Up hill, down dale: quantitative genetics of curvaceous traits

Abstract

'Repeated' measurements for a trait and individual, taken along some continuous scale such as time, can be thought of as representing points on a curve, where both means and covariances along the trajectory can change, gradually and continually. Such traits are commonly referred to as 'functionvalued' (FV) traits. This review shows that standard quantitative genetic concepts extend readily to FV traits, with individual statistics, such as estimated breeding values and selection response, replaced by corresponding curves, modelled by respective functions. Covariance functions are introduced as the FV equivalent to matrices of covariances.Considering the class of functions represented by a regression on the continuous covariable, FV traits can be analysed within the linear mixed model framework commonly employed in quantitative genetics, giving rise to the so-called random regression model. Estimation of covariance functions, either indirectly from estimated covariances or directly from the data using restricted maximum likelihood or Bayesian analysis, is considered. It is shown that direct estimation of the leading principal components of covariance functions is feasible and advantageous. Extensions to multidimensional analyses are discussed.

Comparing Complex Fitness Surfaces: Among‐Population Variation in Mutual Sexual Selection inDrosophila serrata

“…As an alternative to the direct comparison of b vectors and g matrices, it is conceivable that random-coefficient models (Longford 1993;Meyer and Kirkpatrick 2005) could be applied to these data to model the variation among populations in either directional selection (linear slopes) or nonlinear selection (second-order regression coefficients), in a restricted maximum likelihood framework. However, determining the significance of broader-scale patterns among populations, particularly in nonlinear selection, is likely to be a formidable task.…”

Exploring plasticity in the wild: laying date–temperature reaction norms in the common gull Larus canus

“…The use of BLUPs in a two-step approach ignores the uncertainity involved in generating BLUPs, and using such BLUPs in an animal model falsifies the assumption of independence across individuals needed to generate them in the first place. An RRAM has been used to study G!E in the wild (Wilson et al 2005(Wilson et al , 2006, and is a commonly used method in animal breeding (Lynch & Walsh 1998;Schaeffer 2004;Meyer & Kirkpatrick 2005). In the absence of pedigree information, this framework can also be used if one is interested in modelling phenotypic plasticity (I!E) only (equation (2.1), a random regression phenotypic model, Nussey et al (2007)).…”

“…Pedigree information can be used to partition the variance in the parameters describing these functions, and this information can be used to describe the (co)variances across environments (e.g. Meyer & Kirkpatrick 2005).…”

Parent–offspring conflict and co-adaptation: behavioural ecology meets quantitative genetics

“…Kirkpatrick & Heckman 1989;de Jong 1990;Gomulkiewicz & Kirkpatrick 1992) and their evolution can be modelled as evolving function-valued traits (Kingsolver et al 2001;Kirkpatrick & Meyer 2004;Meyer & Kirkpatrick 2005). Reaction norms have rarely been used to conceptualize the evolution of social behaviours (Agrawal 2001), although the expression of many social behaviours would fit the definition of a reaction norm or a function-valued trait.…”

“…Appropriate evolutionary models are now needed to formalize a theory of coevolving behavioural reaction norms in the context of parent-offspring interactions. Such models may be founded around and extended upon negotiation models (McNamara et al 1999;Taylor & Day 2004), or upon reaction norms (Kirkpatrick & Heckman 1989;de Jong 1990;Gomulkiewicz & Kirkpatrick 1992) and other types of function-valued trait models (Kingsolver et al 2001;Kirkpatrick & Meyer 2004;Meyer & Kirkpatrick 2005).…”