On the maintenance of genetic variation: global analysis of Kimura's continuum-of-alleles model

Abstract

Methods of functional analysis are applied to provide an exact mathematical analysis of Kimura's continuum-of-alleles model. By an approximate analysis, Kimura obtained the result that the equilibrium distribution of allelic effects determining a quantitative character is Gaussian if fitness decreases quadratically from the optimum and if production of new mutants follows a Gaussian density. Lande extended this model considerably and proposed that high levels of genetic variation can be maintained by mutation even when there is strong stabilizing selection. This hypothesis has been questioned recently by Turelli, who published analyses and computer simulations of some multiallele models, approximating the continuum-of-alleles model, and reviewed relevant data. He found that the Kimura and Lande predictions overestimate the amount of equilibrium variance considerably if selection is not extremely weak or mutation rate not extremely high. The present analysis provides the first proof that in Kimura's model an equilibrium in fact exists and, moreover, that it is globally stable. Finally, using methods from quantum mechanics, estimates of the exact equilibrium variance are derived which are in best accordance with Turelli's results. This shows that continuum-of-alleles models may be excellent approximations to multiallele models, if analysed appropriately.