Recombination and genetic variation – models and
inference
Simon Myers
Department of Statistics, Oxford
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
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1
Sites
Xq13: 10kb
69 worldwide
Lipoprotein Lipase: 10kb
48 African Americans
Chromosome 22: 1Mb
57 Europeans
1
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Human pairwise association revisited
Data for ENR131, Chromosome 2q, Chinese and
Japanese population sample (The International HapMap
Consortium, Nature 2005)
LD
• 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
Recombination and genealogical history
• Forwards in time
Grandmaternal sequence
Grandpaternal sequence
x
TCAGGCATGGATCAGGGAGCT
TCACGCATGGAACAGGGAGCT
TCAGGCATGG AACAGGGAGCT
• Backwards in time
Non-ancestral genetic material
G
A
G
A
The ancestral recombination graph
• The combined history of recombination, mutation and coalescence is
described by the ancestral recombination graph
Coalescence
Mutation
Coalescence
Coalescence
Mutation
Coalescence
Recombination
Event
Deconstructing the ARG
Time
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?
Ancestral
chromosome
recombines
Recurrent mutation
Recombination
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
Recombination and genetic variation – models and
inference, part II
Simon Myers
Department of Statistics, Oxford
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
HIV Subtype B (2kb segment)
HIV Subtype C (2kb segment)
• 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
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 r/2 per
sequence where r = 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 nr/2
– Mutations on edges added as a Poisson process with rate nq/2
• The time until the next recombination or coalescence event is also a
Poisson process with rate nr /2+ n(n-1)/2, and the probability that this next
event is a recombination is
Pr(rec) 
nr / 2
r

nr / 2  n(n  1) / 2 r  n  1
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
Rh
Rm
r=10, q=10
Rm against log log sample size
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)
DlnL
DlnL
1
R
0
R
0
2
4
6
-1
15
7
1
1
2
7
2
2
4
2
3
1
1
1
1
-3
DlnL
-5
Compositelikelihood
approximation
-6
R
L(ρ)   P( Di , D j | rij ,q )
i j
10
Full likelihood
-2
-4
8
Fitting a variable recombination rate
• Use a reversible-jump MCMC approach (Green 1995)
Cold
Hot
SNP positions
Split blocks
Merge blocks
Change block size
Change block rate
Acceptance rates
  C ( ) q( , )  ( ) ( , u) 
 ( , )  min1,






(

)
q
(

,

)

(

)

(

,
u
)
 C

Composite likelihood ratio
Hastings ratio
Ratio of priors
Jacobian of partial derivatives relating changes
in parameters to sampled random numbers
• 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
200kb region of human HLA
Rates estimated from genetic variation
McVean et al (2004)
Rates estimated from sperm
Jeffreys et al (2001)
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
Across chromosome12
Myers et al. (2005)
• Throughout the genome, human recombination clusters into narrow hotspots
• These explain LD breakdown sites
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, r=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