Probability Models for DNA Sequence Evolution

Our basic question is: Given a collection of DNA sequences, what underlying forces are responsible for the observed patterns of variability? To approach this question we introduce and analyze a number of probability models: the Wright-Fisher model, the coalescent, the infinite alleles model, and the infinite sites model. We study the complications that come from nonconstant population size, recombination, population subdivision, and three forms of natural selection: directional selection, balancing selection, and background selection. These theoretical results set the stage for the investigation of various statistical tests to detect departures from "neutral evolution." The final chapter studies the evolution of whole genomes by chromosomal inversions, reciprocal translocations, and genome duplication.
Throughout the book, the theory is developed in close connection with data from more than 60 experimental studies from the biology literature that illustrate the use of these results. This book is written for mathematicians and for biologists alike. We assume no previous knowledge of concepts from biology and only a basic knowledge of probability: a one semester undergraduate course and some familiarity with Markov chains and Poisson processes.
Rick Durrett received his Ph.D. in operations research from Stanford University in 1976. He taught in the UCLA mathematics department before coming to Cornell in 1985. He is the author of six books and 125 research papers, and is the academic father of more than 30 Ph.D. students. His current interests are the use of probability models in genetics and ecology, and decreasing the mean and variance of his golf.

From the reviews:

This book serves as an introduction to, and a compilation of, results in the new and growing field of probability theory for DNA evolution. The author is a probabilist, but he has made an effort to appeal to biologists and mathematicians alike. Biologists will find in this book those methods used for studying DNA variability that go beyond straightforward applications of statistics and require genuine mathematical insight into the underlying mechanisms. The example data are real and the driving motivation is to understand these truthfully. Mathematicians will find in this book not an attempt to develop some abstract theory, but rather a kaleidoscopic collection of ideas and methods invented to attack a common problem. The emphasis is on catching the main ideas in down-to-earth calculations; continuum models and more sophisticated methods from analysis are avoided. The book starts with the Wright-Fisher model for neutral selection in a constant population, and then builds and varies on this in Chapters 1--3 by adding the effects of mutation, recombination, selection, nonconstant population sizes and spatial structure. Chapter 4 is devoted to statistical tests for detecting departures from the neutral model and Chapter 5 discusses the large-scale structure of the DNA such as the size and number of chromosomes, inversions, translocations and duplications. This book will serve as a useful guide to this new and exciting field and an inspiration even for the more experienced readers.--Mathematical Reviews

"With numerous interspersed example datasets from the primary literature, this book offers a brief introduction to models of both long-term and short-term DNA sequence evolution, with emphasis on the latter…Diverse audiences will find much of the value in this concise book. The author’s rigor and ability to embed material in a broader mathematical context will appeal to quantitative readers with little background inbiology…The book’s strength is in its mathematical content, which revives almost-forgotten theorems, includes new proofs, and is thorough but not overwhelmingly detailed or overburdened with notation."Journal of the American Statistical Association, June 2004

"Stochastic modelling of genetic sequences and their evolution continues to be one of the hottest areas of research in applied probability even today. This book is a useful and interesting addition to a large number of similar texts in this area. ... it is surely a reliable introductory text for beginners taking interest in the subject." (Probal Chaudhuri, Sankhya: The Indian Journal of Statistics, Vol. 65 (4), 2003)

"This is the first book covering the mathematical theory of molecular evolution developed during the last two decades. … A particularly pleasing feature of this text is that most important concepts and results are immediately applied to real data, thus providing many illuminating examples. This book can be highly recommended to students and researchers in applied probability and mathematical biology, especially to those interested in genetics, evolution, or bioinformatics." (R.Bürger, Monatshefte für Mathematik, Vol. 143 (1), 2004)

"The author did an admirable job in providing a concise and up-to-date summary of modern molecular population genetics. The details of derivations are often given, allowing the reader to follow the theory. … Overall, I found the book great fun to read. I recommend it to population geneticists who are interested in a concise modern summary of analytical results in the field, and to mathematicians who would like to work in this exciting area of research." (Z Yang, Heredity, Vol. 92, 2004)

"This book serves as an introduction to, and a compilation of, results in the new and growing field of probability theory for DNA evolution. The author is a probabilist, but he has made an effort to appeal to biologists and mathematicians alike. … Thisbook will serve as a useful guide to this new and exciting field and an inspiration even for the more experienced readers." (Jan M. Swart, Mathematical Reviews, Issue 2003 b)

"The evolution of genetic characteristics of populations over generations is an important scientific frontier to mathematical modelling. It is one where probability models and statistical analysis have a fundamental and important role to play. … The goal of this book is to introduce the reader to probability models which might explain the patterns of variability observable in DNA sequences collected from many individuals in a population. … it would provide a good supplemental source for a graduate course in stochastic modelling or in mathematical genomics." (R.W. Oldford, Short Book Reviews, Vol. 22 (3), 2002)

“This is an beautifully written book by a distinguished probabilist on what used to be called Population Genetics, and now, more appropriately, is called DNA Sequence Evolution. This is a second edition, but the second edition has so much new material that it’s almost a new book. …accessible to readers with a minimal background in Biology and some knowledge of basic probability theory and Markov processes… It is strongly recommended to all readers interested in Population Genetics in a broad sense. …” (International Statistical Review 2009, 77, 2.)