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Adaptive landscapes, genetic distance and the evolution of quantitative characters

Published online by Cambridge University Press:  14 April 2009

N. H. Barton  and
Michael Turelli
N. H. Barton
Affiliation:
Department of Genetics and Biometry, University College, 4 Stephenson Way, London NW1 2HE, U.K.
Michael Turelli
Affiliation:
Department of Genetics, University of California, Davis, California 95616, U.S.A.
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The maintenance of polygenic variability by a balance between mutation and stabilizing selection has been analysed using two approximations: the ‘Gaussian’ and the ‘house of cards’. These lead to qualitatively different relationships between the equilibrium genetic variance and the parameters describing selection and mutation. Here we generalize these approximations to describe the dynamics of genetic means and variances under arbitrary patterns of selection and mutation. We incorporate genetic drift into the same mathematical framework.

The effects of frequency-independent selection and genetic drift can be determined from the gradient of log mean fitness and a covariance matrix that depends on genotype frequencies. These equations describe an ‘adaptive landscape’, with a natural metric of genetic distance set by the covariance matrix. From this representation we can change coordinates to derive equations describing the dynamics of an additive polygenic character in terms of the moments (means, variances, …) of allelic effects at individual loci. Only under certain simplifying conditions, such as those derived from the Gaussian and house-of-cards approximations, do these general recursions lead to tractable equations for the first few phenotypic moments. The alternative approximations differ in the constraints they impose on the distributions of allelic effects at individual loci. The Gaussian-based prediction that evolution of the phenotypic mean does not change the genetic variance is shown to be a consequence of the assumption that the allelic distributions are never skewed. We present both analytical and numerical results delimiting the parameter values consistent with our approximations.

Type
Research Article
Information
Genetics Research , Volume 49 , Issue 2 , April 1987 , pp. 157 - 173
Copyright
Copyright © Cambridge University Press 1987

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