MA,U) C f(A,Ux,M) f(x,M)
• Normalising constant.
• Likelihood of haplotype data given model
  parameters M and location x.
• Prior density of M and x.

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Bayesian framework (3)

• By Bayes Theorem
• f(x,MA,U) C f(A,Ux,M) f(x,M)
• Normalising constant.
• Likelihood of haplotype data given model
  parameters M and location x.
• Prior density of M and x.

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Bayesian framework (3)

• By Bayes Theorem
• f(x,MA,U) C f(A,Ux,M) f(x,M)
• Normalising constant.
• Likelihood of haplotype data given model
  parameters M and location x.
• Prior density of M and x.

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Bayesian framework (3)

• By Bayes Theorem
• f(x,MA,U) C f(A,Ux,M) f(x,M)
• Normalising constant.
• Likelihood of haplotype data given model
  parameters M and location x.
• Prior density of M and x.

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Control chromosomes

• Assumed to carry an ancient normal allele at the
  disease locus.
• Effects of recent shared ancestry of less
  importance, so simple model assumed
• f(A,Ux,M) f(Ax,M) f(Uh)
• The likelihood, f(Uh), depends only on
  population SNP haplotype frequencies, h.
• For many SNPs, the number of possible haplotypes
  is large, so frequencies are parameterised in
  terms of allele frequencies and first-order LD
  between pairs of adjacent loci.

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Gene trees

• Representation of the recent shared ancestry of
  case chromosomes at the disease locus.
• Star shaped tree each case chromosome descends
  independently from founder. Assumes there is too
  much information in sample about ancestral
  recombination and mutation events.
• Bifurcating tree shared ancestral recombination
  and mutation events between chromosomes appear
  only once in their shared ancestry.

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Gene trees

• Representation of the recent shared ancestry of
  case chromosomes at the disease locus.
• Star shaped tree each case chromosome descends
  independently from founder. Assumes there is too
  much information in sample about ancestral
  recombination and mutation events.
• Bifurcating tree shared ancestral recombination
  and mutation events between chromosomes appear
  only once in their shared ancestry.

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Tree specification

• Topology T the branching pattern of the tree.
• Branch lengths, t, determined by the waiting
  times, w, between merging events in the gene
  tree.
• Scaled in units of 2N generations, where N is
  effective population size.

Root
Leaf nodes
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Prior probability model

• Uniform prior probability model for population
  haplotype frequencies, the location of disease
  locus, and the effective population size.
• Each gene tree topology has equal prior
  probability.
• Prior probability model reduces to
• f(x,M) C f(w)
• Need prior probability model for waiting times
  between merging events.

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The coalescent process (1)

• Time between merging event from k to k-1
  lineages.
• Scaled in units of 2N generations.
• Exponential distribution with rate k(k-1)/2.

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The coalescent process (1)

• Time between merging event from k to k-1
  lineages.
• Scaled in units of 2N generations.
• Exponential distribution with rate k(k-1)/2.

Exponential rate 8x7/2 28 Expected time 0.0357
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The coalescent process (1)

• Time between merging event from k to k-1
  lineages.
• Scaled in units of 2N generations.
• Exponential distribution with rate k(k-1)/2.

Exponential rate 7x6/221 Expected time 0.0476
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The coalescent process (1)

• Time between merging event from k to k-1
  lineages.
• Scaled in units of 2N generations.
• Exponential distribution with rate k(k-1)/2.

Exponential rate 2x1/21 Expected time 1
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The coalescent process (2)

• Assumes constant effective population size, N.
• Flexible can allow for exponential population
  growth and population sub-structure.
• Assumes sample is ascertained at random from the
  population. Problem case chromosomes ascertained
  because they carry a copy of the disease
  mutation.
• Assumes sample has single common ancestor.
  Problem genetic heterogeneity.

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The shattered coalescent model

• Generalisation of the coalescent process to allow
  branches of the gene tree to be removed.
• Introduce indicator variable, zb, for each node,
  b, taking the value 1 if b has a parent in the
  gene tree and 0 otherwise.
• Allows for singleton leaf nodes, corresponding to
  sporadic case chromosomes, and disconnected
  sub-trees, corresponding to independent mutation
  events at the same disease locus.
• Assume number of branches of gene tree not
  removed in the shattered coalescent process given
  by binomial distribution, with shattering
  parameter ?.

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Ancestral haplotypes

• Haplotypes, I, carried by internal nodes of the
  gene tree are unknown.
• To calculate posterior probability, need to
  integrate over distribution of possible ancestral
  haplotypes, which depends on gene tree and other
  model parameters.
• Treated as augmented data in Bayesian framework
  enters posterior probability through likelihood
• f(xA,U) ? ? f(x,M,IA,U)dMdI
• and
• f(x,M,IA,U) C f(A,U,Ix,M) f(x,M)

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Likelihood calculations

• If node has no parent in shattered gene tree,
  treat as a random chromosome from the population
  (sporadic or founder for mutation).
• If node has parent in genealogy, depends on
  marker haplotype carried by the parental node,
  and the occurrence of recombination and mutation
  events along the connecting branch.

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Likelihood calculations

• If node has no parent in shattered gene tree,
  treat as a random chromosome from the population
  (sporadic or founder for mutation).
• If node has parent in genealogy, depends on
  marker haplotype carried by the parental node,
  and the occurrence of recombination and mutation
  events along the connecting branch.

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MCMC algorithm (1)

• Need to calculate joint posterior distribution
  f(x,h,T,w,z,N,?,IA,U).
• Parameter space extremely complex, so cannot be
  calculated analytically.
• Markov chain Monte Carlo (MCMC) algorithm
  approximates the posterior distribution by
  sampling from f(x,h,T,w,z,N,?,IA,U).
• Computationally intensive, but becoming more
  practical with improvements in computing power.
• Can handle missing SNP data treat as augmented
  data in the same way as ancestral haplotypes.

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MCMC algorithm (2)

• Let S denote current set of model parameters
  x,h,T,w,z,N,?,I.
• Propose small change to model parameters, S.
• Accept S in place of S with probability
  f(SA,U)/f(SA,U).
• If S is not accepted, the current parameter S is
  retained.
• Initial burn-in to allow convergence of f(SA,U)
  from random starting parameter set.
• Subsequent sampling period, parameter set
  recorded every rth step of the algorithm each
  recorded output represents a random draw from
  f(SA,U).

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MCMC algorithm (3)
Tree height
Location
?
N
101 0.47374 2557.62766 4.24189612
10849.19083 0.78104 -1769.51173 102 0.40629
2112.19993 4.16846454 8804.63049 0.79777
-1788.66623 103 0.46534 1679.71719
4.30423786 7229.90233 0.75364 -1854.19049
104 0.48211 2229.24788 4.33740414
9669.14899 0.78009 -1763.70173 105 0.43808
2402.10599 4.29011844 10305.31919 0.82178
-1760.56671 106 0.44607 2275.33453
4.03331587 9177.14285 0.82601 -1775.90300
107 0.41822 3016.70273 4.39000994
13243.35496 0.77768 -1844.20629 108 0.40934
2534.50113 4.07270615 10322.27832 0.81590
-1861.97411 109 0.41032 3122.91416
4.25386813 13284.46504 0.82479 -1814.27448
110 0.45020 3209.14218 4.34316471
13937.83307 0.78422 -1801.44160
Log posterior probability
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MCMC algorithm (3)
Tree height
Location
?
N
101 0.47374 2557.62766 4.24189612
10849.19083 0.78104 -1769.51173 102 0.40629
2112.19993 4.16846454 8804.63049 0.79777
-1788.66623 103 0.46534 1679.71719
4.30423786 7229.90233 0.75364 -1854.19049
104 0.48211 2229.24788 4.33740414
9669.14899 0.78009 -1763.70173 105 0.43808
2402.10599 4.29011844 10305.31919 0.82178
-1760.56671 106 0.44607 2275.33453
4.03331587 9177.14285 0.82601 -1775.90300
107 0.41822 3016.70273 4.39000994
13243.35496 0.77768 -1844.20629 108 0.40934
2534.50113 4.07270615 10322.27832 0.81590
-1861.97411 109 0.41032 3122.91416
4.25386813 13284.46504 0.82479 -1814.27448
110 0.45020 3209.14218 4.34316471
13937.83307 0.78422 -1801.44160
Log posterior probability
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Cystic fibrosis revisited

• Assume a fixed recombination rate of 0.5cM per Mb
  and a marker mutation rate of 2.5 x 10-5 per
  locus, per generation.
• Each run of MCMC algorithm begins with 20,000
  step burn-in period thrown away.
• Subsequent 200,000 step sampling period, output
  recorded every 50th step of the algorithm 4000
  outputs.
• Two analyses of CF data performed control
  chromosomes (92) and (i) ?F508 case chromosomes
  (62) only (ii) all case chromosomes (94).

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Cystic fibrosis summary statistics
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Cystic fibrosis genetic heterogeneity

• Structure of shattered gene tree provides
  information about genetic heterogeneity at
  disease locus.
• For each output of MCMC algorithm, record
  shattered gene tree.
• For each pair of chromosomes, record whether they
  appear in the same sub-tree.
• Over all outputs, estimate probability that each
  pair of chromosomes carry the same allele at the
  disease locus.
• Cluster chromosomes according to these
  probabilities cladogram to represent genetic
  heterogeneity.

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SNP genotype data

• SNP haplotype rarely available.
• Could infer haplotypes from SNP genotype data
  PHASE, SNPHAP, HAPLOTYPER algorithms.
• Better to treat haplotypes as augmented data in
  Bayesian framework
• f(xG) ? ? ? ? f(x,M,I,A,UG)dMdIdAdU
• and
• f(x,M,I,A,UG) C f(A,U,Ix,M) f(x,M)

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Cystic fibrosis revisited again!

• Create genotype data from original CF haplotype
  data.
• Pair together case chromosmes at random.
• Pair together control chromosomes at random.
• Total sample 46 controls and 47 cases.

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Cystic fibrosis genotypes v haplotypes
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Limitations

• Computationally intensive limited to sample
  sizes 100 cases and controls with up to 20 SNPs.
• Alternative approach do not model gene tree
  explicitly estimate shattered gene tree using
  standard clustering methods.

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Summary

• High density SNP map of the human genome now
  available.
• Fine scale mapping of disease loci requires
  effective modelling of shared ancestry of sample
  of case and control chromosomes.
• Methods exist for haplotype and genotype data
  MCMC algorithms are very computationally
  intensive and are currently limited to relatively
  small sample sizes.
• Further development is necessary