The scope of this book is the field of evolutionary genetics. The book contains new methods for simulating evolution at the genomic level. It sets out applications using up to date Monte Carlo simulation methods applied in classical population genetics, and sets out new fields of quantifying mutation and selection at the Mendelian level. A serious limitation of Wright-Fisher process, the assumption that population size is constant, motivated the introduction of self regulating branching processes in this book. While providing a short review of the principles of probability and its application and using computer intensive methods whilst applying these principles, this book explains how it is possible to derive new formulas expressed in terms of matrix algebra providing new insights into the classical Wright-Fisher processes of evolutionary genetics. Also covered are the development of new methods for studying genetics and evolution, simulating nucleotide substitutions of a DNA molecule and on self regulating branching processes. Components of natural selection are studied in terms of reproductive success of each genotype whilst also studying the differential ability of genotypes to compete for resources and sexual selection. The concept of the gene is also reviewed in this book and it provides a current definition of a gene based on very recent experiments with micro-array technologies. A development of stochastic models for simulating the evolution of model genomes concludes the studies in this book. Deserving of a place on the book shelves of workers in biomathematics, applied probability, stochastic processes and statistics, as well as in bioinformatics and phylogenetics, it will also be relevant to those interested in computer simulation, and evolutionary biologists interested in quantitative methods.

Description-Table Of Contents

1. An introduction to mathematical probability with applications in Mendelian genetics. 1.1. Introduction. 1.2. Mathematical probability in Mendelian genetics. 1.3. Examples of finite probability spaces. 1.4. Elementary combinatorial analysis. 1.5. The binomial distribution. 1.6. The multinomial distribution. 1.7. Conditional probabilities and a Bayesian theorem. 1.8. Expectations and generating functions for binomial and multinomial distributions. 1.9. Marginal and conditional distributions of the multinomial distribution. 1.10. A law of large numbers and the frequency interpretation of probability. 1.11. On computing Monte Carlo realizations of a random variable with a binomial distribution. 1.12. The beta-binomial distribution -- 2. Linkage and recombination at multiple loci. 2.1. Introduction. 2.2. Some thoughts on constructing databases of DNA markers from sequenced genomes of relatives. 2.3. Examples of informative matings for the case of two loci. 2.4. General case of two linked loci. 2.5. General case of three linked loci. 2.6. General case of four or more linked loci. 2.7. Theoretical calculations in statistical and population genetics -- 3. Linkage and recombination in large random mating diploid populations random mating diploid populations. 3.1. Introduction. 3.2. The one locus case. 3.3. The case of many autosomal loci with arbitrary linkage. 3.4. Sex linked genes in random mating populations. 3.5. Comments and historical notes -- 4. Two allele Wright-Fisher process with mutation and selection. 4.1. Introduction. 4.2. Overview of Markov chains with stationary transition probabilities. 4.3. Overview of Wright-Fisher perspective. 4.4. Absorbing Markov chains with a finite state space. 4.5. Distributions of first entrance times into an absorbing state and their expectations and variances. 4.6. Quasi-stationary distribution on the set of transient states. 4.7. Incorporating mutation and selection into two allele Wright-Fisher processes. 4.8. Genotypic selection with no mutation and random mating. 4.9. A computer experiment with the Wright-Fisher neutral model. 4.10. A computer experiment with Wright-Fisher selection model. 4.11. A computer experiment with Wright-Fisher genotypic selection model. 4.12. A computer experiment with a Wright-Fisher model accommodating selection and mutation -- 5. Multitype gamete sampling processes, generation of random numbers and Monte Carlo simulation methods. 5.1. Introduction. 5.2. A Wright-Fisher model with multiple types of gametes - Mutation and selection. 5.3. Examples of multiple alleles and types of gametes involving two chromosomes. 5.4. A genetic theory for inherited autism in man. 5.5. An evolutionary genetic model of inherited autism. 5.6. Multitype gamete sampling processes as conditioned branching processes. 5.7. On the orderly pursuit of randomness underlying Monte Carlo simulation methods. 5.8. Design of software and statistical summarization procedures. 5.9. Experiments in the quantification of ideas for the evolution of inherited autism in populations. 5.10. Comparative experiments in the quantification of two formulations of gamete sampling models. 5.11. An experiment with a three allele neutral model. 5.12. Rapid selection and convergence to a stationary distribution. ; 8 6. Nucleotide substitution models formulated as Markov processes in continuous time. 6.1. Introduction. 6.2. Overview of Markov jump processes in continuous time with finite state spaces and stationary laws of evolution. 6.3. Stationary distributions of Markov chains in continuous time with stationary laws of evolution. 6.4. Markov jump processes as models for base substitutions in the molecular evolution of DNA. 6.5. Processes with preassigned stationary distributions. 6.6. A numerical example for a class of twelve parameters. 6.7. Falsifiable predictions of Markov models of nucleotide substitutions. 6.8. Position dependent nucleotide substitution models. 6.9. A retrospective view of a Markov process with stationary transition probabilities -- 7. Mixtures of Markov processes as models of nucleotide substitutions at many sites. 7.1. Introduction. 7.2. Mixtures of Markov models and variable substitution rates across sites. 7.3. Gaussian mixing processes. 7.4. Computing realizations of a Gaussian process with specified covariance function. 7.5. Gaussian processes that may be computed recursively. 7.6. Monte Carlo implementation of mixtures of transition rates for Markov processes. 7.7. Transition rates based on logistic Gaussian processes. 7.8. Nucleotide substitution in a three site codon. 7.9. Computer simulation experiments -- 8. Computer implementations and applications of nucleotide substitution models at many sites - Other non-SNP types of mutation. 8.1. Introduction. 8.2. Overview of Monte Carlo implementations for nucleotide substitution models with N sites. 8.3. Overview of genographic research project - studies of human origins. 8.4. Simulating nucleotide substitutions in evolutionary time. 8.5. Counting back and parallel mutations in simulated data. 8.6. Computer simulation experiments With a logistic Gaussian mixing process. 8.7. Potential applications of many site models to the evolution of protein coding genes. 8.8. Preliminary notes on stochastic models of indels and other mutations -- 9. Genealogies, coalescence and self-regulating branching processes.9.1. Introduction. 9.2. One type stochastic genealogies. 9.3. Overview of the Galton-Watson process. 9.4. Self-regulating Galton-Watson processes. 9.5. Fixed points and domains of attraction. 9.6. Probabilities of extinction. 9.7. Stochastic genealogies in the multitype case. 9.8. Multitype Galton-Watson processes. 9.9. Self-regulating multitype processes. 9.10. Estimating the most recent common ancestor. 9.11. The deterministic model and branching process. 9.12. Realizations of a Poisson random variable -- 10. Emergence, survival and extinction of mutant types in populations of self replicating individuals evolving from small founder populations. 10.1. Introduction. 10.2. Experiments with the evolution of small founder populations with mutation but no selection. 10.3. Components of selection - Reproductive and competitive advantages of some types. 10.4. Survival of deleterious and beneficial mutations from a small founder populations. 10.5. Survival of mutations with competitive advantages over an ancestral type. 10.6. Chaotic embedded deterministic model with three types. 10.7. Self regulating multitype branching processes in random environments. 10.8. Simulating multitype genealogies and further reading. ; 8 11. Two sex multitype self regulating branching processes in evolutionary genetics. 11. Introduction. 11.2. Gametes, genotypes and couple types in a two sex stochastic population process. 11.3. The parameterization of couple formation processes. 11.4. An example of couple formation process with respect to an autosomal locus with two alleles. 11.5. Genetics and offspring distributions. 11.6. Overview of a self-regulating population process. 11.7. Embedding non-linear difference equations in the stochastic population process. 11.8. On the emergence of a beneficial mutation from a small founder population. 11.9. An alternative evolutionary genetic model of inherited autism. 11.10. Autism in a population evolving from a small founder population. 11.11. Sexual selection in populations evolving from a small founder population. 11.12. Two sex processes with linkage at two autosomal loci -- 12. Multitype self-regulatory branching process and the evolutionary genetics of age structured two sex populations. 12.1. Introduction. 12.2. An overview of competing risks and semi-Markov processes. 12.3. Age dependence and types of singles and couples. 12.4. Altruism and semi-Markovian processes for evolution of single individuals. 12.5. On an age dependent couple formation process. 12.6. A semi-Markovian model for deaths, dissolutions and transitions among couple types. 12.7. Gamete, genotypic and offspring distributions for each couple type. 12.8. Overview of stochastic population process with two sexes and age dependence. 12.9. Overview of non-linear difference equations embedded in the stochastic population process. 12.10. A two sex age dependent population process without couple formation. 12.11. Parametric latent risk functions for death by age. 12.12. Sexual selection in an age dependent process without couple formation. 12.13. Population momentum and emergence of a beneficial mutation. 12.14. Experiments with a version of the age dependent model with couple formation -- 13. An overview of the history of the concept of a gene and selected topics in molecular genetics. 13.1. Introduction. 13.2. A brief history of the definition of a gene. 13.3. Transcription and translation processes. 13.4. Pre-processing messenger RNA. 13.5. Difficulties with current gene concepts. 13.6. Acronyms in tiling array technology. 13.7. Genome activity in the ENCODE project. 13.8. Interpreting tiling array experiments. 13.9. A tentative updated definition of a gene. 13.10. ABO blood group genetics in humans. 13.11. Duffy blood group system in man. 13.12. Regulation of the Shh locus in mice -- 14. Detecting genomic signals of selection and the development of models for simulating the evolution of genomes. 14.1. Introduction. 14.2. Types of selection and genomic signals. 14.3. DNA sequence evolution in large genomic regions. 14.4. Statistics used in genome wide scans. 14.5. Detecting signals of natural selection. 14.6. Simulated genomic data in statistical tests. 14.7. Species and gene trees from mammalian genomic data. 14.8. Overview of Markovian codon substitution models. 14.9. Simulating genetic recombination. 14.10. Modelling gene conversion. 14.11. Nucleotide substitutions during meiosis. 14.12. Simulating insertions and deletions. 14.13. Simulating copy number variation. 14.14. Simulating mutational events and genetic recombination -- 15. Suggestions for further research, reading and viewing. 15.1. Introduction. 15.2. Suggestions for further research on self-regulating branching processes. 15.3. Suggestions for continuing development of stochastic models of genomic evolution. 15.4. A brief list of references on genetics and evolution for further study.