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Reconstructing Sibling Relationships from Genotyping Data

Saad Sheikh Department of Computer Science University of Illinois at Chicago. ?. Brothers!. ?. Reconstructing Sibling Relationships from Genotyping Data. Biological Motivation. Used in: conservation biology, animal management, molecular ecology, genetic epidemiology

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Reconstructing Sibling Relationships from Genotyping Data

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  1. Saad Sheikh Department of Computer Science University of Illinois at Chicago ? Brothers! ? Reconstructing Sibling Relationships from Genotyping Data

  2. Biological Motivation • Used in: conservation biology, animal management, molecular ecology, genetic epidemiology • Necessary for: estimating heritability of quantitative characters, characterizing mating systems and fitness. Lemon sharks, Negaprionbrevirostris • But: hard to sample parent/offspring pairs. Sampling cohorts of juveniles is easier 2 Brown-headed cowbird (Molothrusater) eggs in a Blue-winged Warbler's nest

  3. Basic Genetics • Gene • Unit of inheritance • Allele • Actual genetic sequence • Locus • Location of allele in entire genetic sequence • Diploid • 2 alleles at each locus

  4. Siblings: two children with the same parents Question: given a set of children, find sibling groups allele locus father(.../...),(a /b ),(.../...),(.../...) one from fatherone from mother Diploid Siblings (.../...),(c /d ),(.../...),(.../...) mother recombination (.../...),(e /f ),(.../...),(.../...) child

  5. CACACACA 5’ Alleles CACACACA #1 CACACACACACA #2 #3 CACACACACACACA Genotypes 1/1 2/2 1/2 1/3 2/3 3/3 Microsatellites (STR) • Advantages: • Codominant (easy inference of genotypes and allele frequencies) • Many heterozygous alleles per locus • Possible to estimate other population parameters • Cheaper than SNPs • But: • Few loci • And: • Large families • Self-mating • …

  6. Sibling Groups: 2, 4, 5, 6 1, 3 7, 8 Sibling Reconstruction Problem Animal Locus1 Locus2 allele1/allele2 1 1/2 11/22 2 1/3 33/44 3 1/4 33/55 4 1/3 77/66 5 1/3 33/44 33/77 6 1/3 7 1/5 88/22 8 1/6 22/22 S={P1={2,4,5,6},P2={1,3},P3={7,8}}

  7. Existing Methods

  8. David C. Queller and Keith F. Goodnight. Computer software for performing likelihood tests of pedigree relationship using genetic markers. Molecular Ecology, 8:1231–1234, 1999. KINSHIP

  9. KINSHIP • First software and likelihood measure for sibling/kinship reconstruction • Estimates a ratio of two likelihoods: • Primary vs. Null Hypothesis • Assumes Population Frequencies are known

  10. Probability of sharing allele • R – Probability of alleles being identical by descent • Rp = Probability (Xp = Yp)‏ • Rm = Probability (Xm = Ym)‏

  11. Haploid Likelihood • Two individuals X =<X> and Y=<Y> • If X=Y • Likelihood = Pr(Drawing X) x Pr(X = Y)‏ • =R+(1-R)Px • Otherwise • Likelihood = Pr(Drawing X) x Pr(X  Y)‏ • =Px(1-R)Py

  12. Diploid Individuals • Diploid Individuals X=<Xp/Xm>‌, Y =<Yp/Ym> • Assumptions • We know which alleles are mother's and father's • No Inbreeding • Likelihood = Likelihoodp x Likelihoodm • Loci are independent • Total Likelihood is a product of likelihoods across loci

  13. Calculating Likelihood • Population Frequencies: Pxm,Pxp,Pym,Pyp • Likelihoods:

  14. Likelihood Ratios • Independent Likelihood is not very reliable or meaningful • Different Ratios => Different Loci • Ratio != Statistical Significance • Simulations used to determine P-values

  15. Statistical Significance • Randomly generate an individual X using allele frequencies • Draw Y using Rm and Rp • First Allele: Copy X's allele with Probability Rm or vice versa • Second Allele: Copy X's allele with Probability Rp or vice versa • Draw a large number of such <X,Y> pairs • The value of the ratio that excludes 95% of such pairs is at P=0.05 significance

  16. Jen Beyer and B. May. A graph-theoretic approach to the partition of individuals into full-sib families. Molecular Ecology, 12:2243–2250, 2003. Family Finder

  17. Graph-Theory? • Build a graph of all individuals • Connect individuals with edges representing relationships • Assign Likelihood Ratio Full Sib/Unrelated as distance measure • Filter using likelihood ratio at 0.05 significance level • Find a cut

  18. Algorithm • Calculate LFS/LUR likelihood ratios for all pairs • Build a graph representing the full-sib relationships • Find the connected components in the graph and store them in a queue. • While the queue is not empty do • Remove a component from the queue and calculate its score. • Build a GH cut tree for the component. • For each cut with less than 1/3 the total number of edges in the component do • Score the components that would result if the cut's edges were removed. • If the scores are the best found so far, then store them. • If the best scores found are higher than the score for the original component • then separate the families and put them in the queue for further analysis. • Otherwise save the original component as a result family.

  19. Example Score the components and Keep the best cuts

  20. Conclusion – Family Finder • Some theoretical basis • Efficiently computable • Produces reasonably good results for many loci • A lot of assumptions because of Goodknight & Queller measure • Requires a significant number of loci - 8+ • Works well only when families are almost equal size

  21. Parsimony • Parsimony=Occam’s Razor • "entities must not be multiplied beyond necessity” • "plurality should not be posited without necessity” • “Parsimony is a 'less is better' concept of frugality, economy or caution in arriving at a hypothesis or course of action. The word derives from Middle English parcimony, from Latin parsimonia, from parsus, past participle of parcere: to spare. It is a general principle that has applications from science to philosophy and all related fields. Parsimony is essentially the implementation of Occam's razor.” • Wikipedia • Min Sib groups = Most Parsimonious explanation

  22. Mendelian Constraints 4-allele rule:siblings have at most 4 different alleles in a locus Yes: 3/3, 1/3, 1/5, 1/6 No:3/3, 1/3, 1/5, 1/6, 3/2 2-allele rule: In a locus in a sibling group: a + R ≤ 4 Yes: 3/3, 1/3, 1/5 No: 3/3, 1/3, 1/5, 1/6 Num distinct alleles Num alleles that appear with 3 others or are homozygote

  23. Min Sibgroups Reconstruction • Find the minimum number of Sibling Groups necessary to explain the given cohort • Minimum Set Cover: • Cohort as universe U • Individuals as elements of U • Covering Groups C include all genetically feasible sibling groups • NP-complete even when we know sibsets at most 3 • Hard to approximate (Ashley et al. 09) • ILP formulation (Chaovalitwongse et al. 08)

  24. Given: universe U = {1, 2, …, n} collection of sets S = {S1, S2,…,Sm} where Si subset of U Find: the smallest number of sets in S whose union is the universe U Minimum Set Cover Minimum Set Cover is NP-hard (1+ln n)-approximable (sharp)

  25. 2-Allele Min Set Cover • Generate all maximal feasible sibling groups (sets) that satisfy 2-allele property using “2-Allele Algorithm” [ISMB 2007; Bioinformatics 23(13)] • Use Min Set Cover to find the minimum sibling groupsOptimally using ILP (CPLEX)

  26. 2-Allele Algorithm Overview • Generate candidate sets by all pairs of individuals • Compare every set to every individual x • if x can be added to the set without any affecting “accomodability” or violating 2-allele: • add it • If the “accomodability” is affected , but the 2-allele property is still satisfied: • create a new copy of the set, and add to it • Otherwise ignore the individual, compare the next

  27. 4/1 2/3 2/1 3/1 2/1 1/3 3/2 2/1 3/1 1/1 1/1 1/2 2/2 1/2 1/3 1/4 2/3 2/4 3/1 3/2 4/2 2/1 1/1 1/2 2/1 1/1 1/3 1/3 2/1 2/3 2/1 3/2 Canonical families 1/3 2/2 1/1 1/2 1/4 2/3 2/4 3/4 3/3 4/4

  28. 1/4 1/4 1/4 Examples • Add • New Group Add (won’t accommodate (2/2)) • Can’t add (a+R =4) 3/ 4 1/ 2 3/ 2 1/ 2 3/ 2 3/ 2 1/ 1 1/ 2 1/ 5

  29. Testing and Validation: Protocol • Get a dataset with known sibgroups(real or simulated) • Find sibgroups using our alg • Compare the solutions • Partition distance, Gusfield’03 • Compare results to other sibship methods

  30. Salmon (Salmosalar) - Herbingeret al., 1999 351 individuals, 6 families, 4 loci. No missing alleles Shrimp (Penaeusmonodon) - Jerry et al., 200659 individuals,13 families, 7 loci. Some missing alleles Ants (Leptothoraxacervorum )- Hammond et al., 1999Ants dataset [16] are haplodiploid species. The data consists of 377 worker diploid ants Real Data

  31. Generate F females and M males (F=M=5, 10, 15) Each with l loci (l=2, 4, 6) Each locus with a allelesa[uniform]=5, 10, 15 a[nonuniform]=4 12-4-1-1 Generate f familiesf[uniform]=2, 5, 10 f[nonuniform]=5 For each family select female+male uniformly at random For each parent pair generate o offspringo[uniform]=2, 5, 10 o[nonuniform]=25-10-10-4-1 For each offspring for each locus choose allele outcome uniformly at random Random Data Generation

  32. Results

  33. Summary (Min Sib Groups) • 2-Allele Min Set Cover • First combinatorial • Makes no assumptions other parsimony • Works consistently and comparatively • Sibling Reconstruction • Growing number of methods • Biologists need (one) reliable reconstruction • Genotyping errors • Answer: Consensus

  34. S2 Sk S Consensus Methods • Combine multiple solutions to a problem to generate one unified solution • C: S*→ S • Based on Social Choice Theory • Commonly used where the real solution is not known e.g. Phylogenetic Trees Consensus ... S1

  35. Strict Consensus • Only Pareto Optimality and Anti-Pareto Optimality are enforced • All solutions must agree on equivalence • All disputed individuals go to singletons Si x≡Siy≡ x≡Sy S1 = {{1,2,3},{4,5},{6,7} S2={{1,2,3,4},{5,6,7}} S3={{1,2},{3,4,5},{6,7}} Strict Consensus S={{1,2},{3},{4},{5},{6,7}} 5 Sibling Groups? When 3 can do?

  36. Majority Consensus • Majority of solutions determine the final solution • Two individuals are together if a majority of solutions vote in their favour • Violates Transitivity: A≡B∧B≡C⇒A≡C S1 = {{1,2,3},{4,5},{6,7} S2={{1,2,3,4},{5,6,7}} S3={{1,2},{3,4,5},{6,7}} 1 ≡ 3 AND 3 ≡ 4 BUT 1 ≡ 4

  37. Majority Consensus • Voting Consensus • Majority under closure • Results in large monolithic groups S1 = {{1,2,3},{4,5},{6,7} S2={{1,2,3,4},{5,6,7}} S3={{1,2},{3,4,5},{6,7}} Voting Consensus S={{1,2,3,4,5},{6,7}} 1 ≡5?

  38. Consensus Methods • Commonly used consensus methods don’t work [AAAI-MPREF08] • Strict Consensus produces too many singletons • Majority violates transitivity AND doesn’t work for error-tolerance

  39. fq S S2 S1 Sk Ss fd Distance-based Consensus • Algorithm • Compute a consensus solution S={g1,...,gk} • Search for a good solution near S fq fd Search Consensus ...

  40. Distance-based Consensus • Needs • A Distance Function fd: S x S →R • A Quality Function fq: S → R • What is the Catch? [Sheikh et al. CSB 2008] • Optimization of fd, fq or an arbitrary linearcombination is NP-Complete • Reduction from the 2-Allele Min Set CoverProblem

  41. A Greedy Approach • Algorithm • Compute a strict consensus • While distance is not too large • Merge two nearest sibgroups • Quality: fq=n-|C| • Distance Function • fd(C,C’)=cost of merging groups in C to obtain C’

  42. A Greedy Approach • S1 ={ {1,2,3}, {4,5}, {6,7} } • S2={ {1,2,3}, {4}, {5,6,7} } • S3={ {1,2}, {3,4,5}, {6,7} } Strict Consensus S={ {1,2}, {3},{4},{5},{6,7} } S={ {1,2}, {3,6,7},{4},{5} }

  43. Greedy Consensus • Distance Function(sibgroup, sibgroup)‏ • Cost of assigning all individuals • fd(C,C’)=min(SXPifassign(Pj,X), SXPjfassign(Pi,X) ) • Distance Function (sibgroup, individual)‏ • Benefit: Alleles and allele pairs shared • Cost: Minimum Edit Distance • fassign(PiX)= benefit X can be a member of Pi cost X cannot be a member of Pi`

  44. Greedy Consensus • Algorithm • Compute a strict consensus • While distance is not too large • Merge two sibgroups which will minimize the TOTAL merging cost • Store the new merging cost in the merged set

  45. S2 Sk S Error-Tolerant Approach ... Locus 1 Locus 2 Locus 3 Locus k Sibling Reconstruction Algorithm ... Consensus S1

  46. Results

  47. Results • >90% accuracy for all real data

  48. Results

  49. Results

  50. Impossibility Result • A consensus method CANNOT be all of these [Arrow 1963,Mirkin 1975] • Fair • Independent • Pareto Optimal • Biologically [AAAI-MPREF 2008] • The subset of individuals chosen will impact the consensus considerably

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