Reconstructing Kinship Relationshipsin Wild Populations

I do not believe that the accident of birth makes people sisters and brothers. It makes them siblings. Gives them mutuality of

parentage. Maya Angelou

Isabel CaballeroUIC

Priya GovindanRutgers

Chun-An (Joe) Chou

Rutgers

Saad SheikhEcole

Polytechnique

Alan Perez-Rathkeo

UIC

Mary AshleyUIC

W. Art Chaovalitwong

seRutgers

Ashfaq Khokhar

UIC

Bhaskar DasGupta

UIC

TanyaBerger-Wolf

UIC

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 …

CACACACA5’

AllelesCACACACA

CACACACACACA

CACACACACACACA

#1

#2

#3

Genotypes

1/1 2/2 3/3 1/2 1/3 2/3

Siblings: two children with the same parentsQuestion: given a set of children, find the

sibling groups

Diploid Siblings

locusallele

father (.../...),(a /b ),(.../...),(.../...) (.../...),(c /d ),(.../...),(.../...) mother

(.../...),(e /f ),(.../...),(.../...) child

one from fatherone from mother

Why Reconstruct Sibling Relationships?

Used in: conservation biology, animal management, molecular ecology, genetic epidemiology

Necessary for: estimating heritability of quantitative characters, characterizing mating systems and fitness.

• But: hard to sample parent/offspring pairs. Sampling cohorts of juveniles is easier

The Problem

Ind Locus 1

Locus 2

allele 1/allele 2

1 1/2 1/2

2 1/3 3/4

3 1/4 3/5

4 3/3 7/6

5 1/3 3/4

6 1/3 3/7

7 1/5 8/2

8 1/6 2/2

Sibling Groups:

2, 4, 5, 6

1, 3

7, 8

Existing MethodsMethod Approach Error-

DetectionAssumptions

Almudevar & Field (1999,2003)

Minimal Sibling groups under likelihood

No Minimal sibgroups, representative allele frequencies

KinGroup (2004)

Markov Chain Monte Carlo/ML

No Allele Frequencies etc. are representative

Family Finder(2003)

Partition population using likelihood graphs

No Allele Frequencies etc. are representative

Pedigree (2001)

Markov Chain Monte Carlo/ML

No Allele Frequencies etc are representative

COLONY (2004)

Simulated Annealing/ ML

Yes Monogamy for one sex

Fernandez & Toro (2006)

Simulated Annealing/ ML

No Co-ancestry matrix is a good measure, parents can be reconstructed or are available

Inheritance Rulesfather (.../...),(a /b ),(.../...),(.../...) (.../...),(c /d ),(.../...),(.../...) mother

child 1 (.../...),(e1 /f1 ),(.../...),(.../...)

child 2 (.../...),(e2 /f2 ),(.../...),(.../...)child 3 (.../...),(e3 /f3 ),(.../...),(.../...)

child n (.../...),(en/fn ),(.../...),(.../...)

…4-allele rule: siblings have at most 4 distinct alleles in a locus

2-allele rule: In a locus in a sibling group: a + R ≤ 4

Num distinct alleles

Num alleles that appear with 3 others or are

homozygot

Our Approach: Mendelian Constrains

4-allele rule:siblings have at most 4 different alleles in a locus

Yes: 3/3, 1/3, 1/5, 1/6No: 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, 1/5No: 3/3, 1/3, 1/5, 1/6

Num distinct alleles

Num alleles that appear with 3 others or are

homozygot

Our Approach: Sibling Reconstruction

Given:n diploid individuals sampled at l loci

Find: Minimum number of 2-allele sets that contain all individuals

NP-complete even when we know sibsets are at most 31.0065 approximation gap Ashley et al ’09

ILP formulation Chaovalitwongse et al. ’07, ’10

Minimum Set Cover based algorithm with optimal solution (using CPLEX) Berger-Wolf et al. ’07

Parallel implementation Sheikh, Khokhar, BW ‘10

ID alleles1 1/2

2 2/3

3 2/1

4 1/3

5 3/2

6 1/4

Canonical families

1/1 1/2 1/3 1/4 2/2 2/3 2/4 3/4 3/3 4/4

1/1 1/1

1/2

2/1

2/21/3

1/4

2/3

2/4

3/1

4/1

3/2

4/2

1/1

1/2

2/1

1/1

1/3

2/1

2/3

3/1

2/1

3/2

1/2

1/3

2/1

3/1

ID alleles

1 55/43

2 43/114

3 43/55

4 55/114

5 114/43

6 55/78

1/3

2/1

2/3

2/1

3/2

Aside: Minimum Set Cover

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 Swhose union is the universe U

€

minI⊆[m ]

| I | such that Ui∈ISi =U

Minimal Set Cover is NP-hard

(1+ln n)-approximable (sharp)

Are we done?Challenges No ground truth available Growing number of methods Biologists need (one) reliable

reconstruction Genotyping errors

Answer: Consensus

Consensus is what many people say in chorus but do not believe as individuals

Abba Eban (1915 - 2002), Israeli diplomat In "The New Yorker," 23 Apr 1990

Consensus MethodsCombine 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 S2 Sk S

Error-Tolerant ApproachSheikh et al. 08

Locu

s

1

Locu

s

2

Locu

s

3

Locu

s

l

Sibling Reconstruc

tion Algorithm

...

Consensus...

S1 S2 Sk S

Distance-based Consensus

Consensus...S1 S2 Sk Ss

S

Search

f q

fq fd

Algorithm– Compute a consensus solution

S={g1,...,gk }– Search for a good solution near S

fd

NP-hard for any fd, fq or an arbitrary linear combination Sheikh et al. ‘08

A Greedy Approach - Algorithm Compute a strict consensus While total distance is not too large

Merge two sibgroups with minimal (total) distance

Quality: fq=n-|C| Distance function from solution C to C’

fd(C,C’) =sum of costs of merging groups in C to obtain C’

=sum of costs of assigning individuals to groups

Cost of assigning individual to a group:Benefit: Alleles and allele pairs sharedCost: Minimum Edit Distance

Change costs to average per locus costsCompare max group error on per locus basisTreat cost and benefit independentlyIn order to qualify a merge

Cost <= maxcostBenefit >= minbenefitBenefit = max benefit among possible merges

Auto Greedy Consensus

A Greedy Approach

{1,2} {3} {4} {5} {6,7}

{1,2} 3.5 1.1 2.5 5.1

{3} 0.5 0.3 0.5 0.1

{4} 1.0 3.0 0.6 1.1

{5} 2.0 1.2 3.5 4.9

{6,7} 0.6 0.9 1.2 4.1

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} }

{1,2} {3,6,7} {4} {5} {6,7}

{1,2} 3.5 1.1 2.5 5.1

{3,6,7} 1.7 3.1 2.2 6.1

{4} 1.0 3.0 0.6 1.1

{5} 2.0 1.2 3.5 4.9

{6,7} 0.6 0.9 1.2 4.1

S = { {1,2}, {3}, {4}, {5}, {6,7} }

S={ {1,2}, {3,6,7}, {4}, {5} }

Testing and Validation: Protocol

1. Get a dataset with known sibgroups(real or simulated)

2. Find sibgroups using our alg3. Compare the solutions

Partition distrance, Gusfield ’03 = assignment problem

Compare to other sibship methods Family Finder, COLONY

Salmon (Salmo salar) - Herbinger et al., 1999 351 individuals, 6 families, 4 loci. No missing alleles

Shrimp (Penaeus monodon) - Jerry et al., 200659 individuals,13 families, 7 loci. Some missing alleles

Ants (Leptothorax acervorum )- Hammond et al., 2001Ants are haplodiploid species. The data consists of 377 worker diploid ants

Test Data

Simulated populations of juveniles for a range of values of number of parents, offspring per parent, alleles, per locus, number of loci, and the distributions of those.

Experimental Protocol

Generate F females and M males (F=M=5, 10, 20)

Each with l loci (l=2, 4, 6,8,10)Each locus with a alleles (a=10, 15)

Generate f families (f=5,10,20)For each family select female+male

uniformly at random

For each parent pair generate o offspring(o=5,10)

For each offspring for each locus choose allele outcome uniformly at random

Introduce random errors

Results

Results

ConclusionsCombinatorial algorithms with minimal

assumptionsBehaves well on real and simulated data Better than others with few loci, few large

familiesError tolerantUseful, high demand

New and improved: Efficient implementation Perez-Rathlke et al. (in submission)

Other objectives (bio vs math) Ashley et al. ‘10

Other genealogical relationships Sheikh et al. ‘09, ’10

Different combinatorial approach Brown & B-W, ‘10

Pedigree amalgamation