A fable of four functions

The review by Blows (Blows, 2007) highlights a perspective with which we whole-heartedly agree: greater biological insight is obtained by focusing on fundamental structures of the two matrices G and c than by analysing the individual entries of those matrices. Here, we advocate extending this point of view further for any ‘function-valued trait’, i.e. a trait that is inherently a mathematical function, z(t). The function z(t) can be viewed as an extension of a multivariate trait vector z (Kirkpatrick & Heckman, 1989). We will introduce three more functions G(t,s), c(t,s) and b(t) that are, respectively, extensions of the matrices G and c, and the vector b discussed in the review. Function-valued traits arise naturally in many contexts. For example, Blows discusses a study of stabilizing selection on male song in the cricket Teleogryllus commodus, characterizing the song as a vector of five song attributes (chip number, interpulse interval, trill number, intercall interval and dominant frequency). However, the song itself is energy (or power) as a function of time and is thus a function-valued trait, z(t). Reducing this function-valued trait to a vector of five measurements allows the ‘song’ to be analysed in a familiar quantitative genetics framework for multivariate traits, but doing so has important drawbacks (Kirkpatrick & Heckman, 1989; Ramsay & Silverman, 1997). First and foremost, information is necessarily lost in the data reduction. (Consider the futility of grasping Debussy’s Clair de Lune from just its dominant frequency, duration, average inter-note interval, and a dozen other such elements!) Second, treating such data as functional ab initio leads to data reduction methods that retain functional information more efficiently (like Clair de Lune’s sheet music or its digital representation on a CD) than do ad hoc collections of attributes (Ramsay & Silverman, 1997). There are many other types of function-valued traits, including growth trajectories (size as a function of age), gene expression profiles (product as a function of time), reaction norms (phenotype as a function of environment) and morphological shapes (radial distance as a function of angle). Typically, these traits are measured at a finite number of values of the independent variable and the finite collection of trait measurements is treated as a multivariate trait. There are two problems with this. First, there may be more effective ways to combine the measurements taken (or more effective ways to take the finite set of measurements). Second, if measurements are taken at different index values on different individuals (e.g. sizes measured at different sets of ages for different individuals) then multivariate approaches would require some sort of binning procedure, usually improvised. By treating the trait at the outset as a function, however, these issues can be avoided altogether (Kirkpatrick et al., 1994). A function-centric quantitative genetics framework has been developed for describing function-valued traits, including their genetic variability, selection, and evolution (Kirkpatrick & Heckman, 1989; Kirkpatrick et al., 1990; Gomulkiewicz & Kirkpatrick, 1992; Kirkpatrick & Lofsvold, 1992; Kirkpatrick et al., 1994; Meyer, 1998; Pletcher & Geyer, 1999; Kingsolver et al., 2001). In this framework, which is a direct extension of multivariate quantitative genetics, a trait function z(t) is usually represented mathematically by an ‘orthogonal function series expansion’. These expansions can describe practically any biologically plausible function shape, including nonsmooth and discontinuous functions that cannot be written in terms of the perhaps more familiar Taylor series expansion. Orthogonal function expansions can be used with least-squares or likelihood statistical methods to develop estimates of z(t) based on the data collected rather than some preconceived notion about the form of the function. These estimates provide natural interpolations of the original data. The extension of the genetic variance–covariance matrix G is the genetic variance–covariance function G(t,s). When t „ s, the G-function gives the additive genetic covariance of the trait expressed at index values t and s (like the off-diagonal elements of G) and when s 1⁄4 t it equals the additive genetic variance of the trait expressed at index value t (like the diagonal elements of G). What does a G-function look like? Consider the example of a growth trajectory, size as a function of age, and suppose all individuals in a study were measured at the same five ages. Given an appropriate breeding design, one could estimate a five-dimensional genetic covariance matrix G, the elements of which can be visualized as a three-dimensional histogram. The corresponding G-function would be a smoothed surface enveloping the tops of the histogram. Figures 1 and 2 of Kirkpatrick et al. (1990) show these visualizations of a genetic covariance matrix G and the corresponding G-function. Like G, the G-function can be ‘diagonalized’ and understood in terms of its eigenstructure. The G-function can be decomposed into a series of eigenvalues, each of which is the variance associated with a principal component. Principal components are represented by loading functions (eigenfunctions) rather than loading vectors Correspondence: Joel Kingsolver, Department of Biology, CB-3280, University of North Carolina, Chapel Hill, NC 27599, USA. Tel.: 919 843 6291; fax: 919 962 1625; e-mail: jgking@bio.unc.edu