Simon Myers

Department of Statistics, Oxford

Recombination and genetic variation – models and inference

What does recombination do to genetic variation?

• Informally, recombination shuffles up genetic diversity

• We can see the effect of recombination in how ‘structured’ genetic variation is

Lipoprotein Lipase: 10kb48 African Americans

Chromosome 22: 1Mb57 Europeans

Xq13: 10kb69 worldwide

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Ch

rom

oso

me

s

Sites

Human pairwise association revisited

• What is going on?

• Recombination causes the association breakdown

• Does the uneven pattern reflect– Chance?

– Real strong differences in the underlying recombination rate in meiosis

• We will explore two approaches to find out

Data for ENR131, Chromosome 2q, Chinese and Japanese population sample (The International HapMap

Consortium, Nature 2005)

LD

Recombination and genealogical history

• Forwards in time

• Backwards in time

TCAGGCATGGATCAGGGAGCT TCACGCATGGAACAGGGAGCT

Grandpaternal sequence Grandmaternal sequence

TCAGGCATGG AACAGGGAGCT

x

G A

G A

Non-ancestral genetic material

The ancestral recombination graph

• The combined history of recombination, mutation and coalescence is described by the ancestral recombination graph

Mutation

Mutation

Event

Recombination

Coalescence

Coalescence

Coalescence

Coalescence

Deconstructing the ARG

Tim

e

Learning about recombination

• Just like there is a true genealogy underlying a sample of sequences without recombination, there is a true ARG underlying samples of sequences with recombination

• We can consider nonparametric and parametric ways of learning about recombination

• There are several useful nonparametric ways of learning about recombination which we will consider first– These really only apply to species, such as humans, where we can be fairly

sure that most SNPs are the result of a single ancestral mutation event

– This is formally called the infinite sites model

Why use a non-parametric approach?

• Non-parametric approaches require few assumptions about evolution

• The infinite sites model, and that’s it!

• We can attempt to learn features of the history of a sample based only on this assumption– Robust inference

– Identify – “detect” the recombination events that shaped our sample

– Clustering of multiple events in a region could signal a high underlying rate

• Some drawbacks to this approach

The signal of recombination?

Recurrent mutation Recombination

Ancestral chromosome recombines

Practical: detecting recombination from DNA sequence data

• Look for all pairs of “incompatible” sites• Combine information across the pairs

• Find minimum number of intervals in which recombination events must have occurred (Hudson and Kaplan 1985): Rm

Simon Myers

Department of Statistics, Oxford

Recombination and genetic variation – models and inference, part II

Example: 7q31

These results are based on a non-parametric minimum number of recombination intervals (events) Rh

• Myers and Griffiths (2003) – improvement over Rm but identical assumptions• Results strongly suggest recombination “hotspots”

Example: humans vs. chimpanzees

Winckler et al. (2005)

Why use parametric approaches?

• The infinite-sites model is not applicable to all species

• There are many more recombination events in the history of the sample than the non-parametric methods can ever detect

– Lack of mutations in the right places– Recombination events completely undetectable

HIV Subtype B (2kb segment) HIV Subtype C (2kb segment)

Modelling recombination

• Model-based approaches to learning about recombination allow us to ask more detailed questions than nonparametric approaches

– What is the rate of recombination (as opposed to just the number of events)

– Is the rate of recombination across a region constant?

– Does gene A have a higher recombination rate than gene B?

– What patterns of genetic diversity might I expect to see in other samples from the same (or different) population?

• We need a model!

Adding recombination to the coalescent

• Each generation, the probability of recombination between two loci is r, working in scaled time, this means that recombination occurs at rate /2 per sequence where = 4Ner

• Recombination, mutation and coalescence occur independently:– Coalescence occurs as a Poisson process with rate n(n-1)/2

– Recombination occurs as a Poisson process with rate n/2

– Mutations on edges added as a Poisson process with rate n/2

• The time until the next recombination or coalescence event is also a Poisson process with rate n/2+ n(n-1)/2, and the probability that this next event is a recombination is

12/)1(2/

2/)Pr(

nnnn

nrec

Recombination in non-ancestral material

• Once a region has recombined, further recombination can occur in both ancestral lineages

• However, recombination in non-ancestral DNA cannot in anyway influence patterns of diversity (under a neutral model)

• We usually ignore such recombination events in the coalescent

X

X

Simulating histories with recombination

• www.coalescent.dk

Properties of the ARG

• Unlike the basic coalescent, there are few results about the effects of recombination on gene genealogies that we can derive analytically

• For example, we cannot even calculate the expected number of recombination events in the history of a sequence– Though we can show it is less than infinity!

• There are some useful results about how many recombination events we can see– The key is that only a small minority of recombination events that occur in the

history of the sample can ever be directly detected by nonparametric methods

=10, =10 against log log sample sizemR

mR

hR

Estimating the population recombination rate

• The ideal inference procedure would calculate the likelihood of the data– Need to allow recombination rate to vary

• ….but full-likelihood inference is effectively impossible for anything but the simplest data sets (and models)

• We need alternatives– Calculate the probability of some summary of the data (like ABC)– Approximate the coalescent model– Approximate the likelihood

• The composite likelihood of Hudson (2001) approximates the likelihood of the full data by the product of the likelihoods for pairs of sites– Not the real likelihood!– Fast to calculate– Allows a variable recombination rate

Composite likelihood estimation of 4Ner: Hudson (2001)

1571127224231111

-6

-5

-4

-3

-2

-1

0

1

0 2 4 6 8 10

Full likelihood

Composite-likelihood approximation

RlnL

R

lnL

R

lnL

ji

ijji DDPL ),|,()( ρ

Fitting a variable recombination rate

• Use a reversible-jump MCMC approach (Green 1995)

Merge blocks

Change block size

Change block rate

Cold

Hot

SNP positions

Split blocks

),(

),(

)(

)(

),(

),(

)(

)(,1min),(

u

u

q

q

C

C

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

Broad scale validation: strong concordance between rates estimated from genetic variation and pedigrees

2Mb correlation between “Perlegen” and deCODE rates (Myers et al. 2005)

Fine-scale validation: strong concordance between fine-scale rate estimates from sperm and genetic variation

Rates estimated from sperm Jeffreys et al (2001)

Rates estimated from genetic variationMcVean et al (2004)

200kb region of human HLA

In this region at least, human recombination clusters into 1-2kb wide hotspots

>90% of recombination in 6 hotspots

We have also developed a specific test for hotspots, based on the same composite likelihood (likelihood ratio test)

Fine-scale rates across the human genome

• Throughout the genome, human recombination clusters into narrow hotspots• These explain LD breakdown sites

Across chromosome12

Myers et al. (2005)

Data for ENR131, Chromosome 2q, Chinese and Japanese population sample (The International HapMap

Consortium, Nature 2005)

Summary

• Both non-parametric and model based approaches allow us to ask detailed questions about recombination from population genetic data

• Recombination can be incorporated within the coalescent framework

• The population recombination rate, =4Ner, is the key quantity in determining the effect of recombination on genetic variation

• Efficiently estimating recombination rates within a coalescent framework is difficult, but approximate methods have proved a powerful approach

• Such methods have allowed us to successfully learn about recombination rates in humans and other species, and reveal “hotspots” across genomes