Gene Genealogies, Variation and Evolution. A Primer in Coalescent Theory

279 Pages·2005·1.31 MB·english

by  Jotun Hein, Mikkel H. Schierup, Carsten Wiuf

Gene Genealogies,
Variation and Evolution
A Primer in Coalescent Theory
Jotun Hein
University of Oxford, UK
Mikkel H. Schierup
and
Carsten Wiuf
University of Aarhus, Denmark
1
PublishedintheUnitedStates
byOxfordUniversityPressInc.,NewYork
©JotunHein,MikkelH.SchierupandCarstenWiuf,2005
Firstpublished2005
ISBN 0-19-852995-3(hbk.)
ISBN 0-19-852996-1(pbk.)
TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India
PrintedinGreatBritain
onacid-freepaperby
BiddlesLtd.,King’sLynn,Norfolk
Preface
Coalescenttheoryhasgonefromanobscurecornerofpopulationgenetics
toacentralconceptforanybodythatstudiesvariationatthesequencelevel.
Besides filling the obvious need for a book on this subject, it is also our
wish to present this theory in a straightforward and elementary manner
thatcoulddispelthemisconceptionthatcoalescenttheoryisinherentlyvery
difficultandneedsastrongmathematicalbackgroundtounderstandit.The
key issues needed for data analysis require only basic combinatorics and
probability theory. Despite the present prominence of coalescent theory,
it also belongs to the future. From an application point of view, human
evolution and association mapping/fine scale mapping are two areas that
areboundtogrowenormouslyinthenextfewyears.Andtomakeoptimal
useofthecomingfloodofdata,theoreticaladvanceswillbeneeded.There
areareaswherepresenttheoryfails(orisimpracticallyslow)inthepresence
ofrealdataandifempiricalresearchersaretousecoalescentbasedmethod,
thereareplentyofchallengesforthetheoreticianbothinmodellingandin
improvementofsimulationalgorithms.
Thepresentbookisdefinitelynotexhaustive,butisonlymeanttoprovide
agoodbasisforfurtherstudy.Chapter1providesthebasicsforunderstand-
ing the assumptions behind and derivation of the basic coalescent model.
Chapter 2 introduces the models of alleles and sequences and associated
mutation processes, and Chapter 3 gives some examples of quantities that
canbecalculatedoncoalescentgenealogies.Thebasiccoalescentisnaivefor
manyapplicationsandChapter4and5relaxtheassumptionsofthebasic
coalescenttoallowforpopulationsizechanges,populationstructure,vari-
ousformsofselection,andrecombination.Chapter6changestheemphasis
fromdescribingthecoalescencestructuretoinferringparametersfromdata
using knowledge on this structure. In Chapter 7 and 8 two areas of much
currentinterestandwherecoalescenttheoryislikelytoplayamajorroleare
introduced.Chapter7discussestheusageofcoalescencetheoryinthefield
of linkage disequilibrium mapping that aims at locating genes underlying
common diseases. Chapter 8 relates the potential and use of coalescence
theorytohumanevolution.Inanappendixabriefintroductiontotheweb
based tools developed in connection with this book are presented. The
collectionoftoolscanbefoundathttp://www.coalescent.dk.
Contents
1 Thebasiccoalescent 1
1.1 Introduction 1
1.2 AY-chromosomedataset 5
1.3 Dataandtheory 10
1.4 TheWright–Fishermodel 11
1.4.1 AssumptionsoftheWright–Fishermodel 13
1.4.2 Thenumberofdescendantsofageneinonegeneration 14
1.4.3 Anexample 15
1.5 Thegeometricdistribution 17
1.6 Theexponentialdistribution 19
1.7 Thediscrete-timecoalescent 21
1.7.1 Coalescenceofasampleoftwogenes 21
1.7.2 Coalescenceofasampleofngenes 22
1.7.3 Example:Effectofapproximations 23
1.8 Thecontinuoustimecoalescent 24
1.9 Calculatingsimplequantitiesonacoalescenttree 25
1.9.1 Theheightofatree 25
1.9.2 Thetotalbranchlengthofatree 27
1.9.3 Theeffectofsamplingmoresequences 28
1.10 Theeffectivepopulationsize 29
1.11 TheMoranmodel 31
1.12 Robustnessofthecoalescent 32
2 Fromgenealogiestosequences 33
2.1 Mathematicalmodelsofalleles 33
2.1.1 Theinfiniteallelesmodel 33
2.1.2 Theinfinitesitesmodel 35
2.1.3 Finitesitesmodel 37
2.2 TheWright–Fishermodelwithmutation 39
2.3 Algorithmsforsimulatingsequenceevolution 41
2.4 Theprobabilityofasampleconfiguration 45
2.4.1 Infiniteallelesmodel 46
2.4.2 Infinitesitesmodel 50
2.4.3 Impossibleancestralstates 55
2.5 Quantitiesrelatedtotheinfinitesitesmodel 58
2.5.1 Thenumberofsegregatingsites 58
2.5.2 Haplotypes 60
2.5.3 Pairwisemismatchdistribution 61
2.5.4 Estimatorsofθ andTajima’sD 62
2.6 Evolutionaryversussamplingvariance 63
2.6.1 Example1:ThevariableSn 64
2.6.2 Example2:Tajima’sestimatorπˆ 65
3 Treesandtopologies 67
3.1 Someterminology 67
3.1.1 Thejumpprocessandthewaitingtimeprocess 67
3.1.2 Thecoalescentandphylogenetictrees 67
3.2 Countingtreesandtopologies 70
3.3 Genetrees 72
3.3.1 Howtobuildagenetree 75
3.4 Nestedsubsamples 76
3.5 Hangingsubtrees 78
3.5.1 Unbalancedtrees 81
3.5.2 Example:Neanderthalsequences 81
3.6 Asinglelineage 82
3.7 Disjointsubsamples 83
3.7.1 Examples 86
3.8 Asamplepartitionedbyamutation 87
3.8.1 Unknownancestralstate 89
3.8.2 TheageoftheMRCAfortwosequences 90
3.9 Theprobabilityofgoingfromnancestors
tokancestors 91
4 Extensionstothebasiccoalescent 95
4.1 Introduction 95
4.2 Thecoalescentwithfluctuatingpopulationsize 95
4.2.1 Stochasticandsystematicchanges 95
4.2.2 Howtomodelpopulationchangesinthecoalescent 96
4.3 Exponentialgrowth 99
4.3.1 Thegenealogyunderexponentialgrowth 100
4.4 Populationbottlenecks 104
4.4.1 Genealogicaleffectofbottlenecks 106
4.5 Effectivepopulationsizerevisited 107
4.6 Thecoalescentwithpopulationstructure 108
4.6.1 Thefiniteislandmodel 108
4.6.2 Thecoalescenttreeinthefiniteislandmodel 110
4.6.3 Generalmodelsofsubdivision 114
4.6.4 Non-equilibriummodels 116
4.7 Coalescentwithbalancingselection 118
4.7.1 Twoallelebalancingselection 118
4.7.2 Multiallelicbalancingselection 120
4.8 Coalescentwithdirectionalselection 123
4.8.1 Theancestralselectiongraph 123
4.9 Summary 126
5 Thecoalescentwithrecombination 127
5.1 Introduction 127
5.2 Dataexamplewithrecombination 128
5.3 Modellingrecombination 130
5.3.1 Hudson’smodelofrecombination 130
5.3.2 Biologicalfeaturesofrecombination 132
5.4 TheWright–Fishermodelwithrecombination 137
5.5 Algorithms 139
5.5.1 Theancestralrecombinationgraph 139
5.5.2 SamplingARGs:Notbackintime,butalongsequences 144
5.5.3 Efficiencyofdifferentalgorithms 147
5.6 Theeffectofasinglerecombinationevent 148
5.7 Thenumberofrecombinationevents 152
5.8 Theprobabilityofadataset 153
5.9 Thenumberofsegregatingsites 155
5.10 Thecoalescentwithgeneconversion 156
5.11 Genetreeswithrecombination—fromincompatibilitiesto
minimalARGs 158
5.11.1Recombinationassubtreetransfer 159
5.11.2Recombinationinferredfromhaplotypes 165
5.11.3Fromlocaltoglobalbounds 166
5.11.4MinimalARGs 167
5.11.5Topologies,recombination,andcompatibility 169
6 Gettingparametersfromdata 173
6.1 Introduction 173
6.2 Estimatorsofθ 174
6.2.1 Watterson’sestimator 175
6.2.2 Tajima’sestimator 176
6.2.3 Fu’stwoestimators 178
6.3 Estimatorsofρ 181
6.3.1 Estimatorsbasedonsummarystatistics 183
6.3.2 Pseudo-likelihoodestimators 185
6.4 MonteCarlomethods 187
6.4.1 Thelikelihoodcurve 189
6.4.2 MonteCarlointegrationandthecoalescent 191
6.4.3 MarkovchainMonteCarlo 195
7 LDmappingandthecoalescent 199
7.1 ThepotentialofLDmapping 199
7.2 LinkageversusLDmapping 200
7.3 Complexdiseaseaetiology 202
7.4 Formulatingthetask 205
7.5 Aroleforthecoalescent 206
7.6 Genealogicaltreesaroundadiseasemutation 208
7.6.1 Qualitativemeasures 209
7.6.2 Anexample 210
7.6.3 Quantifyinggenealogicaltreedifferences 213
7.7 Thegenealogicalprocessreflectedindata 216
7.8 Linkagedisequilibrium(LD) 217
7.8.1 TestingforLD 220
7.8.2 Accountingforpopulationadmixture 220
7.8.3 Differencesbetweenhumanpopulations 221
7.9 Measuringassociationusingsinglemarkers 223
7.10 HaplotypeLDmapping 223
7.11 ModelbasedLDmapping 224
7.11.1Starshapedgenealogy 225
7.11.2Coalescentbasedgenealogy 225
7.11.3Anexample 227
7.11.4Furtherchallenges 228
8 Humanevolution 231
8.1 Introduction 231
8.2 Ourphylogeneticpositionandancestralpopulationgenetics 232
8.2.1 Thenumberofgeneticancestorstoagenome 235
8.3 Humanmigrationsandpopulationstructure 240
8.3.1 OurrelationshiptotheNeanderthaler 242
8.3.2 Populationgrowth 244
8.3.3 Structurewithinglobalmodernhumanpopulations 244
8.3.4 Specifichistories 245
8.3.5 Empiricalpedigreesandthecoalescent 246
8.3.6 Othergenealogicalissues 250
8.3.7 Tracinggeneticmaterialwithintheparentgenealogy 252
Appendix:Webbasedtools 255
Bibliography 259
Index 273
The basic coalescent
1
1.1 Introduction
In this chapter, we first motivate the need for a mathematical model that
candescribetheprocessgeneratinggeneticdata,withspecificemphasison
human variation. We assume that genetic data are in the form of DNA
sequence data. The sequences or genes are all homologous copies of the
samegeneticregioninthegenomeofaspecies.Whetheroneorbothcopies
of a gene in an individual is sampled does not matter here, what matters
is their number and genetic type. Such data are collected from one or sev-
eral present-day populations of a single species, and from this sample we
wanttoinferdetailsabouttheevolutionaryprocessesthatcreatedthedata.
(Apopulationisherebestunderstoodasapopulationofgenes,ratherthan
a population of individuals, because we focus at the level of genes.) The
inferential analysis is retrospective; we seek to understand aspects of the
sample’s (and the population’s) evolutionary past through analysis of the
presentdaysample.Belowwesketchananalysisofadatasetfromhumans
to make more clear the different types of questions that coalescent theory
seekstoanswer.
Thehumangenomeconsistsoftwenty-twoautosomalchromosomes,the
twosexchromosomesXandY,andthemitochondrion.Onerepresentative
sequence of the human genome has recently been approximately determ-
ined, and is presently subject to refinement. Additionally, a major effort
hasnowbeendirectedtowardsdeterminingthepopulationvariationinthe
human genome through determination of single nucleotide polymorphism
(SNPs).Thesearepositionsthatvarywithinorbetweenhumanpopulations.
Table1.1showsthesizesofeachchromosomeandthenumberofpositions
wherevariationhavesofarbeendetected.Thehumangenomeandthevari-
ationthatisobservedistheresultofinteractionamongevolutionaryforces,
such as mutational and selectional processes, mixing of variation through
recombination,anddemographicfactors,suchasthesize,history,andgeo-
graphicalstructureofthepopulations.Effectsofdemographyareillustrated
by the major colonisations of the globe (Figure 1.1). The present human
population migrated out of East Africa approximately 100,000 years ago
2 1:Thebasiccoalescent
Table1.1 Thehumangenomea
Chromosome SizeinMb LengthincM Genes SNPs SNPdensity
1 245 293 1,945 426 1.74
2 243 277 1,283 396 1.63
3 199 233 1,049 317 1.59
4 191 212 765 318 1.66
5 181 198 879 323 1.78
6 171 201 1,053 309 1.79
7 158 184 952 282 1.78
8 146 166 717 256 1.75
9 134 167 755 263 1.96
10 135 182 756 250 1.85
11 135 156 1,294 249 1.84
12 133 169 1,006 213 1.60
13 114 118 341 166 1.46
14 105 129 647 148 1.41
15 100 110 592 166 1.66
16 90 131 900 183 2.03
17 82 129 1,121 144 1.76
18 78 124 267 138 1.77
19 64 110 1,303 110 1.72
20 64 97 631 232 3.63
21 47 60 231 88 1.87
22 49 58 485 121 2.47
X 152 198 750 45 1.61
Y 50 1 94 30 1.60
aThesecondcolumnshowsthelengthofeachchromosomeinmillionbases(Mb),
thethirdthelengthincentiMorgans(cM),thefourththeestimatednumberof
genesforeachchromosome,thefifththenumberofcurrentlyidentifiedSNPs
(thousands),andthelastcolumnshowsthecurrentdensityofSNPsperkilobases
(kb).ThedetectednumberofSNPsandthusalsotheSNPdensitywillgoup
withinthenextfewyears.Thedataisfromwww.ensembl.org,release17.33.1
(July2003),exceptforthegeneticmaplengthswhicharetakenfromtheGenethon
geneticmap.
15–35,000
40,000
50–60,000
100,000
>40,000
Figure1.1 Theworldandhistoricalmigrations.Theapproximatedatesofmass
migrations(yearsago)havemainlybeendeterminedbydatingfossilsfoundatdifferent
locations.AdaptedfromCavalli-Sforza(2001).