Gil McVean Department of Statistics, Oxford Approximate genealogical inference

Motivation We have a genome’s worth of data on genetic variation We would like to use these data to make inferences about multiple processes: recombination, mutation, natural selection, demographic history

Example I: Recombination In humans, the recombination rate varies along a chromosome Recombination has characteristic influences on patterns of genetic variation We would like to estimate the profile of recombination from the variation data – and learn about the factors influencing rate

Example II: Genealogical inference The genealogical relationships between sequences are highly informative about underlying processes We would like to estimate these relationships from DNA sequences We could use these to learn about history, selection and the location of disease- associated mutations

Modelling genetic variation We have a probabilistic model that can describe the effects of diverse processes on genetic variation: The coalescent Coalescent modelling describes the distribution of genealogical relationships between sequences sampled from idealised populations Patterns of genetic variation result from mapping mutations on the genealogy

Where do these trees come from? Present day

Ancestry of current population Present day

Ancestry of sample Present day

The coalescent: a model of genealogies time coalescence Most recent common ancestor (MRCA) Ancestral lineages Present day

Coalescent modelling describes the distribution of genealogies

…and data

Generalising the coalescent The impact of many different forces can readily be incorporated into coalescent modelling With recombination, the history of the sample is described by a complex graph in which local genealogical trees are embedded – called the ancestral recombination graph or ARG Ancestral chromosome recombines

Genealogical trees vary along a chromosome

Coalescent-based inference We would like to use the coalescent model to drive inference about underlying processes Generally, we would like to calculate the likelihood function However, there is a many-to-one mapping of ARGs (and genealogies) to data Consequently, we have to integrate out the ‘missing-data’ of the ARG This can only be done using Monte Carlo methods (except in trivial examples)

time

A problem and a possible solution Efficient exploration of the space of ARGs is a difficult problem The difficulties of performing efficient exact genealogical inference (at least within a coalescent framework) currently seem insurmountable There are several possible solutions –Dimension-reduction –Approximate the model –Approximate the likelihood function One approach that has proved useful is to combine information from subsets of data for which the likelihood function can be estimated –Composite likelihood

Example I: Recombination rate estimation We can estimate the likelihood function for the recombination fraction separating two SNPs To approximate the likelihood for the whole data set, we simply multiply the marginal likelihoods (Hudson 2001) The method performs well in point-estimation

Full likelihood Composite- likelihood approximation R  lnL R R

Good and bad things about CL Good things –Estimation using CL can be made very efficient –It performs well in simulations –It can generalise to variable recombination rates Bad things –It throws away information –It is NOT a true likelihood –It typically underestimates uncertainty because of ‘double-counting’

Fitting a variable recombination rate Use a reversible-jump MCMC approach (Green 1995) Merge blocksChange block size Change block rate Cold Hot SNP positions Split blocks

Composite likelihood ratio Hastings ratio Ratio of priors Jacobian of partial derivatives relating changes in parameters to sampled random numbers Acceptance rates Include a prior on the number of change points that encourages smoothing

rjMCMC in action 200kb of the HLA region – strong evidence of LD breakdown

How do you validate the method? Concordance with rate estimates from sperm-typing experiments at fine scale Concordance with pedigree-based genetic maps at broad scales

Strong concordance between fine-scale rate estimates from sperm and genetic variation Rates estimated from sperm Jeffreys et al (2001) Rates estimated from genetic variation McVean et al (2004)

We have generated a map of hotspots across the human genome Myers et al (2005)

We have identified DNA sequence motifs that explain 40% of all hotspots Myers et al (2008)

? Age of mutation Date of population founding Migration and admixture Example II: Estimating local genealogies

The decay of a tree by recombination

0 100 The decay of a tree by recombination

Two sequence case Any pair of haplotypes will have regions of high and low divergence We can combine HMM structures with numerical techniques (Gaussian quadrature) to estimate the marginal likelihood surface at a given position, x We can further approximate the likelihood surface by fitting a scaled gamma distribution –This massively reduces the computational load of subsequent steps –In the case of no recombination the truth is a scaled gamma distribution

Combining surfaces Suppose we have a partially-reconstructed tree We can approximate the probability of any further step in the tree using the composite-likelihood Pr( ) 0 t   } (assumes un-coalesced ancestors are independent draws from stationary distribution)

An important detail Actually, don’t use exactly this construction Use a ‘nearest-neighbour’ construction –Each lineage chooses a nearest neighbour –Choose which nearest-neighbour event to occur –Choose a time for the nearest-neighbour event Still uses composite likelihood

Building the tree We can use these functions to choose (e.g. maximise or sample) the next event The gamma approximation leads to an efficient algorithm for estimating the local genealogy that has the same time and memory complexity of neighbour-joining This mean it can be applied to large data sets

Desirable properties of the algorithm It can be fully stochastic (unlike NJ, UPGMA, ML) It returns the prior in the absence of data: It returns the truth in the limit of infinite data: It is correct for a single SNP: It is close to the optimal proposal distribution (as defined by Stephens and Donnelly 2000) in the case of no recombination It uses much of the available information It is fast – time complexity in n the same as for NJ

Example: mutation rate = recombination rate The true tree at 0.5 Simulations = 10, R = 10

How to evaluate tree accuracy? Specific applications will require different aspects of the estimated trees to be more or less accurate Nevertheless, a general approach is to compare the representation of bi-partitions in the true tree to estimated ones Rather than require 100% accuracy in predicting a bi-partition, we can (for every observed bi-partition) find the ‘most similar’ bi- partition in the estimated tree We should also weight by the branch length associated with each bi-partition

Simple distance weighting (UPGMA) Single sample from posterior Average weighted max r 2 = 0.59Average weighted max r 2 = 0.96 n = 100,  = 20, R = 30 (hotspots)

Open questions Obtaining useful estimates of uncertainty –Power transformations of composite likelihood function Using larger subsets of the data –E.g. quartets

Acknowledgements Many thanks to Oxford statistics –Simon Myers, Chris Hallsworth, Adam Auton, Colin Freeman, Peter Donnelly Lancaster –Paul Fearnhead International HapMap Project