Prologue ...................................................... vii
Acknowledgments ............................................... xix
1 An Introduction to Mathematical Probability with
Applications in Mendelian Genetics ........................... 1
1.1 Introduction ............................................ 1
1.2 Mathematical Probability in Mendelian Genetics .......... 2
1.3 Examples of Finite Probability Spaces ................... 7
1.4 Elementary Combinatorial Analysis ...................... 11
1.5 The Binomial Distribution .............................. 15
1.6 The Multinomial Distribution ........................... 20
1.7 Conditional Probabilities and a Bayesian Theorem ....... 26
1.8 Expectations and Generating Functions for Binomial
and Multinomial Distributions .......................... 29
1.9 Marginal and Conditional Distributions of the
Multinomial Distribution ............................... 32
1.10 A Law of Large Numbers and the Frequency
Interpretation of Probability .......................... 35
1.11 On Computing Monte Carlo Realizations of a Random
Variable with a Binomial Distribution .................. 39
1.12 The Beta-Binomial Distribution ......................... 43
Bibliography
2 Linkage and Recombination at Multiple Loci .................. 53
2.1 Introduction ........................................... 53
2.2 Some Thoughts on Constructing Databases of DNA
Markers From Sequenced Genomes of Relatives ............ 56
2.3 Examples of Informative Matings for the Case of
Two Loci ............................................... 60
2.4 General Case of Two Linked Loci ........................ 65
2.5 General Case of Three Linked Loci ...................... 68
2.6 General Case of Four or More Linked Loci ............... 72
2.7 Theoretical Calculations in Statistical and
Population Genetics .................................... 76
2.8 Appendix: Proof of Theorem 2.6.1 ....................... 80
Bibliography ................................................ 82
3 Linkage and Recombination in Large Random Mating
Diploid PopulationsRandom Mating Diploid Populations ........ 83
3.1 Introduction ........................................... 83
3.2 The One Locus Case ..................................... 84
3.3 The Case of Many Autosomal Loci With Arbitrary
Linkage ................................................ 91
3.4 Sex Linked Genes in Random Mating Populations ......... 100
3.5 Comments and Historical Notes ......................... 107
Bibliography ............................................... 108
4 Two Allele Wright-Fisher Process with Mutation and
Selection .................................................. 110
4.1 Introduction .......................................... 110
4.2 Overview of Markov Chains with Stationary Transition
Probabilities ......................................... 111
4.3 Overview of Wright-Fisher Perspective ................. 113
4.4 Absorbing Markov Chains with a Finite State Space ..... 116
4.5 Distributions of First Entrance Times Into an
Absorbing State and Their Expectations and
Variances ............................................. 122
4.6 Quasi-Stationary Distribution on the Set of
Transient States ...................................... 128
4.7 Incorporating Mutation and Selection Into Two Allele
Wright-Fisher Processes ............................... 132
4.8 Genotypic Selection with no Mutation and
Random Mating ......................................... 136
4.9 A Computer Experiment with the Wright-Fisher Neutral
Model ................................................. 139
4.10 A Computer Experiment with Wright-Fisher Selection
Model ................................................. 142
4.11 A Computer Experiment with Wright-Fisher Genotypic
Selection Model ....................................... 145
4.12 A Computer Experiment with a Wright-Fisher Model
Accommodating Selection and Mutation .................. 147
Bibliography ............................................... 149
5 Multitype Gamete Sampling Processes, Generation of
Random Numbers and Monte Carlo Simulation Methods .......... 150
5.1 Introduction .......................................... 150
5.2 A Wright-Fisher Model with Multiple Types of Gametes
- Mutation and Selection .............................. 151
5.3 Examples of Multiple Alleles and Types of Gametes
Involving Two Chromosomes ............................. 155
5.4 A Genetic Theory for Inherited Autism in Man .......... 157
5.5 An Evolutionary Genetic Model of Inherited Autism ..... 158
5.6 Multitype Gamete Sampling Processes as Conditioned
Branching Processes ................................... 166
5.7 On the Orderly Pursuit of Randomness Underlying
Monte Carlo Simulation Methods ........................ 174
5.8 Design of Software and Statistical Summarization
Procedures ............................................ 178
5.9 Experiments in the Quantification of Ideas for the
Evolution of Inherited Autism in Populations .......... 182
5.10 Comparative Experiments in the Quantification of Two
Formulations of Gamete Sampling Models ................ 188
5.11 An Experiment with a Three Allele Neutral Model ....... 191
5.12 Rapid Selection and Convergence to a Stationary
Distribution .......................................... 192
Bibliography ............................................... 195
6 Nucleotide Substitution Models Formulated as Markov
Processes in Continuous Time ............................... 196
6.1 Introduction .......................................... 196
6.2 Overview of Markov Jump Processes in Continuous
Time with Finite State Spaces and Stationary Laws of
Evolution ............................................. 197
6.3 Stationary Distributions of Markov Chains in
Continuous Time with Stationary Laws of Evolution ..... 203
6.4 Markov Jump Processes as Models for Base
Substitutions in the Molecular Evolution of DNA ....... 209
6.5 Processes with Preassigned Stationary Distributions ... 217
6.6 A Numerical Example for a Class of Twelve Parameters .. 220
6.7 Falsifiable Predictions of Markov Models of
Nucleotide Substitutions .............................. 223
6.8 Position Dependent Nucleotide Substitution Models ..... 225
6.9 A Retrospective View of a Markov Process with
Stationary Transition Probabilities ................... 228
Bibliography ............................................... 233
7 Mixtures of Markov Processes as Models of Nucleotide
Substitutions at Many Sites ................................ 235
7.1 Introduction .......................................... 235
7.2 Mixtures of Markov Models and Variable Substitution
Rates Across Sites .................................... 236
7.3 Gaussian Mixing Processes ............................. 240
7.4 Computing Realizations of a Gaussian Process with
Specified Covariance Function ......................... 245
7.5 Gaussian Processes That May be Computed Recursively ... 248
7.6 Monte Carlo Implementation of Mixtures of
Transition Rates for Markov Processes ................. 255
7.7 Transition Rates Based on Logistic Gaussian
Processes ............................................. 261
7.8 Nucleotide Substitution in a Three Site Codon ......... 265
7.9 Computer Simulation Experiments ....................... 268
Bibliography ............................................... 271
8 Computer Implementations and Applications of Nucleotide
Substitution Models at Many Sites - Other Non-SNP Types
of Mutation ................................................ 272
8.1 Introduction .......................................... 272
8.2 Overview of Monte Carlo Implementations for
Nucleotide Substitution Models with N Sites ........... 273
8.3 Overview of Genographic Research Project - Studies
of Human Origins ...................................... 280
8.4 Simulating Nucleotide Substitutions in Evolutionary
Time .................................................. 282
8.5 Counting Back and Parallel Mutations in Simulated
Data .................................................. 289
8.6 Computer Simulation Experiments With a Logistic
Gaussian Mixing Process ............................... 295
8.7 Potential Applications of Many Site Models to
the Evolution of Protein Coding Genes ................. 298
8.8 Preliminary Notes on Stochastic Models of Indels
and Other Mutations ................................... 300
Bibliography ............................................... 304
9 Genealogies, Coalescence and Self-Regulating Branching
Processes .................................................. 306
9.1 Introduction .......................................... 306
9.2 One Type Stochastic Genealogies ....................... 309
9.3 Overview of the Galton-Watson Process ................. 315
9.4 Self-Regulating Galton-Watson Processes ............... 321
9.5 Fixed Points and Domains of Attraction ................ 324
9.6 Probabilities of Extinction ........................... 327
9.7 Stochastic Genealogies in the Multitype Case .......... 330
9.8 Multitype Galton-Watson Processes ..................... 333
9.9 Self-Regulating Multitype Processes ................... 338
9.10 Estimating the Most Recent Common Ancestor ............ 342
9.11 The Deterministic Model and Branching Process ......... 346
9.12 Realizations of a Poisson Random Variable ............. 351
Bibliography ............................................... 355
10 Emergence, Survival and Extinction of Mutant Types in
Populations of Self Replicating Individuals Evolving
From Small Founder Populations ............................. 357
10.1 Introduction .......................................... 357
10.2 Experiments with the Evolution of Small Founder
Populations with Mutation but no Selection ............ 361
10.3 Components of Selection - Reproductive and
Competitive Advantages of Some Types .................. 367
10.4 Survival of Deleterious and Beneficial Mutations
From a Small Founder Populations ...................... 372
10.5 Survival of Mutations with Competitive Advantages
Over an Ancestral Type ................................ 376
10.6 Chaotic Embedded Deterministic Model with Three
Types ................................................. 382
10.7 Self Regulating Multitype Branching Processes in
Random Environments ................................... 390
10.8 Simulating Multitype Genealogies and Further
Reading .............................................. 397
Bibliography ............................................... 399
11 Two Sex Multitype Self Regulating Branching Processes
in Evolutionary Genetics ................................... 401
11.1 Introduction .......................................... 401
11.2 Gametes, Genotypes and Couple Types in a Two Sex
Stochastic Population Process ......................... 403
11.3 The Parameterization of Couple Formation Processes .... 405
11.4 An Example of Couple Formation Process with Respect
to an Autosomal Locus with Two Alleles ................ 409
11.5 Genetics and Offspring Distributions .................. 411
11.6 Overview of a Self-Regulating Population Process ...... 415
11.7 Embedding Non-Linear Difference Equations in the
Stochastic Population Process ......................... 417
11.8 On the Emergence of a Beneficial Mutation From
a Small Founder Population ............................ 420
11.9 An Alternative Evolutionary Genetic Model of
Inherited Autism ...................................... 423
11.10 Autism in a Population Evolving From a Small
Founder Population .................................... 428
11.11 Sexual Selection in Populations Evolving From
a Small Founder Population ........................... 433
11.12 Two Sex Processes with Linkage at Two Autosomal
Loci .................................................. 439
Bibliography ............................................... 445
12 Multitype Self-Regulatory Branching Process and the
Evolutionary Genetics of Age Structured Two Sex
Populations ................................................ 446
12.1 Introduction .......................................... 446
12.2 An Overview of Competing Risks and Semi-Markov
Processes ............................................. 448
12.3 Age Dependence and Types of Singles and Couples ....... 454
12.4 Altruism and Semi-Markovian Processes for Evolution
of Single Individuals ................................. 457
12.5 On an Age Dependent Couple Formation Process .......... 461
12.6 A Semi-Markovian Model for Deaths, Dissolutions and
Transitions Among Couple Types ........................ 465
12.7 Gamete, Genotypic and Offspring Distributions for
Each Couple Type ...................................... 468
12.8 Overview of Stochastic Population Process with Two
Sexes and Age Dependence .............................. 474
12.9 Overview of Non-Linear Difference Equations
Embedded in the Stochastic Population Process ......... 476
12.10 A Two Sex Age Dependent Population Process Without
Couple Formation ...................................... 479
12.11 Parametric Latent Risk Functions for Death by Age .... 483
12.12 Sexual Selection in an Age Dependent Process
Without Couple Formation .............................. 488
12.13 Population Momentum and Emergence of a Beneficial
Mutation .............................................. 493
12.14 Experiments with a Version of the Age Dependent
Model with Couple Formation ........................... 497
Bibliography ............................................... 504
13 An Overview of the History of the Concept of a Gene
and Selected Topics in Molecular Genetics .................. 505
13.1 Introduction .......................................... 505
13.2 A Brief History of the Definition of a Gene ........... 506
13.3 Transcription and Translation Processes ............... 510
13.4 Pre-processing Messenger RNA .......................... 514
13.5 Difficulties with Current Gene Concepts ............... 518
13.6 Acronyms in Tiling Array Technology ................... 520
13.7 Genome Activity in the ENCODE Project ................. 523
13.8 Interpreting Tiling Array Experiments ................. 529
13.9 A Tentative Updated Definition of a Gene .............. 532
13.10 ABO Blood Group Genetics in Humans ................... 537
13.11 Duffy Blood Group System in Man ...................... 540
13.12 Regulation of the Shh Locus in Mice .................. 541
Bibliography ............................................... 545
14 Detecting Genomic Signals of Selection and the
Development of Models for Simulating the Evolution
of Genomes ................................................. 549
14.1 Introduction .......................................... 549
14.2 Types of Selection and Genomic Signals ................ 551
14.3 DNA Sequence Evolution in Large Genomic Regions ....... 556
14.4 Statistics Used in Genome Wide Scans .................. 562
14.5 Detecting Signals of Natural Selection ................ 569
14.6 Simulated Genomic Data in Statistical Tests ........... 574
14.7 Species and Gene Trees From Mammalian Genomic Data .... 581
14.8 Overview of Markovian Codon Substitution Models ....... 586
14.9 Simulating Genetic Recombination ...................... 594
14.10 Modelling Gene Conversion ............................ 601
14.11 Nucleotide Substitutions During Meiosis .............. 606
14.12 Simulating Insertions and Deletions .................. 612
14.13 Simulating Copy Number Variation ..................... 621
14.14 Simulating Mutational Events and Genetic
Recombination ......................................... 624
Bibliography ............................................... 627
15 Suggestions for Further Research, Reading and Viewing ...... 631
15.1 Introduction .......................................... 631
15.2 Suggestions for Further Research on Self-Regulating
Branching Processes ................................... 632
15.3 Suggestions for Continuing Development of
Stochastic Models of Genomic Evolution ................ 634
15.4 A Brief List of References on Genetics and
Evolution for Further Study ........................... 637
Bibliography .................................................. 641
Index ......................................................... 645
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