Mathematical Population Genetics

This is the first of a planned two-volume work discussing the mathematical aspects of population genetics with an emphasis on evolutionary theory. This volume draws heavily from the author’s 1979 classic, but it has been revised and expanded to include recent topics which follow naturally from the treatment in the earlier edition, such as the theory of molecular population genetics.

This is the first of a planned two-volume work discussing the mathematical aspects of population genetics with an emphasis on evolutionary theory. This volume draws heavily from the author’s 1979 classic, but it has been revised and expanded to include recent topics which follow naturally from the treatment in the earlier edition, such as the theory of molecular population genetics.

A definitive account of the origins of modern mathematical population genetics, first published in 2000.

A basic method of analyzing particulate gene systems is the proba bilistic and statistical analyses. Mendel himself could not escape from an application of elementary probability analysis although he might have been unaware of this fact. Even Galtonian geneticists in the late 1800's and the early 1900's pursued problems of heredity by means of mathe matics and mathematical statistics. They failed to find the principles of heredity, but succeeded to establish an interdisciplinary area between mathematics and biology, which we call now Biometrics, Biometry, or Applied Statistics. A monumental work in the field of popUlation genetics was published by the late R. A. Fisher, who analyzed "the correlation among relatives" based on Mendelian gene theory (1918). This theoretical analysis over came "so-called blending inheritance" theory, and the orientation of Galtonian explanations for correlations among relatives for quantitative traits rapidly changed. We must not forget the experimental works of Johanson (1909) and Nilsson-Ehle (1909) which supported Mendelian gene theory. However, a large scale experiment for a test of segregation and linkage of Mendelian genes affecting quantitative traits was, prob ably for the first time, conducted by K. Mather and his associates and Panse in the 1940's.

The present monograph develops a versatile and profound mathematical perspective of the Wright--Fisher model of population genetics. This well-known and intensively studied model carries a rich and beautiful mathematical structure, which is uncovered here in a systematic manner. In addition to approaches by means of analysis, combinatorics and PDE, a geometric perspective is brought in through Amari's and Chentsov's information geometry. This concept allows us to calculate many quantities of interest systematically; likewise, the employed global perspective elucidates the stratification of the model in an unprecedented manner. Furthermore, the links to statistical mechanics and large deviation theory are explored and developed into powerful tools. Altogether, the manuscript provides a solid and broad working basis for graduate students and researchers interested in this field.

This work reflects sixteen hours of lectures delivered by the author at the 2009 St Flour summer school in probability. It provides a rapid introduction to a range of mathematical models that have their origins in theoretical population genetics. The models fall into two classes: forwards in time models for the evolution of frequencies of different genetic types in a population; and backwards in time (coalescent) models that trace out the genealogical relationships between individuals in a sample from the population. Some, like the classical Wright-Fisher model, date right back to the origins of the subject. Others, like the multiple merger coalescents or the spatial Lambda-Fleming-Viot process are much more recent. All share a rich mathematical structure. Biological terms are explained, the models are carefully motivated and tools for their study are presented systematically.

This volume contains pedagogical and elementary introductions to genetics for mathematicians and physicists as well as to mathematical models and techniques of population dynamics. It also offers a physicist''s perspective on modeling biological processes. Each chapter starts with an overview followed by the recent results obtained by authors. Lectures are self-contained and are devoted to various phenomena such as the evolution of the genetic code and genomes, age-structured populations, demography, sympatric speciation, the Penna model, Lotka-Volterra and other predator-prey models, evolutionary models of ecosystems, extinctions of species, and the origin and development of language. Authors analyze their models from the computational and mathematical points of view.

In this book, mathematical aspects of a population genetics are considered. On the basis of the Hardy - Weinberg law, the standard approach to population genetics problems is stated. Along with the standard approach, the necessity of separate research of family tree genetics and population genetics, which represent set of the family trees, is shown. Family trees are investigated by methods of discrete mathematics in a discrete time scale which is defined by alternation of generations. It is necessary to transit to a continuous time scale, continuous functions, therefore the Hardy-Weinberg law is written down in the form of the differential equation of the second order. Transition to continuous functions has allowed us to receive new and certainly not trivial results in population genetics. In particular, a new approach to problems of a mutations occurrence under radiation is discussed, of a new growths occurrence, and migrations of populations under various conditions to reveal nonlinear character of inbreeding and natural selection. The book can be useful to geneticists, students-biologists, post-graduate students and everyone who is interested in problems of population genetics.