# Stochastic Processes in Genetics and Evolution

The interest of this book is in the use of stochastic tools in the field of evolutionary genetics and, more particularly, in the use of computer-intensive methods to study models where biologists incorporate a considerable level of detail into the evolutionary genetic description. A common aspect is related to the simulation of a single species during short periods of time. To that end, the key element is a sample of Monte Carlo waiting times until a new beneficial mutation arises and becomes predominant in a population. The final result is a set of computer simulation experiments that are appropriately derived from a variety of Monte Carlo simulation models. These models are rooted in stochastic processes, for which the authors make a serious attempt to present the mathematics in sufficient detail, in such a way that if a reader were interested, it would be straightforward to duplicate the computer experiments implemented in APL 2000 in the book in terms of any programming language.

The book consists of fifteen chapters. To begin with, the authors present basic elements on finite probability spaces, random variables and a few discrete distributions, such as the binomial, multinomial and Poisson distributions, which are applied extensively throughout the book. The focus in Chapter 2 is on the parameterization of the gametic distribution by using recombination probabilities for some arbitrary number of linked loci. In Chapter 3, the authors discuss on a mathematical structure that allow them to accommodate a large number of linked loci, with a finite but arbitrary number of alleles at each locus. The starting point of Chapter 4 is a set of formulas for conditional absorption probabilities of finite absorbing Markov chains and the related quasi-stationary distribution, which are then applied to Wright-Fisher processes with respect to a single autosomal locus. In analysing the Wright-Fisher process with multiple alleles and a single autosomal locus, the authors use Monte Carlo simulation methods.

In Chapters 6, 7 and 8, the authors deal with the mutational process of nucleotide substitutions. They first present an overview on continuous-time Markov jump processes with a finite state space. In constructing the rate processes of Markov jump processes, they use simple Gaussian processes based on first- and second-order autoregressive processes. As an application of the resulting construction, simulation experiments on nucleotide substitutions in the D loop of the human mitochondrial genome are presented.

Up to Chapter 9 the authors do not take demographic factors into account. As a first step towards correcting this omission, they present in Chapter 9 one-type Galton-Watson processes and self-regulating branching processes, and Chapter 10 is devoted to experiments in the quantification of mutation and selection within a framework of self-regulating multi-type branching processes evolving on a discrete-time scale expressed in terms of generations. Chapters 11 and 12 consider self-regulating two-sex population models, as well as an age-structured population in the development of self-regulating two-sex models.

The ultimate goal of the book is to include the evolutional dynamics of a genome at the molecular level. To that end, the authors first describe in Chapter 13 the concept of gene and the coding and regulatory regions of DNA, and in Chapter 14 they then present a review of recent literature on developing computer models to simulate the dynamics of model genomes. The focus is on human haplotype data, and a discussion on the problem of constructing stochastic models to simulate various types of mutations occurring at the genomic level. The book ends with Chapter 15, which is devoted to open problems and a helpful review of books and materials in the field of evolutionary genetics.

The first author, Charles J. Mode, has published well-known books in mathematical biology, so that this book on evolutionary genetics can be seen as a new addition to a collection of publications showing how evolutionary biologists need enough mathematical training, specially stochastic processes, to be able to develop more accurate theories and models.

**Submitted by antonio_gomez@mat.ucm.es |

**8 / Feb / 2016