1 Models of nucleotide substitution ............................ i
1.1 Introduction ............................................ 1
1.2 Markov models of nucleotide substitution and distance
estimation .............................................. 4
1.2.1 The JC69 model ................................... 4
1.2.2 The K80 model .................................... 7
1.2.3 HKY85, F84, TN93, etc. ........................... 9
1.2.4 The transition/transversion rate ratio .......... 13
1.3 Variable substitution rates across sites ............... 15
1.4 Maximum likelihood estimation of distance .............. 17
1.4.1 The JC69 model .................................. 18
1.4.2 The K80 model ................................... 22
1.4.3 Likelihood ratio test of substitution models .... 22
1.4.4 Profile and integrated likelihood methods ....... 24
1.5 Markov chains and distance estimation under general
models ................................................. 26
1.5.1 Markov chains ................................... 26
1.5.2 Distance under the unrestricted (UNREST) model .. 27
1.5.3 Distance under the general time-reversible
model ........................................... 29
1.6 Discussions ............................................ 32
1.6.1 Distance esdmation under different
substitution models ............................. 32
1.6.2 Limitations of pairwise comparison .............. 32
1.7 Problems ............................................... 33
2 Models of amino acid and codon substitution ................. 35
2.1 Introduction ........................................... 35
2.2 Models of amino acid replacement ....................... 35
2.2.1 Empirical models ................................ 35
2.2.2 Mechanistic models .............................. 39
2.2.3 Among-site heterogeneity ........................ 39
2.3 Estimation of distance between two protein sequences ... 40
2.3.1 The Poisson model ............................... 40
2.3.2 Empirical models ................................ 41
2.3.3 Gamma distances ................................. 41
2.4 Models of codon substitution ........................... 42
2.4.1 The basic model ................................. 42
2.4.2 Variations and extensions ....................... 44
2.5 Estimation of dS and dN ................................ 47
2.5.1 Counting methods ................................ 47
2.5.2 Maximum likelihood method ....................... 55
2.5.3 Comparison of methods ........................... 57
2.5.4 More distances and interpretation of the dN/dS
ratio ........................................... 58
2.5.5 Estimation of S and dN in comparative
genomics ........................................ 61
2.5.6 Distances based on the physical-site
definition ...................................... 63
2.5.7 Utility of the distance measures ................ 65
2.6 Numerical calculation of the transition probability
matrix ................................................. 65
2.7 Problems ............................................... 68
3 Phylogeny reconstruction: overview .......................... 70
3.1 Tree concepts .......................................... 70
3.1.1 Terminology ..................................... 70
3.1.2 Species trees and gene trees .................... 79
3.1.3 Classification of tree reconstruction methods ... 81
3.2 Exhaustive and heuristic tree search ................... 82
3.2.1 Exhaustive tree search .......................... 82
3.2.2 Heuristic tree search ........................... 82
3.2.3 Branch swapping ................................. 84
3.2.4 Local peaks in the tree space ................... 86
3.2.5 Stochastic tree search .......................... 88
3.3 Distance matrix methods ................................ 88
3.3.1 Least-squares method ............................ 89
3.3.2 Minimum evolution method ........................ 91
3.3.3 Neighbour-joining method ........................ 91
3.4 Maximum parsimony ...................................... 95
3.4.1 Brief history ................................... 95
3.4.2 Counting the minimum number of changes on
a tree .......................................... 95
3.4.3 Weighted parsimony and dynamic programming ...... 96
3.4.4 Probabilities of ancestral states ............... 99
3.4.5 Long-branch attraction .......................... 99
3.4.6 Assumptions of parsimony ....................... 100
3.5 Problems .............................................. 101
4 Maximum likelihood methods ................................. 102
4.1 Introduction .......................................... 102
4.2 Likelihood calculation on tree ........................ 102
4.2.1 Data, model, tree, and likelihood .............. 102
4.2.2 The pruning algorithm .......................... 103
4.2.3 Time reversibility, the root of the tree, and
the molecular clock ............................ 107
4.2.4 A numerical example: phylogeny of apes ......... 108
4.2.5 Amino acid, codon, and RNA models .............. 110
4.2.6 Missing data, sequence errors, and alignment
gaps ........................................... 110
4.3 Likelihood calculation under more complex models ...... 114
4.3.1 Mixture models for variable rates among sites .. 114
4.3.2 Mixture models for pattern heterogeneity
among sites .................................... 122
4.3.3 Partition models for combined analysis of
multiple datasets .............................. 123
4.3.4 Nonhomogeneous and nonstationary models ........ 125
4.4 Reconstruction of ancestral states ................... 125
4.4.1 Overview ....................................... 125
4.4.2 Empirical and hierarchical Bayesian
reconstruction ................................. 127
4.4.3 Discrete morphological characters .............. 130
4.4.4 Systematic biases in ancestral reconstruction .. 131
4.5 Numerical algorithms for maximum likelihood
estimation ............................................ 133
4.5.1 Univariate optimization ......................... 134
4.5.2 Multivariate optimization ....................... 136
4.6 ML optimization in phylogenetics ...................... 138
4.6.1 Optimization on a fixed tree ................... 138
4.6.2 Multiple local peaks on the likelihood
surface for a fixed tree ....................... 139
4.6.3 Search in the tree space ....................... 140
4.6.4 Approximate likelihood method .................. 143
4.7 Model selection and robustness ........................ 144
4.7.1 Likelihood ratio test applied to rbcL dataset .. 144
4.7.2 Test of goodness of fit and parametric
bootstrap ...................................... 146
4.7.3 Diagnostic tests to detect model violations .... 147
4.7.4 Akaike information criterion (AIC and AICc) .... 148
4.7.5 Bayesian information criterion ................. 149
4.7.6 Model adequacy and robustness .................. 150
4.8 Problems .............................................. 151
5 Comparison of phylogenetic methods and tests on trees ...... 153
5.1 Statistical performance of tree reconstruction
methods ............................................... 153
5.1.1 Criteria ....................................... 154
5.1.2 Performance .................................... 156
5.2 Likelihood ............................................ 157
5.2.1 Contrast with conventional parameter
estimation ..................................... 157
5.2.2 Consistency .................................... 158
5.2.3 Efficiency ..................................... 159
5.2.4 Robustness ..................................... 163
5.3 Parsimony ............................................. 165
5.3.1 Equivalence with misbehaved likelihood models .. 165
5.3.2 Equivalence with well-behaved likelihood
models ......................................... 168
5.3.3 Assumptions and justifications ................. 169
5.4 Testing hypotheses concerning trees ................... 171
5.4.1 Bootstrap ...................................... 172
5.4.2 Interior-branch test ........................... 177
5.4.3 K-H test and related tests ..................... 178
5.4.4 Example: phylogeny of apes ..................... 179
5.4.5 Indexes used in parsimony analysis ............. 180
5.5 Problems .............................................. 181
6 Bayesian theory ............................................ 182
6.1 Overview .............................................. 182
6.2 The Bayesian paradigm ................................. 183
6.2.1 The Bayes theorem .............................. 183
6.2.2 The Bayes theorem in Bayesian statistics ....... 184
6.2.3 Classical versus Bayesian statistics ............ 189
6.3 Prior ................................................. 197
6.3.1 Methods of prior specification ................. 197
6.3.2 Conjugate priors ............................... 198
6.3.3 Flat or uniform priors ......................... 199
6.3.4 The Jeffreys priors ............................. 200
6.3.5 The reference priors ............................ 202
6.4 Methods of integration ................................ 203
6.4.1 Laplace approximation .......................... 203
6.4.2 Mid-point and trapezoid methods ................ 204
6.4.3 Gaussian quadrature ............................ 205
6.4.4 Marginal likelihood calculation for JC69
distance estimation ............................ 206
6.4.5 Monte Carlo integration ........................ 210
6.4.6 Importance sampling ............................ 210
6.5 Problems .............................................. 212
7 Bayesian computation (MCMC) ................................ 214
7.1 Markov chain Monte Carlo .............................. 214
7.1.1 Metropolis algorithm ........................... 214
7.1.2 Asymmetrical moves and proposal ratio .......... 218
7.1.3 The transition kernel .......................... 219
7.1.4 Single-component Metropolis-Hastings
algorithm ...................................... 220
7.1.5 Gibbs sampler .................................. 221
7.2 Simple moves and their proposal ratios ................ 221
7.2.1 Sliding window using the uniform proposal ...... 222
7.2.2 Sliding window using the normal proposal ....... 223
7.2.3 Bactrian proposal .............................. 223
7.2.4 Sliding window using the multivariate normal
proposal ....................................... 224
7.2.5 Proportional scaling ........................... 225
7.2.6 Proportional scaling with bounds ............... 226
7.3 Convergence, mixing, and summary of MCMC .............. 226
7.3.1 Convergence and tail behaviour ................. 226
7.3.2 Mixing efficiency, jump probability, and step
length ......................................... 230
7.3.3 Validating and diagnosing MCMC algorithms ...... 241
7.3.4 Potential scale reduction statistic ............ 242
7.3.5 Summary of MCMC output ......................... 243
7.4 Advanced Monte Carlo methods .......................... 244
7.4.1 Parallel tempering (MC3) ....................... 245
7.4.2 Trans-model and trans-dimensional MCMC ......... 247
7.4.3 Bayes factor and marginal likelihood ........... 256
7.5 Problems .............................................. 260
8 Bayesian phylogenetics ..................................... 263
8.1 Overview .............................................. 263
8.1.1 Historical background .......................... 263
8.1.2 A sketch MCMC algorithm ........................ 264
8.1.3 The statistical nature of phylogeny
estimation ..................................... 264
8.2 Models and priors in Bayesian phylogenetics ........... 266
8.2.1 Priors on branch lengths ....................... 266
8.2.2 Priors on parameters in substitution models .... 269
8.2.3 Priors on tree topology ........................ 276
8.3 MCMC proposals in Bayesian phylogenetics .............. 279
8.3.1 Within-tree moves .............................. 279
8.3.2 Cross-tree moves ............................... 281
8.3.3 NNl for unrooted trees ......................... 284
8.3.4 SPR for unrooted trees ......................... 287
8.3.5 TBR for unrooted trees ......................... 289
8.3.6 Subtree swapping ............................... 291
8.3.7 NNI for rooted trees ........................... 292
8.3.8 SPR on rooted trees ............................ 293
8.3.9 Node slider .................................... 294
8.4 Summarizing MCMC output ............................... 295
8.5 High posterior probabilities for trees ................ 296
8.5.1 High posterior probabilities for trees or
splits ......................................... 296
8.5.2 Star tree paradox .............................. 298
8.5.3 Fair coin paradox, fair balance paradox, and
Bayesian model selection ....................... 300
8.5.4 Conservative Bayesian phylogenetics ............ 305
8.6 Problems .............................................. 306
9 Coalescent theory and species trees ........................ 308
9.1 Overview .............................................. 308
9.2 The coalescent model for a single species ............. 309
9.2.1 The backward time machine ...................... 309
9.2.2 Fisher-Wright model and the neutral
coalescent ..................................... 309
9.2.3 A sample of n genes ............................ 312
9.2.4 Simulating the coalescent ...................... 315
9.2.5 Estimation of 9 from a sample of DNA
sequences ...................................... 316
9.3 Population demographic process ........................ 320
9.3.1 Homogeneous and nonhomogeneous Poisson
processes ...................................... 321
9.3.2 Deterministic population size change ........... 322
9.3.3 Nonparametric population demographic models .... 323
9.4 Multispecies coalescent, species trees and gene
trees ................................................. 325
9.4.1 Multispecies coalescent ........................ 325
9.4.2 Species tree-gene tree conflict ................ 331
9.4.3 Estimation of species trees .................... 335
9.4.4 Migration ...................................... 343
9.5 Species delimitation .................................. 349
9.5.1 Species concept and species delimitation ....... 349
9.5.2 Simple methods for analysing genetic data ...... 351
9.5.3 Bayesian species delimitation .................. 352
9.5.4 The impact of guide tree, prior, and
migration ...................................... 355
9.5.5 Pros and cons of Bayesian species
delimitation ................................... 358
9.6 Problems .............................................. 359
10 Molecular clock and estimation of species divergence
times ...................................................... 361
10.1 Overview .............................................. 361
10.2 Tests of the molecular clock .......................... 363
10.2.1 Relative-rate tests ............................ 363
10.2.2 Likelihood ratio test .......................... 364
10.2.3 Limitations of molecular clock tests ........... 365
10.2.4 Index of dispersion ............................ 366
10.3 Likelihood estimation of divergence times ............. 366
10.3.1 Global clock model ............................. 366
10.3.2 Local clock model .............................. 367
10.3.3 Heuristic rate-smoothing methods ............... 368
10.3.4 Uncertainties in calibrations .................. 370
10.3.5 Dating viral divergences ....................... 372
10.3.6 Dating primate divergences ..................... 373
10.4 Bayesian estimation of divergence times ............... 375
10.4.1 General framework .............................. 375
10.4.2 Approximate calculation of likelihood .......... 376
10.4.3 Prior on evolutionary rates .................... 377
10.4.4 Prior on divergence times and fossil
calibrations ................................... 378
10.4.5 Uncertainties in time estimates ................ 382
10.4.6 Dating viral divergences ....................... 384
10.4.7 Application to primate and mammalian
divergences .................................... 385
10.5 Perspectives .......................................... 388
10.6 Problems .............................................. 389
11 Neutral and adaptive protein evolution ..................... 390
11.1 Introduction .......................................... 390
11.2 The neutral theory and tests of neutrality ............ 391
11.2.1 The neutral and nearly neutral theories ........ 391
11.2.2 Tajima's D statistic ........................... 393
11.2.3 Fu and Li's D, and Fay and Wu's H statistics ... 394
11.2.4 McDonald-Kreitman test and estimation of
selective strength ............................. 395
11.2.5 Hudson-Kreitman-Aquade test .................... 397
11.3 Lineages undergoing adaptive evolution ................ 398
11.3.1 Heuristic methods .............................. 398
11.3.2 Likelihood method .............................. 399
11.4 Amino acid sites undergoing adaptive evolution ........ 400
11.4.1 Three strategies ............................... 400
11.4.2 Likelihood ratio test of positive selection
under random-site models ....................... 402
11.4.3 Identification of sites under positive
selection ...................................... 405
11.4.4 Positive selection at the human MHC ............ 406
11.5 Adaptive evolution affecting particular sites and
lineages .............................................. 408
11.5.1 Branch-site test of positive selection ......... 408
11.5.2 Other similar models ........................... 409
11.5.3 Adaptive evolution in angiosperm phytochromes .. 410
11.6 Assumptions, limitations, and comparisons ............. 411
11.6.1 Assumptions and limitations of current
methods ........................................ 412
11.6.2 Comparison of methods for detecting positive
selection ...................................... 413
11.7 Adaptively evolving genes ............................. 414
11.8 Problems .............................................. 416
12 Simulating molecular evolution ............................. 418
12.1 Introduction .......................................... 418
12.2 Random number generator ............................... 418
12.3 Generation of discrete random variables ............... 420
12.3.1 Inversion method for sampling from a general
discrete distribution .......................... 420
12.3.2 The alias method for sampling from a discrete
distribution ................................... 421
12.3.3 Discrete uniform distribution .................. 422
12.3.4 Binomial distribution .......................... 423
12.3.5 The multinomial distribution ................... 423
12.3.6 The Poisson distribution ....................... 423
12.3.7 The composition method for mixture
distributions .................................. 424
12.4 Generation of continuous random variables ............. 424
12.4.1 The inversion method ........................... 425
12.4.2 The transformation method ...................... 425
12.4.3 The rejection method ........................... 425
12.4.4 Generation of a standard normal vanate using
the polar method ............................... 428
12.4.5 Gamma, beta, and Dirichlet variables ........... 430
12.5 Simulation of Markov processes ........................ 430
12.5.1 Simulation of the Poisson process .............. 430
12.5.2 Simulation of the nonhomogeneous Poisson
process ........................................ 431
12.5.3 Simulation of discrete-time Markov chains ...... 433
12.5.4 Simulation of continuous-time Markov chains .... 435
12.6 Simulating molecular evolution ........................ 436
12.6.1 Simulation of sequences on a fixed tree ........ 436
12.6.2 Simulation of random trees ..................... 439
12.7 Validation of the simulation program .................. 439
12.8 Problems .............................................. 440
Appendices .................................................... 442
Appendix A. Functions of random variables ..................... 442
Appendix B. The delta technique ............................... 446
Appendix C. Phylogenetic software ............................. 448
References .................................................... 450
Index ......................................................... 488
Reader note: The asterisk next to a heading indicates
a more difficult or technical section/problem.
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