exploring recent relatedness ibd and biparental ancestry

EXPLORING RECENT RELATEDNESS

IBD AND BIPARENTAL ANCESTRY

Peter Ralph and Graham Coop

Department of Evolution and EcologyUC Davis

June 11th, 2012, CIRMProbability, Population Genetics and Evolution

OUTLINE

INTRODUCTION

PROBABILITY: INHERITANCE THROUGH THE PEDIGREE

DATA: COMMON GENETIC ANCESTORS

WHEN DID THESE COMMON ANCESTORS LIVE?

NEXT STEPS

POPULATION HISTORY AND SPATIAL DYNAMICS

(from Feldman & Cavalli-Svorza)

Humans:I range expansion(s)I admixtureI adaptation locally and to local conditions

GENOMIC SIGNALS OF MIGRATION

● ●●●●●●●●●● ●

●● ●●●● ● ●● ●●●●

●● ●●● ● ●●●

● ●●●● ● ●●●

●●● ●

● ●

●

●

●

●

5000 10000 15000 20000 25000

0.0

0.2

0.4

0.6

0.8

1.0

Distance from Addis Ababa (km)

Het

eroz

ygos

ity

(Ramachandran et al 2005)

HISTORY FROM GENOMES

Goal: infer recent migrations and population structure.

Method:I Infer rates of shared ancestryI by identifying close relatives (10th–100th cousins)I How can we hope to do this?

I Unlikely that any given pair are 10th cousins, butI many ways to be related, andI between thousands of samples there are millions of possibly

related pairs.

MEIOSIS & RECOMBINATION (SEX)

I You have two copies of each chromosome,one from each parent.

I When you make a gamete, the copies recombine.I genetic distance: such that recombination rate is unityI units of centiMorgans (cM) ≈ 106 bp in humans

IBD: “IDENTITY BY DESCENT”

Definition: A block is IBD between two chromosomes if inherited fromthe same ancestor, without intervening recombinations.

Everyone is IBD everywhere, but the blocks are mostly short and old.

IBD: “IDENTITY BY DESCENT”

Definition: A block is IBD between two chromosomes if inherited fromthe same ancestor, without intervening recombinations.

IBD

Everyone is IBD everywhere, but the blocks are mostly short and old.

THE PEDIGREE AND IBD

Fragmentation-coalescencein the pedigree

I number ofgenealogical ancestorsfrom n generations agois 2n

I number of geneticancestors growslinearly

I since n meiosesdivides a 1Mchromosome into ∼ nblocks


IBDIBD





WHAT WE ACTUALLY OBSERVE: MUTATIONSInfer IBD from long regions of agreement (relative to everyone else).

BLACK BOX IBD FINDING

I fastIBD inBEAGLE(Browning &

Browning)

I Fits a variablelength Markovchain to phasedata and inferIBD blocks.

I Power analysisI False positive

rate

ON TO SOME DATA

Data from POPRES:(Nelson et al 2008)

I 2257 Europeansafter removingoutliers and closerelatives

I have country andlanguage data: 40populations

I ∼500,000 SNPs

PLENTY OF IBD BLOCKSI 1877114 blocksI 831 blocks per indiv, 0.737 per pairI covering 30–250% of each individual

1

Ireland

United Kingdom

Swiss French

Swiss German

Spain

Portugal

Italy

Yugoslavia

PLENTY OF IBD BLOCKSI 1877114 blocksI 831 blocks per indiv, 0.737 per pairI covering 30–250% of each individual

11

Ireland

United Kingdom

Swiss French

Swiss German

Spain

Portugal

Italy

Yugoslavia

MEAN WITHIN-COUNTRY IBD RATES

Mean # blocks > 1cM

Italy 0.44France 0.62Belgium 0.97Swiss.German 1.32Swiss.French 1.01Germany 1.01Spain 1.14Portugal 1.40United.Kingdom 1.04Ireland 2.15Poland 3.40Yugoslavia 3.59

Ireland

*

Germany

*

Sweden

*

Poland

*

Albania

*

Turkey

*

Spain

*

Italy

*

DECAY OF IBD WITH DISTANCE

0 1000 2000 3000 4000

0.00

50.

020

0.05

00.

200

0.50

0(1,3]cM

# bl

ocks

per

pai

r

0 1000 2000 3000 4000

(3,5]cM

dist (km)0 1000 2000 3000 4000

>5cM

Decay with distance is steeper for longer (older) blocks.(circle size is sample size)

COMMON ANCESTORS ACROSS GEOGRAPHY

10 gens

distance

20 gens

r

Geographic distribution ofnth generation co-locatedancestors:

more recent⇒ more localized

Gaussian distribution⇒coancestry at distance r∝ C

n exp(− r2

nσ2

)

DECAY OF IBD WITH DISTANCE

0 1000 2000 3000 4000

0.00

50.

020

0.05

00.

200

0.50

0(1,3]cM

# bl

ocks

per

pai

r(3,5]cM

dist (km)0 1000 2000 3000 4000

>5cM

within Easternto Turkeyto Italy except Turkeywithin Western and to Eastern

. . . and differs by location.Italians have much slower decay (older?).

WHEN DID THESE COMMON ANCESTORS LIVE?

0 1000 2000 3000 4000

0.00

50.

020

0.05

00.

200

0.50

0

(1,3]cM#

bloc

ks p

er p

air

(3,5]cM

dist (km)0 1000 2000 3000 4000

>5cM

within Easternto Turkeyto Italy except Turkeywithin Western and to Eastern

BACK TO THEORY: EXPECTED IBD

I Fix pedigree (not recombinations), andtwo (sampled) chromosomes of length G.

I N(x) = #{ IBD blocks of length ≥ x}I Decompose N(x) =

∑T NT (x)

by paths T through the pedigreeI E[NT (x)] = K (|T |, x) 4−|T |, whereI |T | = # of meioses along T ,I 0 = R0 ≤ R1 ≤ . . .Rk = G

locations of recombinationsI K (t , x) = E[#{j : Rj − Rj−1 > x}].

EXPECTED NUMBER OF LONG IBD BLOCKS

Coalescent distribution: choose path T withprobability 4−|T |. Let |T | = τ .

Expected block length distribution is a linearfunction of coalescent distribution:

E[N(x)] =∑

t

P{τ = t}K (t , x)

If recombinations are Poisson,

K (t , x) = (1 + t(G − x))e−tx

MEAN BLOCK RATE AND COALESCENT DISTRIBUTION

Actually: mean IBD length distribution is a linear function of thecoalescent distribution, so with

µ(t) := P{τ = t},

N(x) is Poisson with mean density

− ddx

E[N(x)] =∑

t

µ(t)K (t , y)γ(y)∫ G

0f (y , x)dy + ξ(x)

=∑

t

µ(t)K̃ (t , x) + ξ(x)

with: power γ, false positive rate ξ, and error kernel f .

. . . maximum likelihood?

EXPLORING THE LIKELIHOOD RIDGE

● ● ● ● ● ●● ●

●● ●

●● ●

●●

●●

●●

●●

●●

●●

●●

●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 10 20 30 40 50

1e−

111e

−05

1e+

01

eige

nval

ue

0 20 40 60 80 100

−0.

3−

0.1

0.1

0.3

generations

2 4 6 8 10

−0.

4−

0.2

0.0

0.2

0.4

block length (cM)

(Poisson) log likelihood function

L(N|µ)

is very flat in many directions(“ridged”).

We’re doing inference, so: need toexplore it.

We do this by finding maximizers to

L(N|µ)− γ(µ)

for various penalizations γ.

Call µ′ feasible ifL(N|µ′) + 2 ≥ maxν L(N|ν).

COALESCENT DISTRIBUTION WITHIN THE UK:Grey is “best” solution; red is “smoothest” solution(differing by no more than 2 units of log likelihood).

0.0

1.0

UK−IE

0.0

1.0

UK−FR

coal

rat

e *

1000

00.

01.

0

UK−IT

0.0

1.0

UK−TK

0 1000 2000 3000 4000years ago

NUMBERS OF GENETIC COMMON ANCESTORS:Grey is “best” solution; red is “smoothest” solution(differing by no more than 2 units of log likelihood).

010

25

UK−IE

010

25

UK−FR

# ge

netic

anc

esto

rs0

1025

UK−IT

010

25

UK−TK

0 1000 2000 3000 4000years ago

COMMON ANCESTRY WITH THE BALKANS:

(“Balkans” is peninsula except Albanian speakers.)0

1025

Balkans

010

25

Balk−Alb

# ge

netic

anc

esto

rs0

1025

Balk−IT

010

25

Balk−FR

0 1000 2000 3000 4000years ago

SUMMARIES OF COMMON ANCESTRY

Box: “best” & “smoothest”; whiskers: most & least0

515

0−540 years ago

020

0

555−1500 years ago

# ge

netic

anc

esto

rs

010

0020

00

1515−2535 years ago

S/C PL RO/BG FR Bal UK IT

FROM WHENCE EASTERN IBD?Numbers of common ancestors shared with:Serbo-Croatian speakers

0−540 yaS/C 555−1500 ya 1515−2535 ya

Poland0−540 yaPL 555−1500 ya 1515−2535 ya

SIGNS OF PAST INVASIONS?

Hunnic empire, 1550ya:

credit: wikipedia

WHAT ABOUT THE GERMANIC MOVEMENTS?

WHAT ABOUT THE GERMANIC MOVEMENTS?0−540 yaFR 555−1500 ya 1515−2535 ya

0−540 yaIT 555−1500 ya 1515−2535 ya

0−540 yaIber 555−1500 ya 1515−2535 ya

FURTHER DIRECTIONS

I Further work on fragmentation-coalescence (in a pedigree?)I Look at the process along the genome.I Lack of fit at short lengths: improve the model.I Geographic method of coalescent distribution inference – more

than pairwise?I What does the coalescent distribution tell us, anyhow?

A PROBLEM WITH SMALL BLOCKS

Some assumptions of the model break down at short lengths:0.

00.

20.

40.

60.

8IB

D r

ate

1 2 3 4 5 6block length (cM)

UK−IE

false pos

15−540

555−1500

1515−2535

2550−4335

4350−7935

(we used blocks > 2cM for timing inference)

GEOGRAPHY: MORE THAN PAIRWISE

We find coalescent distribution: pairwise, nonparametric.Could fit more than pairwise in parametric model. Other ideas?

COALESCENT TIME DISTRIBUTIONS?

We can∗ infer distribution ofcoalescent time in thepedigree, across samples.

What does this tell us aboutshared history?

Intuition is mostly from:I n-deme migration

(usually, n = 2;rigorous)

I cartoons (not so much)∗ maybe

10 gens

distance

20 gens

r

SUMMARY

Patterns of recent relatedness between Europeans are shapedprimarily by:

I continuous, local gene flow – isolation by distanceI large population expansions – Slavs, Huns?I other historical factors – language, stability, . . .

Summaries of long shared tracts of genome from recent commonancestors:

I have lots of signal about recent history in modern datasetsI can be used to infer statistical properties of the recent pedigreeI but may have fundamental drawbacks

THANKS

Graham Coop

Steve Evans, Charles Langley, Yaniv Brandvain, Torsten Günther

Funding: NIH

exploring recent relatedness ibd and biparental ancestry

Documents