10 Short - term Changes in the Mean : The Breeders ’ Equation

Our examination of selection of quantitative characters, and its consequences, comprises the remainder of this book. We start by discussing the simplest models — changes in the mean of a single character following a single generation of selection. As the reader will see over the next several chapters, even these simple models are not without complications. Over subsequent chapters we build on these models, making them more complex and biologically realistic. Selection changes the distribution of both phenotypic and genotypic values from one generation to the next. While it is convenient to assume that only the means of these distributions change, this is generally not the case. Rigorous prediction of the expected change in the mean over several generations requires predictions of how higher moments (such as the variance and skew) of the genotypic distribution change as well (Chapters 13, 24). Unfortunately, this requires considerable knowledge about the underlying distributions of allelic effects and frequencies, both of which are almost always unknown. Despite this gloomy initial assessment, the change in mean over a few generations of selection can often be accurately projected using estimates of genetic variances from an unselected base population. This chapter considers the basis for much of this success: models that assume a linear parent-offspring regression and short-term stability of the genetic variance (i.e., over short time scales, selection-induced changes in variance are assumed to be negligible). We will have much to say about selection-induced changes in the genetic variances and covariances in subsequent chapters (Chapters 13, 24). For now, the assumption of the infinitesimal model (a very large number of underlying loci, each with very small effect) ensures that negligible change in allele frequencies will occur over only a few generations of selection. However (Chapter 13), even under the infinitesimal model, genotypic frequencies can change, due to selection generating gametic-phase disequilibrium, even while allele frequencies remain essentially unchanged. Over longer time scales, changes in variances can be substantial and are extremely difficult to predict. Thus, while the results presented here hold for a single generation, they are expected to become increasingly less accurate as the number of generations of selection increases. Besides the constancy of variances and linearity of parent-offspring regression, the third major assumption required here is that there is no selection acting on characters correlated to our character of interest. Breeders have the luxury of artificial selection and hence fairly precise control over the nature of selection, while evolutionary biologists face the additional problem of estimating the form of selection acting on the character(s) under consideration. However, even under highly controlled breeding designs, natural selection can also be acting on the character of interest (either directly, or through correlations with other characters under selection). Accounting for selection on multiple characters is discussed in Chapters 30–36. There is a huge literature on different selection schemes that exploit special features of