Coalescent Theory: An Introduction

Abstract

“The Coalescent” is a powerful extension of classical population genetics because it is a collection of mathematical models that can accommodate biological phenomena as reflected in genomic data. The theory was initially developed by Kingman (1982) in 3 papers published in probability theory journals, which outline the foundation of coalescent theory as a suite of probability models. Recent publications continue to address the mathematical development of the coalescent in mathematical journals (e.g., Sagitov and Jagers 2005), as well as addressing questions in real biological systems. The 2 important points to take from these facts are that, first, coalescent theory is still an active and exciting topic that continues to be developed in its fundamentals and applications and, second, that any book on coalescent theory is going to be heavy on the mathematics. The new book by John Wakeley aims to summarize the fundamentals of coalescent theory, and it certainly fits the second expectation. The primary audience is population geneticists who are interested in gaining a mathematical understanding of the coalescent and the theory behind “black box” computer applications. This book provides a walk through the mathematical derivations of major aspects of coalescent theory and some specific applications in population genetics. The text primarily stays in the realm of the theoretical, with only a few points illustrated with specific biological examples. This book offers a challenging, but rewarding, introduction to coalescent theory, and it will remain an indispensable text for some time to come. The emphasis in coalescent thinking is to view populations backwards in time, using the divergence observable in a population to estimate the time to a most recent common ancestor (MRCA); this ancestor is the point where gene genealogies come together, or `coalesce', in a single biological organism. Coalescent theory is dependent on tree-based (genealogical) thinking familiar to systematists (Harding 1996). Unlike phylogenetic methods, coalescent theory assumes that genealogies are random variables. This follows from an assumption of mutational neutrality; in the basic model no particular ancestor is either more or less fit or likely to produce descendants within the genealogy. Also, phylogenetic trees are measured in terms of substitutions or state changes, without an intrinsic time constraint. In contrast, coalescent trees are calculated in terms of time, set by a fixed mutation rate, and coalescent analyses therefore assume a molecular clock. The main parameters that are estimated in coalescence analyses are the coalescence time (the number of generations that have passed since samples share an ancestor) and theta, the scaled product of the mutation rate and effective population size. Therefore, more diverse populations have longer coalescence times and larger coalescent effective population sizes than less diverse populations (assuming the same mutation rate). The coalescent model can be manipulated to explore other subsidiary questions about population diversity through time, such as changes in population structure and size and the total length of the genealogy (further back in time than the MRCA under immediate study). The models use the starting point of neutrality to simulate and test the magnitude and influences of population-level events such as population fluctuation, migration, recombination, and selection. Any mathematical model, by virtue of its weakness or simplifications, illuminates the unquantified complexities of biological systems. Wakeley's text is organized into 8 chapters, in 2 halves. The first 4 chapters present the basic models or Kingman coalescent. The second half includes newer work and more complex applications. It explicitly requires a good understanding of calculus and probability theory (particularly stochastics), and any potential reader who is not comfortable with those subjects and the mathematical notation involved is advised to keep a probability theory textbook or mathematically inclined colleague close at hand. Anyone who is comfortable in this realm but has not actively used stochastic models recently is advised to take the time to work through the derivations in order to absorb them fully. Chapter 1 concerns gene genealogies and the general nature of population-level genetic processes. (If you do not know what coalescent theory really is, and you did not read my summary above, you have to wait until Chapter 3.) The chapter includes detailed background of the model assumptions and a review of relevant literature. In particular, the sections focus on background information about genealogical thinking and set up the vocabulary for coalescent theory: mutations and discussion of the fundamental assumption in the basic models that variation in population genetics is selectively neutral. The third section concerns polymorphisms, which may be the most important piece of coalescent theory back