1. FINE SCALE MAPPING ANDREW MORRIS Wellcome Trust Centre for Human Genetics March 7, 2003

2. 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.

3. 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.

4. 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.

8. 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.

9. 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.

10. 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, ΔF508, has been identified in the CFTR gene at ~0.88Mb into the candidate region.  Now known that ΔF508 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 ΔF508.

13. Challenges…  The ΔF508 locus does not lie at the centre of the region of high LD.  Non-ΔF508 case chromosomes are not expected to share the same founder marker haplotype.  Useful test-data set for fine-scale mapping methods…

15. Challenges…  The ΔF508 locus does not lie at the centre of the region of high LD.  Non-ΔF508 case chromosomes are not expected to share the same founder marker haplotype.  Useful test-data set for fine-scale mapping methods…

16. Published methods…

17. 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.

18. 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(x|A,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(x|A,U) = ∫ f(x,M|A,U)dM

19. Bayesian framework (3)  By Bayes’ Theorem… f(x,M|A,U) = C f(A,U|x,M) f(x,M)  Normalising constant.  Likelihood of haplotype data given model parameters M and location x.  Prior density of M and x.

20. Bayesian framework (3)  By Bayes’ Theorem… f(x,M|A,U) = C f(A,U|x,M) f(x,M)  Normalising constant.  Likelihood of haplotype data given model parameters M and location x.  Prior density of M and x.

21. Bayesian framework (3)  By Bayes’ Theorem… f(x,M|A,U) = C f(A,U|x,M) f(x,M)  Normalising constant.  Likelihood of haplotype data given model parameters M and location x.  Prior density of M and x.

22. Bayesian framework (3)  By Bayes’ Theorem… f(x,M|A,U) = C f(A,U|x,M) f(x,M)  Normalising constant.  Likelihood of haplotype data given model parameters M and location x.  Prior density of M and x.

23. 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,U|x,M) = f(A|x,M) f(U|h)  The likelihood, f(U|h), 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.

24. 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.

29. 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.

30. Tree specification  Topology T: the branching pattern of the tree.  Branch lengths, τ, 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. Leaf nodes Root

31. 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.

32. 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.

33. 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

34. 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 7x6/2=21 Expected time: 0.0476

35. 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 2x1/2=1 Expected time: 1

36. 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.

37. The shattered coalescent model  Generalisation of the coalescent process to allow branches of the gene tree to be removed.  Introduce indicator variable, z b, 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 ρ.

41. 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(x|A,U) = ∫ ∫ f(x,M,I|A,U)dMdI and… f(x,M,I|A,U) = C f(A,U,I|x,M) f(x,M)

42. 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.

43. 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.

44. MCMC algorithm (1)  Need to calculate joint posterior distribution f(x,h,T,w,z,N,ρ,I|A,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,ρ,I|A,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.

45. MCMC algorithm (2)  Let S denote current set of model parameters {x,h,T,w,z,N,ρ,I}.  Propose “small” change to model parameters, S*.  Accept S* in place of S with probability f(S*|A,U)/f(S|A,U).  If S* is not accepted, the current parameter S is retained.  Initial burn-in to allow convergence of f(S|A,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(S|A,U).

46. MCMC algorithm (3) 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 Location N Tree height ρ Log posterior probability

47. MCMC algorithm (3) 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 Location N Tree height ρ Log posterior probability

48. 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 50 th step of the algorithm: 4000 outputs.  Two analyses of CF data performed: control chromosomes (92) and (i) ΔF508 case chromosomes (62) only; (ii) all case chromosomes (94).

50. Cystic fibrosis: summary statistics ParameterΔF508 subsetAll cases Location x (Mb) 0.864 0.654-1.040 0.851 0.650-1.003 Shattering parameter ρ 0.935 0.857-0.985 0.829 0.746-0.892 Time to MRCA (generations) 595 183-1877 824 246-3257

51. 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.

54. 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(x|G) = ∫ ∫ ∫ ∫ f(x,M,I,A,U|G)dMdIdAdU and… f(x,M,I,A,U|G) = C f(A,U,I|x,M) f(x,M)

55. 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.

57. Cystic fibrosis: genotypes v haplotypes ParameterGenotypesHaplotypes Location x (Mb) 0.855 0.625-1.137 0.851 0.650-1.003 Shattering parameter ρ 0.842 0.771-0.901 0.829 0.746-0.892 Effective population size N 375 107-871 846 367-1657

58. 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.

59. 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…