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

Contents......Page 10
1.1 Introduction......Page 16
1.2 A Y-chromosome data set......Page 20
1.3 Data and theory......Page 25
1.4 The Wright–Fisher model......Page 26
1.4.1 Assumptions of the Wright–Fisher model......Page 28
1.4.2 The number of descendants of a gene in one generation......Page 29
1.4.3 An example......Page 30
1.5 The geometric distribution......Page 32
1.6 The exponential distribution......Page 34
1.7.1 Coalescence of a sample of two genes......Page 36
1.7.2 Coalescence of a sample of n genes......Page 37
1.7.3 Example: Effect of approximations......Page 38
1.8 The continuous time coalescent......Page 39
1.9.1 The height of a tree......Page 40
1.9.2 The total branch length of a tree......Page 42
1.9.3 The effect of sampling more sequences......Page 43
1.10 The effective population size......Page 44
1.11 The Moran model......Page 46
1.12 Robustness of the coalescent......Page 47
2.1.1 The infinite alleles model......Page 48
2.1.2 The infinite sites model......Page 50
2.1.3 Finite sites model......Page 52
2.2 The Wright–Fisher model with mutation......Page 54
2.3 Algorithms for simulating sequence evolution......Page 56
2.4 The probability of a sample configuration......Page 60
2.4.1 Infinite alleles model......Page 61
2.4.2 Infinite sites model......Page 65
2.4.3 Impossible ancestral states......Page 70
2.5.1 The number of segregating sites......Page 73
2.5.2 Haplotypes......Page 75
2.5.3 Pairwise mismatch distribution......Page 76
2.5.4 Estimators of θ and Tajima’s D......Page 77
2.6 Evolutionary versus sampling variance......Page 78
2.6.1 Example 1: The variable S[sub(n)]......Page 79
2.6.2 Example 2: Tajima's estimator π......Page 80
3.1.2 The coalescent and phylogenetic trees......Page 82
3.2 Counting trees and topologies......Page 85
3.3 Gene trees......Page 87
3.3.1 How to build a gene tree......Page 90
3.4 Nested subsamples......Page 91
3.5 Hanging subtrees......Page 93
3.5.2 Example: Neanderthal sequences......Page 96
3.6 A single lineage......Page 97
3.7 Disjoint subsamples......Page 98
3.7.1 Examples......Page 101
3.8 A sample partitioned by a mutation......Page 102
3.8.1 Unknown ancestral state......Page 104
3.8.2 The age of the MRCA for two sequences......Page 105
3.9 The probability of going from n ancestors to k ancestors......Page 106
4.2.1 Stochastic and systematic changes......Page 110
4.2.2 How to model population changes in the coalescent......Page 111
4.3 Exponential growth......Page 114
4.3.1 The genealogy under exponential growth......Page 115
4.4 Population bottlenecks......Page 119
4.4.1 Genealogical effect of bottlenecks......Page 121
4.5 Effective population size revisited......Page 122
4.6.1 The finite island model......Page 123
4.6.2 The coalescent tree in the finite island model......Page 125
4.6.3 General models of subdivision......Page 129
4.6.4 Non-equilibrium models......Page 131
4.7.1 Two allele balancing selection......Page 133
4.7.2 Multiallelic balancing selection......Page 135
4.8.1 The ancestral selection graph......Page 138
4.9 Summary......Page 141
5.1 Introduction......Page 142
5.2 Data example with recombination......Page 143
5.3.1 Hudson’s model of recombination......Page 145
5.3.2 Biological features of recombination......Page 147
5.4 The Wright–Fisher model with recombination......Page 152
5.5.1 The ancestral recombination graph......Page 154
5.5.2 Sampling ARGs: Not back in time, but along sequences......Page 159
5.5.3 Efficiency of different algorithms......Page 162
5.6 The effect of a single recombination event......Page 163
5.7 The number of recombination events......Page 167
5.8 The probability of a data set......Page 168
5.9 The number of segregating sites......Page 170
5.10 The coalescent with gene conversion......Page 171
5.11 Gene trees with recombination—from incompatibilities to minimal ARGs......Page 173
5.11.1 Recombination as subtree transfer......Page 174
5.11.2 Recombination inferred from haplotypes......Page 180
5.11.3 From local to global bounds......Page 181
5.11.4 Minimal ARGs......Page 182
5.11.5 Topologies, recombination, and compatibility......Page 184
6.1 Introduction......Page 188
6.2 Estimators of θ......Page 189
6.2.1 Watterson’s estimator......Page 190
6.2.2 Tajima’s estimator......Page 191
6.2.3 Fu’s two estimators......Page 193
6.3 Estimators of ρ......Page 196
6.3.1 Estimators based on summary statistics......Page 198
6.3.2 Pseudo-likelihood estimators......Page 200
6.4 Monte Carlo methods......Page 202
6.4.1 The likelihood curve......Page 204
6.4.2 Monte Carlo integration and the coalescent......Page 206
6.4.3 Markov chain Monte Carlo......Page 210
7.1 The potential of LD mapping......Page 214
7.2 Linkage versus LD mapping......Page 215
7.3 Complex disease aetiology......Page 217
7.4 Formulating the task......Page 220
7.5 A role for the coalescent......Page 221
7.6 Genealogical trees around a disease mutation......Page 223
7.6.1 Qualitative measures......Page 224
7.6.2 An example......Page 225
7.6.3 Quantifying genealogical tree differences......Page 228
7.7 The genealogical process reflected in data......Page 231
7.8 Linkage disequilibrium (LD)......Page 232
7.8.2 Accounting for population admixture......Page 235
7.8.3 Differences between human populations......Page 236
7.10 Haplotype LD mapping......Page 238
7.11 Model based LD mapping......Page 239
7.11.2 Coalescent based genealogy......Page 240
7.11.3 An example......Page 242
7.11.4 Further challenges......Page 243
8.1 Introduction......Page 246
8.2 Our phylogenetic position and ancestral population genetics......Page 247
8.2.1 The number of genetic ancestors to a genome......Page 250
8.3 Human migrations and population structure......Page 255
8.3.1 Our relationship to the Neanderthaler......Page 257
8.3.3 Structure within global modern human populations......Page 259
8.3.4 Specific histories......Page 260
8.3.5 Empirical pedigrees and the coalescent......Page 261
8.3.6 Other genealogical issues......Page 265
8.3.7 Tracing genetic material within the parent genealogy......Page 267
Appendix: Web based tools......Page 270
Bibliography......Page 274
D......Page 288
L......Page 289
R......Page 290
Y......Page 291