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Module de Master 2 BioStat: Mod´ elisation en g´ en´ etique des populations
Inferences of demographic history from whole genomes using approximations of the ancestral recombination graph Rapha¨el Leblois Centre de Biologie pour la Gestion des populations (CBGP, UMR INRA)
Janvier 2018
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Introduction ARG Methods HMM PSMC Simuls Conclusion
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Context and typical biological question : • Good genomic data : - Reference genome → polymorphism ”spatial” arrangement (physical / genetic distance) - Or many long runs of sequences → ≥ 10 Mb • ”Not too recent” demographic history
inference : - Detailed ancestral population size variations - Inference of gene flow / admixture during speciation process
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Demographic inferences in population genetics/genomics Before the genomic area, inferences based on • Low number of independent loci (e.g. 5-100 loci) • Short DNA sequences, with low polymorphism levels ( few
Kb, with 1-10 haplotypes) • Fast mutating DNA sequences, e.g. microsatellites, with high
polymorphism levels (up to 50 alleles) Now, we have more and more access to • Whole genomes assemblies for many species • Or many long sequences (> few Mb) with better and better
genome assemblies for less studied organisms → Genome wide sequence data contains rich information about evolutionary processes
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Demographic inferences in population genomics : what’s really new ? 1 - More polymorphisms (mutation events)... • More alleles in 10,000 SNPs than in 20 microsatellites • Simple mutational process
But ascertainment bias is often important with SNPs, lots of sequencing errors, problem with phasing haplotype data 2- With a good genome assembly, we now have access to spatial arrangement of markers →
Information about recombination events
→ Genome wide sequence data contains rich information about evolutionary processes, which remains unaccessible with independent markers.
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The coalescent with recombination : The ancestral recombination graph (ARG)
ARG = modeling recombination within or between genetic markers recombination events must be considered within long DNA sequences or between spatially close SNPs
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The coalescent with recombination : The ancestral recombination graph (ARG)
M more complex genealogical space to explore than for non-recombining loci Not a tree anymore, but a graph → much longer computation times & maybe untractable for long sequences and large sample sizes
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Several methods developed since 2007 Several recent inference methods uses the information of linkage disequilibrium present in genomic data : e.g. Mailund et al. 2011 ; Palamara et al. 2012 ; Harris & Nielsen 2013 ; MacLeod et al, 2013 ; Sheehan et al, 2013 but analyses are often limited to n = 2 sequences
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Several methods developed since 2007 Several recent inference methods uses the information of linkage disequilibrium present in genomic data but analyses are often limited to n = 2 sequences 2 main approaches • Distributions of homozygous sequence length (Runs of Homozygosity RoH, identity by state IBS tract length)
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Several methods developed since 2007 Several recent inference methods uses the information of linkage disequilibrium present in genomic data but analyses are often limited to n = 2 sequences 2 main approaches • Distributions of homozygous sequence length (Runs of Homozygosity RoH, identity by state IBS tract length) • Approximation of the Coalescent with recombination : the sequential coalescent with Hidden Markov Models (PSMC, coaHMM)
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Ideas under the coalescent HMM Approximation of the sequential coalescent with recombination based on the following ideas
• Effects of population size variation on the coalescent • ...
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Ideas under the coalescent HMM Approximation of the sequential coalescent with recombination based on the following ideas
• Effects of population size variation on the coalescent • ...
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Ideas under the coalescent HMM
• The population size at time t is inversely proportional to the
rate of coalescence at time t • Coalescence times are unknown, but are correlated with the
number of mutations between two leaves (and thus allelic frequencies)
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Ideas under the coalescent HMM
• Genealogies of nearby loci are correlated with each other • Use mutations to learn about the ancestral recombination
graph
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Ideas under the coalescent HMM Analysis of two long DNA sequences
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the coalescent HMM : A generative model for two DNA sequences
1 Transitions between trees What is the probability for a tree change from height s to height t between two successive sites (nucleotides) ? 2 Actual observed sequence Given a tree height t : what is the probability of being heterozygous or homozygous at one site between the two sequences ?
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The Sequential Markov Coalescent (SMC)
Simplifications on the coalescent with recombination between two sequences (SMC/SMC’) 1 Consider a single recombination event between 2 sites 2 Consider that the coalescent along the sequence is markovian (i.e. only depends on what happened on the previous site) P(tl ∣tl−1 , tl−2 , ..., t1 ) = P(tl ltl−1 )
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The Sequential Markov Coalescent (SMC) Transitions between trees : From a tree of height s to a tree of height t
• Wiuf and Hein (1999) : Coalescent with recombination along two DNA sequences • McVean and Cardiin (2005) : Approximating the coalescent with two possible recombination events. SMC=(a)+(b) • Marjoram and Wall (2006) : with two possible recombination events. SMC’=(a)+(b)+(c)
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The Sequential Markov Coalescent (SMC) Transitions between trees : From a tree of height s to a tree of height t
Probability of transitions are ”relatively” simple to compute : Recomb. event follows an exponential distribution of rate ρ = 2Nr Prob. of one recombination event between 0 and s : (1 − e −ρs ) Prob. of no recombination event between 0 and s : e −ρs
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The Sequential Markov Coalescent (SMC) Transitions between trees : From a tree of height s to a tree of height t
Given a recombination event happened, it is uniformly distributed between 0 and s → Pr(Rec. at u∣ Rec)=1/s
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The Sequential Markov Coalescent (SMC) Transitions between trees : From a tree of height s to a tree of height t
Given a recombination event happened, the conditional probability for a tree change from height s to height t (for a stable population size) ⎧ t ⎪ ⎪ 1 e −(t−u) du, q(t∣s) = ⎨∫0s s1 −(t−u) ⎪ du, ⎪ ⎩∫0 s e
if t ⩽ s, if t ⩾ s.
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the coalescent HMM : A generative model for two DNA sequences
1 Transitions between trees What is the probability for a tree change from height s to height t ? 2 Emmision probabilities Given a tree height t, is the site het or him ? Transition probabilities (with variable population size) : P(t∣s) = (1 − e −ρs )q(t∣s) + e −ρs δ(t − s) with q(t∣s) =
1 λ(t)
min(s,t) 1 − ∫ut e s
∫0
dv λ(v )
du
and λ(t) = N(t)/N0 the relative population size at t. and δ(⋅) is the Dirac delta function
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the coalescent HMM : A generative model for two DNA sequences
1 Transitions between trees What is the probability for a tree change from height s to height t ? 2 Emmision probabilities Given a tree height t, is the site het or him ? Mutations are distributed along branches at rate θ = 2Nµ → CDF of the time to the most recent mutation between two sites : t H(t) = ∫0 θe −θv dv = 1 − e −θt Emmision probabilities : P(Hom∣t) = e −θt P(Het∣t) = 1 − e −θt
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the coalescent HMM : A generative model for two DNA sequences
1 Transitions between trees What is the probability for a tree change from height s to height t ? 2 Emmision probabilities Given a tree height t, is the site het or him ? Transition probabilities (with variable population size) : P(t∣s) = (1 − e −ρs )q(t∣s) + e −ρs δ(t − s) Emmision probabilities : P(Hom∣t) = e −θt P(Het∣t) = 1 − e −θt
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the coalescent HMM : A generative model for two DNA sequences Posterior of TMRCA from simulations
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The PSMC : Pairwise Sequential Markov Chain (Li & Durbiin 2011) The model • Coalescent with mutation and recombination • Single panmictic isolated population (i.e. no population
structure) • Piecewise constant effective population size = ”skyline” model
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The PSMC : Pairwise Sequential Markov Chain (Li & Durbiin 2011)
• Markov chain for the TMRCA based on the SMC / SMC’ • Estimation through a coalescent Hidden Markov Model • Limited to two sequences (i.e. two haploid genomes, one diploid individual)
→ not efficient for recent times
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The PSMC : Pairwise Sequential Markov Chain (Li & Durbiin 2011)
• Population size variation modeled with ”Skyline” = discrete changes with many phases of constant population size (ad-hoc)
• Rich information about past population sizes compared to few
microsatellites loci !
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The PSMC : Pairwise Sequential Markov Chain (Li & Durbiin 2011) Population size variability in Great Apes
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The PSMC : Pairwise Sequential Markov Chain (Li & Durbiin 2011) Population size variability in Great Apes
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The PSMC : Pairwise Sequential Markov Chain (Li & Durbiin 2011) The ”skyline” model (BEAST, Drummond et al.) is very attractive but has some drawbacks • Need to define the number of phases and their lengths • Difficult to draw correct confidence intervals / credibility
intervals around the maximum
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The PSMC : Pairwise Sequential Markov Chain (Li & Durbiin 2011)
Limits of the current PSMC • the ad-hoc ”Sky-line” model • Need to fix some parameter values to get unscaled parameters →pop sizes, times in generations
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The PSMC : Pairwise Sequential Markov Chain (Li & Durbiin 2011)
Limits of the current PSMC • the ad-hoc ”Sky-line” model • Need to fix some parameter values to get unscaled parameters →pop sizes, times in generations • influence of sequencing errors →stop the spatial correlation along the genome, cut RoH
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The PSMC : Pairwise Sequential Markov Chain (Li & Durbiin 2011)
Limits of the current PSMC • the ad-hoc ”Sky-line” model • Need to fix some parameter values to get unscaled parameters →pop sizes, times in generations • influence of sequencing errors →stop the spatial correlation along the genome, cut RoH • No migration
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The PSMC : Test using simulation (From Sheehan, Harris & Song 2013)
Simple simulation tests of the PSMC (Done by S. Sheehan, PhD)
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The PSMC : Test using simulation (From Sheehan, Harris & Song 2013)
Simple simulation tests of the PSMC (Done by S. Sheehan, PhD) • Strong influence of the sample size for recent times
(reminder : often limited to 2 sequences)
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PSMC
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The PSMC : Test using simulation (From Sheehan, Harris & Song 2013)
Simple simulation tests of the PSMC (Done by S. Sheehan, PhD) • Strong influence of the sample size for recent times
(reminder : often limited to 2 sequences)
Introduction
ARG
Methods
HMM
PSMC
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The PSMC : Test using simulation (From Sheehan, Harris & Song 2013)
Simple simulation tests of the PSMC (Done by S. Sheehan, PhD) • Strong influence of the sample size for recent times
(reminder : often limited to 2 sequences)
Introduction
ARG
Methods
HMM
PSMC
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The PSMC : Test using simulation (From Sheehan, Harris & Song 2013)
Simple simulation tests of the PSMC (Done by S. Sheehan, PhD) • Strong influence of the sample size for recent times
(reminder : often limited to 2 sequences)
Introduction
ARG
Methods
HMM
PSMC
Simuls
Conclusion
The PSMC : Test using simulation (From Sheehan, Harris & Song 2013)
Simple simulation tests of the PSMC (Done by S. Sheehan, PhD) • Strong influence of the sample size for recent times
(reminder : often limited to 2 sequences)
Introduction
ARG
Methods
HMM
PSMC
Simuls
Conclusion
The PSMC : Test using simulation (From Sheehan, Harris & Song 2013)
Simple simulation tests of the PSMC (Done by S. Sheehan, PhD) • Strong influence of the sample size for recent times
(reminder : often limited to 2 sequences)
Introduction
ARG
Methods
HMM
PSMC
Simuls
Conclusion
The PSMC : Test using simulation (From Sheehan, Harris & Song 2013)
Simple simulation tests of the PSMC (Done by S. Sheehan, PhD) • Strong influence of the sample size for recent times
(reminder : often limited to 2 sequences) • Biais when low stable population size ? (→ CI ?)
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The PSMC : Test using simulation (From Sheehan, Harris & Song 2013) probably a problem with the ”skyline” parametrization : e.g. number and length of the different demographic phases
Simple simulation tests of the PSMC (Done by S. Sheehan, PhD) • Strong influence of the sample size for recent times
(reminder : often limited to 2 sequences) • Biais when low stable population size ? (→ CI ?)
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Conclusions • Genomic data contains much more information than classical
independent markers • They often cannot be analyzed with previous methods,
especially coalescent ones. • Methods that can analyze LD information are very promising • but models are still too ”simple” (but see CoaHMM with the
IM model) • Those new methods need to be implemented in user-friendly
software because they are difficult to run. • and to be clearly tested...
