Stability of the G-Matrix in a Population Experiencing Pleiotropic Mutation, Stabilizing Selection, and Genetic Drift

Abstract

Abstract. Quantitative genetics theory provides a framework that predicts the effects of selection on a phenotype consisting of a suite of complex traits. However, the ability of existing theory to reconstruct the history of selection or to predict the future trajectory of evolution depends upon the evolutionary dynamics of the genetic variancecovariance matrix (G-matrix). Thus, the central focus of the emerging field of comparative quantitative genetics is the evolution of the G-matrix. Existing analytical theory reveals little about the dynamics of G, because the problem is too complex to be mathematically tractable. As a first step toward a predictive theory of G-matrix evolution, our goal was to use stochastic computer models to investigate factors that might contribute to the stability of G over evolutionary time. We were concerned with the relatively simple case of two quantitative traits in a population experiencing stabilizing selection, pleiotropic mutation, and random genetic drift. Our results show that G-matrix stability is enhanced by strong correlational selection and large effective population size. In addition, the nature of mutations at pleiotropic loci can dramatically influence stability of G. In particular, when a mutation at a single locus simultaneously changes the value of the two traits (due to pleiotropy) and these effects are correlated, mutation can generate extreme stability of G. Thus, the central message of our study is that the empirical question regarding Gmatrix stability is not necessarily a general question of whether G is stable across various taxonomic levels. Rather, we should expect the G-matrix to be extremely stable for some suites of characters and unstable for others over similar spans of evolutionary time. Modern quantitative genetics theory provides points of connection between microevolution and macroevolution (Arnold et al. 2001). For a phenotype comprising multiple traits, the single-generation response to selection is given by the multivariate version of the breeder's equation (Lande 1979), ⌬z ϭ G␤, where z is a vector of population trait means, ␤ is a vector of directional selection gradients, and G is the genetic variance-covariance matrix (the G-matrix). Hence, the response to selection depends upon the intensity and direction of selection, as well as upon the amount of genetic variation and the nature of genetic correlations among traits. This equation for the change in the mean phenotype can be extrapolated over multiple generations to reconstruct the history of selection or to predict the future trajectory of the phenotype as a consequence of selection. This potential for extrapolation provides a connection between microevolutionary processes and macroevolutionary patterns (Lande 1979; but see Zeng 1988). However, such an extrapolation is possible only if the G-matrix remains relatively constant over long spans of evolutionary time. An extremely unstable G-matrix would render the goal of understanding selection over evolutionary time unachievable within the ...

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“…Comparisons of G matrices sampled from conspecific populations and related species often show that the matrices have one or more principal components in common or are even proportional Steppan et al 2002). Computer simulations suggest that G matrix stability is promoted by large effective population size and mutational correlation (Jones et al 2003). Persistent stabilizing (and correlational) selection may also produce stability in G by promoting alignment of G with the major axes of the adaptive landscape (Jones et al 2003(Jones et al , 2004(Jones et al , 2007.…”

“…Empirical and simulation research suggests that different aspects of G evolve in different ways and in response to multiple forces (Phillips et al 2001;Steppan et al 2002;Jones et al 2003Jones et al , 2004Jones et al , 2007. Consequently, it is useful to reparameterize G into what can be considered size, shape, and orientation parameters, as suggested by Jones et al (2003).…”

“…Consequently, it is useful to reparameterize G into what can be considered size, shape, and orientation parameters, as suggested by Jones et al (2003). A symmetrical m-bym G matrix contains m variance terms and (m 2 − m)/2 covariance terms, representing a total of (m 2 − m)/2 parameters.…”

“…Each G matrix can then be decomposed into a single size parameter (5a) a vector ε containing m − 1 shape parameters (5b) and a vector ϕ containing (m 2 − m)/2 orientation parameters ϕ i . The orientation parameters describe the angles between the eigenvectors and the trait axes, and their formulas depend on m. Note that we calculate ε i differently from Jones et al (2003) in order to facilitate estimation procedures and handle cases in which m > 2. Because G is a positive definite matrix, the size parameter Σ also equals the trace of G, or the sum of the genetic variances.…”

“…In the case of a 2-by-2 G matrix, there is conveniently only one parameter each for size, shape, and orientation. In this two-trait case, we can represent the parameters simply as scalars (Σ, ε, and ϕ), with ϕ representing the angle between the leading eigenvector (g max ) and the axis of the first trait (Jones et al 2003). A 2-by-2 G matrix is then parameterized as (6) G is expected to change over time by neutral genetic drift in a population of finite size in the absence of natural selection.…”