Kinship in the era of genome-wide data: what does it mean and what use is it?

Transcription

1 Kinship in the era of genome-wide data: what does it mean and what use is it? David Balding (with much help from Doug Speed, funding: MRC) Institute of Genetics University College London From 1/11/14: Departments of MathStats and Genetics University of Melbourne. Statistical and computational methods for relatedness and relationship inference from genetic marker data ICMS Edinburgh, 23 September, 2014

2 Kinship, heritability and prediction For over a century we ve tried to measure the extent to which phenotypic similarity between pairs of individuals can be explained by their relatedness. Underlying mathematical model involves latent random effects, expressed in terms of components of variance e.g. Var[Y ] = σ 2 aa + σ 2 d D + σ2 cc + σ 2 i I where Y is the phenotype, and A and D are matrices of kinship coefficients corresponding to Additive and Dominance genetic effects, while C and I represent Common and Individual environmental effects (I is the identity matrix). Narrow sense heritability h 2 is σ 2 a/(σ 2 a+σ 2 d +σ2 c+σ 2 i ). Same model underlies prediction using BLUP.

3 Relatedness: what is it? how do we measure it? A B θ(a, B) = X (1+f X )2 g X. Sum is over common ancestors X of A and B within the pedigree, f X = θ(m(x ), F (X )) C Traditionally, relatedness (a general concept) has been measured by kinship coefficients (numerical measures of relatedness) computed from Identity by Descent (IBD) probabilities under Mendelian inheritance in known pedigrees. Most important is coancestry θ(a, B), the probability that a random allele from A is IBD with one from B.

4 Additive kinship coefficient based on pedigrees Actually, 16 possible IBD states among 4 alleles of 2 diploid individuals; reduces to 9 ignoring within-individual ordering. Also ignoring inbreeding: 3 IBD states (IBD = 0, 1, 2). Also ignoring dominance: 1 additive kinship (coancestry) coefficient θ = E[IBD]/4 = P[IBD=1]/4 + P[IBD=2]/2. 9 IBD states: Individual 1 Individual 2 Individual 1 Individual 2 Individual 1 Individual 2 circles = alleles, arcs = IBD. θ used in the A matrix of additive genetic correlations. Works well in some applications, but serious problems have been overlooked in the past, because there wasn t much choice.

5 Problem 1: θ depends on the pedigree you happen to have available For diploids, there is no such thing as a complete pedigree. As more ancestors are added, θ among original pedigree members can only increase and eventually converges to one; so if a complete pedigree were possible, it would be useless. There is also no ideal pedigree in any other sense. Similarly for inbreeding (θ between parents): an inbreeding coefficient depends on the available pedigree, and always increases with increasing pedigree information. Didn t matter much in the past because we could only make use of close relatedness, but with genome-wide date now we can see relatives separated by 10 or more meioses.

6 Problem 2: θ only captures expected, and not realised, genome-sharing θ for half-sibs is 0.125, but 95% CI is (0.092,0.158). Just 6 parent-child transmissions can result in no DNA remaining from the first parent. Two children may share no DNA from their common great-grandparent. Conversely, θ = 0 for many pairs of individuals, yet the levels of genome-sharing among unrelateds can vary substantially; this has been exploited e.g. for prediction or to estimate SNP heritability.

7 Statistics of IBD sharing (update of Donnelly 1983) # # θ(a, B) P[IBD E[# E[rl] Relationship G A E[IBD]/4 95% CI >0] sr] (Mb) Sibling (0.204,0.296) /2-sib (0.092,0.158) Cousin (0.039,0.089) /2-cuz (0.012,0.055) nd-cuz (0.004,0.031) /2-2nd-cuz (0.001,0.020) rd-cuz (0.000,0.012) (0.000,0.005) (1/2) 14 (0.000,0.001) (1/2) G: generations; A: ancestors; sr = shared regions; rl = region length

8 Kinships based on unobserved pedigrees A C Gene Pool Allele fractions p and 1!p A A C A A A A A A A A C C Many pop gen models incorporate: p aa = θp a + (1 θ)p 2 a i.e. 2 alleles are either IBD or random draws from a gene pool. Leads to θ = p aa p 2 a p a (1 p a ) so θ is a correlation coefficient, only +ve correlations possible. If individuals come from a finite pedigree with unrelated founders, and if allele probabilities in founders are known, then average allelic correlation of markers gives an unbiased and efficient estimator of θ, without knowing the pedigree.

9 Interpretation problems In natural populations (finite pedigree + unrelated founders). What use is an MVUE if unbiased for a meaningless parameter? Not only do we not know allele fractions in founders, we can t usually estimate them and generally use estimates from current population, resulting in: downward bias for θ estimates negative estimates are frequent yet θ is positive by definition.

10 IBD genome segments Homologous segments from two haploid genomes are (recombination-sense) IBD if there has been no recombination within the segment since their MRCA (mutation is ignored). Advantages: No need for an explicit pedigree and no founder population. Problems: Recombinations cannot always be inferred. Easy to identify if shared segment is large, almost impossible if short; most shared segments are short, even for close relatives. Limited use as measure relatedness: two haploid genomes are entirely IBD, relatedness is reflected in distribution of IBD fragment lengths, which is hard to infer. Inferred IBD is used for inferences of demographic parameters: but is it needed? or an optimal approach?

11 Fragment lengths IBD from 1, 5, 9 and 11 generations ago Generation 1 Mean Length 30.3 Generation 5 Mean Length 7.6 Frequency Frequency Chunk Length (Mb) Chunk Length (Mb) Generation 9 Mean Length 4.4 Generation 11 Mean Length 3.6 Frequency Frequency Chunk Length (Mb) Chunk Length (Mb)

12 Distribution of TMRCA given IBD fragment length G>20 G=6 G=5 G=4 G=3 G=2 G= Region Length (Mb)

13 Consumer genetics and IBD Large consumer genetics companies have 10 6 customers genotyped at 10 6 SNPs. They are interested to identify IBD segments in order to infer (remote) pedigree relationships. The relationship is usually expressed in terms of the shortest lineage path (e.g. 3rd cousin, path length = 8) but these cannot be distinguished from many other relationships e.g. involving multiple lineage paths. Why should a customer prefer a poorly-inferred pedigree relationship to a direct measure of genome similarity?

14 Conclusion so far Is it time to ditch pedigrees and related concepts from most scientific discussions? Felsenstein s dismal theorem: given full genomes, history is bunk. Corollary: only actual genome similarity matters, not pedigree relationships and not IBD status. We should formulate our conservation/evolutionary/demographic/disease models and analyses in terms of genome similarity allows us to exploit genome-sharing from all common ancestors.

15 How to measure genomic similarity? There is a ton of ways to measure genetic similarity of two individuals from genome-wide genetic markers (SNPs), no obvious canonical SNP-based alternative to θ. One difficulty in humans is that we are all closely related: Any two haploid human genomes share over 99.9% sequence identity due to shared ancestry. This isn t evident for SNPs because they are highly polymorphic, but measures of similarity can depend sensitively on the Minor Allele Fraction (MAF) spectrum. more low-maf sites more similarity.

16 SNP-based kinships: two approaches Genome-wide average of a single-snp measure. Easiest approach to implement. Ignores information in lengths of shared DNA segments, Better for remote relatedness. Average haplotype sharing. Identify (recombination-sense) IBD segments using e.g. FastIBD, or copied haplotypes using Chromopainter or the positional Burrows-Wheeler transform (PBWT). Kinship coefficient is fraction of all genome in these segments/haplotypes. Ascertainment bias: most IBD is in short segments that are inferred poorly, relatively few, longer segments inferred well. OK if close relatedness (one or more short lineage paths) is of primary interest.

17 SNP-based kinship coefficient 1: Average allele-sharing Define in same way as θ: the fraction of random alleles from A that match a random allele from B. Code SNP genotypes as 0,1 and 2. Then (0, 0) or (2, 2) 1 (0, 1), (1, 1) and (1, 2) 1/2 (0, 2) 0 Two heterozygotes (1,1) are consistent with both IBD=2 and IBD=0, but because the former is often of greater interest some authors (and the highly-influential software PLINK) code (1,1) as 1, rather than 0.5. Most published values do not state which coding is used.

18 SNP kinships 2: Average allelic correlation Suggested by correlation form of θ. Upweights sharing of rare shared alleles more evidence for recent common ancestor. Write G Ai is genotype of A at the ith SNP, then 1 m m i=1 (G Ai 2p i )(G Bi 2p i ) 4p i (1 p i ) is a genome-wide average of single-snp sample-size-1 correlation estimates. The p i are in practice the sample fractions from the same individuals downward bias, often negative ve kinships are the work of the devil to those trained on pedigree ideas; interpretation: B and C have less allele sharing than expected if alleles randomly assigned with probabilities given by the p i.

19 More general SNP-based kinships Researchers still regard pedigree-kinships as gold standard but they aren t always very good; pedigrees were useful when we didn t have genome data, now should be consigned to dustbin of history. Unbiased and efficient properties of allelic-correlation-kinships are not meaningful in practice. There is no true measure of kinship between two individuals and there seems no reason in principle e.g. to prefer allele-sharing kinships to allelic-correlation kinships or haplotype-sharing kinships. We are free to invent new ways to measure genome similarity that best fit the application: explain the most variance (i.e. maximise ĥ 2 )? or to provide the best predictive performance (using BLUP)?

20 MAF and effect sizes A possible 1-parameter family of kinship coefficients is given by: 1 m m i=1 (G Ai 2p i )(G Bi 2p i ) [p i (1 p i )] α α = 0 centred genotypes; popular in plant & animal breeding. α = 1 allelic correlation; upweights low-maf SNPs; popular for SNP-based h 2 estimation in human genetics. Different values of α correspond to different assumptions about the genome-wide MAF effect size relationship. It now becomes an empirical question of which α is best the answer is likely to be trait-specific; reflects true effect sizes for that trait.

21 In the following two slides we consider Allelic-correlation-like kinships for α = 2, 1, 0, 1, Allele-sharing kinships (PLINK). average IBD sharing as computed by fastibd with a liberal significance threshold.

22 Heritability of 139 mouse traits, various kinship matrices Heritability pow 2 (0.29) pow 1 (0.29) pow 0 (0.29) pow 1 (0.29) PLINK (0.32) IBD (0.31) Phenotype

23 Prediction of 139 mouse traits, various kinship matrices Phenotype Prediction (r^2) pow 2 (0.16) pow 1 (0.17) pow 0 (0.17) pow 1 (0.17) PLINK (0.17) IBD (0.14)

24 Conclusions Many definitions of kinship and hence of heritability. Genome similarity is the key concept, no need for pedigree ideas (they don t work for most natural populations) or IBD. Is there a useful canonical definition of relatedness? likely candidates are based on coalescent concepts, such as the genome-wide distribution of times since most recent common ancestor; Rousset (2002) has an interesting idea based on coalescent ideas: excess of TMTCA density at short times, where excess is based on an asymptotic fit; no marker-based estimator so no use in practice. We can choose whatever measure of genome similarity best suits our purpose; e.g. choose to optimise model likelihood or predictive accuracy. There is a huge space of possible genomic similarity matrices so overfitting is potentially a serious problem.

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